Virtual 3D Bone Fracture Reconstruction via Inter-fragmentary SurfaceAlignment

Beibei Zhou†, Andrew Willis†∗, Yunfeng Sui†, Donald D. Anderson‡, Thomas D. Brown‡, Thaddeus P. Thomas‡

†University of North Carolina at Charlotte, Charlotte, NC 28223‡The University of Iowa, 2181 Westlawn Building, Iowa City, IA 52242

arwillis@uncc.edu

Abstract

This paper presents a system for virtual reconstructionof comminuted bone fractures. The system takes as inputa collection of bone fragment models represented as sur-face meshes, typically segmented from CT data. Users in-teract with fragment models in a virtual environment to re-construct the fracture. In contrast to other approaches thatare either completely automatic or completely interactive,the system attempts to strike a balance between interactionand automation. There are two key fracture reconstructioninteractions: (1) specifying matching surface regions be-tween fragment pairs and (2) initiating pairwise and globalfragment alignment optimizations. Each match includes twofragment surface patches hypothesized to correspond in thereconstruction. Each alignment optimization initialized bythe user triggers a 3D surface registration which takes asinput: (1) the specified matches and (2) the current positionof the fragments. The proposed system leverages domainknowledge via user interaction, and incorporates recent ad-vancements in surface registration, to generate fragment re-constructions that are more accurate than manual methodsand more reliable than completely automatic methods.

1. IntroductionExtremity injuries that involve highly comminuted bone

fractures almost always occur as a result of high-energytrauma such as ballistic penetrations, vehicular accidents, orfalls from a height. They are especially a major concern inmilitary conflicts. Treatment goals include achieving expe-ditious bony union in a position of acceptable limb align-ment, and avoiding post-traumatic osteoarthritis (PTOA)when there is involvement of an articular joint such as thehip, knee, or ankle. As a point of reference, for axial “pi-lon” fractures of the distal tibial articular surface, the inci-

∗This work was supported by NIH grants P50AR055533 and1R21AR054015.

dence of PTOA of the ankle is in the range of 60% to 80%[1]. Accurate restoration of the articular surface is criticalin avoiding PTOA, but in many comminuted articular frac-tures, this task can be quite challenging. Often dozens ofindividual fragments are involved, sometimes displaced ap-preciably from their site of anatomic origin, and often inter-spersed in a complex geometric pattern.

The trauma surgeon reconstructing a comminuted frac-ture faces a problem very much akin to puzzle solving, al-beit with clear additional complexities. Muscle forces, com-bined with complex displacements, intervening soft tissues,and fragment fracture surface interactions all make it diffi-cult to reposition the fragments. To partially or completelyrestore the osseous anatomy, traditional surgical treatment(open fracture reduction) exposes the fragments by a sur-gical approach through the damaged soft tissue envelope,so that the surgeon can directly access and reposition thefragments. This often requires considerable force, and in-volves trial and error. Unfortunately each “error” prolongsthe surgery and adds yet more trauma to the fragments andthe surrounding soft tissues. The fracture is considered tobe “reduced” when the surgeon judges that an optimal fithas been obtained between all relevant fragments.

The more extensive the surgical dissection, the better thefracture is visualized and the easier it is to execute accuratefragment reduction. Fragment reduction accuracy is a par-ticularly important consideration for peri-and intra-articularcomminuted fractures. Residual geometric incongruity ofan articular bearing surface strongly predisposes to painfuland debilitating PTOA, a poor clinical outcome regardlessof osseous union. However, wide surgical exposure comesat a significant price, including increased risk for woundhealing failure, infection, joint stiffness, delayed fracturehealing, and damage to articular surfaces. These risks pro-vide tremendous impetus for developing less invasive tech-niques to surgically restore displaced comminuted fractures.

