Computational study of the use of microfluidics

for the synthesis of the radiopharmaceutical

[177Lu]Lu-DOTA-TATE

Kevin Schaap

Background image on cover page is retrieved from http://www.microfluidicseng.com

[accessed on 15-07-2017]

Computational study of the use of microfluidics

for the synthesis of the radiopharmaceutical

[177Lu]Lu-DOTA-TATE

Kevin Schaap

in partial fulfilment of the requirements for the degree of

Master of Science

in Chemical Engineering

at the department Radiation Science and Technology (RST)

at the Delft University of Technology

to be defended publicly on Wednesday September 20, 2017 at 15:00 hrs

Supervisors: Prof. Dr. E. Oehlke

Ir. Z. Liu

Graduation Committee: Prof. Dr. E. Oehlke

Prof. Dr. J. L. Kloosterman

Dr. Ir. V. van Steijn

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Preface This report presents my master thesis titled "Computational study of the use of microfluidics for

the synthesis of the radiopharmaceutical [177Lu]Lu-DOTA-TATE". This document marks the end of

my study at the faculty of Applied Sciences at the Delft University of Technology and serves as

the final element of my master in Chemical Engineering. This master has trained me in the world

of chemical engineering and process engineering and serves as great enrichment to my back-

ground in management of technology. The past two years attending the master program Chemi-

cal Engineering has allowed me to acquire both the scientific research skills and analytical prob-

lem-solving skills that were needed to successfully complete this master thesis.

Acknowledgement

I would like to thank the members of my graduation committee for guiding me to through this

graduation process. Foremost, I would like to thank my first supervisor Prof. Dr. Elisabeth

Oehlke for her continuous support, guidance and motivation. Your feedback and insightful com-

ments have helped me in the process of research and writing this thesis.

I would like to thank Zheng Liu for his advice and helpful discussions of the computational mod-

els that were built in COMSOL.

Finally, I want to thank my parents for their unconditional support in this intensive and chal-

lenging period.

Kevin Schaap

Delft, September 2017

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Abstract The radionuclide 177Lu is commonly used in peptide receptor radionuclide therapy (PRRT) by

attaching it to [DOTA0,Tyr3]octreotate to form the complex [177Lu]Lu-[DOTA0,Tyr3]octreotate

([177Lu]Lu-DOTA-TATE). This thesis takes a computational approach to investigate the potential

of microfluidic systems for the synthesis of the radiopharmaceutical [177Lu]Lu-DOTA-TATE, and

also provides an evaluation for reactions with faster and slower kinetics.

The computational models used in this thesis were made in Comsol Multiphysics 5.2. Both 2-

dimensional as well as 3-dimensional models were simulated and a direct comparison between

the two models revealed that the 2-dimensional model is a sufficient approximation for the sys-

tems considered in this thesis.

In this thesis, the impact of channel dimensions on the mixing process and reaction in a micro-

fluidic channel has been studied. This was accomplished by making a comparison between a

microfluidic system in which the reactants are fully mixed and a T-shaped microfluidic system

where the reactants are added through two separate inlets for channel diameters ranging from

100 to 1500 μm. The evaluation of these systems has revealed that heat transfer and diffusion

limitations will lead to slower reactions for increasing channel diameters. The fully mixed sys-

tem realizes a reaction yield of 99% after a residence time of 44.6 seconds for a 100 μm channel

and 53.9 seconds for a 1500 μm channel.

Diffusion limitations that occur as a result of the reactants entering through separate inlets lead

to significantly slower reactions for larger diameters in the case of the T-shaped model com-

pared to the fully mixed system. For channel diameters below 500 μm, the residence time to

achieve the required reaction yield of 99% in the T-shaped system is very similar to the fully

mixed system (the difference in required residence time is within 10%). Therefore, enhanced

mixing methods to ensure that the reactants are fully mixed at the entrance of the main reaction

channel are not required for channel diameters below 500 μm.

The simulation results have shown that a simple T-shaped system is capable to process clinical

amounts of 3 mL reaction solution with 99% reaction yield within 13 minutes for channel diam-

eters of 500 μm and higher. This is a significant improvement compared to the conventional

reaction vessel approach used during clinical labelling that resulted in a process time of 30 min-

utes. This indicates that a microfluidic system would be a promising alternative to the conven-

tional, batch-wise technique that is currently used during clinical radiolabelling.

List of abbreviations

Abbreviation or acronym

Definition

CFD Da DOTA DOTA-TATE FEM Le Ln 177Lu 177Lu- DOTA-TATE MCSs NETs Nu PDEs PRRT Re s SSTRs SVR

Computational fluid dynamics Diffusion coefficient Damkӧhler number 1,4,7,10 – tetra-azacyclododecane-N,N’,N”,N”’-tetraacetic acid [DOTA0,Tyr3]octreotate Finite element method Lewis number Lanthanide Isotope Lutetium-177 177Lu- [DOTA0,Tyr3]octreotate Microchannel systems Neuroendocrine tumours Nusselt number partial differential equations Peptide receptor radionuclide therapy Reynolds number Seconds Somatostatin receptors Surface-to-volume ratio

Table of contents Preface ................................................................................................................................................................. 5

Abstract ............................................................................................................................................................... 7

List of abbreviations ....................................................................................................................................... 8

List of Figures ................................................................................................................................................. 11

List of Tables ................................................................................................................................................... 13

Chapter 1. Introduction .......................................................................................................................... 14

1.1 Medical background ................................................................................................................................... 14

1.2 Microfluidics .................................................................................................................................................. 15

1.3 Research questions ..................................................................................................................................... 17

Chapter 2. Theory ..................................................................................................................................... 18

2.1 Flow in microfluidic channels................................................................................................................. 18

2.2 Governing equations .................................................................................................................................. 20

2.2.1 Fluid motion ......................................................................................................................................... 20

2.2.2 Mass transport .................................................................................................................................... 20

2.2.3 Heat transport ..................................................................................................................................... 21

2.3 Dimensionless numbers ............................................................................................................................ 21

2.3.1 Reynolds number ............................................................................................................................... 21

2.3.2 Damkӧhler number ........................................................................................................................... 22

2.3.3 Nusselt number ................................................................................................................................... 22

2.3.4 Lewis number ...................................................................................................................................... 22

2.4 Formation of [177Lu]-DOTA-TATE and its reaction kinetics ...................................................... 23

2.5 Finite element method .............................................................................................................................. 26

Chapter 3. Computational models ...................................................................................................... 28

3.1 Y-shaped models .......................................................................................................................................... 28

3.1.1 Y-shaped 2-dimensional model .................................................................................................... 28

3.1.2 Y-shaped 3-dimensional model .................................................................................................... 30

3.2 T-shaped 2-dimensional models ........................................................................................................... 31

3.3 Fully mixed model ....................................................................................................................................... 32

3.4 Four-inlet model .......................................................................................................................................... 32

3.5 Boundary conditions and input parameters .................................................................................... 33

3.5.1 Input parameters 2D and 3D Y-shaped computational models...................................... 33

3.5.2 Input parameters 2D T-shaped computational model. ...................................................... 34

Chapter 4. Results & Discussion .......................................................................................................... 35

4.1 2D and 3D model Y-shaped model ....................................................................................................... 35

4.1.1 Mixing of reactants ............................................................................................................................ 36

4.1.2 Reaction ................................................................................................................................................. 38

4.1.3 Influence of diffusion coefficient ................................................................................................. 40

4.1.4 Comparison 3D and 2D model ...................................................................................................... 41

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic

channel ........................................................................................................................................................................... 44

4.2.1 Heating of reaction channel ........................................................................................................... 44

4.2.2 T-shaped 2-dimensional model .................................................................................................... 48

4.2.3 Fully mixed model ............................................................................................................................. 51

4.2.4 Comparison of fully mixed model and T-shaped model .................................................... 54

4.2.5 Influence of reaction kinetics ........................................................................................................ 55

4.2.6 Difference in residence time between the fully mixed and T-shaped model as a

function of the Damkӧhler number ............................................................................................................... 58

4.2.7 Four inlet model ................................................................................................................................. 61

4.3 Up-scaling to clinical relevant quantities ........................................................................................... 63

Chapter 5. Conclusion ............................................................................................................................. 65

Recommendations ..................................................................................................................................................... 66

Appendix I ........................................................................................................................................................ 67

