Copyright

by

Mario Carneiro Faustini

2004

The Dissertation Committee for Mario Carneiro Faustini Certifies that this is the

approved version of the following dissertation :

MODELING AND FABRICATION OF PROSTHETIC SOCKETS

USING SELECTIVE LASER SINTERING

Committee:

Richard H. Crawford, Supervisor

Richard R. Neptune, Co-Supervisor

Kristin L. Wood

Nicolas E. Walsh

S. V. Sreenivasan

MODELING AND FABRICATION OF PROSTHETIC SOCKETS

USING SELECTIVE LASER SINTERING

by

Mario Carneiro Faustini, BSME, MSME

Dissertation

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

The University of Texas at Austin

May, 2004

Dedication

To my parents.

v

Acknowledgements

The author would like to acknowledge the inestimable guidance from his

advisors, Dr. Richard H. Crawford and Dr. Richard R. Neptune, the support for this

research provided by the Department of Veterans Affairs and the personal support

provided by CNPq.

vi

Modeling and Fabrication of Prosthetic Sockets using Selective Laser

Sintering

Publication No._____________

Mario Carneiro Faustini, Ph.D.

The University of Texas at Austin, 2004

Supervisors: Richard H. Crawford and Richard R. Neptune

Successful rehabilitation of transtibial amputees involves effective fitting of

prosthetic components. However, conventional techniques used to produce sockets with

suitable characteristics are labor intensive and expensive, and depend on the work of

skilled prosthetists that are relatively scarce compared to the number of transtibial

amputees. Selective Laser Sintering (SLS) is a very promising technique for producing

subject-specific sockets for transtibial amputee prostheses due to its inherent ability to

create complex three-dimensional objects directly from digital shape information without

the need for specific tooling, molds or human labor.

This dissertation presents a framework for the design, analysis, manufacture and

testing of SLS sockets for transtibial prostheses, including the development of a

computer-aided design model of the socket with compliant features to enhance comfort,

structural analysis using a Finite Element Method (FEM) model, fabrication of a

functional prototype using SLS, and experimental validation of the FEM analysis. The

validation involved quantifying the failing conditions of sockets manufactured using the

vii

framework during destructive tests. The experimental failure loads for the sockets were

within a 3% range of the FEM results and were considered satisfactory. The specific

design of orthogonally compliant features for socket was also analyzed. This process

included the preliminary evaluation of design alternatives using FEM with validation

through experimental measurements, definition of specific design methodology for the

best alternatives, incorporation of solution within the socket and refinement of design

using auxiliary features obtained through topology optimization. Finally, to investigate

the structural response of the SLS socket during gait, a FEM model (acquired from

Computed Tomography data of a transtibial amputee) composed of a socket, liner and

residual limb under quasi-static loading derived from typical ground reaction forces was

employed. Three different compliant designs were evaluated to assess their ability to

locally relieve local pressure during the stance phase of gait, as well as their structural

integrity to ensure safety. The design with a compliant feature consisting of spiral slots

within the socket wall was determined to produce a local average relief of 65.8% in the

interface pressure, reducing the peak pressure from 172 kPa to 66.4 kPa.

viii

Table of Contents

List of Tables ......................................................................................................... xi

List of Figures ....................................................................................................... xii

List of Figures ....................................................................................................... xii

Chapter 1: INTRODUCTION..................................................................................1 1. State-of-Practice review.........................................................................2 2. Previous Work .......................................................................................4 3. Dissertation Structure.............................................................................5

Chapter 2: MEASUREMENT OF PATELLAR-TENDON BEARING SOCKET/RESIDUAL LIMB INTERFACE PRESSURE AT SENSITIVE SITES.........................................................................................................................7 Abstract ...........................................................................................................7 1. Introduction............................................................................................7 2. Material and methods.............................................................................9

2.1) The F-Socket transducers............................................................10 2.2) Subject data and socket instrumentation.....................................11 2.3) Walking test and step up/down test ............................................12 2.4) Post-processing of results ...........................................................12

3. Results..................................................................................................13 3.1) Contact pressures during the walking test ..................................13 3.2) Pressure differences during the walking test ..............................13 3.3) Contact pressures during step up/down test................................14

4. Discussion............................................................................................15

Chapter 3: AN EXPERIMENTAL AND THEORETICAL FRAMEWORK FOR MANUFACTURING PROSTHETIC SOCKETS FOR TRANSTIBIAL AMPUTEES ..................................................................................................17 Abstract .........................................................................................................17 1. Introduction..........................................................................................17 2. Material and methods...........................................................................19

ix

2.1) Overall Surface Model Development .........................................20 2.2) Design Constraints ......................................................................21 2.3) Socket Computer-Aided Design (CAD) Model Development ...22 2.4) FEM Analysis .............................................................................24 2.5) Load Conditions..........................................................................24 2.6) SLS Fabrication ..........................................................................26 2.7) Experimental Validation .............................................................28

3. Results..................................................................................................30 4. Discussion............................................................................................33

Chapter 4: DESIGN OF ORTHOGONALLY COMPLIANT FEATURES FOR LOCAL CONTACT PRESSURE RELIEF IN DURAFORMTM/SLS MANUFACTURED TRANSTIBIAL PROSTHESES ......................................................................36 Abstract .........................................................................................................36 1. Introduction..........................................................................................37 2. Design Constraints ...............................................................................40 3. Material and Methods ..........................................................................41

3.1) Material Properties......................................................................41 3.2) Evaluation of General Design Alternatives ................................42 3.3) Design of spiral slot compliant feature .......................................47 3.4) Refinement of designed discs: FEM and experimental validation49 3.5) Implementation of spiral-slot features on socket ........................54 3.6) Design of features with Topology Optimization for controlled

compliance ..................................................................................56 4. Conclusion ...........................................................................................64

Chapter 5: INVESTIGATION OF QUASI-STATIC RESPONSE OF COMPLIANT PROSTHETIC SOCKETS FOR TRANSTIBIAL AMPUTEES USING FINITE ELEMENT METHOD..................................................................................66 Abstract .........................................................................................................66 1. Introduction..........................................................................................66 2. Material and methods...........................................................................69

2.1) Generation of FEM from CT data...............................................69 2.2) Boundary Conditions ..................................................................72

x

3. Results..................................................................................................74 3.1) FEM results for vertical 800 N load ...........................................74 3.2) Socket stresses, pressure at fibula head and deformation of compliant

features using quasi-static loading conditions from stance phase75 4. Discussion............................................................................................78

Chapter 6: CONCLUSION ....................................................................................81 1. Summary ..............................................................................................81 2. Future Work .........................................................................................82

Appendices.............................................................................................................85 Appendix 1: Patellar-Tendon Bearing (PTB) Socket Rectifications ............85 Appendix 2: Typical gait cycle .....................................................................86 Appendix 3: Validation of socket wall thickness estimation using FEM.....87 Appendix 4: Considerations about the reuse of DuraformTM PA powder ....88 Appendix 5: Determination using FEM of the compliant wall thickness in the

socket with thinner walled sections .....................................................90 Appendix 6: Curved Beam Analysis.............................................................92 Appendix 7: Matlab code for Topology Optimization .................................96

References..............................................................................................................99

Vita 105

xi

List of Tables

Table A: Material properties achieved from tensile test of DuraformTM PA

specimens made from powder used more than 10 times. .................89

xii

List of Figures

Figure 1.1: Square film being lowered on the positive mold of the socket...........3

Figure 1.2: (a) Double-wall socket; (b) Detail of compliant feature composed of 3

intercepting slots. ................................................................................5

Figure 2.1: Calibration bench for transducers .....................................................10

Figure 2.2: (a) Runway with force plates to collect GRF data; (b) example of socket

instrumented with transducers. .........................................................11

Figure 2.3: Pressure map example from F-Scan software (with values in kPa)

displaying circular sampling region and central region for pressure

difference determination. ..................................................................12

Figure 2.4: (a) Pressure output from transducers for the walking test; (b) Profiles

measured over the stance phase. .......................................................13

Figure 2.5: Pressure at the fibula head and from points in a 25 mm dia circular line

around the fibula head.......................................................................14

Figure 2.6: (a) Pressure difference around fibula head and (b) Pressure difference

around the distal tibia: the black line is the average and the red lines

describe the standard deviation.........................................................14

Figure 2.7: Pressure results for the step up/down experiment. ..............................15

Figure 3.1: Framework for subject-specific socket design, analysis and

manufacturing. ..................................................................................20

Figure 3.2: (a) laser scanned surface with shape information about remaining limb;

(b) rectified and trimmed inner surface for socket with points marking

areas where more compliance is desired...........................................21

xiii

Figure 3.3: Initial estimation of wall thickness using a thin-walled cylinder

approximation for the socket shape. The wall thickness was evaluated

using contact pressure (q) and shear stress (p) taken from literature [8-

19]. ....................................................................................................22

Figure 3.4: (a) Anterior and (b) Posterior views of the inner socket profile

displaying the points used in the definition of the pressure distribution

boundary condition. ..........................................................................26

Figure 3.5: (a) Final CAD model of the socket; (b) Ready-to-wear prosthesis with

the SLS produced DuraformTM Socket. ............................................27

Figure 3.6: (a) Socket, pylon and attachment model; (b) Top view of the socket

filled with plaster and with a rod in the middle; (c) Experimental set-up

for the heel strike test........................................................................29

Figure 3.7: a) Set-up for compression test: socket attached to a hydraulic test

machine and filled with ZTA media to be compressed by piston; b)

travel indicator to measure deformation. ..........................................30

Figure 3.8: (a) FEM model of the socket; (b) FEM results for stresses under peak

loading conditions found during gait (in Pa). ...................................31

Figure 3.9: (a) FEM results for the stresses in the socket with bending moment

boundary conditions (in Pa); (b) a more detailed view of the failure area

(highlighted by a white square on the left); (c) output from the MTS

machine, showing the load applied at the pylon vs. its displacement

during the heel strike test; (d) and (e) broken socket after completion of

the test. ..............................................................................................32

xiv

Figure 3.10: (a) FEM results for the stresses in the socket with a compressive force

boundary condition (in Pa); (b) Comparison between measured

deformation at bottom of compliant socket with FEM results. ........33

Figure 4.1: (a) and (b) A patient testing an SLS fabricated socket; (c) the description

of the parts of a prosthesis for below-the-knee amputees: i – socket, ii –

attachment fitting, iii – pylon and iv – prosthetic foot......................38

Figure 4.2: Example of a current SLS fabricated socket with locally thinned walls in

areas where compliance is needed. ...................................................39

Figure 4.3: Test disc to evaluate compliance options (dimensions in meters)....42

Figure 4.4: FEM deformation results (in meters) for control disc with no special

compliant features.............................................................................43

Figure 4.5: FEM deformation results (in meters) for test disc with 4 void spaces.44

Figure 4.6: FEM deformation results (in meters) for test disc with 16 radial slots.

45

Figure 4.7: FEM deformation results (in meters) for test disc with two spiral slots.

46

Figure 4.8: FEM deformation results (in meters) for test disc with four spiral slots.

47

Figure 4.9: Diagram with the geometric representation of the half turn spiral beam

(under a load W)................................................................................48

Figure 4.10: Parts used in the experiment of test discs. ........................................50

Figure 4.11: Experimental set-up for the test of compliant discs..........................50

Figure 4.12: Comparison between experimental and FEM results for the central

displacement of test discs with two spiral slots. ...............................52

xv

Figure 4.13: FEM results for disc with two spiral slots and a normal load of 10 kPa:

displacement (in meters) on the left, stresses (in Pa) on the right. ...53

Figure 4.14: FEM results for disc with four spiral slots and a normal load of 10 kPa:

displacement (in meters) on the left, stresses (in Pa) on the right. ...53

Figure 4.15: Comparison between experimental and FEM results for the central

displacement of test discs with four spiral slots................................54

Figure 4.16: Example of orthogonally compliant features incorporated in a socket.55

Figure 4.17: On the left, a detailed view of spiral slot compliant feature and back wall

(unseen); on the right, section view showing void space between slotted

wall and back wall. ...........................................................................56

Figure 4.18: Design domain for the Homogenization Method..............................57

Figure 4.19: Schematic model of a cut view of a test disc: orthogonal deflection of

center portion is desired, while the border is fixed...........................59

Figure 4.20: Definition of design domain Γ for the topology optimization problem.

Material presence is mandatory in black areas, material is forbidden in

white areas, and grey areas indicate the sections to be optimized. ...59

Figure 4.21: Results for topology optimization of orthogonally compliant test disc for

different input volume fractions, and their theoretical compliances.61

Figure 4.22: FEM results for test disc with compliant structure: on the left,

displacement (in meters); on the right, stresses (in Pa). ...................62

Figure 4.23: On the left, implementation of compliant structure on top of test disc; on

the right, a section view of this model. .............................................62

Figure 4.24: Definition of design domain Γ for the topology optimization problem

with curved domain to be optimized.................................................63

xvi

Figure 4.25: Result for topology optimization of orthogonally compliant curved

domain...............................................................................................63

Figure 4.26: Implementation of compliant feature on socket................................64

Figure 5.1: FEM model of (a) bones, (b) soft tissues and (c) liner. ....................71

Figure 5.2: FEM models of the four different socket designs simulated: (a) plain

socket with no compliant feature; (b) socket with thin-walled compliant

feature; (c) socket with spiral slots for compliance; (d) socket with spiral

slots and compliant support. .............................................................71

Figure 5.3: Free-body diagram of a prosthetic leg, where the plane of the drawing

(x-y) is the sagittal plane...................................................................73

Figure 5.4: (a) Anterior and (b) Posterior view of pressures on the residual limb

using the plain socket under a downward vertical load of 800 N.....74

Figure 5.5: Structural FEM results displaying the stresses (in Pa) on each compliant

socket at 35% of stance, where the largest stresses were encountered.75

Figure 5.6: Comparison of resulting FEM values for (a) pressure at fibula head and

(b) local deformation at compliant features among the different socket

designs simulated. .............................................................................77

Figure A1: Areas of the lower limb where rectifications are needed for the PTB

socket: darker red areas are appropriate for weight bearing, while lighter

red areas must have pressure relieved [57]. ......................................85

Figure A2: Typical gait cycle illustrating the nomenclature of its phases and main

events [56].........................................................................................86

Figure A3: (a) FEM results (in meters) for displacement; (b) FEM results for

stresses (in Pa)...................................................................................87

xvii

Figure A4: Tensile specimens used to evaluate the material properties of older

Duraform powder..............................................................................88

Figure A5: (a) FEM stress results (in Pa) for the thinner walled section of the

socket; (b) Results for reaction forces (in N)....................................91

Figure A6: Diagram based on Figure 4.9, with the addition of distributed load w.92

1

Chapter 1: INTRODUCTION

Successful rehabilitation of transtibial amputees involves effective fitting of

several different type of prosthetic components and one of these components is the

prosthetic socket. However, conventional techniques used to produce these sockets with

suitable characteristics are labor intensive and expensive, and depend on the work of

skilled prosthetists that are relatively scarce compared to the number of transtibial

amputees. With more than 500,000 lower extremity amputees in the United States alone,

and at least 60,000 new ones every year [1], there is an urgent demand for improved

methods of prosthetic limb fabrication that would offer faster and more cost effective

rehabilitation and improved patient care. The inherent ability of SLS techniques to create

complex physical three-dimensional objects directly from subject-specific digital shape

information without the need for specific tooling, molds or human labor makes it

extremely suitable to fabricate prosthetic sockets. Previous research has established

Selective Laser Sintering (SLS) as a very promising technique for producing subject-

specific sockets for transtibial amputee prostheses [1].