1

(a) (b) (c)

Figure 1: (a-c) This summary of the system for interactive bone fragment fracture reconstruction. (a) shows a collection of11 fragments loaded into the virtual environment (each fragment is colored differently and is numbered in the image), (b) avirtual environment allows inter-fragmentary surface matches to be specified, (c) automatic alignment algorithms minimizethe surface alignment error between the fragments for all (or any desired subset) of the specified surface matches.

2. Contribution of this workThere is a substantial body of work in the field of generic

surface alignment, involving methods for generic alignmentof multiple surface models (e.g. [2, 3]). Yet, these meth-ods are not directly applicable to the problem of bone re-construction because they do not enforce two importantdomain-specific constraints:

1. Bone fragment surface matches consist only of pointscoming from one or more fracture surfaces.

2. Fracture surfaces from large bone fragments may sharematches with many different fragments.

These constraints restrict the problem to something morespecific than generic surface alignment and cast the problemas a puzzle solving problem (constraint 1), with the addi-tional restriction that inter-fragmentary matches may not be1-to-1 (constraint 2). There exist a number of methods forreconstructing puzzle-like geometric problems. Yet, thesemethods are designed to work with materials such as pot-tery and stone, which have different fracture mechanics thanbone, they rely on highly-accurate laser-scan data, and theyrestrict their matching assumptions to have a 1-to-1 corre-spondence, i.e., solutions for these systems require that eachfragment fracture surface matches with only one other frag-ment fracture surface.

Early efforts toward computer-aided bone fracture re-construction included a system for reconstructing a sim-ple two-fragment bone fracture [4]. Completely interac-tive approaches for bone fragment manipulation and align-ment have since been proposed by Scheuering et al.[5]and by Harders et al.[6]. Scheuering et al.[5] empha-sized development of a realistic bone manipulation systemthat performs real-time collision detection to prevent frag-ment inter-penetration, and that involved pairwise fragment

alignment. In [6], the authors developed an interactive sys-tem demonstrated to reconstruct a 5-fragment comminutedbone fracture of the proximal humerus, which relied onbone positioning via a haptic interface. Work by Willis et al.[7] involved an automatic system capable of reconstructingbone fractures. However, determination of fragment cor-respondences was left to a user interaction, which was notdiscussed in detail within their work.

The proposed approach departs from previous methods,as it is neither completely interactive nor completely auto-matic. User interactions in the new approach provide coarsesolutions to the puzzle-solving and fracture surface match-ing constraints listed above in the form of surface matches(Figure 1(b)). These matches are input for alignment al-gorithms that optimize the fragment positions to obtain anaccurate reconstruction of the unbroken bone (Figure 1(c)).The approach avoids the problem of painstakingly makingsmall refinements to fragment positions which can occur insystems that rely completely on interactive methods [5, 6].It also avoids having to search the very large space of possi-ble fragment matches faced by fully automatic fracture re-construction systems[7]. Hence, the proposed system lever-ages domain knowledge through user-specified matches,and it uses automatic surface registration techniques to gen-erate fragment reconstructions that are more accurate thanmanual methods and more reliable than completely auto-matic methods. Throughout this process, expert user in-teraction provides a basis for guiding the alignment towardrestoration of the pre-fracture anatomy.

3. MethodologyDiscussion of the system is separated into two parts:

1. A user interface where the user can manipulate thefragment models and specify gross-scale fragment

matches (see §3.1).

2. Alignment algorithms that take the fragments, asplaced by the user, and the specified matches to gen-erate reconstructions that seek to minimize the globalmatch alignment error (see §3.2.1).

3.1. System Interactions

The system developed for bone fragment alignment in-cludes three different user interactions.

1. Manipulation: each bone fragment may be freely ro-tated and translated.

2. Specifying Fragment Matches: surface patches maybe selected between fragment pairs to indicate regionsthat coarsely correspond.