Appendix II ...................................................................................................................................................... 68

Appendix III ..................................................................................................................................................... 70

Bibliography ................................................................................................................................................... 85

List of Figures Figure 1: Microfluidic channels in a glass lab-on-a-chip device [19] ........................................................ 15

Figure 2: Motion of dye filament in a straight tube showing different flow regimes [33]. .............. 18

Figure 3: Velocity profile for laminar flow in a pipe [36]. .............................................................................. 19

Figure 4: Laminar flow of two differently dyed aqueous streams in a microfluidic channel [39].19

Figure 5: Qualitative dependence of the Nusselt number on axial position (z) in confined flow

[43]. ....................................................................................................................................................................................... 22

Figure 6: Chemical structure of DOTA (a) and Ln-DOTA (b) [50]. ............................................................. 23

Figure 7: Chemical structure of 177Lu-DOTA-TATE, where 177Lu-DOTA (white background) is

conjugated to [Tyr3]octreotate (TATE, green background). Figure based on [51]. ............................ 23

Figure 8: Dependence of reaction rate constant for the formation of [177Lu]Lu‐DOTA‐TATE on

temperature [58] ............................................................................................................................................................. 25

Figure 9: Arbitrary mesh consisting of different element shapes. ............................................................. 26

Figure 10: Different basic element shapes to mesh a 3-dimensional geometry. .................................. 26

Figure 11: Mesh consisting of quadrilateral elements and nodes. ............................................................. 27

Figure 12: Mesh used for 2D Y-shaped channel. ................................................................................................ 28

Figure 13: Model geometry for the 3D Y-channel. ............................................................................................ 30

Figure 14: Mesh of finite elements used in the 3D computational model. Top view xy-plane (left

figure) and cross-section of main channel yz-plane (right figure) ............................................................. 30

Figure 15: Mesh used for 2D T-shaped channel ................................................................................................. 31

Figure 16: Mesh used for 4-inlet model. ................................................................................................................ 32

Figure 17: Velocity profile at the main channel entrance of the 2D Y-channel model ....................... 36

Figure 18: Lutetium concentration at the main channel entrance of the 2D Y-channel model ...... 36

Figure 19: Concentration profile at the end of the main reaction channel (10 cm length) of

lutetium (a) and DOTA-TATE (b) ............................................................................................................................. 37

Figure 20: Concentration profile Lu-DOTA-TATE at the end of the main reaction channel

(channel length: 10 cm) for different reaction rate constants: k=10 mol/m3 s (Figure 20(a)), k=1

mol/m3 s (Figure 20(b)), k=0.1 mol/m3 s (Figure 20(c)) ............................................................................... 38

Figure 21: Concentration profile of Lu-DOTA-TATE at the main reaction channel outlet for

varying reaction rate constants. All three cases were performed for a channel diameter of 150

µm and a flow rate of 125 µl/min. ............................................................................................................................ 39

Figure 22: Velocity profile (a) and concentration profile of Lu-DOTA-TATE (b) at the outlet of

main reaction channel. .................................................................................................................................................. 41

Figure 23: Concentration profile of Lu-DOTA-TATE at outlet of microfluidic channel for 2D and

3D model ............................................................................................................................................................................. 42

Figure 24:Differential volume element used in heat balance [40] ............................................................. 45

Figure 25: Thermal entrance length and developing temperature profile (based on [76]) ............ 46

Figure 26: Simulation results of the temperature profile at 0.5 cm down the main reaction

channel for a channel diameter of 100 μm (Figure 24a) and 1500 μm (Figure 24b) with an

applied average velocity of 0.2 m/s......................................................................................................................... 47

Figure 27: Velocity profile for the T-shaped 2D model ................................................................................... 48

Figure 28: Influence of channel diameter on reaction yield. ........................................................................ 49

Figure 29: Residence time needed for 99% conversion for T-shaped model as a function of

channel diameter............................................................................................................................................................. 50

Figure 30: Concentration Lu-DOTA-TATE at the outlet of the channel for a channel diameter of

1000 μm and a velocity of 0.05 m/s, which equals a residence time of 20 seconds. .......................... 51

Figure 31: Influence of channel diameter on reaction yield when the reactants are fully mixed. 52

Figure 32: Comparison of reaction yield of experimental results vs simulation results in a 100

μm channel. Experimental conditions: concentration lutetium inlet = 0.260±0.003 mM and

concentration DOTA-TATE inlet= 0.303±0.003 mM. Flow rate ratio set to 2:1 (DOTA-TATE:Lu).

................................................................................................................................................................................................ 52

Figure 33: Influence of channel diameter on the reaction yield of [177Lu]Lu-DOTA-TATE [58]. ... 53

Figure 34: Ratio of the required residence time to achieve a reaction yield of 99% for the T-

shaped model (tT-shaped model) and the fully mixed model (tfully mixed model) as a function of the

Damkӧhler number. ....................................................................................................................................................... 59

Figure 35: Concentration profile of lutetium (a) and DOTA-TATE (b) at the start of main reaction

channel of the four inlet model. The lutetium and DOTA-TATE solution are introduced into the

main channel in an alternating order. Similarly to the T-shaped model, the flow rate ratio was set

to 2:1 (DOTA-TATE:Lu) ................................................................................................................................................ 62

Figure 36: Dependence of highest possible flow rate in the fully mixed and T-shaped geometry to

achieve 99% yield on channel diameter, and time required for the production of 3 mL reaction

solution. ............................................................................................................................................................................... 64

Figure 37: COMSOL model of H chip ....................................................................................................................... 68

Figure 38: Diffusion coefficient [177Lu]Lu-DOTA-TATE for different flow rates determined using

COMSOL............................................................................................................................................................................... 69

List of Tables Table 1: Inlet parameters for 3D and 2D Y-shaped computational models. .......................................... 33

Table 2: Inlet parameters for 2D T-shaped, fully mixed and four-inlet computational models ..... 34

Table 3: Influence of diffusion coefficients of lutetium and DOTA-TATE on the formation of Lu-

DOTA-TATE ....................................................................................................................................................................... 40

Table 4: Comparison 2D and 3D model ................................................................................................................. 42

Table 5: Residence time needed for 99% conversion for T-shaped model. ........................................... 49

Table 6: Residence time needed for 99% conversion for fully mixed model ......................................... 53

Table 7: Reaction yield for T-shaped model and fully mixed model for a channel diameter of

100μm and 1000μm ....................................................................................................................................................... 54

Table 8: Reaction yield for T-shaped model and fully mixed model for a channel diameter of

1000 μm with reaction 10 times faster than the reaction of Lu-DOTA-TATE. ...................................... 55

Table 9: Reaction yield for T-shaped model and fully mixed model for a channel diameter of

1000 μm with reaction 10 times slower than the reaction of Lu-DOTA-TATE ..................................... 56

Table 10: Residence time needed for 99% conversion for fully mixed model and T-shaped model

in a 1000 μm channel for different reaction rates. ............................................................................................ 57

Table 11: Damkӧhler number for three different channel diameters and the corresponding ratio

of required residence times of the T-shaped model and fully mixed model obtained for the

formation of Lu-DOTA-TATE ( = 7.3E-9 m2/s) ............................................................................................... 58

Table 12: Different Damkӧhler numbers for a 500 μm channel and the corresponding ratio of

required residence times of the T-shaped model and fully mixed model ............................................... 59

Table 13: Reaction yield for the four inlet model compared to the T-shaped model and the fully

mixed model for a channel diameter of 1000 μm .............................................................................................. 61

1.1 Medical background 14

Chapter 1. Introduction

1.1 Medical background

In most western countries, cancer is one of the main causes of death. In the Netherlands,

105,844 people were diagnosed with cancer in the year 2015 and 44,408 people (262 per

100,000 population) died from cancer [1]. In addition, a report by the European Heart Journal in

2016 has shown that cancer has overtaken heart diseases as the number one cause of death in

12 Western European countries [2]. The high share of death due to cancer is exemplified by

mortality statistics in 2015 which indicate that 591,700 U.S. residents (184 per 100,000 popula-

tion) died of cancer making it the second leading cause of death in America [3].