With the intent of developing a framework for this application, collaboration

between the Department of Mechanical Engineering at The University of Texas at Austin

and the Department of Rehabilitation Medicine of the University of Texas Health Science

Center at San Antonio (UTHSCA) was established. A framework for the design, analysis,

manufacture and testing of SLS sockets for transtibial prostheses was created, and its

development forms the basis of this dissertation.

2

1. STATE-OF-PRACTICE REVIEW

Interviews with prosthetists at the UTHSCSA and visits to their socket fabrication

laboratory were conducted to gain insight into the current state-of-the-art in socket

fabrication. Several different methods are currently used in the production of

conventional sockets. However, the most common technique is also highly dependent on

the skill of experienced prosthetists and technicians. The procedure can be summarized as

follows. A negative mold of the residual limb is acquired. This portion is completed using

a moistened cast sock that is pulled over the residual limb. Using an indelible pencil, the

important sites of the residual limb are marked on the cast sock to help with their future

identification. The residual limb is then wrapped with a flexible plaster bandage that

becomes rigid with time. Once the cast is hard, it is then removed from the residual limb.

A positive mold is made by filling the negative mold with molding plaster mixed with

water and allowing it to harden. After it has hardened, the bandage is stripped off.

The positive mold is then modified by the prosthetist, using the pencil marked

sites as a reference, to relieve areas that are pressure sensitive by adding plaster, while

weight-bearing areas that are pressure tolerant are altered by removing plaster from the

positive mold – this procedure is commonly referred to as socket rectification. The end

result is a pattern for socket fabrication by either by vacuum forming or lamination. In the

case of lamination, the positive mold is then sealed using a poly-vinyl alcohol (PVA) bag

placed over the positive mold. Several pieces of stockinette (made of nylon, fiberglass,

carbon graphite, kevlar or a combination of these materials) are then pulled over the

positive mold and sealed by another PVA bag. This PVA bag has an opening at the top to

allow the bag to be filled with a liquid resin. In order to improve the flow of the resin,

maintain the shape of the positive mold and to help saturate the stockinette evenly, a

negative pressure is placed on the inside of the outer bag drawing the resin into the

stockinette. Once the resin has hardened the plaster pattern is broken out. After the plaster

has been removed, the socket is trimmed and the edges are smoothed. Alternatively, the

socket can be vacuum formed. A sheet of thermoplastic is heated in an oven to achieve a

softer consistency, and then lowered over the mold (with the mold positioned upside

down). Vacuum is applied in order to conform the heated plastic to the pattern’s outer

surface (Figure 1.1). Again, trimming and smoothing of the edges is still necessary. The

mating features to connect the socket with the pylon are added to the socket, which is

then wrapped with strips of carbon fiber and epoxy resin. The socket is then ready for

fitting by the patient. If the fit is considered adequate by the patient and the prosthetist,

then the patient may start to use it for ambulation. Otherwise the socket may undergo

further manual trimming or may even be completely rejected, with an altogether new

socket being fabricated.

Figure 1.1: Square film being lowered on the positive mold of the socket.

3

4

This method, as previously stated, is extremely time consuming, laborious and

ultimately depends on the skills of an experienced prosthetist and technician. If both the

prosthetist and technician dedicate exclusive attention to the production of one socket, the

work can be completed in about 8 hours. Moreover, the resulting socket must yet be

tested by the patient for long time comfort and fit to avoid the onset of pain and

circulatory problems. Thus, the socket fabrication may become an iterative process if the

socket is found to be unsatisfactory. As such, an alternative method capable of providing

solutions to these limitations is greatly needed.

2. PREVIOUS WORK

The first design approach for the fabrication of SLS sockets consisted of a double-

walled prosthetic socket [3]. The double-walled socket (Figure 1.2a) consisted of a

thicker outer wall and a thinner inner wall, which contacts the residual limb. The socket

incorporated design features to allow greater compliance at high-pressure spots. The

compliant features were intercepting slots (Figure 1.2b). The double-wall allowed

compliance and strength to be dealt with in separate mechanical structures. However, the

socket was heavier and bulkier than traditional sockets and lacked a natural look. In

addition, during clinical trials the sock worn on the patient’s limb interfered with

compliant features of the inner wall. Thus, additional work was needed to improve this

initial approach.

compliant feature

ba

Figure 1.2: (a) Double-wall socket; (b) Detail of compliant feature composed of 3 intercepting slots.

3. DISSERTATION STRUCTURE

This dissertation is focused on developing a framework for producing improved

subject-specific sockets. Chapter 2 analyzes the interface mechanics between the socket

and residual limb by measuring the contact pressure distribution between the socket and

residual limb, which is a prerequisite for integrating design features to improve patient

comfort. This analysis is presented, including a description of the instrumentation of a

socket to provide contact pressure measurements in two common interface locations

where pressure relief is needed.

Chapter 3 presents details of the framework for fabrication of prosthetic sockets

using SLS. There, the main tasks are described, including the acquisition of shape

information, surface model modifications to incorporate the rectifications of the patellar-

5

6

tendon bearing (PTB) method, socket computer-aided design (CAD) model development,

structural analysis using a Finite Element Method (FEM) model to assess structural

integrity, fabrication of a functional prototype using SLS, and experimental validation of

the FEM analysis.

Chapter 4 presents the methodology for the design and integration of orthogonally

compliant features in the socket structure to reduce contact pressure. The methodology

includes the preliminary evaluation of compliant design alternatives on test coupons

using FEM, validation of computed results through experimental measurements,

definition of design method for the most appropriate solution, incorporation of the

solution within the socket geometry and refinement of the design solution using auxiliary

features obtained through topology optimization.

Chapter 5 presents an alternative analysis of the structural response of the socket

to typical loading conditions that occur during gait. With the addition of the FEM model

of the residual limb derived from computed tomography (CT) data to the socket model,

the evaluation of different socket designs and the corresponding effects on interfacing

tissues using quasi-static loads is performed. The integration of the socket and residual

limb models was shown to eliminate much of the need for extensive prototyping and

socket instrumentation.

Finally, Chapter 6 presents a summary of the research performed and highlights

areas for future work.

7

Chapter 2: MEASUREMENT OF PATELLAR-TENDON BEARING SOCKET/RESIDUAL LIMB INTERFACE PRESSURE AT SENSITIVE

SITES

ABSTRACT

In a patellar-tendon bearing (PTB) prosthesis, the inner profile of the socket is

modified with respect to that of the residual limb so that the load is sustained in specific

pressure resistant tissues, while relieving this contact pressure in other sensitive sites.

Still, it is not uncommon for many patients to feel discomfort and pain in certain regions.

Further pressure relief can be achieved through the incorporation of compliant features

within the socket, which became possible with the development of rapid prototyping

techniques for the fabrication of prosthetic sockets. In order to adequately design such

features, an investigation of the magnitude of contact pressures at sites where relief is

needed must be performed. This paper describes the acquisition of interface pressure data

at the fibula head and the distal tibia during a walking test and during a step up/step down

test, using thin-film tactile pressure sensors. The average maximum pressures obtained at

the fibula head and distal tibia were 120 kPa and 75 kPa, respectively, for the walking

test. For the step up/down test, the values for the peak pressures at these sites were 125

kPa and 90 kPa, respectively. Both results were considered consistent throughout the

trials, with deviations of 2.3% for the walking test and 6% for the step up/down test.

1. INTRODUCTION

8

The socket of a patellar-tendon bearing (PTB) prosthesis is designed to reproduce

the overall shape of the patient’s stump, but undergoes typical local modifications

(rectifications) to cause most of the load distribution to occur in specific areas (e.g. the

patellar tendon and the gastrocnemius muscle belly) that have a greater tolerance for load

and pressure [4]. However, some sensitive areas may still be affected by enough contact

pressure to cause discomfort to the patient. In addition, integrating larger local

rectifications to relieve pressure may deteriorate residual limb/socket fit and compromise

structural integrity. The introduction of local compliant features may provide an

alternative solution to relieving concentrated high pressure areas. Although in

conventional hand cast sockets such features are hard to implement, the production of

sockets using freeform fabrication methods (to be discussed in Chapter 3) provides to the

prosthetist with a fabrication technique to easily incorporate complex compliant

topologies in the socket with no production cost penalty.

To properly design such features, it is necessary to quantify the magnitude of the

pressure in such sensitive areas during common workload conditions (to be addressed in

Chapters 3 and 4). Several methods have previously been used to measure such pressures

[7], including the use of fluid-filled sensors, pneumatic sensors and diaphragm strain

gauges. The greatest challenge with such transducers is their placement in the socket

without significantly modifying the interface conditions. Another issue is allowing

freedom of movements for the patient during testing while avoiding discomfort due to the

presence of such transducers within the instrumented sockets. Thus, a promising

alternative is the use of pressure transducers with arrays of pressure cells, which are thin

enough to have minimal impact in the interface conditions. This technique has been

previously reported [6] for the measurement of PTB socket/limb interface pressure

distributions during gait. There, four transducer arrays were used simultaneously in the

9

socket/limb interface, covering a considerable area of the socket. However, this might

lead to a change in the friction coefficient between the socket and the residual limb in all

such areas. A previous study reported that changes in the friction parameters of the

interface will cause ensuing changes in the shear stress distributions, thus significantly

affecting, as a consequence, the pressure distribution [11].

The focus of this paper is to use transducer arrays of pressure sensing resistors

and measure the contact pressure magnitudes in two specific sensitive sites of the residual

limb, the fibula head and the distal tibia at the anterodistal area. From a designer’s

viewpoint, it is necessary to obtain not only the value of the pressure at the center of the

site where compliance is needed, but also to predict the expected pressure gradient in

immediately surrounding areas. Furthermore, pressure values for workload situations

other than gait are also important considerations. Our goals were 1) obtain the peak

pressure at the specific sites during the stance phase of gait; 2) obtain the pressure

gradient during the stance phase of gait in the areas immediately surrounding the specific

sites; 3) obtain the peak pressure at the specific sites during a step up/step down test.

2. MATERIAL AND METHODS

The tests were performed by instrumenting the socket of a chosen patient with

two calibrated transducers. The patient was first asked to walk on a runway as pressure

data was collected for the walking test. Then he was asked to step up and down on a stool

for acquisition of the step up/down pressure data. The results were processed using the

software bundled with the transducers.

2.1) The F-Socket transducers

The two transducers used to measure the pressure were F-Socket 9811 sensors,

manufactured by Tekscan1. Each sensor was 76.2 mm x 203.2 mm, with a thickness of

0.28 mm, pressure range between 0 and 517 kPa, and had a total of 96 sensels (each with

161.3 mm2), arranged in 6 columns and 16 rows. They were connected to a computer by

an interface cable. F-Scan software was used to provide the interface between the sensors

and the data acquisition hardware. The software also provided the equilibration and

calibration capabilities as well as visualizing and recording pressure data taken during the

test.

The sensors were equilibrated and calibrated, using the equipment shown

schematically in Figure 2.1, prior to being placed in the socket for testing. The sensors

were each put into the pressure bladder, and a pressure of 200 kPa (raised from 0) was

used for calibration. To dissipate the effects of signal drifting, 120 seconds were allowed

to pass after application of pressure before calibration was performed [20].

Computer

Pressure Gauge

Needle ValveDigital Pressure Gauge

Pressure Bladder

Figure 2.1: Calibration bench for t

1 South Boston, MA.

Signal Generator

Pressure Regulator

Air Compressor

Solenoid Valve

ransducers

10

2.2) Subject data and socket instrumentation

The patient chosen for the contact pressure measurement was a 54 years old male,

1.68 m tall, weighing 65 kg, being an amputee for 23 years (with amputation caused by

trauma), with no relevant clinical issue, in very good health and fairly active. He was

allowed to get familiar with the environment of the gait lab where the test was held.

Simultaneously, ground reaction force data was collected from four AMTI2 force plates,

which compose the runway of the lab, to evaluate uniformity of his bilateral gait pattern.

Each plate was 508 mm by 457 mm, and they are disposed side by side in the runway

(Figure 2.2a).

The sensors were positioned on the inner surface of the subject’s socket (Figure

2.2b) using a non-aggressive bonding agent at the fibular head and the distal tibia. The

sensor on the fibular head was placed just far enough below the socket’s rim to cover the

site to be evaluated in order to minimize changes in interface friction. The sensor mat on

the distal tibia was placed in the orientation that minimized curvature of the transducer in

order to reduce the associated depletory effects [5].

a b

Figure 2.2: (a) Runway with force plates to collect GRF data; (b) example of socket instrumented with transducers.

112 Watertown, MA.

2.3) Walking test and step up/down test

The patient was then allowed to walk around and reported to feel no difference

from the presence of the sensors. Pressure data was collected at 60 Hz for 10 seconds as

the subject walked along a straight path at a self-selected pace. Six trials were recorded.

Finally, the subject was asked to step up on a bench that was 200 mm tall, and then step

down from it. The motion was repeated six times while pressure data was recorded at a

60 Hz rate.

2.4) Post-processing of results

The pressure data was processed using the F-Scan software. All pressures

measured were calibrated and the peak contact pressures at the fibula head and the distal

tibia were identified in the output data during the walking and step up/down trials.

Results for the local surrounding pressure difference to evaluate the gradients in the

surrounding area of the fibula head and the distal tibia were obtained in a similar way,

with the software interface being used to sample the pressure values along a 25 mm circle

around the center point of the site (Figure 2.3). The readings from these points were

averaged and the pressure difference between the mean surrounding values and the

central values was calculated.

Fibula head

Figure 2.3: Pressure map example from F-Scan software (with values in kPa) displaying circular sampling region and central region for pressure difference determination.

12

3. RESULTS

The transducers produced consistent profiles during both the walking and step

tasks. Representative profiles of each task will be presented and discussed below.