3. Initiating match alignments: the user can initiate align-ment of the collection of specified matches.

Manipulation: It is widely known that 3D alignment algo-rithms, especially those based on the iterative closest point(ICP) algorithm, may converge to local minima. Hence,matches for fragment surfaces that have been displaced farfrom each other may not converge to a correct alignmenteven with correctly specified surface matches. For this rea-son, the instantaneous pose of fragments in the virtual en-vironment is taken as the initial pose of the fragment. Bymanipulating fragments in the virtual environment, the usercan “help” the ICP algorithm to converge for fragments thatare far from their correct positions. This interaction is par-ticularly important for fragments that have rotated signifi-cantly.

Specifying Fragment Matches: This is the most time-consuming aspect of the reconstruction system, and it is alsothe most important aspect for obtaining good reconstructionresults. This interaction allows the user to generate frag-ment matches. A button is selected to indicate that a newmatch is to be specified. The user then selects portions ofthe fracture surface from two fragments that coarsely cor-respond. Selections are made using a 3D mouse cursor orby selecting a bounding box on the screen than enclosesthe desired surface region. The selected surface positionsfrom each fragment are stored in the interactively generatedmatch. This interaction avoids a complex (and often unre-liable) computational search necessary to identify fracturesurface sub-regions on each fragment that match with onlyone other fragment. Specified matches are appended to arunning match-list that tracks all of the (provisional) frag-ment matches for later reconstruction.

Initiating match alignments: At any point in the assem-bly process, the user can trigger an automatic bone recon-struction which is animated for the user, based upon the sys-tem’s instantaneous optimization of the global alignment er-ror (see §3.2.1 and §3.2.2).

Solutions for cases involving large numbers of fragmentsare possible using this environment, whereas solutions tosmaller fracture cases (e.g., 5 or so fragments) may be com-puted quickly. These interactive tools also provide capa-bilities typically offered by pre-operative planning softwarepackages, from which the user can gain insights on howbest to perform the actual surgical reconstruction. A screencapture of the interface can be seen in Figure 1(b).

As with all human-computer interaction systems, thereare several issues that can effect the time needed to obtaina satisfactory reconstruction. These include: (i) the finalsolution can vary significantly depending upon the manip-ulated pose of the fragments and the complexity and accu-racy of the specified selections, (ii) selection of surfaces onsmall fragments which require fine-scale motions can be-come time consuming, and may provide diminishing returnsin terms of clinical importance. We anticipate ongoing im-provements to cope with these issues.

3.2. Alignment Optimization

The problem of aligning the fracture fragments in orderto restore the bone to its original anatomy involves bothpair-wise and group-wise issues. While an overall align-ment, including all fragments, is the final goal, the systemaccomplishes this goal based upon a pair-wise alignmentapproach.

3.2.1 Fragment Groups

Given a collection of user-specified matches, automatic so-lution of the multi-fragment alignment problem is non-trivial for two reasons: (i) the user may specify several sur-face correspondences for a given fragment pair, and (ii) asingle fragment may include matches with many other frag-ments. Our solution consists of two pre-processing actionson the match-list, followed by an algorithm that optimizesthe global alignment error functional. The first groups to-gether match-list entries that include the same fragmentpair. This grouping is necessary to prevent oscillatory be-havior in the pairwise fragment alignment steps that resultsif these matches are independently optimized. The secondcomputes a graph G(M,F ) where each node (F) of thegraph is a fragment and each edge (M) corresponds to one ormore items from the match-list. The fracture reconstructionalgorithm uses this graph to determine the merge sequence,i.e., the order in which fragments are aligned into the globalreconstruction solution.

Fracture reconstruction involves iteratively merging in-dividual fragments into a group of aligned fragments us-ing the geometric data from the match-list and the graphcomputed in the planning steps. Following each fragmentmerge, the pose of all fragments in the reconstruction areadjusted to accommodate the newly added fragment data.

The algorithmic steps for the reconstruction are listed be-low and closely follow those described in [8]:

1. Compute a graph G(M,F ) where the ith graph node,Fi, corresponds to fragment i in the fracture and eachedge Mk represents a user-specified surface match(correspondence) between two fragments.Note that under this graph model two graph nodes maybe connected by more than one, and possibly many,edges. These situations occur when the user has spec-ified several different matches between the same twofragments.