Several different methods exist to treat cancer. The most important methods consist of surgery,

chemotherapy, and radiotherapy. Two main types of radiotherapy exist: external beam radiation

and internal radiation therapy. This study will focus on the synthesis of a radiopharmaceutical

used for internal radiotherapy. Radiopharmaceuticals are a type of drugs used in internal radia-

tion therapy that contain radioisotopes and are injected inside the human body. The ionizing

radiation of the radioisotope will damage the DNA of cancer tissue leading to the death of the

cancer cells.

Recent advances in molecular biology and in the area of labelling peptides that target tumour

cells have sparked interest in radiopharmaceuticals and led to an emerging treatment method

called peptide receptor radionuclide therapy (PRRT)[4]. This treatment method is mainly used

to treat neuroendocrine tumours (NETs). In addition, it is currently under investigation as a po-

tential treatment method for prostate and pancreatic tumours [5].

The cells of neuroendocrine tumours are characterized by the overexpression of somatostatin

receptors (SSTRs). This provides an opportunity for tumour imaging or treatment by targeting

the tumour cells with a molecule that specifically binds to the receptor. PRRT takes advantage of

this attribute of the tumour cells by coupling a radionuclide emitting beta radiation to a peptide

that has a strong affinity to somatostatin receptors subtype 2 (sstr2) [6]. In this way, the applied

radionuclide is able to specifically target the tumour cells while minimizing radiation exposure

to surrounding healthy tissue.

Lutetium-177 (177Lu) and Yttrium-90 (90Y) are the most commonly used radionuclides in current

PRRT [7]. This thesis will primarily focus on 177Lu which is a medium energy beta-emitter

(maximum energy of 498 keV) [8]. In addition, 177Lu emits low energy gamma radiation which

offers the additional possibility to visualize the biodistribution of the radiopharmaceutical.

In PRRT, 177Lu is attached to the somatostatin analogue [DOTA0,Tyr3]octreotate (DOTA-TATE)

[9]. The macrocyclic molecule DOTA is used as a chelating agent that binds to the 177Lu radionu-

clide. The peptide [Tyr3]octreotate (TATE) is the functional group that acts as a somatostatin

analogue and is conjugated to DOTA to form DOTA-TATE [10].

[177Lu]Lu-DOTA-TATE is currently produced by introducing the reactants in a small vial acting

as a batch reactor. The mixture is heated at 80˚C for 30 minutes [11]. However, this method en-

counters several limitations. The low concentrations of the reactants used in radiopharmaceuti-

cal preparation lead to slow reaction kinetics [12] . In order to speed up the reaction, the peptide

1.2 Microfluidics 15

is applied in excess (Lu:DOTA-TATE ratio range from 1:1.2 to 1:4) [13]. Taking into account the

high price of the peptide DOTA-TATE ($310/mg) [14], the use of excess DOTA-TATE results in a

problematic and significant increase in cost. In addition, the use of excess DOTA-TATE is unde-

sired as the amount of non-labelled DOTA-TATE that binds to the receptors should be minimized

for effective PRRT. Since the labelling should be executed with high yield in a cost efficient man-

ner, it is desired to increase the control over the reaction and reduce the reaction time and con-

sumption of material.

Microfluidics has shown to be an useful approach to reduce reaction time due to the advantages

of micro-scale systems such as fast thermal heating rates and mass transfer rates [15, 16].

1.2 Microfluidics

The field of microfluidics takes it origin in the 1990s, has rapidly emerged during the past dec-

ades [17], and has found many applications at the interface of biology, chemistry and engineer-

ing [18]. Figure 1 shows an example of a microfluidic chip [19].

Figure 1: Microfluidic channels in a glass lab-on-a-chip device [19]

In recent years, there has been increasing interest in the application of microfluidics for the syn-

thesis of radiopharmaceuticals [15], since this has been shown to solve some of the problems

conventional methods are faced with [16, 20]:

I. The high surface-to-volume ratio (SVR) of microfluidic systems enhances mass transport

rates and leads to more efficient heat transfer due to smaller characteristic length scales

for diffusion and conduction [15, 16]. This results in improved control of reaction condi-

tions, such as temperature, while at the same time it offers the advantage of shorter reac-

tion times or significant reductions of precursor consumption.

II. The reduction of the size of the set-up has the additional benefit of requiring less shield-

ing, which reduces both the weight and the cost of possible future devices [20].

III. Another advantage of the use of microfluidics for the synthesis of radiopharmaceuticals

is the reduction of radiolysis. Radiolysis refers to the degradation of molecules by ioniz-

ing radiation. In this way, the reactants and the radio-labeled product could be degraded

resulting in a reduction of the yield of the radiopharmaceutical. Due to the small dimen-

sions of microfluidic systems, a large fraction of the radiation energy is not deposited in

the solution [21, 22]. As a result, the reactants and the radiopharmaceutical itself are less

prone to radiolytic effects.

1.2 Microfluidics 16

Several studies have demonstrated the successful application of microfluidics for the synthesis

of 18F- and 11C-based PET tracers [15, 23]. To a lesser extent also metal containing radiopharma-

ceuticals for therapeutic purposes have been studied, and shorter overall reaction times at lower

temperatures have been observed [20]. The design of the microfluidic reactors used in these

studies has been, to the best of my knowledge, typically optimized based on experimental work,

thereby increasing the radiation burden of the researcher. Only one recent study from Haroun et

al. reports the application of computational methods to optimize the design of the reaction

channels [24]. They show that the use of simulations can be very beneficial in designing a new

microfluidic system by gaining a better understanding of the processes that play a role and the

kind of channel dimensions that are needed for an optimal design. By performing simulation

studies prior to resorting to radioactive experiments, the design process of a microfluidic sys-

tems becomes more efficient and radiation burden can be reduced.

Differences in the design are for example the channel or vessel systems, width and height of the-

se systems, mixing devices at the beginning of the channel, but also internal structures that en-

hance mixing within the microsystem [20, 25]. Especially the mixing behavior of reactants in the

channel can be important as a slow mixing process could be a limiting factor for the reaction

yield that is achieved.

Due to the laminar flow profile in microfluidic channels, mixing occurs through molecular diffu-

sion. Numerous micromixers have therefore been developed to accelerate the mixing process in

microfluidic channels [25]. These micromixers can rely on different techniques to enhance mix-

ing. For example, the use of a special geometrical designs can improve mixing significantly by

reducing the diffusion distance or inducing chaotic advection. Another technique to enhance

mixing is by the use external forces [26]. Several studies have used computational fluid dynam-

ics (CFD) to investigate the mixing performance of various micromixers [27-29].

However, the combination of this knowledge on the mixing behaviour of reactants with knowl-

edge about chemical reaction kinetics is scarce. Only a few studies combined these two, and in

this way determined the optimal parameters for a particular chemical reaction in a microfluidic

system [30, 31]. The study by Buddoo et al. uses CFD simulations to evaluate the pyrolysis of cis-

2-pinanol to linalool in a multi-channel microfluidic system compared to the conventional tubu-

lar reactor and report that higher conversions and selectivities are possible using a microfluidic

system [31].

Systematic knowledge on how reaction kinetics and system dimensions influence the necessity

for the incorporation of different mixing designs in microchannels is missing.

1.3 Research questions 17

1.3 Research questions

This thesis will investigate the potential of microfluidic systems for radiopharmaceutical synthe-

sis. As mentioned above, it is currently unknown if mixing processes occurring through molecu-

lar diffusion is limiting the yield of the desired product in a microfluidic system. An investigation

will be presented that studies at what channel dimensions and reaction kinetics, diffusion will be

a limiting factor and alternative designs are necessary to accelerate the mixing process.

In this study, computational methods are applied to investigate the synthesis of radiopharma-

ceuticals in microchannels. The main focus of this thesis is on the reaction applied for the pro-

duction of the radiopharmaceutical [177Lu]Lu-DOTA-TATE. However, also reactions with faster

and slower kinetics are evaluated.

In order to use computational methods for the design of an optimized microfluidic reactor for

the synthesis of radiopharmaceuticals, several questions need to be investigated:

I. How do the simulation results of a 2-dimensional and 3-dimensional model for reactions

in a microfluidic channel compare? Is it possible, to use 2-dimensional models (less com-

putational time) to predict reaction behaviour?

II. What is the impact of channel dimensions on the mixing process and reaction in a micro-

fluidic channel?

III. Are mixing structures needed at the beginning of the channel or within it? How does this

depend on the reaction kinetics of the investigated radiopharmaceutical?