3.1) Contact pressures during the walking test

For the walking test, the average maximum pressure verified at the fibula head

was 120 kPa. At the distal tibia, the average maximum pressure was 75 kPa (Figure 2.4).

From the time data of the results and the average length of the patient’s stride, the

subject’s average walking speed was determined to have been consistently near 1 m/s.

Contact Pressure

0

20

40

60

80

100

120

140

0 2 4 6 8 10

Time (s)

Pres

sure

(kPa

)

Fibula head

Distal Tibia

Contact Pressure during Stance

0

20

40

60

80

100

120

140

0 20 40 60 80 100

% of Stance

Pressure (kPa)

Fibula Head

Distal Tibia

a b

Figure 2.4: (a) Pressure output from transducers for the walking test; (b) Profiles measured over the stance phase.

3.2) Pressure differences during the walking test

The data sampled from points along the circular line surrounding the high

pressure sites was also very consistent (Figure 2.5), which allowed reliable calculation of

the radial pressure drop around the sites (Figure 2.6). The output data revealed that the

peak pressure drop at a 25 mm circle around the fibula head was 37 kPa. For the distal

tibia this value was 28 kPa.

13

Pressure

0

20

40

60

80

100

120

140

0 2 4 6 8 10

Time (s)

Pres

sure

(kPa

)

Pressure at surroundingsampled point

Pressure at surroundingsampled point

Pressure at surroundingsampled point

Pressure at fibula head

Figure 2.5: Pressure at the fibula head and from points in a 25 mm dia circular line around the fibula head.

Pressure difference (around Fibula Head)

0

5

10

15

20

25

30

35

40

45

0 20 40 60 80 100

% of Stance

Pres

sure

(kPa

)

Pressure difference (around Distal Tibia)

0

5

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100

% of Stance

Pre

ssur

e (k

Pa)

a b

Figure 2.6: (a) Pressure difference around fibula head and (b) Pressure difference around the distal tibia: the black line is the average and the red lines describe the standard

deviation.

3.3) Contact pressures during step up/down test

The step up/step down pressure measurements were also consistent. The peak

pressure on the fibula head was similar to the one from the walking test, 125 kPa.

14

However, on the distal tibia this value increased to 90 kPa, or 20% higher than the one

obtained in the walking test (Figure 2.7).

Step Up/Down Test

0

20

40

60

80

100

120

140

0 20 40 60 80 100

% of Motion

Pres

sure

(kPa

)

Fibula HeadDistal Tibia

StabilizingStepdown

Step up Stabilizing

Figure 2.7: Pressure results for the step up/down experiment.

4. DISCUSSION

The goal of this study was to determine peak limb/socket contact pressures at the

fibula head and distal tibia during walking and a step up/down motion. Another objective

was to evaluate the pressure gradient around these two points during walking. The

pressure arrays provided consistent results across trials. For the walking and step

up/down experiments the deviations of the peak pressures were 2.3% and 6%,

respectively. The measured pressure magnitudes were similar to profiles found in the

literature for the fibula head and the distal tibia [6,12].

It should be noted that, although the profiles presented on Figures 2.4, 2.5 and 2.7

imply that the contact pressures drop to zero, in reality the contact pressures never 15

16

actually drop to zero since there is never any loss of contact between the socket and the

residual limb at the investigated sites. This effect in the results is due to the threshold

setting of the measuring equipment to filter out noise for low loads. This filtering causes

the plot to indicate zero contact pressure every time the measured pressure was below 20

kPa. It should also be noted that the values measured and presented are subject and socket

specific, and may vary even for the same subject depending on many factors, such as

time, socket/foot alignment [8-10].

The data presented in this article is important because, with the development of

new techniques for socket design involving rapid prototyping and freeform fabrication,

many different design solutions are possible to provide relief to the contact pressure at

sensitive areas (as will be addressed in Chapter 4). Thus, the measured values for the

pressure differences on areas surrounding the sensitive sites allow a better understanding

of the load conditions to be considered in the design pressure relieving features.

However, a challenging aspect of the socket design process is that such pressure

data needs to be predicted a priori, in order to serve as an input parameter for the design

itself, instead of being acquired a posteriori. Thus, using FEM as a tool to predict such

pressures is a promising alternative [12-14,17,18], allowing in situ pressure

measurements, such as described in this article, to be used for design validation (as will

be described in Chapter 5).

17

Chapter 3: AN EXPERIMENTAL AND THEORETICAL FRAMEWORK FOR MANUFACTURING PROSTHETIC SOCKETS

FOR TRANSTIBIAL AMPUTEES

ABSTRACT

An attractive application of Solid Freeform Fabrication, given its ability to easily

generate parts with customized complex geometric design, is to produce sockets for

transtibial amputee prostheses. The present work focuses on developing a manufacturing

framework to produce subject-specific sockets made of DuraformTM PA material using

Selective Laser Sintering. The framework includes establishing an overall socket design

(using the patellar-tendon bearing approach), performing a structural analysis using the

Finite Element Method (FEM) to ensure reliability during patient use, and validating the

results by comparing the model output with experimental data. The validation included

quantifying the failing conditions of sockets built using the framework through bending

moment and compression tests. The experimental failure loads for the socket were within

3% of the FEM results and were hence considered satisfactory.

1. INTRODUCTION

For lower extremity amputees, a well fitting socket is a prerequisite for successful

rehabilitation [1]. The socket provides the interface between the prosthesis and residual

limb, which is designed to allow appropriate load transmission, comfort and efficient

motion control. Attaining these objectives is extremely challenging, with up to 55% of

18

lower limb amputees reporting dissatisfaction with socket comfort, residual limb pain,

and/or skin breakdown. Moreover, current techniques used to produce sockets with

suitable characteristics are labor intensive and expensive, and depend on the work of

skilled prosthetists that are relatively scarce compared to the number of lower limb

amputees (more than 500,000 in the United States alone, with 60,000 new ones every

year [1]). Thus, an objective and systematic method of fabricating prosthetic sockets is

needed to improve efficiency in prosthetic care, enhance comfort and fit, and reduce time

and cost to ensure the optimal amputee rehabilitation. Previous systematic approaches

have not centered on the engineering design and structural analysis of the socket and

lacked experimental validation of the FEM analysis [17,21]. In addition, no study has

utilized Selective Laser Sintering (SLS) technology for the socket fabrication.

SLS technology has proven to be a very powerful manufacturing technique with

several advantages over traditional methods [22-25]. One advantage is that SLS can

directly create sockets from digital subject-specific shape information, which eliminates

the need for extensive molds, hand lamination and finishing procedures. Second, SLS has

the ability to create complex geometries with minimal cost penalty in manufacturing.

From a design perspective, this significantly expands the options for developing and

exploring alternate socket designs, including geometric variants of traditional socket

shapes, and for incorporating variable compliant elements in selected locations to relieve

contact pressure between the limb and socket. Third, the integration of additional

prosthetic components/features directly into the socket, such as an integrated pylon

mounting plate at the attachment site is straightforward.

The overall goal of this paper was to develop a manufacturing framework to

produce subject-specific sockets made of DuraformTM PA material using SLS. The

framework included defining the overall socket design using the patellar-tendon bearing

19

approach, performing a structural analysis using FEM to ensure reliability during patient

use, and validating the FEM results with experimental data.

2. MATERIAL AND METHODS

The framework for design and fabrication of prosthetic sockets is presented

below, and its principal steps are discussed in detail. Then, two tests for experimental

validation performed are presented. The chart shown in Figure 3.1 describes the overall

framework for the production of sockets using SLS. The main steps of the framework

are: 1) surface model development using the geometry of the residual limb acquired

through laser scanning, 2) CAD model development using appropriate design constraints,

3) FEM analysis of the socket using related boundary conditions and 4) SLS fabrication.

Figure 3.1: Framework for subject-specific socket design, analysis and manufacturing.

Pylon and foot mounting Prosthesis ready

to be used

Prototype Socket

Socket Fabrication with SLS

Mold cast directly from patient’s limb

Surface model from digital shape information

Laser scanning of mold

CAD Model Development

STL file

Mesh Generation

FEM analysis

Redesign if needed

Else proceed to fabrication

Boundary conditions:

e.g. patient’s weight,

interface pressure

2.1) Overall Surface Model Development

The first step of the framework is to acquire the patient’s residual limb shape

information through laser scanning of a positive mold cast directly from the patient’s

limb. For the socket designed and analyzed in this article, data was taken from a 54 year

old male, 1.68 m tall, weighing 65 kg, being a transtibial amputee for 23 years (with

amputation caused by trauma), with no relevant clinical issue, in very good health and

fairly active. A Seattle Limb Systems ShapeMaker 3000 laser scanner (built using

technology licensed from UTHSCSA) was used to provide a point cloud that described

the exact shape of the limb with an angular resolution of 3 degrees and a z-axis resolution

20

of 3 millimeters. The point cloud was converted to a surface model (Figure 3.2a) that was

then used by the prosthetist to define the appropriate trimming line at the top of the

socket and perform rectifications to allow most of the load to be carried in the patellar-

tendon area [4,12], which provides relief to sensitive areas on the patient’s limb.

GordoScan3 software was used to perform the rectifications. The surface model was also

used by the prosthetist to indicate areas of needed compliance to help reduce socket-limb

interface pressure (Figure 3.2b).

At this stage, the surface model described precisely the desired inner shape of the

socket and was ready for engineering design.

Trimming line

a b

Needed compliance

Figure 3.2: (a) laser scanned surface with shape information about remaining limb; (b) rectified and trimmed inner surface for socket with points marking areas where more compliance is desired.

2.2) Design Constraints

The primary design constraint was that the subject-specific inner shape of the

surface model must be preserved. However, additional design considerations included

minimizing the weight of the socket, adding area-specific compliance to relieve socket-

21

3 From Dept. of Rehabilitation Medicine, University of Texas Health Science Center at San Antonio, San Antonio, TX.

limb interface pressure and strength to withstand the normal loads applied during human

gait. In addition, aesthetic considerations that produce a socket shape that mimics the

profile of a leg anatomy were also important. From this set of design criteria, the overall

topology of a traditional thin-walled shell socket satisfied the considerations above.

2.3) Socket Computer-Aided Design (CAD) Model Development

The initial calculation for the wall thickness of the socket was estimated using a

thin-walled cylinder as an approximation (Figure 3.3). The radius of the cylinder was

estimated by the mean radius of the inner surface, and the contact pressure and shear

stress were estimated using the peak values found in the literature [8-19].

Figure 3.3: Initial estimation of wall thickness using a thin-walled cylinder approximation for the socket shape. The wall thickness was evaluated using contact pressure (q) and shear stress (p) taken from literature [8-19].

The equation used to define the overall thickness of the socket wall is given by

[31,32]:

max

2

2RE

LRpRqt

∆⋅

⋅⋅⋅+⋅=

ν ( 1 )

where q is the normal pressure load, p is the transferred shear stress, R is the

radius of the cylinder, L is its length, E is Young’s Modulus and ∆Rmax is the maximum

22

radial expansion desired. To ensure that the socket fit is preserved during applied

workload, a cylinder radius expansion of 1% or less was desired.

To preserve structural integrity, the resulting wall thickness should provide

stresses within boundaries of the strength of the material:

t

RqpSy⋅+

> ( 2 )

where Sy is the value for tensile strength of DuraformTM. Validation of this initial

value for thickness with FEM analysis can be found in Appendix 3. Once the appropriate

initial value for wall thickness was identified from ( 1 ), the inner surface profile was

modified in the CAD software (Rhinoceros 3D4) to produce the final socket shape. First,

an “outwards” offset operation was applied to the inner surface of the socket. Then, a

“blend” operation was performed at the top free edges of the inner surface and offset

surface to turn the model into a closed volume model. Features to facilitate the pylon

attachment were added to the bottom of the model [27].

The final design features to be incorporated were the compliant regions to relieve

contact pressure (Figure 3.2b), which were implemented using reduced wall thickness at

the compliance-needed sites. This was performed using “offset” operations on the outer

surface sections surrounding the identified sites. Once these sections were defined, they

were rejoined to the overall outer surface using “blend” operations, which preserved the

closed volume format of the model. The structural integrity of the wall thickness was

refined through FEM analysis of these compliant sections using the desired deformation

defined by the prosthetist as target and contact pressure data around these specific sites as

input (such as the pressure data measured in Chapter 2).

234 From Robert McNeel & Associates, Seattle, WA.

24

2.4) FEM Analysis

In order to evaluate the structural behavior of the socket under expected applied

loads, a FEM model was created from the socket CAD model using SDRC I-deas5. The

socket CAD model was imported into I-deas using the IGES file format. A Delaunay-

based mesh generator incorporated in I-deas was used to create the initial model mesh.

The elements used were 10-node parabolic tetrahedra, and the initial element size was

chosen to be less than the overall socket wall thickness. The automated mesh generator

was set to create a model based on sections, which allows specific areas and geometric

features to be further refined. The resulting mesh was interactively inspected to identify

highly distorted or stretched elements that may lead to increased errors in the analysis.

When such areas were identified, individual elements were locally refined. The complete

FEM mesh for the socket had 7582 elements and 22570 nodes.

Since the SLS material used in this study was DuraformTM PA6, a type of nylon

12 which has properties suitable for sockets, the material properties used were 44 MPa

for tensile strength, 1600 MPa for tensile modulus and 1285 MPa for flexural modulus.

2.5) Load Conditions

The FEM analysis of the socket was performed under different loading and

boundary conditions in order to evaluate the structural integrity of the socket under

typical loading conditions found during gait. First, a simulation was run to analyze the

behavior of the socket under the peak limb-socket pressure profile and shear stresses

found in the literature for individuals with similar body weight as the patient modeled in

the present study. These measured values were then applied to the socket as boundary

conditions on the inner socket surface (where contact occurs). Values for contact pressure

5 From EDS, Plano, TX. 6 http://www.3dsystems.com/products/datafiles/lasersintering/datasheets/DURAFORMHigh.pdf

and shear stress were defined at ten areas (Medial and Lateral Supracondyle, Medial and

Lateral Tibia, Medial and Lateral Gastrocnemius, Patellar Tendon, Popliteal Depression,

Distal Tibia and Fibula Head – Figure 3.4). Also, the top edge of the inner surface was

defined as having a zero load. The data surfaces were then created through refined

inverse distancing interpolation, leading to smooth and first order continuous pressure

distribution functions at contact surfaces. Such functions were mathematically defined as

follows:

- Given a set of pressure values p1...pn at n discreet points nPPr

Kr

1 of a

parametric surface of the socket model defined in terms of the parametric

space (u,v) as ( )v,uS ;

- Then for each discreet point iPr

, a local interpolation function was created: ( ) ( )( )

( ) ( ) ( )iii

iiiii

vvauuavva

vvuuauuapQ

−+−+−+

−−+−+=

542

3

22

1 ( 3 )

where ui and vi are the u and v components of iPr

and a1...a5 are the pressure

values at the five points of the set nPPr

Kr

1 that were closest to iPr

;

- Also for each discreet point iPr

, a weighing factor was calculated:

( )2ii

iii dr

drW⋅−

= ( 4 )

where ri is the radius of influence in the parametric space (u,v) around point

iPr

(defined to be equal to 0.05) and di is the parametric distance between

(u,v) and (ui,vi);

- Thus, the pressure distribution on the surface ( )v,uS was defined by:

( )∑

∑

=

=

⋅= n

ii

n

iii

W

QWv,up

1

1 ( 5 )

25

ba

Medial Gastrocnemius

Lateral Gastrocnemius

Popliteal Depression

Distal Tibia

Fibula Head

Lateral Tibia

Medial Tibia

Patellar Tendon

Lateral Supracondyle

Medial Supracondyle

Figure 3.4: (a) Anterior and (b) Posterior views of the inner socket profile displaying the points used in the definition of the pressure distribution boundary condition.