2. Construct an active set and a dormant set, both ofwhich consist of fragment models. Initially, all frag-ments are placed in the dormant set.

3. Select the fragment in the dormant set that has thehighest number of edges to fragments from the activeset. Initially, we simply choose the fragment with thehighest number of edges.

4. Merge all match-list data shared between the selectedfragment and fragments in the active set.

5. Holding the fragments of the active set fixed, the se-lected fragment is aligned with the fragments in theactive set as specified in §3.2.2.

6. All aligned fragment matches within the active set arere-aligned to accommodate the new fragment.

7. If the dormant set is not empty, return to step 3. Other-wise, exit the alignment algorithm.

Steps 1-3 are initialization steps for the algorithm. Thechoice of the initial fragment in step 3 is arbitrary. How-ever, we find that the algorithm tends to converge morequickly by choosing the fragment with most matches first.Yet, there may be reason to select a different initial frag-ment. One such situation exists when one of the measured“fragments” is still in its proper anatomic position, by virtueof belonging to that portion of the bone that was not frac-tured (e.g., the proximal portion of the tibia in pilon frac-tures). Here, one may choose this “fragment” as the initialfragment, such that the reconstruction algorithm will alignpieces to the remnant intact portion of the bone.

3.2.2 Fragment Pairs

Steps (5) and (6) of the algorithm for aligning fragmentgroups require a second alignment algorithm, which modi-fies the pose of a fragment to minimize its match error withall of the surfaces in the active set. We refer to this problemas pairwise alignment since the fragments of the active setare fixed. As mentioned in §3.1, we use a variant of the ICPalgorithm, following [9].

Generic Surface Alignment Generic pairwise surfacealignment proceeds by fixing one of the two surfaces,and subsequently transforming the second surface untila surface-matching performance functional is optimized.Given two collections of measurements and absolutely noinformation regarding the measurement overlap, this hasproven to be a difficult problem to reliably solve. The twomost significant difficulties associated with this approachare: (1) identifying the subset of overlapping points fromeach surface and (2) reliably finding the correspondence be-tween points within the overlapping point sets. Given thatboth (1) and (2) can be solved, classical work by Horn [10]or a closely related variant, [11], may be used to computethe Euclidean transformation, i.e., the translation and rota-tion, that minimizes the alignment error in closed form (inthis case the alignment error taken to be the squared dis-tances between corresponding points).

Let X and Y denote a pair of surface patches interac-tively specified as a fragment match, and let xi denote theith point from surface X and yj denote the jth point fromsurface Y. Without loss of generality we fix surface X andseek to find the hypothesized correspondence between thetwo pointsets as specified by a sequence of (i, j) pairs heldin a set C, where each pair contains indices of points fromthe overlapping regions of X and Y that correspond. Thealignment problem is then a matter of finding the transfor-mation T = {R, t} that minimizes (1)

T = minR,t

∑(i,j)∈C

f(xi,yj ,R, t) (1)

where f() is an objective function that measures thegoodness-of-fit between the matched surface-point pairs. Ifthe correspondence of the points is unknown, then T de-pends non-linearly on the values of the measured pointsand requires iterative non-linear minimization techniquesto solve. Let k denote the iteration number in the itera-tive minimization technique. We can then denote the initialvalue of the unknown transformation as T0 and the esti-mate of the transformation at the end of iteration k is thenTk = {Rk, tk}.