The simulations are done using the Multiphysics platform COMSOL, combining flow dynamics,

heat and mass transfer, and reaction kinetics.

In addition, the practical aspects of a microfluidic system for the synthesis of radiopharmaceuti-

cals intended for clinical use should be taken into consideration. For the synthesis of [177Lu]Lu-

DOTA-TATE, the final system should meet the following requirements in order for it to be suita-

ble for clinical use in hospitals:

Capacity to process 3 mL in less than 30 mins

Minimized reaction time

Yield of [177Lu]Lu-DOTA-TATE approaching 100%

2.1 Flow in microfluidic channels 18

Chapter 2. Theory This chapter presents the reader with the theoretical background needed to understand the

concepts that play a role in this study. In section 2.1, the flow profile in microfluidic channels is

discussed. The next section introduces the governing equations that need to be taken into ac-

count when building a computational model that describes the flow profile, reactions and diffu-

sion in microfluidic channels. Subsequently, section 2.3 covers the dimensionless numbers that

are relevant for this study. Section 2.4 provide the molecular background about the chemical

species involved in this study and explains the reaction kinetics for the formation of [177Lu]Lu-

DOTA-TATE.

The final section of this chapter, section 2.5, elaborates on the numerical technique (finite ele-

ment method) that is used to compute approximate solutions presented in the previous sections.

2.1 Flow in microfluidic channels

Three different regimes of flow can be identified in fluid dynamics: laminar flow, turbulent flow

and transitional flow [32]. Laminar flow occurs when the fluid moves along smooth paths or

layers while turbulent flow is characterized by an irregular or chaotic flow pattern. A transi-

tional flow pattern exhibits both laminar and turbulent flow characteristics. The different flow

regimes are illustrated in Figure 2 where a thin filament of dye is introduced at the centre of the

main stream [33]. For laminar flow, the dye stays as a well-defined line in the centre of the

stream. In this case, mixing is slow and occurs through molecular diffusion. In a turbulent flow

regime, on the other hand, the dye is rapidly mixed throughout the fluid.

Figure 2: Motion of dye filament in a straight tube showing different flow regimes [33].

2.1 Flow in microfluidic channels 19

The flow regime can be characterized through the evaluation of the Reynolds (Re) number. This

and other dimensionless numbers are discussed in more detail in section 2.3. In general, laminar

flow is stable when the Reynolds number is below a critical value that is dependent on the ge-

ometry. In a pipe, laminar flow is observed for a Reynolds number below 2100 [34]. The Rey-

nolds number for flow in microchannels is usually well below 100 due to the small dimensions

of microfluidic systems, and therefore the flow is generally laminar [35]. The velocity profile for

laminar flow in a pipe has a parabolic shape, as shown in Figure 3, where the velocity varies

from zero at the walls to a maximum along the centreline of the pipe [36]. The radial velocity

profile can be expressed by equation (1) [37]:

(1)

Figure 3: Velocity profile for laminar flow in a pipe [36].

In laminar flows, mixing occurs only through molecular diffusion which is several orders of

magnitude slower than mixing through turbulence [38]. Figure 4 shows two streams that are

combined in a microfluidic channel [39]. Both streams consist of water with a dye added to be

able to make a distinction between the two streams. This figure shows that the two streams do

not instantly mix when the streams meet. Instead, a flow pattern consisting of two layers flowing

side-by-side is established.

Figure 4: Laminar flow of two differently dyed aqueous streams in a microfluidic channel [39].

2.2 Governing equations 20

2.2 Governing equations

In order to describe the behaviour of fluids and chemical species in microfluidic channels using

computational modelling, several equations need to be taken into account. These equations de-

fine the velocity and temperature of the fluid, and the concentration of the different chemical

species in the fluid.

2.2.1 Fluid motion

The equations that describes the motion of the fluid are derived by the application of a mass

balance and a momentum balance over an infinitely small control volume [40]. The motion of a

Newtonian fluid is governed by the conservation of mass given in equation (2) and the Navier-

Stokes equation (conservation of momentum) given in equation (3):

(2)

where is the density of the fluid and is the velocity field vector of the fluid, and

μ

(3)

where μ is the dynamic viscosity of the liquid, is the pressure and represents any other ex-

ternal forces acting upon the fluid. For a 3-dimensional problem in a Cartesian coordinate sys-

tem (x, y, z), the (Nabla differential operator) is defined as

.

For stationary incompressible laminar flows, equations (2) and (3) can be further simplified to

equations (4) and (5), respectively:

(4)

μ

(5)

The term in equation (4) expresses the transport of momentum by convection, the term

can be interpreted as diffusion of momentum due to viscous stress, and , the pressure

gradient, expresses the pressure force.

2.2.2 Mass transport

The differential equation that describes the transport of chemical diluted species follows from

the conservation of chemical species and can be obtained by setting up a mole balance over an

infinitely small control volume. The steady state differential equation that governs transport of

chemical species is the convection-diffusion-reaction equation given in equation (6) [40].

(6)

where i is the diffusion coefficient (m s-2) and ci is the concentration of chemical species i.

The term in equation (6) describes transport due to molecular diffusion and the

term captures convective transport. The final term is a source (sink) term that takes into ac-

count the formation (consumption) of chemical species in possible chemical reactions that oc-

cur. In this study, this term is captured by the reaction rate discussed in section 2.4.

2.3 Dimensionless numbers 21

2.2.3 Heat transport

Analogous to mass transport, a differential equation can be obtained that describes the transport

of heat by setting up a micro balance over an infinitely small control volume. The heat transfer

process is governed by equation (7) [40]:

(7)

where T is the temperature (K), is the specific heat at constant pressure (J kg-1 K-1), is the

thermal conductivity coefficient (J m-1 s-1 K-1), and q is the rate of heat production (J m-3 s-1).

For a steady state problem, the heat equation in equation (7) simplifies to equation (8):

(8)

In equation (8), the first term represent the transfer of heat by convection, the term is the

term that captures diffusive transfer of heat (conduction), and the final term q is a source term

(e.g. a heating element).

By solving this differential equation, a temperature profile of the fluid in the microchannel can

be obtained. The temperature profile serves as an input for the reaction rate constant, which is

temperature dependent (Section 2.4), and is subsequently used in the mass transfer equation.

2.3 Dimensionless numbers

Dimensionless numbers are helpful to solve problems in fluid mechanics, mass transfer and heat

transfer. This section discusses three important dimensionless numbers that are relevant for

this thesis: the Reynolds number, the Damkӧhler number, the Nusselt number and the Lewis

number.

2.3.1 Reynolds number

The flow regime in a tube can be characterized through the evaluation of the Reynolds (Re)

number. The Reynolds number is a dimensionless quantity that takes the ratio of inertial forces

to viscous forces and is defined as [40]:

(9)

where is the average velocity of the fluid and L is the characteristic length dimension. For pipe

flow, the flow is considered to be laminar for Re < 2100. The flow is turbulent for Re > 3000 and

the flow is in the transitional regime between laminar and turbulent flow for 2100 < Re < 3000

[34]. Due to the small dimensions of microfluidic systems, this study will deal with flows at low

Reynolds numbers that fall within the laminar flow region.

2.3 Dimensionless numbers 22

2.3.2 Damkӧhler number

Another important dimensionless number in this study is the Damkӧhler number (Da). This

number embodies the competition between reaction and diffusion. The Damkӧhler number is

the ratio of reaction rate over diffusion rate and is defined for a second-order reaction in equa-

tion (10):

(10)

where k is the reaction rate, L is the characteristic length scale, and Ɗ is the diffusion constant.

The Damkӧhler number can be interpreted as the characteristic mixing or diffusion time (L2 / Ɗ)

divided by the characeteristic reaction time (1/ k c) [41]. For Da<<1, reaction is slow relative to

diffusion. In this case, the reaction is the controlling step and the process is said to be reaction

limited. For Da>>1, the reaction is fast relative to diffusion, thus the process is controlled by

mass transfer and is said to be diffusion limited [34, 42] .

2.3.3 Nusselt number

The Nusselt number (Nu) is the dimensionless form of the heat transfer coefficient ( ) and is

defined as:

(11)

where L is the characteristic length (e.g. tube diameter) and is the thermal conductivity of the

fluid [40]. The Nusselt number is used for the heat transfer problem in this study in section 4.2.