Once the load values were constructed and applied to the elements of the inner

surface of the socket, the pylon attachment fitting features at the bottom of the socket

were spatially restrained.

To investigate the structural response of the socket, the resulting stresses obtained

from the FEM analysis were used, assessing the integrity of the socket by comparing

these stresses with the range tolerated by the material.

2.6) SLS Fabrication

To experimentally validate the FEM analysis results, a prototype of the socket

was manufactured using SLS (Figure 3.5). The socket CAD model was exported using

the STL (or stereolithography) format, which is used as input into any SLS system. SLS

is a type of freeform fabrication technique where any computer solid model forming a

closed volume in space is manufactured in layers. Adjacent planar layers of a powdery

26

material are placed in a part bed, melted in the desired cross-section of the model by a

laser beam and solidified, thus adding one layer to the part itself. The bed is then lowered

by one layer thickness and covered again with another layer of the powdery material.

When all the cross-sections of the model are processed, the final part will possess the

same shape and dimensions as the computer model.

The socket in the present study was produced using a 3DSystems (Valencia, CA)

Sinterstation 3500. The model was fabricated in a vertical position for optimal sintering

and thermal conditions. The total fabrication time, including the warming up and cooling

down phases, was 15 hours, during which the machine operated completely unassisted.

Once the fabrication was finished, the socket was ready to be assembled into a prosthesis.

a b

Figure 3.5: (a) Final CAD model of the socket; (b) Ready-to-wear prosthesis with the SLS produced DuraformTM Socket.

27

28

2.7) Experimental Validation

The socket validation consisted of two experiments to assess the structural

responses of the socket and compare them with FEM predictions generated with

boundary conditions similar to the experimental conditions.

The first experiment was set to simulate the bending moment observed during a

heel strike (Figure 3.6) [28]. The socket was filled with sand and plaster and a steel rod

was placed in the center of the filling plaster. Once the plaster cured, the pylon and

attachment were mounted at the bottom of the socket. The entire assembly was placed in

a horizontal position, with the end of the rod firmly gripped by a vice attached to a fixed

table and the end of the pylon being positioned onto an MTS hydraulic testing machine.

The piston of the hydraulic machine applied a gradually increasing perpendicular force at

the end of the pylon while a load cell recorded the force magnitude and displacement that

was transferred to a data acquisition computer. The bending moment was increased until

the socket failed.

a

socket

29

Testing machine

pylon

rod

plaster

b c

Figure 3.6: (a) Socket, pylon and attachment model; (b) Top view of the socket filled with plaster and with a rod in the middle; (c) Experimental set-up for the heel strike test.

A second validation test was used to assess the maximum compressive force that

could be applied to the inner surface of the socket, as well as the deformation at one of

the compliant features with the increase of the compressive force. At first, a pilot study

was performed that demonstrated [29] the top of the socket could be modified to a

cylindrical shape (in order to contain the filling media while compression by a piston was

taking place) without significant changes in its stress and displacement distribution at the

remainder of the socket. With this verification, a top-modified socket was manufactured

(while maintaining all other dimensions) and filled with 1 mm diameter Zirconia-

Toughened Alumina (ZTA) beads. This medium was chosen for being strong enough to

withstand the compression load associated with the test while minimizing the effects of

friction. This created a pressure distribution profile over the inner surface of the socket

that could be replicated as a boundary condition for the FEM model. The MTS hydraulic

test machine was then used to gradually compress the medium through a piston until the

socket failed (Figure 3.7a). A travel indicator was used to measure the deformation of the

socket wall at the bottom compliant feature (Figure 3.7b) as the applied load increased.

travel indicatorsocket

piston

a b

Figure 3.7: a) Set-up for compression test: socket attached to a hydraulic test machine and filled with ZTA media to be compressed by piston; b) travel indicator to measure deformation.

3. RESULTS

The results for stresses within the socket under peak loading conditions during

gait (Figure 3.8) revealed that the maximum value (42.5 MPa), found at the bottom of the

socket, was within the range of the properties of commercially produced Duraform parts.

30

ba

Peak stress

Figure 3.8: (a) FEM model of the socket; (b) FEM results for stresses under peak loading conditions found during gait (in Pa).

For the heel strike test, the FEM simulation predicted that the socket would fail

when the applied perpendicular force exceeded 690 N, which produces a stress value of

45.2 MPa. This is just above the failure stress of DuraformTM PA of 44 MPa. During the

experimental validation, the socket failed with an applied force of 710 N. In addition, the

socket failed in the area of peak stress predicted by the FEM analysis (Figure 3.9).

31

ba

Peak stress

Socket test - Heel Strike

-200

-100

0

100

200

300

400

500

600

700

800

-20 0 20 40 60 80 100

Deformation (mm)

Load

(N)

c

d e

Figure 3.9: (a) FEM results for the stresses in the socket with bending moment boundary conditions (in Pa); (b) a more detailed view of the failure area (highlighted by a white square on the left); (c) output from the MTS machine, showing the load applied at the pylon vs. its displacement during the heel strike test; (d) and (e) broken socket after completion of the test.

32

The compression test found that the force at failure was 26.25 kN, which was

similar to the force (26.47 kN) predicted by the FEM analysis (Figure 3.10a). This peak

force produced a maximum stress of 45.3 MPa, which is just above the failure stress of

DuraformTM PA (44 MPa). The measured orthogonal deformation at the bottom

compliant feature was found to be within a 25% interval of the values predicted by the

FEM (Figure 3.10b) for loads under 78% of the failing force.

0

5

10

15

20

a

25

30

0 0.2 0.4 0.6 0.8 1

Orthogonal deformation at bottom compliant feature (mm)

Forc

e (k

N)

MeasuredFEM

b

Peak stress

Figure 3.10: (a) FEM results for the stresses in the socket with a compressive force boundary condition (in Pa); (b) Comparison between measured deformation at bottom of compliant socket with FEM results.

4. DISCUSSION

In this paper, a framework for the production of subject-specific prosthetic

sockets for transtibial amputees using SLS was described (Figure 3.1). The framework

included the acquisition of the residual limb shape through laser scanning, a methodology

for socket design, structural analysis using FEM to ensure reliability under normal

33

34

workload during gait, and socket fabrication using SLS. An experimental validation of

the FEM model was performed through tests of SLS built sockets. The validation results

achieved were considered quite satisfactory, with the FEM predictions for failure loads in

both tests being within 3% of the actual measured values. However, it should be noted

that the criterion to consider these results satisfactory was an outcome within 25% range

from the actual measured value. Such criterion was also used for the deformation data of

the compression test, and the results were valid for loads under 78% of the failing load.

For values above, the discrepancy between the predicted and measured results could be

explained by the transition of the FEM model into a large deformation regime, which

leads to higher errors in FEM results.

A limitation of the present FEM analysis procedure was the determination of the

pressure profile applied to the prosthesis, which can greatly vary among patients [8-16].

Ideally, the pressure profile should be measured in the actual socket being analyzed to

ensure the loading conditions are precise. However, obtaining actual pressure loads from

the socket being analyzed would require an iterative process that is not feasible.

Furthermore, previous analysis has shown that instrumenting a socket for contact

pressures measurement may considerably change the interface conditions from their

regular state [7]. However, another promising possibility [11-14,16-19] is to assess the

load transfer directly from an FEM model of the residual limb, including the bones, soft

tissues and liner, where the dimensional data to generate the corresponding meshes come

from a CT or MRI scan. Future work will be directed at integrating FEM models of the

residual limb and socket to obtain pressure data for design analysis (and will be addressed

in Chapter 5).

The underlying goal of this work is to provide an efficient framework for

prosthetists to produce effective, subject-specific sockets. However, the current

35

framework involves many steps and experience using CAD and FEM analysis software.

Future work will be directed at providing an integrated user-friendly system that

automatically performs all the CAD operations needed to produce the solid model of the

socket (including the thickening of the wall, incorporation of compliant features and

bottom pylon fitting attachment). This task is very feasible, since the involved parameters

can all be easily stored in a database within the program.

Nevertheless, one advantage of the presented framework is that it allows for

systematic and controlled design changes in socket shape or volume, which may permit

exploration of the relative advantages of various socket design philosophies [4,26] (e.g.

specific-area weight bearing/patellar-tendon bearing or PTB vs. total surface bearing;

smooth vs. abrupt contouring of socket features, etc). A second advantage is that

compliant features can be easily integrated into the socket design. Although the present

design integrated one form of compliance (i.e. thin-walled sections), future work will be

directed at examining other forms of compliant features (which will be addressed in

Chapter 4).

36

Chapter 4: DESIGN OF ORTHOGONALLY COMPLIANT FEATURES FOR LOCAL CONTACT PRESSURE RELIEF IN DURAFORMTM/SLS

MANUFACTURED TRANSTIBIAL PROSTHESES

ABSTRACT

A very attractive advantage of manufacturing prosthetic sockets through Solid

Freeform Fabrication is the freedom to introduce design solutions that would be

otherwise difficult to implement. Such is the case with compliant features embedded in

the structure of the prosthetic sockets for relief of contact pressure at specific sites. This

article discusses the design of orthogonally compliant elements within the context of

prosthetic sockets. In particular, we describe a process for designing compliant elements

to be incorporated into a model of transtibial sockets, which will be produced with

Selective Laser Sintering using DuraformTM, a material derived from nylon, and

exploring some different alternatives. This design process includes the preliminary

evaluation of design alternatives on test coupons using Finite Element Method, validation

of computed results through experimental measurements, definition of design

methodology for the most appropriate solution, incorporation of solution within the

socket geometry and refinement of design solution using auxiliary features obtained

through topology optimization. Using this methodology, a compliant feature consisting of

spiral beams and a supporting external structure was developed to provide pressure relief

to specific areas of the prosthetic socket.

37

1. INTRODUCTION

In many applications involving mechanical structures, compliance is very

important for proper function of a part. In other situations, it is simply a much more

elegant and functional design solution where the joints and links of mechanisms are

replaced by the incorporation of controlled flexibility within the structure itself.

However, very well defined constraints need to be considered and evaluated in order to

proceed with the design of compliant features [34]: the part material and its mechanical

properties must be understood, boundary conditions for the structure must be well

defined, the target deformation under certain loads must be set, and analysis to predict the

limits of performance should be executed.

This article focuses on the design of orthogonally compliant features associated

with thin walls (where the load/displacement direction of the feature is normal to the

wall), and the evaluation of options for a given set of design goals. The techniques are

described within the context of relieving limb-socket interface contact pressure in

transtibial prosthetic sockets. The solutions are meant to be incorporated as compliant

features in sockets produced with selective laser sintering (SLS).

The production of sockets for prostheses using SLS (Figure 4.1) has several

potential advantages compared to common, more labor intensive and time-consuming

fabrication methods [22]. One of the main advantages is that SLS directly creates sockets

from digital shape information, eliminating the need for molds, hand lamination and

finishing procedures. Another advantage of SLS is the ability to create complex

geometries with minimal cost penalty in manufacturing, which significantly expands the

options for developing and exploring alternate socket designs.

i

ii

iii

ivca b

Figure 4.1: (a) and (b) A patient testing an SLS fabricated socket; (c) the description of the parts of a prosthesis for below-the-knee amputees: i – socket, ii – attachment fitting, iii – pylon and iv – prosthetic foot.

Our socket design/fabrication framework was developed through collaboration

between the Laboratory for Freeform Fabrication at The University of Texas at Austin

and the Rehabilitation Engineering Laboratory of The University of Texas Health

Science Center at San Antonio [1]. In our socket fabrication framework, we create a

computer model of the socket from laser digitization of the patient’s residual limb. The

shape of the socket is designed so that specific regions of the limb, which can safely

resist contact pressures generated when the patient walks, will support most of the load

that arises during gait. This design is called the Patellar Tendon Weight Bearing (PTB)

socket.

Other regions of the residual limb are not suitable for high contact pressures either

because they are too sensitive to pain, are susceptible to developing sores, or due to other

health issues. High pressures at these sites lead to discomfort for the patient during gait

(thus affecting rehabilitation) and, sometimes, pose a health risk for the patient. Thus, it is 38

necessary to incorporate means of relieving contact pressure in these regions. One

solution to this problem is to make the socket wall orthogonally compliant at these high

contact pressure sites.

Currently, local compliance is achieved by reducing the wall thickness of the

socket at pressure sensitive areas (Figure 4.2). However, this approach provides limited

compliance and pressure relief. A more effective design may be achieved by taking

advantage of the ability of SLS to easily create geometric variants of traditional socket

shapes to allow controlled expansion of the socket volume through deformation of the

wall at sites where pressure relief is needed, thus adapting to limb change during gait.

This chapter is structured as follows. First, the set of design constraints is defined

for the problem. Then, the methods used for design of the compliant features are

discussed as well as the material used for the fabrication of sockets. Results achieved

with the use of these methods are presented, as well as their incorporation in the socket

geometry. Finally, the achieved methodology is discussed.

39

Figure 4.2: Example of a current SLS fabricated socket with locally thinnedareas where compliance is needed.

compliantfeatures

walls in

40

2. DESIGN CONSTRAINTS

In order to design orthogonally compliant features to be incorporated into the

socket structure, a target deformation for the compliant region when subjected to the

normal steady-load pressure needs to be specified. The optimum value of the desired

displacement at the center of the compliant site is difficult to set precisely, since it

depends on several parameters that may vary from patient to patient, such as the specific

condition of the limb, existence of localized sores, circulatory issues, body fat, geometry

of underlying bones, and tissue specific sensitivity to pressure, among others. Many of

these parameters are highly subjective. From Chapter 2 and [6], the contact pressure at

sensitive sites on the limb can be evaluated and used as a guide in determining the desired

displacement values. These target displacement values are determined by the prosthetist

based on subject-specific conditions from clinical evaluation of the patient.