As an example, the ICP algorithm defines the surfacecorrespondence pairs using the closest point criterion, e.g.,for each xi we define the correspondence (i, j), by find-ing the index of the closest point in surface Y as shown inequation (2) the nearest neighbor. In practice, the number ofpoints selected from each surface may not be equal. We ob-tain a 1-to-1 mapping by defining a mapping of the smallerpointset onto the larger pointset.

j = minj‖xi −Rk (yj − tk)‖2 (2)

The ICP algorithm then computes Tk+1 using the closedform solution from [10] assuming that the computed corre-

(a)

(b)

Figure 2: In each panel (a,b), the left image shows a fragment pair for alignment, the selected surface match for the pair (inblue) and the points selected by the geometrically-stable sampling algorithm (green on blue). The middle image shows theoptimization-aligned fragment pair. The right plot shows the alignment error observed when using uniform sampling to alignthe matched surfaces, and when using geometrically stable sampling to align the matched surfaces.

spondence is correct. Since this seminal work, there havebeen a number of notable variants of the ICP algorithm. Thetwo key modifications included in our surface alignment ap-proach are:

1. Defining the error metric as the sum of point-to-planedistances instead of point-to-point distances [12].

2. Selectively sub-sampling the surface such that moresamples for alignment are taken from regions of sig-nificant geometric surface variation such as ridges andvalleys [9].

The point-to-plane error metric respects the true nature ofthe surface alignment problem, i.e., matching whole sur-faces, rather than the specific samples from each of the sur-faces. This modification to ICP makes the alignment resultrobust to variations in the sampling density on the surface,allows surfaces to slide against each other in flat regions tofind better global minima, and has better convergence be-havior than standard ICP.

Sub-Sampling Alignment Surfaces Surface alignmentproblems often include many measurements that encode re-dundant information and, as a consequence, do not mean-

ingfully contribute to the end alignment solution. To ad-dress this problem, researchers often sub-sample the align-ment surfaces to match a smaller set of measurements thatrequires less computational cost. For nearest neighbor com-putation, the cost savings can be significant. Early ap-proaches suggested selecting points uniformly distributedacross the surface or selecting a random subset of the align-ment points. Work in [13] and [9] propose new methods thatidentify specific points on the alignment surfaces that, whenused for alignment, improve the accuracy and performanceof geometric surface alignments.

Geometrically Stable Sampling for Fragment AlignmentWe use a modified version of the method proposed in [9],which computes a set of geometrically stable points on thealignment surface. The geometrically stable collection ofpoints is determined by finding the collection of alignmentpoints that cause the largest change in alignment error foran infinitesimal change in the fragment pose, i.e., by find-ing the subset of points that tend to dominate the alignmenterror when the fragment undergoes a change in its 3D pose.This geometrically-biased sampling of the alignment sur-faces enhances the stability, convergence and accuracy ofthe alignment, particularly for surface regions whose curva-

(a) (b) (c) (d)

Figure 3: (a) shows an eleven-fragment fracture case. (b,c) show two views of the reconstruction result. (d) shows theincrease in the global match alignment error as more fragments are merged into the fracture reconstruction.

ture is zero in one or more principal directions (e.g., planes,cones, and cylinders) or for surfaces having two equal prin-cipal curvatures, e.g., spheres. This is particularly importantfor bone fracture surfaces.

Selection of point samples within matching surfaces isobtained through the following steps:

1. Compute the eigenvectors and eigenvalues of C as de-fined in equation (3). Eigenvectors of C associatedwith large eigenvalues correspond to directions of theunknown transformation that are highly constrained bythe data, and eigenvectors associated with small eigen-values correspond to directions of the unknown trans-formation that are weakly constrained by the data.

C = FFT

=

»p1 × n1 . . . pk × nk

n1 . . . nk

–2664(p1 × n1)

T nT1

......

(pk × nk)T nT

k

3775(3)

2. For each point, pi, and associated normal, ni, withinthe matched surface region, compute the 6-vector vi =[

(pi × ni) ni

]. The vector vi is then projected

onto each of the six eigenvectors of C and stored in asorted list associated with each of the six eigenvectors.

3. The set of geometrically stable samples is formed byselecting the collection of points having the largest val-ues from each of the six sorted lists. Samples are se-lected until each of the unknown eigenvectors of thetransformation have been sufficiently constrained, i.e.,until the total projection of the selected pointset ontoeach of the six eigenvectors is large enough to providea stable solution to the alignment problem.