In this study, we focus on a confined laminar flow where the wall temperature increase from T0

to Tw at z=0 and remains at this constant temperature for z>0. As shown in Figure 5, Nusselt is

largest near z=0 and declines with increasing z in the thermal entrance region. For larger dis-

tances, at positions where the fluid enters the fully developed region, Nu typically approaches a

constant.

Figure 5: Qualitative dependence of the Nusselt number on axial position (z) in confined flow [43].

2.3.4 Lewis number The Lewis number (Le) is a dimensionless number that is frequently used for systems with si-

multaneous heat and mass transfer. The Lewis number is defined as the ratio of the thermal dif-

fusivity to the mass diffusion coefficient [44]. In general, Le~1 for gases. For liquids, Le is

normally between 10-100, indicating that heat diffuses faster than mass [45].

(12)

2.4 Formation of [177Lu]-DOTA-TATE and its reaction kinetics 23

2.4 Formation of [177Lu]-DOTA-TATE and its reaction kinetics

The macrocyclic molecule 1,4,7,10 – tetra-azacyclododecane-N,N’,N”,N”’-tetraacetic acid (DOTA)

is an organic compound that can be used as a chelating agent for various metal ions [46]. Figure

6a shows the chemical structure of DOTA .

DOTA is frequently used in nuclear medicine as it is able to encapsulate trivalent metal isotopes

that can be used for imaging purposes or radiotherapy such as gallium-68, indium-111, luteti-

um-177 and yttrium-90 [47].

This study will primarily focus on the complex that is formed between a DOTA derivative and

lutetium that is a member of the lanthanide (Ln) series. Members of the lanthanide series exhibit

comparable chemical behavior as they share a similar electronic configuration and all lantha-

nides are stable in oxidation state III (LnIII) [48]. The chemical structure of a complex of lantha-

nide (Ln) metal ions with DOTA is shown in Figure 6b. The R1-symbol in Figure 6b indicates the

position where the DOTA chelator is commonly linked to a peptide, such as octreotate or

octreotide derivatives, that serves as a cancer-specific targeting agent [49, 50].

Figure 6: Chemical structure of DOTA (a) and Ln-DOTA (b) [50].

An example of a peptide used in PRRT is the somatostatin analog [Tyr3]octreotate. When linked

to DOTA, it forms the DOTA derivative [DOTA0,Tyr3]octreotate or DOTA-TATE. The

[Tyr3]octreotate group is needed as this is the functional group acting as an somatostatin analog

that binds to the somastostatin receptors that are overexpressed on neuroendocrine tumours.

The chemical structure of [177Lu]-DOTA-TATE is shown in Figure 7.

Figure 7: Chemical structure of 177Lu-DOTA-TATE, where 177Lu-DOTA (white background) is conjugated to

[Tyr3]octreotate (TATE, green background). Figure based on [51].

2.4 Formation of [177Lu]-DOTA-TATE and its reaction kinetics 24

DOTA can exist in various forms depending on the protonation status of the carboxylic groups

which is dependent on the pH value of the solution. The pKa values have been determined pre-

viously: pKa1=4.14, pKa2=4.50, pKa3=9.70 and pKa4=12.60 [52]. As DOTA loses one or more hy-

drogen ions, the molecule becomes increasingly negatively charged. As a result, it is more at-

tracted to the positively charged metal ion which has an accelerating effect on the speed of reac-

tion. This explains the positive relation between the pH and the observed reaction rate found in

existing research [53].

The formation of complexes of DOTA with lanthanides (Ln3+) is a process involving several steps

before the final product is formed. A common pathway described in literature indicates that lan-

thanide-DOTA complexes are formed via a diprotonated intermediate followed by consecutive

deprotonation steps [54, 55]. Therefore, the formation of a complex between DOTA and luteti-

um, that is part of the lanthanide series, can be represented by:

(13)

(14)

(15)

A more elaborate reaction scheme that takes all different protonated forms of DOTA and multi-

ple reversible intermediate forms into account can be found in Appendix I.

The overall reaction has been shown to follow second order kinetics [56] and will be treated as

such in this thesis. The overall reaction between lutetium and DOTA is expressed as:

(16)

As shown in equation (16), the reaction between a lutetium metal ion and DOTA proceeds ac-

cording to a 1:1 molar ratio. This means that one DOTA molecule coordinates to one lutetium

ion. The rate of formation of the product (Lu-DOTA) and the consumption of reactants for a se-

cond order reaction is governed by the reaction rate given in equation (17).

(17)

In equation (17), , and represent the concentration of the metal-

DOTA complex Lu-DOTA, the concentration of the metal ion Lu3+ and the concentration of DOTA.

k represents the second order reaction rate constant (mol m-3 s-1). In this thesis, it is assumed

that the reaction between metal ions and DOTA-TATE occurs through a similar mechanism as

the reaction between metal ions and DOTA and is therefore also treated as a second order reac-

tion. The reaction equation for the formation of Lu-DOTA-TATE is specified as:

(18)

2.4 Formation of [177Lu]-DOTA-TATE and its reaction kinetics 25

Although the formation of [177Lu]Lu‐DOTA‐TATE has been studied extensively before, infor-

mation on the kinetic rate constants in existing literature is not available [57]. A recent study

performed in conjunction with my thesis has determined, for the first time, the kinetic rate con-

stants for the formation of [177Lu]Lu‐DOTA‐TATE using a microfluidic system [58]. The reaction

rate constants were obtained for different temperatures and fitted to the Arrhenius equation:

(19)

where A is the pre-exponential factor (M-1 s-1), EA is the activation energy (J mol-1), R is the ideal

gas constant (R=8.3145 J mol-1 K-1), and T is the absolute temperature (K). The Arrhenius pa-

rameters for the formation of [177Lu]Lu‐DOTA‐TATE were determined as A=1.24E19 M-1 s-1 and

Ea=109.5E3 J mol-1. The resulting plot which shows the temperature dependence of the reaction

rate constant is presented in Figure 8 [58].

Figure 8: Dependence of reaction rate constant for the formation of [177Lu]Lu‐DOTA‐TATE on temperature [58]

2.5 Finite element method 26

2.5 Finite element method

The partial differential equations (PDEs) describing 3-dimensional systems or 2-dimensional

systems are often too complex to be solved analytically. Instead, numerical techniques are used

to find an approximate solution. COMSOL Multiphysics, the software package used in this thesis,

uses the finite element method (FEM) to compute approximations of partial differential equa-

tions. Compared to other numerical techniques such as the finite difference method, this method

is more memory intensive, but has the advantage that it is more accurate and better able to deal

with irregular structures [59]. Another numerical technique regularly used in computational

fluid dynamics simulations is the finite volume method (FVM) which is very similar to FEM.

The finite element method discretizes the domain into smaller parts (a finite number of ele-

ments) [60]. In other words, the entire geometry is divided into a number of subregions. The

collection of all elements has to cover the entire geometry and is called the mesh. It is structured

in such a way that there are no gaps between elements and no existence of overlapping elements

[61]. Figure 9 shows an example of how a 2D geometry of random shape can be modelled by

elements of various sizes and shapes.

Figure 9: Arbitrary mesh consisting of different element shapes.

In general, four different 3D element types can be identified: tetrahedrons, hexahedrons, prisms

and pyramids:

Figure 10: Different basic element shapes to mesh a 3-dimensional geometry.

These elements can be used in various combinations to mesh any 3D geometry. Similarly, 2D

geometries can be meshed using triangular or quadrilateral shaped elements.

2.5 Finite element method 27

The points that define the geometrical coordinates of these elements, are called 'nodes' or 'nodal

points', and are usually found on the corners and edges of an element. An example of a mesh

containing nodes and elements is illustrated in Figure 11. The field variables appearing in the

governing PDEs are calculated at these nodes. The values of the variables between nodes and

within an element are then calculated by means of an interpolation function. This interpolation

function is called the shape function or basis function. By combining all the element solutions, a

final approximation of the exact solution can be assembled for the entire system.

Figure 11: Mesh consisting of quadrilateral elements and nodes.

The accuracy of the solution acquired by finite element analysis is governed by the number of

elements. A more accurate solution can be achieved by using smaller elements such that the

mesh density is increased. Although smaller element sizes yield more accurate solutions, this

benefit comes at the cost of increased computation time and memory demand as the total num-

ber of equations to be solved increase due to a larger number of total elements. For this reason,

it is a good practice to use small element sizes in regions where large gradients are anticipated

while using larger elements in regions where the variation of a variable is anticipated to be small

[60].