Second, compliant features on the walls of a socket should not compromise

structural safety during gait. Moreover, such features should not affect the overall shape

of the inside of the socket, should follow the limb shape during deformation as closely as

possible, and should have minimal impact on the socket’s outer appearance. The latter

condition, which aims at keeping the socket outer appearance close to that of a natural

leg, is subjective but important for the patient to feel positive about using the prosthesis.

Third, void areas on the socket wall, such as slots and holes, should be as small as

possible. A constant minimal pressure must be applied to all areas of the residual limb,

even to sensitive areas, so large holes in the wall should be avoided altogether. Such

features would completely eliminate contact pressure at these areas, but would also cause

the load to be redistributed to surrounding regions, mainly along the border of the hole.

41

This would lead to high localized contact pressure gradients and concentrations that

should also be avoided, since they can create circulatory problems for the patient.

Finally, compliant features should not produce spikes or sharp corners when

deformed. Hence, slots on the socket wall should not self-intersect.

3. MATERIAL AND METHODS

This section discusses the material used for manufacture of the socket and

presents the design methodology used in the development of the desired compliant

features. Several commercial software tools were used to implement the methodology.

MatLab7 was used for design calculations and optimization. Solid models were generated

using Rhinoceros8. SDRC I-deas9 was used for Finite Element Method (FEM) analysis

and design refinement. All FEM models used 10-node parabolic tetrahedral elements.

3.1) Material Properties

The material utilized in fabrication of the sockets is DuraFormTM Polyamide

(PA). This material is a type of nylon 12 processed into a powder form by 3D Systems

(Valencia, CA) for utilization in SLS machines. Our experience shows that the material

properties of DuraFormTM PA parts vary based on a number of conditions. Alterations in

build parameters lead to parts with altered properties. Furthermore, the age of the

DuraFormTM PA powder, proportion between virgin and used powder in a blend, the

number of times it has been reused, and even machine dependent maintenance issues

7 MathWorks, Natick, MA. 8 Robert McNeel & Associates, Seattle, WA. 9 EDS, Plano, TX.

such as the cleanliness of the laser lenses and heater sensors, all are factors that severely

the quality of the parts produced.

For these reasons we built tensile specimens with each compliant prototype in

order to measure the properties associated with the build conditions. All parts were

produced in a SinterStation 2000 System (3DSystems). The tensile specimens were

tested in a MTS hydraulic testing machine following the ASTM Standard D638–02a

procedures. The value for Young’s Modulus for these parts was close to 800 MPa (with a

Tensile Strength of 30 MPa) for stresses in the direction of the layer plane – which was

also the main direction of the lager surfaces of the solid models produced as they were

positioned for SLS fabrication. The value for Poisson’s Ratio used in the model was 0.4

[30].

3.2) Evaluation of General Design Alternatives

For the evaluation of various design alternatives that can be incorporated into the

socket wall and satisfy the design constraints described in the section 2, computer

modeled test discs were created in order to define a planar approximation of the geometry

of the wall around the sites where pressure relief is needed. They were dimensionally

similar to compliant areas on the socket, as shown in Figure 4.3. Geometric features such

as slots and holes would then be added to each disc to assess their ability to provide

compliance through FEM analysis, and some of these alternatives are presented next.

Figure 4.3: Test disc to evaluate compliance options (dimensions in meters)

42

The boundary conditions for FEM analysis were defined as follows: the borders

of the discs were fully restrained and a dummy test force similar to the one expected at

the contact area of the socket in a sensitive site was applied to the center of the disc.

The first disc to be analyzed was a solid control disc with no compliant feature,

which could provide a reference value. The deformation results revealed that the

displacement at the center was 5.54 x 10-5 m (Figure 4.4).

Figure 4.4: FEM deformation results (in meters) for control disc with no special compliant features.

43

For a disc with 4 larger void spaces, the central displacement reached a value of

8.52 x 10-5m (Figure 4.5), which is far from an acceptable value. In addition, there is a

potential problem with localized pressure gradients (due to contact discontinuity cause by

the large empty areas) at the borders of the holes that may affect the peripheral

circulation in the residual limb at these sites.

Figure 4.5: FEM deformation results (in meters) for test disc with 4 void spaces.

44

Another possible solution involves the addition of slots along the socket wall. Due

to the geometry of the problem in the way that the contact pressure is applied to the

socket at the relief needed sites, a concentric distribution of the slots is intuitively more

adequate to be used in the design. The first option studied was the use of straight slots

radially distributed on the discs. The number of slots tested was varied, with a maximum

of 16 in order to preserve maximum stresses under a value that could start to compromise

structural integrity. The results for the case of 16 straight slots revealed that the central

displacement achieved was 7.20 x 10-5m (Figures 4.6), which was also still far from a

suitable deformation.

Figure 4.6: FEM deformation results (in meters) for test disc with 16 radial slots.

A different option for the slot approach is curved slots that spiral towards the

center of the disc. This shape increases the influence of the slots (by increasing their

length) while still keeping them bounded within the area of the disc. The resulting shape

actually creates a curved beam (also in a spiral shape) that acts like the coil of an

45

orthoplanar spring. More than one spiral slot is advisable to ensure symmetric

deformation.

The results for designs with spiral slots are displayed in Figures 4.7 and 4.8 (two

and four radially distributed spirals, respectively). For the case with two spirals, a central

displacement of 4.38 x 10-3 m was achieved, which is very satisfactory. The case with 4

spirals, on the other hand, results in a central displacement of 8.88 x 10-3 m, which was

considered to be too large – meaning that for a smaller input displacement, a low reaction

pressure will be applied by the feature on the contacting region, which could lead to

undesirable pressure concentration in surrounding areas.

Figure 4.7: FEM deformation results (in meters) for test disc with two spiral slots.

46

Figure 4.8: FEM deformation results (in meters) for test disc with four spiral slots.

3.3) Design of spiral slot compliant feature

The design of compliant features with spiral slots is parametrized by the number

of slots, the thickness of the wall at the feature and by the area of the feature. The area

needs to remain as small as possible around the compliant region defined by the

prosthetist (as discussed in Chapter 3), or the overall fit of the socket may suffer. This

area, together with the number of slots, will define the pitch of the spiral slot, which, in

turn, defines the width of the curved beam. For instance, in the example of Figure 4.7

where there are two full turn slots, the width of the beam is half the pitch of the spiral

slot. In the case of Figure 4.8, with four slots, it is one fourth of the pitch, and so on.

47

The thickness of the wall (and of the beam) should remain close to that of the

overall socket wall to avoid adding unnecessary mass (and bulkiness) to the socket and to

maintain the poor aesthetic effect. Thus, if more stiffness is needed to refine the

compliance, this should be done using supporting structures from topology optimization,

to be described in section 3.6 of this Chapter, rather than by increasing the wall thickness.

A diagram to model the beam width and thickness for the case with two full turn

spiral slots (Figure 4.7) is displayed in Figure 4.9.. These two slots lead to two half-turn

spiral beams.

Figure 4.9: Diagram with the geometric representation of the half turn spiral beam (under a load W).

Hence, deformation at the point of application of the load (end of beam) can be

approximated by [35]:

⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅=

AGL

GKLRWf

562

( 1 )

where

48

tlA

lt

lttlK

⋅=

⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

−−⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛= 4

43

12136.3

316

22 ⎥⎦

⎤ and

R is the mean radius of curvature of beam, L is the developed arc length of the beam, G is

the torsional modulus of rigidity, l is the width of the beam, and t is the thickness of the

beam (height of cross section).

An estimation for shear stress can be calculated by [36]:

⎟⎠⎞

⎜⎝⎛ +++

⋅+⋅= 3222

5.056.02.11)2/()2/(8

)2/8.12/3(ccclt

tlTτ ( 2 )

where , and the twist moment T can be approximated by: lRc /2=

RWT 2⋅= ( 3 )

where W is the load. The results found for the parameters of the beam can be refined

analytically using formulas for computation of deformations and loads along curved

beams, as seen in Appendix 6, or by using FEM.

49

3.4) Refinement of designed discs: FEM and experimental validation

An experiment based on the one described in [30] was conducted to verify the

FEM model, which was iteratively used for the refinement of the design of the compliant

discs. The set-up is shown in Figure 4.10. The SLS test discs were tightly secured

between the flange and the main structure. An elastic rubber sheet was placed between

the disc and the main structure to prevent air leakage, both sideways and through the slots

of the discs. The whole set-up was fixed using clamps on a test table, then air pressure

was introduced into the main cylinder while the orthogonal displacement at the center of

the disc was measured using a dial indicator (Figure 4.11).

Figure 4.10: Parts used in the experiment of test discs.

Figure 4.11: Experimental set-up for the test of compliant discs.

50

51

The dial indicator had a resolution of 0.001 in, and due to friction and hysteresis

the accuracy for this measurement was determined to be 0.002 in. Also, the pressure

gauge was accurate to 0.1 psi. The rubber sheet was chosen to be as thin as possible to

eliminate the effects of compressing it against the disc. However, since it is a highly

deformable and elastic material, a significant amount of the mechanical energy provided

by the input air pressure to the system was stored in the deformation of the rubber. To

adjust the results of the discs to this effect, the displacement of the rubber sheet under the

effect of air pressure, in the absence of any test disc, was also measured, and the results

were processed to determine the pressure-displacement correlation of the rubber in the

experimental set-up, allowing the later results for the test discs to be corrected using

superposition.

The measured results and FEM predictions (obtained with boundary conditions

similar to those of the experiment) for a disc with two spiral slots are seen in Figure 4.12.

As expected, the FEM results are consistent with the measured values for small

displacements. When the deflections become large (in this case, greater than the disc

thickness itself), then non-linear effects start to appear and dominate, introducing a

discrepancy between the experimental and FEM results. This discrepancy was considered

to be irrelevant for this design, since the compliant features are expected to operate only

within the linear small-deflections range. FEM data for displacement and stresses for the

same test disc configuration, under a load of 10 kPa is shown in Figure 4.13.

-2.5

0

2.5

5

7.5

10

12.5

15

17.5

20

22.5

25

27.5

30

32.5

35

37.5

40

-1 0 1 2 3 4 5 6 7 8 9 10

Displacement at center (mm)

Pres

sure

(kPa

)

Measurement set 1Measurement set 2Measurement set 3FEM

Figure 4.12: Comparison between experimental and FEM results for the central displacement of test discs with two spiral slots.

Similarly, the same procedure was used on SLS test discs with four spiral slots.

FEM data for displacement and stresses for the test disc configuration, under a load of 10

kPa, is seen in Figure 4.14. The graph with the results for the measurements and FEM

predictions is seen in Figure 4.15. Again, for deflections smaller than the thickness of the

disc, the FEM results can be considered very satisfactory when compared to the measured

values.

52

Figure 4.13: FEM results for disc with two spiral slots and a normal load of 10 kPa: displacement (in meters) on the left, stresses (in Pa) on the right.

Figure 4.14: FEM results for disc with four spiral slots and a normal load of 10 kPa: displacement (in meters) on the left, stresses (in Pa) on the right.

53

-2.5

0

2.5

5

7.5

10

12.5

15

17.5

20

22.5

25

27.5

30

32.5

35

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Displacement at center (mm)

Pres

sure

(kPa

)

Measurement set 1Measurement set 2Measurement set 3FEM

Figure 4.15: Comparison between experimental and FEM results for the central displacement of test discs with four spiral slots.

3.5) Implementation of spiral-slot features on socket

The incorporation of the compliant features using spiral slots can be performed in

a relatively straightforward fashion. The profile curves of the spiral slots can actually be

stored in a design library within the CAD environment. Once the feature is needed, it can

54

be placed onto an auxiliary plane, scaled to the size required to cover the compliant area,

and then stamped onto the socket wall using a pass-through projection tool. The same

socket from Figure 4.2 using spiral slots for compliance is shown in Figure 4.16.

Figure 4.16: Example of orthogonally compliant features incorporated in a socket.

An outer support wall can also be added to the socket around compliant spiral-

slotted regions (Figure 4.17). This wall protects the compliance feature by providing an

end-of-course limitation to the deflection of the feature. Furthermore, if more stiffness is

needed in the early stages of deformation, the void space between the feature and the

support wall (also seen in Figure 4.17) can be filled with a cushioning material.

55

Figure 4.17: On the left, a detailed view of spiral slot compliant feature and back wall (unseen); on the right, section view showing void space between slotted wall and back wall.

3.6) Design of features with Topology Optimization for controlled compliance

In some cases, the compliant feature incorporated to the socket wall itself may

provide enough deflection range but without a sufficient reaction pressure being kept

applied to the limb from the early stages of deformation (necessary to avoid high pressure

gradients in relation to the surrounding areas where supporting pressure has been rerouted

to, as discussed in the end of section 2 of the Chapter). In this situation, a compliant

structure may be added to the feature in order to add some stiffness to the solution and

control compliance. The design of such structure may be achieved using topology

optimization.

Various approaches for topology optimization are found in the literature [39]. Due

to the nature of the design problem at hand, where the freeform fabrication process gives

almost complete freedom geometrically (albeit constrained to a spatial design domain),

56

optimization approaches tuned for discrete (finite element) domains are most appropriate.

In particular, we have applied the Homogenization Method [37,44,48], which considers a

design domain described by a 2D microstructure of unit cells, each one parametrized by

void space, with sides defined by values a and b, which vary from 0 – no material – to 1,

and orientation (defined by angle θ) (Figure 4.18). In general terms, the method

calculates homogenized elasticity coefficients for the design domain, calculates

sensitivities with FEM, optimizes for compliance with respect to a and b, filters the

results to eliminate “checkerboard” patterns, updates angle θ and repeats all steps until

convergence is achieved.

Unit cellDesign domain: microstructure

θb

a

Figure 4.18: Design domain for the Homogenization Method.

However, evaluation of optimal microstructures and their orientations is still

problematic. Because no dimensional scale is associated with the microstructures, often

the resulting structures simply cannot be built [47]. An alternative approach is the Solid

Isotropic Material with Penalization, or “power-law”, where material properties are

assumed to be constant within each element of the discretized design domain.

57

Furthermore, the variables are the relative densities of the elements, and their material

properties are estimated as: 0)( ijkl

pijkl CC ρρ = ( 4 )

where ρ is the density, p is the penalization power, and is the material tensor of the

solid. Such a material is physically feasible as long as the following condition is obeyed

for the power p in a 2D design domain [38]:

0ijklC

( )⎭⎬⎫

⎩⎨⎧

+−=≥

υυυ

14,

12max*pp ( 5 )

where υ is the value of Poisson’s ratio.