While this approach to alignment requires more computa-tion than do random and uniform subsampling, the set ofsamples selected using this approach provide improved ac-curacy (see §4). The eigenvectors of C and the projec-tion of the vectors vi onto the eigenvectors provide a lin-ear approximation of the change in alignment error for a

change in the transformation parameters. The approach pro-posed in [9] re-computes the sampling after each iterationof the alignment algorithm. We modified the approach tobe Euclidean-invariant by subtracting the mean sample po-sition p from each of the surface point locations, i.e., sub-stituting p′i = pi − p for pi in equation (3) and for thecomputation of the vectors vi which makes re-computingthe sampling unnecessary. The authors of [9] also note thattheir sampling technique may be adversely effected by sur-face noise. To alleviate this problem, the matched bone sur-faces are smoothed using the method proposed in [14].

4. Results

Puzzle solution computations are performed for drop-tower-created fractures of anatomically realistic bony surrogatesfabricated from a specialty high-density polymeric foammaterial that has two important attributes: (1) it exhibits me-chanical behavior nominally comparable to that of humancortical bone, and (2) it has bone-similar radiographic ap-pearance on x-ray and CT. Fracture fragments are scannedto provide a set of virtual 3D fragment models, used in theexperiments shown in Figures 1, 2, and 3. Images on theleft side of Figure 2(a,b) show fragment pairs and their user-specified surface match as a blue surface region. Points onone of the two surfaces have been sub-sampled using thegeometrically stable sampling approach, and they are indi-cated in green. The center images in Figure 2(a,b) showthe fragment positions after running the geometrically sta-ble pairwise surface alignment. Line plots on the right sideof Figure 2(a,b) show the alignment error for each align-ment iteration when using uniform surface sampling versusgeometrically stable sampling. In each case, geometricallystable sampling provides improved accuracy in the align-ment. Figure 3(a-d) shows images of a complete fracturecase involving 11 fragments, (a), two views of the recon-struction, (b,c), and a plot of the total alignment error as

each new fragment is added to the reconstruction (d). (d)shows a significant improvement in the global alignmentaccuracy when using the geometrically stable surface align-ment approach.

Figure 4 shows reconstructions of other fractures. Thefractures shown in the top and middle row were generatedby the experimental drop tower and the third example showsa result for clinical data, i.e., a real-world fracture case. Im-ages in the top row (a,b,c) show an alignment result for frag-ments generated by a high-energy fracture of a thick-walledcylinder surrogate. Images in the middle row (d,e,f) showan alignment result for a cadaveric bovine tibia fractured viathe drop tower. Images in the bottom row (g,h,i) show thereconstruction result for the clinical fracture case.

Many fracture cases involve numerous (>15) fragments.Yet, many fragments are too small to be considered for re-construction. For example, the fracture in Figure 4(g) in-cluded 18 fragments and only 7 fragments were of suffi-cient size to be clinically important for restoring alignment.In practice, smaller fragments are often left in place, as theyaid in the bone healing process. However, their exact posi-tion and orientation are not considered to be significant forsuccessful fracture reduction.

5. ConclusionsWe have discussed a prototype system to aid reconstruc-

tion of comminuted bone fractures, that takes as input a col-lection of bone fragment models. Reconstruction is accom-plished via a collaboration between the user (who specifiesfragment surface matches) and the computer (which pro-cesses the specified surface matches to compute fine-detailalignments between the fracture fragments). In contrast toexisting approaches, which tend to emphasize completelyautomatic or completely interactive solutions, we seek tostrike a balance between interaction and automation. Theproposed system leverages domain knowledge via user in-teraction, incorporating results from recent surface regis-tration techniques to generate fragment reconstructions thatare more accurate than manual methods, and more reliablethan completely automatic methods.

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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 4: (a,d,g) show collections of fragments for three different high-energy fractures and two views of each fracturereconstruction is shown in (b,c), (e,f), and (h,i) respectively (see §4 Results for details).