3.1 Y-shaped models 28

Chapter 3. Computational models This chapter will discuss the computational models that have been built in this study. The

software package COMSOL Multiphysics 5.2 was used to build different computational models to

simulate the formation of [177Lu]Lu-DOTA-TATE in a microfluidic system. This system involves

the reactants 177Lu3+ and DOTA-TATE entering through seperate inlets, and the model describes

the mixing and reaction of 177Lu3+ and DOTA-TATE in a microchannel.

Section 3.1 will present a 3-dimensional and 2-dimensional model of a Y-shaped microfluidic

channel. Section 3.2 will describe a 2-dimensional model of a T-shaped microfluidic channel.

Subsequently, section 3.3 will describe a 2-dimensional model where the reactants are fully

mixed. Section 3.4 will move away from the simple 2-inlet geometries described in sections 3.1 &

3.2 and will use a geometry with four inlets. The final section of this chapter 3.5 will present a

summary of the input parameters that are used in the computational models.

3.1 Y-shaped models

The models discussed in this section consist of two inlet channels connected through a Y-shaped

connector to a single outlet channel (main reaction channel). Section 3.1.1 will present the 2-

dimensional model of the Y-channel while the 3-dimensional model of the Y-channel is discussed

in section 3.1.2.

3.1.1 Y-shaped 2-dimensional model

The streams containing the reactants were connected through a Y connector where the two inlet

channels join at an angle of 90˚ before they enter the main reaction channel. The main reaction

channel length is specified as distance between the point where the two inlets meet and the out-

let. This model used a main reaction channel length of 10 cm. The diameter of the inlet and out-

let channels has been set to 150 μm. The flow rates of the inlets were set to a 1:1 ratio. The inlet

concentration for both Lu3+ and DOTA-TATE has been set to 1 mol m-3.

The mesh that was constructed for this 2-dimensional geometry is shown in Figure 12. In total,

61442 elements consisting of 842 triangular elements and 60600 quadrilateral elements were

used to mesh the entire geometry.

Figure 12: Mesh used for 2D Y-shaped channel.

3.1 Y-shaped models 29

The Navier-Stokes equations were enforced to the system in COMSOL by implementing the

'Laminar flow' physics module. In addition, the 'Transport of diluted species' and the 'Chemistry'

modules have been implemented to model the diffusion, convection, and reaction of the dis-

solved species. The model assumes that fluid has a constant temperature throughout the entire

microfluidic channel and no heating occurs. Therefore, the model does not take heat transfer

into account.

3.1 Y-shaped models 30

3.1.2 Y-shaped 3-dimensional model

A 3-dimensional model of the geometry described in section 3.1.1 was constructed in order to

directly compare the simulation results of the 3-dimensional and 2-dimensional model. The

channels in this 3D model have a cylindrical shape. The predefined part 'Y-channel 3D, Circular

Cross-section' has been imported from the microfluidic module to build the geometry. Due to the

existence of a symmetry plane, only half of the geometry was modelled for the purpose of saving

computational time. The geometry is shown in Figure 13.

In contrast to the 2D model, the 3D model also considers variations in the z direction shown in

Figure 13.

Figure 13: Model geometry for the 3D Y-channel.

The mesh that was constructed to solve the governing equations with the finite element method

is shown in Figure 14. The complete mesh consists of a total of 833929 elements. The total num-

ber of elements is composed of 14633 tetrahedral elements, 2070 pyramid elements and

817226 hexahedral elements.

The 'Laminar flow', the 'Transport of diluted species', and the 'Chemistry' physics modules have

been used in the 3D model, similarly to the 2D model.

Figure 14: Mesh of finite elements used in the 3D computational model. Top view xy-plane (left figure) and cross-section of main channel yz-plane (right figure)

3.2 T-shaped 2-dimensional models 31

3.2 T-shaped 2-dimensional models

In conjunction with this thesis, an experimental study was conducted that used a microfluidic

system to determine the reaction kinetics of the formation of [177Lu]Lu-DOTA-TATE. For a more

detailed description of the experimental setup, see Liu et al. [58] (included in Appendix III).

The T-shaped 2-dimensional model was used to simulate the behaviour of this experimental

setup, which used a normal tee connector to combine the reactants as they enter the main

channel. The normal tee connector is a type of connector that merges two inlet streams at an

angle of 180˚ and connects to one outlet stream (IDEX). The resulting meshed T-shaped

geometry with two separate inlet streams is shown in figure 13. The main channel length used in

this model was set to 100 cm. In total, 92700 quadrilateral elements were used to mesh the en-

tire geometry. This geometry was simulated for channels with varying diameters. Simulations

were conducted to investigate channels with diameters of 100, 150, 500, 1000, and 1500 μm.

Figure 15: Mesh used for 2D T-shaped channel

In contrast to the computational models described in section 3.1, this model also takes heat

transfer into account as it is used to simulate a situation where the main reaction channel is

heated. The temperature of the solutions containing the reactants was set to 20˚C as they enter

the system. The temperature of the wall of the heated main reaction channel was set to 80˚C. The

'Heat transfer'physics module available in COMSOL has been used to calculate the temperature

profile of the fluid in the microfluidic channel.

3.3 Fully mixed model 32

3.3 Fully mixed model

An additional computational model was built to simulate the behaviour for the situation when

the reactants are fully mixed as they enter the main reaction channel. In this way, the formation

of [177Lu]Lu-DOTA-TATE for a situation is studied for a situation where perfect mixing is accom-

plished and diffusion limitations do not play a role. This model consists of a simple straight

channel (length: 1 meter, diameter: 100-1500 μm).

The channel is meshed identically to the main reaction channel in the T-shaped model using

rectangular elements.

3.4 Four-inlet model

One way of reducing the mixing time in a microfluidic channel is to use a parallel lamination

technique with multiple inlets [62, 63]. Depending on the number of inlets, multiple layers or

laminae can be formed. As a result, the diffusion distance can be reduced to achieve faster

mixing. In the Y- and T-shaped microfluidic systems described above, two laminae are formed.

The model in this section uses four inlets, instead of the 2 inlets used in the Y- and T-shaped

models, and forms a total of 4 laminae consisting of DOTA-TATE and Lutetium in an alternating

fashion. In this case, the diffusion distance is reduced by a factor 2 and the mixing time is

reduced by a factor 22=4 compared to a Y- or T-shaped system [64].

A channel diameter of 1000 μm was used for the four-inlet model and the simulation results will

be compared with the T-shaped model with an equivalent channel diameter. The meshed

geometry is shown in Figure 16 and consists of 108400 rectangular elements.

Figure 16: Mesh used for 4-inlet model.

3.5 Boundary conditions and input parameters 33

3.5 Boundary conditions and input parameters

The steady-state Navier-Stokes equation was solved to find the velocity profile of the fluid in the

microchannel. A no-slip boundary condition was imposed at the microchannel walls, and at the

outlet a pressure outlet boundary condition with zero relative pressure was used.

For the mass transport process, the walls of the microchannel were set to no flux and the outlet

was set at the end of the main reaction channel.

The fluid in the simulations was defined as water for all cases. The material properties of water

were imported from the built-in material library in COMSOL.

Several input parameters such as the reaction rate constant, diffusion coefficients, and dimen-

sions of the system needed to be predefined in the computational model. The following section

will first specify the input parameters used in the models described in section 3.1. Subsequently,

the input parameters of the model described in section 3.2, 3.3 and 3.4 will be presented.

3.5.1 Input parameters 2D and 3D Y-shaped computational models.

The simulation results of the 2D and 3D Y-shaped computational models described in section 3.1

are directly compared to each other to determine the validity of the simplified 2D model relative

to the more complex 3D model. In order to make a direct comparison between the 2D and 3D

model possible, the models need to have identical input parameters. The parameters that have

been used to construct the first two computational models in COMSOL are summarized in Table

1.

The diffusion coefficient of Lu3+ ( Ɗ =2.14E-9 m2s-1 at 20˚C) was calculated by using the Stokes-

Einstein equation [65]. The diffusion coefficients for DOTA-TATE or Lu-DOTA-TATE, however,

were not available at the time of these simulations. Therefore, it was decided to use the value of

the diffusion coefficient of DOTA (Ɗ =3.5∙E-10 m2s-1 at 20˚C) as an approximation of the diffusion

coefficient of DOTA-TATE [66].