Given the design task of creating a structure that will provide extra stiffness to the

compliant feature achieved through incorporation of spiral slots into the socket wall, the

target is to minimize the compliance c of the structure. Hence, the objective function for

the problem can be formulated as follows:

[ ]( ) [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ] [ ]10

:subject to :min

min

*

≤≤<≤

=⋅

⋅=

xxVV

fdK

dKdxc T

( 6 )

where [d] is the displacement vector, [f] is the force vector, [K] is the stiffness matrix, V

is the volume of material in design domain Γ, V* is the total volume of design domain Γ,

and [x] is the design variable vector. In this case [x] stores the densities of the elements,

which are constrained to be between a positive minimum value (to avoid singularities)

and 1.

58

The definition of the design domain Γ can be simplified to a 2D domain due to the

approximate radial symmetry of the compliant feature. A radial section of a disc that

represents this approximation is displayed in Figure 4.19.

Figure 4.19: Schematic model of a cut view of a test disc: orthogonal deflection of center portion is desired, while the border is fixed.

Again, based on symmetry (through the central vertical plane), the problem can be

locally defined by the design domain Γ of Figure 4.20. The rounded corner of the grey

area is defined to ensure smoothness of the structure with respect to the fixed geometry.

Figure 4.20: Definition of design domain Γ for the topology optimization problem. Material presence is mandatory in black areas, material is forbidden in white areas, and grey areas indicate the sections to be optimized.

59

The objective function compliance can thus be rewritten in terms of a sum of the

individual compliances of each 4-noded finite element of the discretized design domain

(based on Figure 4.20) following the power-law approach:

[ ]( ) ( ) [ ] [ ]∑=

elementselementelement

Telement

pelement dKdxxc ][ ( 7 )

where p is the penalization power.

To ensure convergence the design domain must be restrained with appropriate

boundary conditions and filtering techniques must be applied. Such techniques are aimed

at resolving two numerical problems. First, the occurrence of checkerboard patterns in the

solution due to the artificially high stiffness of such checkerboard structures when

compared with ones with uniform material distribution. Second, the mesh-dependency of

the solution, which can be described by different topologies being achieved in the

solution from meshes with different degrees of discretization. These problems can be

solved with the application of an appropriate convolution operator to the element

sensitivities elementxc ∂∂ [49].

A previous work [47] presents a MatLab program to solve a problem with basic

boundary conditions (single constraint) using the Optimality Criteria. To allow for its use

with multiple constraints, the optimizer was modified to implement the Method of

Moving Asymptotes, discussed in [50]. Also, a geometric finite element mesh

representing the problem posed in Figure 4.20 was created, with modified elements to

ensure a fixed base structure and a desired degree of smoothness in the solution (so that it

can be applied directly to the socket following the design constraints established in

section 2 of the Chapter). An example of this modified code is found on Appendix 7.

60

The resulting structures for different input values of the initial volume fraction

V/V* are presented in Figure 4.21. By altering this value, the resulting compliance of the

structure can be changed. The displacement results of the FEM model of the test disc

with 2 spiral slots, with a compliant structure added to it, are shown in Figure 4.22. The

reaction pressure for a central deflection of approximately 4 mm was 75 kPa, 7.5 times

the reaction pressure for this deflection in Figure 4.14 (10 kPa).

The compliant structure can also be constructed in a different way, for greater

stiffness, by revolving the structure resulting from topology optimization on top of the

compliant area, taking advantage of its radial symmetry. The model for this version (on a

slotted test disc) is displayed in Figure 4.23.

vol. fraction = 0.3, compliance = 15.2 vol. fraction = 0.4, compliance = 9.6

vol. fraction = 0.5, compliance = 7.8 vol. fraction = 0.6, compliance = 6.5

Figure 4.21: Results for topology optimization of orthogonally compliant test disc for different input volume fractions, and their theoretical compliances.

61

Figure 4.22: FEM results for test disc with compliant structure: on the left, displacement (in meters); on the right, stresses (in Pa).

Figure 4.23: On the left, implementation of compliant structure on top of test disc; on the right, a section view of this model.

The topology optimization code was also applied to a design domain with a

curved boundary (as seen in Figure 4.24) to evaluate the effect of curvature (which will

be present on the socket wall) on the final optimized topology. The resulting structures

did not present a significant topological difference (Figure 4.25).

62

Figure 4.24: Definition of design domain Γ for the topology optimization problem with curved domain to be optimized.

Figure 4.25: Result for topology optimization of orthogonally compliant curved domain.

The implementation of the compliant structure on the socket can be performed

with a sequence of well-defined CAD operations, adapting it to the outer surface of the

socket for a smooth transition. The complete compliant feature incorporated in the socket

geometry of Figure 4.16 is displayed in Figure 4.26.

63

Figure 4.26: Implementation of compliant feature on socket.

4. CONCLUSION

This work developed the strategies for the design of orthogonally compliant

features for SLS-produced transtibial prosthetic sockets. The design constraints were

discussed, and a procedure for achieving the target compliance under these conditions

was established. In particular, given the design constraints associated with the target

application of providing local compliance to prosthetic sockets, general design solutions

implemented in disc coupons were evaluated with FEM and the results validated through

experimental testing. The approach considered more adequate, involving spiral slots, had

its design parameters studied and defined. A parallel form of controlling compliance

using auxiliary structures obtained through topology optimization was also studied and

implemented.

64

65

One limitation of this work is that the analysis methods assume a static load.

However, the loads in a real socket are cyclic in nature. Thus, our approach must be

modified in the future to consider fatigue of the socket. This will require experimental

determination of the fatigue strength of SLS parts produced with DuraformTM. The

durability of compliant features can then be included as a design parameter.

Another approach for providing compliance in sockets is varying the porosity

within the SLS build. As described in section 3.6 of this Chapter, the variables for

optimization of the objective function are the densities of the elements in the design

domain. In the current work, the final target value for each element density is either the

minimum value (for no material) or one (for the presence of material). However, since it

may be possible to change the density of the part (by introducing porosity) during the

SLS build through local variation of build parameters, other target values for element

density might be acceptable in the topology optimization, leading to more alternatives for

compliant structures.

Our goal in this research is to establish a rapid, customized fabrication process for

prosthetic sockets. In this vision, a practicing prosthetist would define a compliant socket

and transmit electronic data (STL files) to an SLS service bureau. The completed socket

would then be delivered to the prosthetist for patient fitting in a matter of days (instead of

weeks or months, as in current practice). This process will require a targeted CAD

environment that allows a prosthetist to incorporate compliant features, possibly from a

library of features, into the socket design without requiring intensive CAD training. The

prosthetist should be able to analyze the design and receive appropriate feedback on its

performance. Development of this environment is the subject of ongoing research.

66

Chapter 5: INVESTIGATION OF QUASI-STATIC RESPONSE OF COMPLIANT PROSTHETIC SOCKETS FOR TRANSTIBIAL

AMPUTEES USING FINITE ELEMENT METHOD

ABSTRACT

Finite element method (FEM) is a very powerful tool for studying the behavior of

structures, especially when their geometry and mechanics are too complex to be modeled

with analytical methods. This article focuses on the analysis of patellar-tendon bearing

prosthetic sockets with compliant features that were incorporated to relieve contact

pressure between the residual limb and socket. We employed a FEM model of the system

composed of a socket, liner and residual limb under quasi-static loading derived from

ground reaction forces. The geometry of the residual limb was acquired from CT data of

a transtibial amputee. Three different compliant designs were analyzed using finite

element analysis to assess their ability to relieve local pressure at the fibula head during

normal walking and the structural integrity of the socket to ensure safety. The compliant

features consisted of thinned walled sections and variations of spiral slots integrated

within the socket wall, which were found to produce an average reduction in the interface

pressure of 65.8% (peak pressure was reduced from 172 kPa to 66.4 kPa) compared to

the pressure within a socket with no compliant features.

1. INTRODUCTION

67

The design of patellar-tendon bearing (PTB) prosthetic sockets for transtibial

amputees has been improved with the integration of state-of-art freeform fabrication

techniques. These techniques allow the socket to be designed within a computer-aided

design (CAD) environment directly from digital shape information of the patient’s

residual limb, ensuring an effective design and swift fabrication of the sockets while

eliminating the need for extensive hand procedures involved in the production of

conventional sockets (as discussed in Chapter 3). Furthermore, this approach allows

flexibility for the designer to incorporate compliant features to enhance socket comfort by

relieving high pressures at specific sensitive sites of the residual limb. However, in order

to design efficient compliant features and evaluate their structural and dynamic behavior,

appropriate load transfer information between the socket and residual limb is necessary.

Estimating a priori the load transfer between the human body and the prosthetic

elements in terms of magnitude and distribution over time is a significant challenge in the

design of prosthetic devices. A common procedure for such analyses is to utilize load

transfer data measured during clinical trials using instrumented prostheses. However,

when one of the design objectives is to change, either locally or globally, the load transfer

itself, then a potentially costly and time consuming iterative design process is required: a

prototype needs to be manufactured and instrumented to measure the resulting load

transfer, which in turn is used as the new load case for redesign, and the process is

repeated until convergence is achieved.

A promising solution for estimating the load transferred to the socket is to use

finite element method (FEM) analysis to model the interaction. The FEM functions as a

bridge to combine data from ground reaction force measurements with complex geometry

and material data to determine the load distribution on the socket structure. Thus, FEM is

a very powerful approach for evaluating different socket designs.

68

The development of a FEM model of the lower extremity of a transtibial amputee

is, however, a challenging task that has been progressing for almost two decades.

Comprehensive surveys on early models and approaches can be found in [14] and [18].

Important criteria to be considered when using FEM in such applications include: 1)

defining the material properties of the soft tissues, which exhibit highly non-linear and

non-uniform behavior; 2) modeling the contact interface between the residual limb and

the socket liner and between the liner and the socket; 3) modeling the pre-stresses that

arise due to wearing the socket. Recent work by two different research groups ([12,13,16]

and [17,19]) addressed some of these questions. Both groups focused on the clinical

study of the interface pressure over the soft tissues by modeling the socket as either a

completely rigid and restrained structure, or as a uniform thickness shell with a constant

stiffness associated with it, instead of concentrating on the socket structure and its

topology.

The overall objective of this work was to focus on the analysis of the structural

response of four different socket designs using FEM. Three of these designs include

compliant features integrated within the socket wall to relieve pressure in the area of the

fibula head, which is an area where discomfort and pain due to usage of a PTB prosthesis

are commonly reported by patients. The sockets were designed to be fabricated using a

form of Solid Freeform Fabrication method called Selective Laser Sintering [51] with

DuraformTM PA. Thus, the specific objectives of this study were to use an FEM model of

the residual limb, liner and socket to evaluate: 1) the stresses within the socket wall, 2)

the resulting deformation of various concepts for integrating compliance within the

socket wall, and 3) the resulting pressure relief provided by these compliant features at

the fibula head.

69

2. MATERIAL AND METHODS

The approach used for the analysis involved the generation of finite element

models from computed tomography (CT) data acquired from the residual limb of a

transtibial amputee patient wearing the prosthesis and liner. Three geometrically similar

models of the prosthetic socket were also created, with different designs of compliant

features integrated within each socket. The original FEM model was first subjected to a

simple downward force of 800 N applied at the knee level for FEM model validation

against contact pressure values reported in the literature. In subsequent analyses, the

models were subjected to quasi-static loading boundary conditions obtained from ground

reaction force (GRF) data. From the FEM results, the structural stresses in the sockets

were computed, as well as the contact pressure at the fibula head and the displacement of

the compliant feature for the compliant sockets. The local deformations at the compliant

features were defined as the difference between the displacement at the center of the

compliant feature and the average displacement from the periphery of the feature.

2.1) Generation of FEM from CT data

The source for the shape information of the residual limb bones, tissues and inner

socket wall was taken from CT data of an amputee. The subject was a 70 years old male,

1.82 m tall and weighing 90 kg, diabetic and relatively active. The CT scan was adjusted

to produce one image every 2 millimeters. At the time of the scan, the patient was

wearing a Pelite liner around his residual limb and wearing a PTB socket at the time of

the CT scan. Thus, the shape of the limb was inherently altered to adjust to the local

modifications of the inner socket shape that defines the PTB method for socket design.

During the scan, the socket bottom was supported using a small force to assure the socket

remained fixed during the CT scan. The CT data was converted into separated point

70

clouds for each main surface area (bones, skin, outer liner/inner socket) using Mimics

(Materialise10). Each point cloud was then processed using Raindrop Geomagic11 in order

to construct a CAD surface model. This data processing resulted in a CAD model

composed of structured first-order continuous four-sided NURBS patches. These surfaces

were then organized into solids using Rhinoceros12, which were imported into I-deas13 for

meshing and FEM analysis. The boundary conditions between the different structure

volumes (i.e., bone, tissue, liner and socket) were defined as allowing zero displacement.

Hence, the contact interface was modeled with zero frictional slip or displacement

between structures. The volumes were meshed using the I-deas automated mesh

generator using 10-node parabolic tetrahedral and the Delaunay mesher option. The

resulting meshes were checked for distorted or stretched elements that may lead to

numerical errors in the FEM analysis. When such elements were identified, they were

locally modified to reduce their distortion.

The resulting models of the bones (distal femur, patella and remaining fibula and

tibia), soft tissues and liner had 8625 elements and 14680 nodes, 19888 elements and

32696 nodes, and 10342 elements and 19482 nodes, respectively (Figure 5.1).

Four socket models were designed from the inner PTB socket shape acquired

from the CT data: one with a constant 6 mm wall thickness, and three with integrated

compliant features. Three compliant feature were designed to relieve pressure at the

fibula head (as discussed in Chapters 3 and 4): 1) a locally thinner wall (1.5 mm), 2) plain

spiral slots and 3) spiral slots with a back compliant support to increase stiffness. The

FEM socket models had 6516 elements and 12657 nodes, 5364 elements and 10644

10 Leuven, Belgium. 11 Research Triangle Park, NC. 12 From Robert McNeel & Associates, Seattle, WA. 13 From EDS, Plano, TX.

nodes, 11100 elements and 21528 nodes, and 13909 elements and 26773 nodes,

respectively (Figure 5.2).

b ca

Figure 5.1: FEM model of (a) bones, (b) soft tissues and (c) liner.

71

cb a

d

Figure 5.2: FEM models of the four different socket designs simulated: (a) plain socket with no compliant feature; (b) socket with thin-walled compliant feature; (c) socket with spiral slots for compliance; (d) socket with spiral slots and compliant support.

72

All structural materials used in the analysis were assumed to be linearly elastic,

homogeneous and isotropic. For the bones and socket (fabricated with DuraformTM

PA14), the Young’s moduli were 10 GPa and 1.6 Gpa, and Poisson’s ratio were 0.3 and

0.39, respectively. For the Pelite liner, Young’s modulus was 380 kPa and Poisson’s ratio

was 0.39 [16]. The Young’s modulus for the soft tissues was assumed to be 200 kPa [13].

However, to account for the pre-stresses due to wearing the socket and the typical

characteristic of the soft tissues of stiffening when compressed [54,55], the Young’s

modulus value in areas where such pre-stresses are expected in the PTB socket were

slightly increased, to a maximum of 250 kPa on the patellar-tendon region. Poisson’s

ratio for the soft tissues was 0.49 [12].

2.2) Boundary Conditions

Initially, the load values used for the boundary conditions were modeled as a

simple downward vertical force of 800N (for the plain socket only) so that the FEM

analysis could be validated against results found in the literature. Afterward, for all

socket designs, the loading conditions were quasi-static approximations of the

experimental forces and moments measured using a force platform during clinical trials

over the stance phase. These forces and moments were then transferred from the

laboratory reference frame to the reference frame fixed at the top surface of the bones

(Figure 5.3). Six points from the stance phase of GRF data were used.

14 http://www.3dsystems.com/products/datafiles/lasersintering/datasheets/DURAFORMHigh.pdf

Figure 5.3: Free-body diagram of a prosthetic leg, where the plane of the drawing (x-y) is the sagittal plane.

The set of quasi-static equations that transfer the load from the reference frame of

GRF data, which is the contact point C on the ground, to the application point A at the

knee joint level of the bone, is given by: ( )

ZAZ

yXAY

yXAX

FF

cosFsinFF

sinFcosFF

−=

θ−θ=

θ+θ−=

( ) lsinFcosFMM

lFM

yXAZ

AY

ZAX

⋅θ+θ−=

=⋅=

0 ( 1 )

where FX, FY and FZ are the ground reaction forces at point C; FAX, FAY, FAZ, MAX, MAY

and MAZ are the forces and moments applied to the bone at point A; l is the perpendicular

distance between points A and C; and θ is the angle between the axis of the prosthesis

and the direction normal to the ground (for simplification purposes, such angular

displacement was considered to occur only within the sagittal plane).

73

The final boundary condition set for the FEM analysis was a complete

displacement restraint of the bottom of the socket, where the pylon of the prosthesis is

normally attached.

3. RESULTS

3.1) FEM results for vertical 800 N load

The first set of results (Figure 5.4), evaluated the pressure distribution on the soft

tissues, with the limb wearing the plain socket, under a simple downward vertical load of

800 N. The maximum pressure was computed at the patellar tendon (250 kPa). The value

found at the popliteal depression for this same the same loading conditions was 109 kPa.

a b

109 kPa

205 kPa 250 kPa

Figure 5.4: (a) Anterior and (b) Posterior view of pressures on the residual limb using the plain socket under a downward vertical load of 800 N.

74

3.2) Socket stresses, pressure at fibula head and deformation of compliant features using quasi-static loading conditions from stance phase

For the FEM simulations using the quasi-static loads during the stance phase,

three sets of results were obtained. First, the structural behavior of the three compliant

sockets was assessed with respect to the stresses produced within the models. The largest

stresses occurred at 35% of the stance phase. The largest stresses were, in all cases,

encountered within the compliant features (Figure 5.5). For the socket with a thinner

walled region, the largest stress was 16.9 MPa (at the anterior most portion of the thinner

region); for the design with spiral slots, 44 MPa (concentrated at the bottommost

beginning of the slots); for the design with supported spiral slots, 35 MPa (also at the

bottommost beginning of the slots).

Peak stresses

Figure 5.5: Structural FEM results displaying the stresses (in Pa) on each compliant socket at 35% of stance, where the largest stresses were encountered.

75

76

Next, the contact pressure at the fibula head was identified. The maximum

pressure occurred at 75% of stance for all designs (Figure 5.6a). For the plain socket, the

maximum pressure was 172 kPa; for the thinner wall design, 135 kPa; for the spiral slot

design, 66.4 kPa; and for the supported spiral slot design, 78.5 kPa. The average pressure

relief (compared to the plain socket) computed between 20% and 80% of stance was

27.4%, 65.8% and 58.5%, respectively.

Finally, the local deformation was computed. The largest deformations for all

socket designs occurred at 35% of stance (Figure 5.6b). At this instant, the computed

local deformation at the socket with a thinner walled region was 1.34 mm; the socket

with spiral slots, 3.73 mm; and at the socket with supported spiral slots, 3.55 mm.

Pressure at Fibula Head

0

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80 100

% of stance

Pres

sure

(kPa

)

Thinner Wall

Spiral Slots

Supported Spiral Slots

No Feature

a

Local Deformation at Compliant Feature

0

0.5

1

1.5

2

2.5

3

3.5

4

0 20 40 60 80 100

% of stance

Def

orm

atio

n (m

m)

Thinner Wall

Spiral Slots

Supported Spiral Slots

b

Figure 5.6: Comparison of resulting FEM values for (a) pressure at fibula head and (b) local deformation at compliant features among the different socket designs simulated.

77

78

4. DISCUSSION

The ability created by FEM to simulate a priori the structural response of

prototype socket designs gives the designer the ability to optimize the solution both

globally and locally within the CAD environment, without the need for time consuming

and costly intermediate prototypes and experimental measurements. However, providing

realistic boundary conditions is critical to the accuracy of FEM analyses. As such, a

model of the residual limb was incorporated into the analysis. This allowed an improved

representation of the load transfer between the residual limb and the socket through the

geometry of the mating structures. A few aspects of the results and the model of the

limb/liner set should be discussed.

The first set of results obtained with a downward vertical 800 N load (Figure 5.4)

was generated to permit an overview of the contact pressure distribution on the skin,

which was consistent with the theoretical distribution in a PTB socket [4]. Also, the

computed contact pressures at typical sites (such as the patellar tendon and the popliteal

depression) produced by the FEM model were within the commonly reported range

encountered in previously published works (both FEM based [12,13,16,17,19] and

experimental [6,15]).

Nevertheless, some approximations were judged necessary to create a

computationally efficient FEM model and require comment and justification. Primarily,

the soft tissues and the liner possess nonlinear, viscoelastic characteristics, rather than the

linear approximations used in this study. However, linear approximations were used

because non-linear material formulations increase exponentially the complexity of the

FEM model. In addition, it is difficult to identify a consistent formulation for the soft

tissues. Previous studies propose a Mooney-Rivlin formulation for these materials

79

[52,53]. The implementation of this approach and the assessment of the sensitivity of the

results with respect to this change remain as potential future work.

The contact between the soft tissues and the liner, and between the liner and the

socket were modeled with no sliding or gaps allowed. Since the present study focuses on

the stance phase of gait, the gapless assumption can be considered valid for most of the

high load portion of the stance phase (20% to 80% stance). As for the no sliding

assumption, previous work has shown that the use of automated contact elements in FEM

analysis of transtibial amputees with a “no frictional slip” model underestimates the mean

normal contact stress by 16% (and the peak normal stress by 2%) when compared to a

model with frictional slip (with the coefficient of friction equal to 0.675), while

overestimating the mean shear stress by a factor of two [19]. Although the shear stresses

are important in the clinical study of socket/limb interface mechanics, this is not the focus

of our work, which does concentrate on the structural behavior of the socket and the

effects of compliant features on normal contact stresses in specific areas of the soft tissue.

Thus, the present no slip model was considered adequate for this study.

The quasi-static approximation for the loading in the boundary conditions also

requires some discussion. An approximation model to account for dynamic effects in the

load formulation from GRF data was recently proposed [13], showing that the main

vertical load derived from FY would not be influenced at a significant level during the

stance phase. The horizontal force derived from FX would be increased by a maximum of

19% (at 35% of stance). Since this component of the force accounts only for a small

portion of the total force, it would have little effect on the normal contact stresses.

However, the moment MAZ would be increased by a maximum of 27% (at 75% of

stance), which would be enough, at that instant of stance, to decrease the normal stresses

80

by 50% at the patellar tendon and increase it by 14% at the popliteal depression. Hence,

including such dynamic effects in the current study is another area of future work.

One concern with the generation of FEM models is the need to expose the

amputee to CT radiation. However, the CT data was necessary to generate the residual

limb model, and an opportunity for that occurred when the patient used in this study had

to go through a CT scan for clinical reasons. One alternative would be to use MRI data

instead, which nonetheless is not always feasible primarily due to lack of availability and

cost. In the literature, different solutions have been proposed including using cylindrical

and conical solid primitives to approximate the structure of the bones [53]. However,

these estimations were quite general and do not accurately represent the residual limb

geometry. Another alternative would be to store, within a CAD environment, a set of

possible geometric variants of the topology of the bones acquired from existing CT data

and, whenever needed, scale them to match, as close as possible, direct measurements

taken at the residual limb. Development and validation of such a finite element model

remains an area of future work.

Still, the work presented here is considered an important advancement in the

structural comparison of different socket designs, as well as their design refinement. In

the designs analyzed, the ability to relieve contact pressure at the specific site of the

fibula head using spiral slots is considerably greater than that of the socket with a thinner

walled region (Figure 5.6a). Conversely, the solution with spiral slots alone reaches a

stress concentration value that approaches the tensile strength of the DuraformTM PA

material (44 MPa). Thus, the design with supported spiral slots, although allowing a

slightly lower compliance, will ensure the structural integrity of the socket is preserved.

81

Chapter 6: CONCLUSION

1. SUMMARY

This dissertation presented the development of a framework for modeling and

fabricating prosthetic sockets using SLS technology. In order to define the loading

conditions needed for integrating compliant features into the socket, the interface

pressure at the fibula head and the distal tibia (two sites where compliance is commonly

needed) during a walking test and a step up/step down test using thin-film tactile pressure

sensors was measured. The average maximum pressures obtained at the fibula head and

distal tibia were 120 kPa and 75 kPa, respectively, for the walking test. For the step

up/down test, the values for the peak pressures at these sites were 125 kPa and 90 kPa,

respectively. Both results were considered consistent throughout the trials, with

deviations of 2.3% for the walking test and 6% for the step up/down test. Once the

loading conditions were established, a design and manufacturing framework to produce

subject-specific sockets made of DuraformTM PA material using SLS was presented. The

framework included establishing an overall socket design (using the patellar-tendon

bearing approach), performing structural analysis using the Finite Element Method

(FEM) to ensure reliability during patient use, and validating the results by comparing the

model output with experimental data. The validation included quantifying the failing

conditions of sockets manufactured using the framework during bending moment and

compression tests. The experimental failure loads for the socket were within 3% of the

FEM results and were hence considered satisfactory. Next, the design of orthogonally

compliant elements to be integrated into the socket designs was analyzed. The design

82

process included the preliminary evaluation of design alternatives using FEM, validation

of FEM results through experimental measurements, definition of a design methodology

for the alternative considered most appropriate, incorporation of the solution within the

socket geometry and refinement of the design solution using auxiliary features obtained

through topology optimization. The design alternative considered most appropriate to

provide compliance involved the incorporation of spiral slots in the socket wall. The

stiffness of this feature could be refined using compliant structures with optimized

topology. Finally, to investigate the structural response of the SLS socket during gait, a

FEM model of the system composed of a socket, liner and residual limb under quasi-

static loading derived from typical ground reaction forces was employed. The geometry

of the structures involved was acquired from CT data of a transtibial amputee. Three

different compliant designs were evaluated to assess their ability to relieve local pressure

at the fibula head during the stance phase of gait, as well as their structural integrity to

ensure safety. The design with a compliant feature consisting of spiral slots within the

socket wall was determined to produce an average relief of 65.8% in the interface

pressure at the fibula head, reducing the peak pressure from 172 kPa to 66.4 kPa, when

compared to the pressures taken from a socket with no compliant feature.

2. FUTURE WORK

The underlying goal of this work was to provide an efficient framework for

prosthetists to produce effective, subject-specific sockets. However, the current

framework involves many steps and experience using CAD and FEM analysis software.

Future work will be directed at providing an integrated user-friendly system that

automatically performs all the CAD operations needed to produce the solid model of the

83

socket, including thickening of the wall, incorporation of compliant features and bottom

pylon fitting attachments. Such a task is very feasible, since the involved parameters can

all be easily stored in a database within the program.

One limitation of this work was that the analysis methods assumed static loading

conditions. However, the loading conditions in vivo are dynamic and cyclic in nature.

Future work should be directed at investigating socket fatigue associated with such

cyclical loading. This will require experimental determination of the fatigue strength of

DuraformTM, which will allow the durability of compliant features to be included as a

design parameter.

To further improve the socket compliant features, a fruitful area of research may

be to vary the porosity within the SLS build. As described in Chapter 4, the density of the

resulting compliant feature was used as the objective function for the topology

optimization. In the present work, the final target value for local density was either the

minimum value (for no material) or one (for the presence of material). However, since it

may be possible to change the density of the part (by introducing porosity) during the

SLS build through local variation of build parameters, other target values for density may

be acceptable in the topology optimization. Different density values would lead to

additional alternatives for the compliant structures.

Further development of the residual limb, liner and socket interface

approximations presented in Chapter 5 is also needed. For instance, an implementation of

the Mooney-Rivlin formulation for the non-linear soft tissue material properties and liner

may be performed in order to assess the sensitivity of the results with respect to

alternative formulations. In addition, the inclusion of dynamic loading conditions for the

boundary conditions may be implemented.

84

Finally, the concern of patient exposure to CT radiation in order to generate the

residual limb FEM model should be addressed. One alternative would be the use of MRI

data instead. Another alternative, more cost efficient, would be to store, within a CAD

environment, a set of possible geometric variants of the topology of the bones acquired

from existing CT data. Then, the most similar bone set could be scaled to match direct

measurements taken from the residual limb. Development and validation of such a Finite

Element model remains as a potential future project.

Appendices

APPENDIX 1: PATELLAR-TENDON BEARING (PTB) SOCKET RECTIFICATIONS

Patellar tendon

Anterior compartment

Distal fibula

Fibular shaft

Fibula head

Common peroneal nerve

Medial tibia

Anterodistal tibia (kick point)

Figure A1: Areas of the lower limb where rectifications are needed for the PTB socket: darker red areas are appropriate for weight bearing, while lighter red areas must have pressure relieved [57].

85

APPENDIX 2: TYPICAL GAIT CYCLE

Figure A2: Typical gait cycle illustrating the nomenclature of its phases and main events [56].