Table 1: Inlet parameters for 3D and 2D Y-shaped computational models.

Name Expression Description

L 10 [cm] Length of main channel

L_in 5 [mm] Length of inlet channel W 150 [um] Diameter of channel

T_in 20 [˚C] Inlet temperature

u_Lu Range of values Flow rate 177Lu3+

u_DOTATATE Range of values Flow rate DOTA-TATE

u_Lu: u_DOTATATE 1:1 Ratio flow rates of Lu and DOTA-TATE c0Lu 1 [mol/m3] Inlet concentration 177Lu3+

c0DOTATATE 1 [mol/m3] Inlet concentration DOTA-TATE

Ɗ _Lu 2.14E-9 [m2/s] Diffusion coefficient 177Lu3+ at 20˚C

Ɗ _DOTATATE 3.5E-10 [m2/s] Diffusion coefficient DOTA-TATE at 20˚C

k Range of values Reaction rate constant

3.5 Boundary conditions and input parameters 34

3.5.2 Input parameters 2D T-shaped computational model.

The input parameters that have been used in the 2D T-shaped, fully mixed and four-inlet compu-

tational models in COMSOL are summarized in Table 2.

In contrast to the previous models, these models use a diffusion coefficient for DOTA-TATE

based on the diffusion coefficient for [177Lu]Lu‐DOTA‐TATE, which has been determined as part

of a bachelor thesis project [67]. The diffusion of [177Lu]Lu‐DOTA‐TATE was measured using

a method described by Miložič et al [68]. In this method, two solutions of which only one con-

tained 0.26mM [177Lu]Lu‐DOTA‐TATE were pushed at flowrates of 1-20 μl/min through an H-

channel chip (TOPAS, 75/150, Microfludic ChipShop). The concentrations of

[177Lu]Lu‐DOTA‐TATE, measured at the two outlets, are dependent on the diffusion of the com-

pound and the applied flow rates. The bachelor thesis by Laura Ballemans has focused on ob-

taining the experimental diffusion data. As part of this master thesis, a computational model of

the H-chip in COMSOL was used to determine the diffusion coefficient of [177Lu]Lu‐DOTA‐TATE

by fitting the outcome of the simulation to the experimental results. A more detailed description

can be found in Appendix II. The diffusion coefficient of DOTA-TATE was approximated to be

identical to the diffusion coefficient of [177Lu]Lu‐DOTA‐TATE. All diffusion coefficients were ad-

justed to the appropriate temperature by use of the Stokes-Einstein equation. This calculation

results in the diffusion coefficient of Lu3+ and DOTA-TATE at 80 ˚C with a value of 7.3 E-9 m2/s

and 6.5 E-10 m2/s respectively.

The computational models were tested by checking if it supports experimental results described

in Liu et al. (Appendix III) [58]. Therefore, the concentrations, temperature, flow rates and di-

mensions used in this model were set according to the experimental set-up.

Table 2: Inlet parameters for 2D T-shaped, fully mixed and four-inlet computational models

Name Expression Description

L 100 [cm] Length of main reaction channel

W Varying values 100-150-500-1000-1500 [um]

Diameter of channel

T_in 20 [˚C] Inlet temperature T_wall 80 [˚C] Wall temperature main reaction channel

u_lu Range of values Flow rate 177Lu3+

u_DOTATATE Range of values Flow rate DOTA-TATE

u_Lu: u_DOTATATE 1:2 Ratio flow rates of Lu and DOTA-TATE

c0Lu 0.26 [mol/m3] Inlet concentration 177Lu3+ c0DOTATATE 0.303 [mol/m3] Inlet concentration DOTA-TATE

Ɗ _Lu20 2.14 E-9 [m2/s] Diffusion coefficient 177Lu3+ at 20˚C

Ɗ _DOTATATE20 1.9 E-10 [m2/s] Diffusion coefficient DOTA-TATE at 20˚C

Ɗ _Lu80 7.3 E-9 [m2/s] Diffusion coefficient 177Lu3+ at 80˚C

Ɗ _DOTATATE80 6.5 E-10 [m2/s] Diffusion coefficient DOTA-TATE at 80˚C EA 109.5[kJ/mol] Activation energy

A 1.24 E-10 Pre-exponential factor

4.1 2D and 3D model Y-shaped model 35

Chapter 4. Results & Discussion This chapter will discuss the outcomes of the simulation results of the different computational

models.

The first three sections of this chapter discuss the relevant processes that play a role in the for-

mation of Lu-DOTA-TATE in a microfluidic system: diffusion, reaction, and heat transfer. Section

4.1 of this chapter will analyse the simulation results of the 2D Y-channel (introduced in section

3.1.1). This section will explore the influence of the diffusion coefficients and the reaction rate

constant on the concentration profile in the channel. Furthermore, this chapter will present a

comparison of the 3-dimensional and 2-dimensional simulation results. Subsequently, section

4.2 will discuss the influence of the channel dimensions on the mixing and reaction process in a

microfluidic channel. Finally, section 4.3 will make the connection of the simulation results to

the relevant factors that are required for a microfluidic system used for the formation of

[177Lu]Lu‐DOTA‐TATE intended for clinical applications.

4.1 2D and 3D model Y-shaped model

In order to find out if 2D models, which would require less computational time, could be used for

this study, the simulation results of a 2-dimensional and a 3-dimensional model were compared.

To achieve better understanding, the following sub-chapters will discuss first the 2-dimensional

Y-shaped model to show the process of mixing (Chapter 4.1.1) and reaction (Chapter 4.1.2) step-

by-step, followed in Chapter 4.1.3 by an evaluation of the influence of diffusion coefficients. Fi-

nally, in Chapter 4.1.4 the 2D and 3D model will be compared.

Both models describe a Y-shaped channel system. The models consist of separate streams con-

taining lutetium and DOTA-TATE that are combined in an Y-shaped connector where lutetium

enters through the top inlet and DOTA-TATE enters through the bottom inlet (Figure 17). As the

two streams combine and enter the main channel that has a diameter of 150 µm and is 10 cm

long, the two species mix through diffusion. Active mixing methods that rely on external forces

such as magnetic microstirrers, acoustic waves or periodic fluid pulsation to induce mixing [69-

71] are not considered in this thesis. Simultaneously to the diffusion of the chemical species, the

reaction DOTA-TATE + LuCl3 ---> Lu-DOTA-TATE takes place.

As a result, both the diffusion coefficients of lutetium and DOTA-TATE and the reaction rate con-

stant of the reaction DOTA-TATE + Lu3+ ---> Lu-DOTA-TATE will influence the reaction profile of

the chemical species in the microfluidic channel. Due to the small channel diameter, no heat

transfer limitations are expected. The absence of heat transfer limitations in microfluidic chan-

nel with a diameter of 150 µm is discussed in more detail in section 4.2.1.

4.1 2D and 3D model Y-shaped model 36

Figure 17: Velocity profile at the main channel entrance of the 2D Y-channel model

4.1.1 Mixing of reactants

As a first step, a 2D model has been built that focuses only on the mixing of lutetium and

DOTA-TATE. Mixing occurs mainly through diffusion and the chemical reaction is not yet

incorporated in the model.

The simulation results of this model, which describes the mixing of Lu and DOTA-TATE, are

shown in Figure 18 and Figure 19. Figure 18 depicts the concentration of lutetium at the en-

trance of the main channel. The figure shows that lutetium stays mostly confined to the top half

of the stream in the main channel. Thus the stream containing lutetium and the stream contain-

ing DOTA-TATE flow side-by-side.

Figure 18: Lutetium concentration at the main channel entrance of the 2D Y-channel model

4.1 2D and 3D model Y-shaped model 37

As discussed in chapter Chapter 2, mixing is largely dominated by molecular diffusion in laminar

flows. The diffusion time t required for a species to diffuse scales quadratically with the diffusion

distance and is inversely proportional to the diffusion coefficient .