86

APPENDIX 3: VALIDATION OF SOCKET WALL THICKNESS ESTIMATION USING FEM

Using initial approximation for the overall wall thickness of the socket shown in

section 2.3 of Chapter 3, a result of 6 mm was obtained for the wall thickness in the

example of Figure 3.2. This result can be evaluated and fine-tuned using a thin-shell FEM

model, where 8-node quadrilateral elements were used and similar load values for

pressure (220 kPa) and shear stress (50 kPa) were applied as boundary conditions. The

FEM result for this analysis shows a maximum displacement equal to 0.676 mm which is

close to the desired value for maximum radial expansion of the socket (1% of its radius)

to preserve fit. Also, the resulting FEM prediction for maximum stress (9.42 MPa) was

within the range tolerated by the DuraformTM PA material (Figure A3).

a b

Figure A3: (a) FEM results (in meters) for displacement; (b) FEM results for stresses (in Pa).

87

APPENDIX 4: CONSIDERATIONS ABOUT THE REUSE OF DURAFORMTM PA POWDER

The material properties of DuraformTM PA may differ a lot from the published

values in different conditions. Variations in build parameters lead to parts with altered

properties. Furthermore, the age of the DuraformTM powder, the number of times it has

been reused, and even machine dependent maintenance issues such as the cleanliness of

the laser lenses and heater sensors are all factors that may dramatically influence in the

quality of the parts. To evaluate the impact such powder age in the mechanical properties,

some tensile specimens were in an MTS hydraulic testing machine. These specimens

(Figure A4) followed the ASTM Standard D638 – 02a – and so did the testing

procedures.

Figure A4: Tensile specimens used to evaluate the material properties of older DuraformTM powder.

The results varied dramatically, depending on the source of the part, with the ones

built with virgin powder being the closest to the published values. But for older, many

times reused DuraformTM powder, some properties were highly degraded, as low as

shown in Table 1. This is an important consideration to be made, since it is common

88

89

practice, due to the cost of the virgin DuraformTM, to reuse the powder several times.

Thus, caution must be exercised with such practice, aiming to at least abide to a blending

proportion of 1/3 virgin powder, 1/3 from part cake and 1/3 from overflow bins [33].

Experience shows that, if no blending is performed, the material properties of the parts

(not to mention other geometric issues such as variable shrinkage and surface

imperfections) will degrade significantly after 6-8 uses of the powder.

Result\Direction of Stress Any direction within

layer plane Normal to layer plane

Modulus of elasticity 255 MPa 160 MPa Tensile strength 27 MPa 8 MPa Elongation at break 15% 5%

Table A: Material properties achieved from tensile test of DuraformTM PA specimens made from powder used more than 10 times.

90

APPENDIX 5: DETERMINATION USING FEM OF THE COMPLIANT WALL THICKNESS IN THE SOCKET WITH THINNER WALLED SECTIONS

The FEM model of the thinner walled sections of the socket was iteratively used

to help determine the needed wall thickness to achieve the target displacement. The FEM

model was generated from the partition of the initial socket CAD model where

compliance is needed, and an initial guess of 1 mm for the wall thickness was assumed.

The 10-node parabolic tetrahedral elements were generated using a Dalaunay-based

mesher available in SDRC I-deas. For the case of the socket from Chapter 3, the desired

socket deformation at the fibula head was set to be around 1 mm. FEM results for stresses

and reaction forces, obtained with the imposition of this orthogonal displacement at the

center of the model while restraining the border, are shown in Figure A5. The predicted

stresses are within the range tolerated by the material (23.3 MPa). The reaction forces, if

averaged over the application area, can be translated into a pressure of 50 kPa, which is

the value needed to produce such deformation in the model. In Chapter 2, the pressure

measurement results show that such value is within the range of the pressure gradient

measured surrounding the fibula head. Thus, the value of the socket wall was considered

satisfactory.

a b

Figure A5: (a) FEM stress results (in Pa) for the thinner walled section of the socket; (b) Results for reaction forces (in N).

91

APPENDIX 6: CURVED BEAM ANALYSIS

To check the accuracy of the initial approximation for the dimensions of the half-

turn spiral beam of the orthoplanar spring, used in the design of compliant features with

spiral slots (section 3.3 of Chapter 4), the following equations were derived from the

general equations of a curved circular beam [31]. The boundary conditions are

summarized in Figure A6, for a half-turn beam. Similar equations can be obtained for

other cases.

Figure A6: Diagram based on Figure 4.9, with the addition of distributed load w.

The shear force, bending moment and twist moment, respectively, are given by:

( )( )( ) )sin()cos1(cossin

)cos1(sinsincos2

2

xxRwxRWxTxMxT

xRwxRWxTxMxMxRwWxV

AA

AA

−⋅−−⋅−+=

−⋅−⋅−−=

⋅⋅−−=

The linear and angular displacements are:

92

( )

( )

( ) ′−−+=

′−−+=

′−−++=

9

3

9

2

87

6

3

6

2

54

3

4

3

3

2

2

1

2

FEI

wRFEI

WRFEI

RTFEI

RMx

FEI

wRFEI

WRFEI

RTF

EIRM

x

FEI

wRFEI

WRFEIRT

FEI

RMyxy

AA

AA

AAA

ψ

θ

where

)cos22sin(2

1

)2

cos1(

)sin(

)sincos(2

1

)cos1(sin2

1

31

31

2

3

23

2

1

xxxF

FxxF

FxxF

xxxF

xxxF

+−+

=′

′−−−=′

−−−=

−+

=

−−+

=

β

β

β

β

ββ

57

36

16

5

4

sin2

1

sin2

1cos2

1

FFFF

FF

xxF

xxxF

==′=

+−=

−+

+=

β

ββ

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

−−⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛=

=

=

′=′=

−+

+−=

4

43

3

319

29

8

12136.3

316

22

121

sin2

1cos2

1

lt

lttlK

ltI

GKEIFF

FF

xxxF

β

ββ

93

The following boundary conditions can be extracted from the supports:

( )( ) 00

00====

ψψθθ

A

A

( )( )( ) 0

00

======

πψψπθθπ

B

B

B yy

The description of the variables is as follows:

W: load at the beginning of the beam (point A);

w: distributed load;

x: angle from point A;

R: mean radius;

V: shear force;

M: bending moment in the plane tangent to the curved axis;

T: twist moment in the plane normal to the curved axis;

y: out-of-beam-plane displacement;

θ : slope of the beam axis in the plane of moment M;

ψ: roll of the beam cross section in the plane of twist moment T;

E: Young’s modulus;

G: Torsional modulus of rigidity;

l: width of the beam;

t: thickness of the beam (height of cross section).

94

95

The last three boundary conditions, applied to the equations of linear and angular

displacements, lead to a three by three system. This system, if resolved, defines the close

approximation of vertical displacement y at point A (which would be the deflection of the

center of the compliant feature) as well as the reaction moments needed to build the

bending moment and twist moment diagrams from their corresponding equations for

stress analysis.

96

APPENDIX 7: MATLAB CODE FOR TOPOLOGY OPTIMIZATION

Below the Matlab code used for Topology Optimization, discussed in section 3.6

of Chapter 4, is presented (implemented loads and boundary conditions follow the

schematics of Figure 4.24):

% TOPOLOGY OPTIMIZATION CODE function TopOpt(nelx,nely,volfrac,penal,rmin); % INITIALIZING VECTOR OF DENSITIES x(1:nely,1:nelx) = volfrac; % INITIALIZING THE NODE VECTOR AND PASSIVE ELEMENTS for ely = 1:nely for elx = 1:nelx iter=elx+(ely-1)*nelx; n1 = (nely+1)*(elx-1)+ely; n2 = (nely+1)* elx +ely; edofvect(1,8*iter-7:8*iter)= ... [2*n1-1 2*n1 2*n2-1 2*n2 2*n2+1 2*n2+2 2*n1+1 2*n1+2]; % INITIALIZING REGIONS WITH MATERIAL if (((elx <= 40)&((ely >= 25)&(ely <= 36)))|((elx >= 91)&(ely >= 37))) passive_s(ely,elx) = 1; x(ely,elx) = 1.0; else passive_s(ely,elx) = 0; end % INITIALIZING EMPTY REGIONS if (((elx <= 90)&(ely >= 36))|... ((((elx - nelx/2)^2/(nelx/2)^2 + (ely - nely + 12)^2/(nely - 12)^2) > 1)&(elx > (nelx/2))&(ely < (nely - 12)))|... ((((elx - 40)^2/(50)^2 + (ely - nely + 12)^2/(12)^2) < 1)&(elx > 40)&(ely < (nely - 12)))) passive(ely,elx) = 1; x(ely,elx) = 0.001; else passive(ely,elx) = 0; end end end loop = 0; change = 1.; % START ITERATION for loop = 1:100 loop = loop + 1; xold = x; % FE-ANALYSIS [U]=FE(nelx,nely,x,penal,edofvect); % OBJECTIVE FUNCTION AND SENSITIVITY ANALYSIS [KE] = lk; c = 0.; for ely = 1:nely for elx = 1:nelx n1 = (nely+1)*(elx-1)+ely; n2 = (nely+1)* elx +ely; dc(ely,elx) = 0.; for i = 1:30 Ue = U([2*n1-1;2*n1; 2*n2-1;2*n2; 2*n2+1;2*n2+2; 2*n1+1;2*n1+2],i); c = c + x(ely,elx)^penal*Ue'*KE*Ue; dc(ely,elx) = dc(ely,elx) - penal*x(ely,elx)^(penal-1)*Ue'*KE*Ue;

97

end end end % FILTERING OF SENSITIVITIES [dc] = check(nelx,nely,rmin,x,dc); % DESIGN UPDATE BY THE MMA METHOD [x] = MMA(nelx,nely,x,volfrac,dc,passive,passive_s); % PRINT RESULTS change = max(max(abs(x-xold))); disp([' It.: ' sprintf('%4i',loop) ' Obj.: ' sprintf('%10.4f',c) ... ' Vol.: ' sprintf('%6.3f',sum(sum(x))/(nelx*nely)) ... ' ch.: ' sprintf('%6.3f',change )]) % PLOT DENSITIES colormap(gray); imagesc(-x); axis equal; axis tight; axis off;pause(1e-6); end %%%%%%%%%% MMA CRITERIA UPDATE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [xnew]=MMA(nelx,nely,x,volfrac,dc,passive,passive_s) xlow=0.001; xhigh=1; L=x-0.1*(xhigh-xlow)*ones(nely,nelx); high=(x-L).^2.*-dc./(xlow-L).^2; low=(x-L).^2.*-dc./(xhigh-L).^2; l2 = min(min(high)); l1 = max(max(low)); for i=1:50 lmid = 0.5*(l2+l1); xnew=max(xlow,min(xhigh,L+abs(x-L).*sqrt(-dc./lmid))); xnew(find(passive)) = 0.001; xnew(find(passive_s)) = 1.0; if sum(sum(xnew)) - volfrac*nelx*nely > 0; l2 = lmid; else l1 = lmid; end end %%%%%%%%%% MESH-INDEPENDENCY FILTER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [dcn]=check(nelx,nely,rmin,x,dc) dcn=zeros(nely,nelx); for i = 1:nelx for j = 1:nely sum=0.0; for k = max(i-floor(rmin),1):min(i+floor(rmin),nelx) for l = max(j-floor(rmin),1):min(j+floor(rmin),nely) fac = rmin-sqrt((i-k)^2+(j-l)^2); sum = sum+max(0,fac); dcn(j,i) = dcn(j,i) + max(0,fac)*x(l,k)*dc(l,k); end end dcn(j,i) = dcn(j,i)/(x(j,i)*sum); end end %%%%%%%%%% FE-ANALYSIS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [U]=FE(nelx,nely,x,penal,edofvect) F = sparse(2*(nely+1)*(nelx+1),30); U = sparse(2*(nely+1)*(nelx+1),30); [KE]=reshape(lk,1,64); xrad=reshape(x',1,nelx*nely); columnindex= reshape(kron(reshape(edofvect,8,nelx*nely), ... ones(1,8)), 1,64*nelx*nely); rowindex=kron(edofvect,ones(1,8)); K=sparse(rowindex,columnindex,kron(xrad.^penal,KE)); % DEFINE LOADS AND SUPPORTS for i = 1:30 F(2*(36)*i,i) = 1./3; end fixeddofs = union([1:2:2*(nely+1)],[2*(nelx)*(nely+1)+2*(nely - 12):2*(nelx+1)*(nely+1)]); alldofs = [1:2*(nely+1)*(nelx+1)]; freedofs = setdiff(alldofs,fixeddofs);

98

% SOLVING U(freedofs,:) = K(freedofs,freedofs) \ F(freedofs,:); U(fixeddofs,:)= 0; %%%%%%%%%% ELEMENT STIFFNESS MATRIX FOR DURAFORM %%%%%%%%%%%%%%%%%%%%%%%%%%%% function [KE]=lk E = 1600.; nu = 0.4; k=[ 1/2-nu/6 1/8+nu/8 -1/4-nu/12 -1/8+3*nu/8 ... -1/4+nu/12 -1/8-nu/8 nu/6 1/8-3*nu/8]; KE = E/(1-nu^2)*[ k(1) k(2) k(3) k(4) k(5) k(6) k(7) k(8) k(2) k(1) k(8) k(7) k(6) k(5) k(4) k(3) k(3) k(8) k(1) k(6) k(7) k(4) k(5) k(2) k(4) k(7) k(6) k(1) k(8) k(3) k(2) k(5) k(5) k(6) k(7) k(8) k(1) k(2) k(3) k(4) k(6) k(5) k(4) k(3) k(2) k(1) k(8) k(7) k(7) k(4) k(5) k(2) k(3) k(8) k(1) k(6) k(8) k(3) k(2) k(5) k(4) k(7) k(6) k(1)];

99

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105

Vita

Mário Carneiro Faustini was born in Rio de Janeiro, Brazil, on February 14, 1974,

the son of Mário Francisco Antônio Faustini and Angélica Carneiro Faustini. After

completing his work at the Colégio de São Bento, Rio de Janeiro, Brazil, in 1992, he

entered the Escola Politécnica of the Universidade de São Paulo, Brazil, where he

received the degree of Bachelor of Science in Mechatronics Engineering in January 1997.

The following semester he entered the Graduate School of the University of São Paulo,

where he received the degree of Master of Science in Mechanical Engineering in May

1999. In August 1999 he entered the Graduate School of The University of Texas at

Austin.

Permanent address: Rua General Barbosa Lima, 99 Apt 401, Copacabana, Rio de

Janeiro, RJ, CEP 22011-060, Brazil

This dissertation was typed by Mario Carneiro Faustini.