(20)

Figure 19 shows the concentration of lutetium and DOTA-TATE at 10 cm downstream at the end

of the main channel. It can be clearly seen in Figure 19(a) that lutetium has almost fully diffused

throughout the channel width as the concentration profile is nearly uniform. DOTA-TATE, on the

other hand, has not fully diffused across the channel which is represented by the separate layers

observed in Figure 19(b). This difference in mixing is due to the fact that the diffusion coefficient

of DOTA-TATE ( = 3.5 E-10 m2 s-1) is a factor 7 smaller than the diffusion coefficient of lutetium

( = 2.14E-9 m2 s-1). Based on equation (20), we can infer that the diffusion of DOTA-TATE is a

factor 7 slower than the diffusion of Lutetium. This means that DOTA-TATE needs a diffusion

time that is sevenfold the diffusion time Lutetium to reach the same diffusion distance. As a

result, the distance that DOTA-TATE has to travel downstream to diffuse throughout the full

width of the channel is sevenfold the distance that Lutetium requires.

Based on the concentration profile of lutetium and DOTA-TATE in Figure 19, it can be expected

that most of product is formed in the bottom half of the reaction channel as the majority of the

DOTA-TATE stays confined to this region in the channel.

Figure 19: Concentration profile at the end of the main reaction channel (10 cm length) of lutetium (a) and DOTA-TATE (b)

4.1 2D and 3D model Y-shaped model 38

4.1.2 Reaction

The next step is to add the reaction taking place between Lu and DOTA-TATE to the model. Fig-

ure 20 shows the concentration profile of Lu-DOTA-TATE at the end of the channel for different

reaction rate constants (10, 1.0, and 0.1 mol/ m3 s). These values of the reaction rate constants

were chosen as it was expected that the reaction rate for the formation of Lu-DOTA-TATE falls

within this region.

The figure shows that due to the faster diffusion of lutetium (compared to DOTA-TATE) , the

majority of the product is formed at the bottom half of the stream where the DOTA-TATE

solution enters the main reaction channel.

In Figure 20, a shift of the concentration profile of the formed product, Lu-DOTA-TATE, can be

observed if the reaction rate constant is altered by a factor 10. The concentration profile shifts

more towards the wall of the DOTA-TATE inlet when the reaction rate constant (k) is decreased.

This can be explained by the fact that the decrease of the reaction rate constant will result in an

increase of the reaction time scale as the reaction occurs slower. This means that the Damkӧhler

number, which is specified as the ratio of the characteristic mixing time and characteristic

reaction time, will decrease from 26.3 for k=10 mol/m3 s to 0.26 for k=0.1 mol/m3 s.

Figure 20: Concentration profile Lu-DOTA-TATE at the end of the main reaction channel (channel length: 10 cm) for different reaction rate constants: k=10 mol/m3 s (Figure 20(a)), k=1 mol/m3 s (Figure 20(b)), k=0.1

mol/m3 s (Figure 20(c))

4.1 2D and 3D model Y-shaped model 39

For Da>>1, the reaction will occur very fast compared to diffusion of reactants. In other words,

the reactants are consumed immediately after the species have diffused into each other

layer.Therefore, the product is formed at the centre of the channel at the interface of the Lu and

DOTA-TATE streams which can be seen in Figure 20(a) for a reaction constant of 10 mol/m3 s

(Da= 26.3). As the Damkӧhler number is reduced by a factor 10 and 100 (the reaction rate

constant is increased tenfold in Figure 20(b) and hunderdfold in Figure 20(c)), the lutetium ions

have more time to diffuse before the reaction takes place. Thus the concentration profile of the

formed product (Lu-DOTA-TATE) shifts more towards the wall of the DOTA-TATE inlet as can be

seen in Figure 20(b) and Figure 20(c).

The concentration of Lu-DOTA-TATE at the outlet of the main reaction channel is plotted in

more detail in Figure 21. This figure shows that the maximum concentration of Lu-DOTA-TATE

is reached near the interface where the two inlet streams meet for a reaction rate constant of 10

mol/ m3 s. For slower reactions (k=0.1 mol/ m3 s and k= 1.0 mol/ m3 s) , however, the maximum

concentration of Lu-DOTA-TATE is found near the wall where the fluid has a longer residence

time due to the laminar velocity profile.

Figure 21: Concentration profile of Lu-DOTA-TATE at the main reaction channel outlet for varying reaction

rate constants. All three cases were performed for a channel diameter of 150 µm and a flow rate of 125 µl/min.

4.1 2D and 3D model Y-shaped model 40

4.1.3 Influence of diffusion coefficient

As stated above in section Mixing of reactants4.1.1, mixing of the reactants occurs through

molecular diffusion. The diffusion coefficients of lutetium and DOTA-TATE used in this thesis

are estimates and not known with precise certainty. Therefore, the diffusion coefficients of

lutetium and DOTA-TATE were varied to study the impact of the mixing rate on the formation of

Lu-DOTA-TATE. Three cases are definined to investigate the influence of the diffusion

coefficients on the formation of Lu-DOTA-TATE.

The base case uses the diffusion coefficients for lutetium and DOTA-TATE that were used in the

previous simulation models ( = 2.14E-9 m2 s-1 = 3.5E-10 m2 s-1). In case ii, the

diffusion coefficient of lutetium is reduced by 50% while the diffusion coefficient of DOTA-TATE

is kept at its original value. In case iii, the diffusion coefficient of DOTA-TATE is reduced by 50%

and the diffusion coefficient of lutetium is brought back to its original value of the base case (

= 2.14E-9 m2 s-1 ). The decision to use a variation of 50% for the diffusion coefficients originates

from the obtained uncertainty of the diffusion coefficient of DOTA-TATE that were determined

by fitting a computational model to experimental results (Appendix II).

The concentration of Lu-DOTA-TATE at the channel outlet for the three cases described above is

shown in Table 3. The table shows that a reduction of the diffusion coefficients of the reactants

will cause a decrease in product concentration. In both case ii and case iii, the mixing process

occurs slower compared to the base case. Furthermore, Table 3 shows that a reduction of the

diffusion coefficient of lutetium has a bigger impact than a proportional decrease of the diffusion

coefficient of DOTA-TATE. This can be explained by the fact that lutetium diffuses faster than

DOTA-TATE as shown in Figure 20 and thus the majority of the product is formed at the bottom

half of the channel (especially for high flow rates where the reaction yield does not approach

100%). As a result, the diffusion of lutetium plays a bigger role in this system and therefore has a

bigger impact on the formation of Lu-DOTA-TATE.

Table 3: Influence of diffusion coefficients of lutetium and DOTA-TATE on the formation of Lu-DOTA-TATE

Flow rate Concentration Lu-DOTA-TATE (mol/m3)

= 2.14E-9 m2 s-1

= 3.5E-10 m2 s-1

(Base case)

reduced by 50%

(Case ii)

reduced by 50%

(Case iii) 10 ul/min 0.4111 0.4040 0.4077 25 ul/min 0.3072 0.2814 0.3009 100 ul/min 0.1014 0.07588 0.09836

4.1 2D and 3D model Y-shaped model 41

4.1.4 Comparison 3D and 2D model

Both 3-dimensional as well as 2-dimensional models have been used in computational studies

for the design of a microfluidic system. For example, a computational study for the design of a

microfluidic system for the synthesis of a 11C-based PET tracer by Haroun et al. takes a 2D ap-

proach [24]. Other studies, however, have taken a 3D approach [31, 72, 73] . Although some

computational studies provide a comparison of the simulation results of a 2D and 3D model of

their specific geometry (e.g. for microfluidic systems with rectangular channels [74]), this

knowledge is not available for the microfluidic system under investigation in this thesis .

Therefore, a 3-dimensional model has been built to simulate the reaction-diffusion process in

the same microfluidic system and compared to the 2-dimensional version of the model.

Due to the existence of a symmetry plane, only the top half of the cylindrical channel was simu-

lated. The velocity profile and the concentration profile of Lu-DOTA-TATE at the outlet of the

main reaction channel are shown in Figure 22. Similarly to the concentration profile of Lu-

DOTA-TATE in the 2D model, the majority of the product is formed at y<0 in Figure 22(b) at the

side of the stream where the DOTA-TATE solution enters the main reaction channel. In contrast

to the 2D model, Figure 22(a) shows that the Lu-DOTA-TATE concentration does not only varies

in the y-direction, but also in the z-direction. This is due to the 3-dimensional laminar flow pro-

file.

If we consider the laminar flow profile in Figure 22(a), we can observe that the velocity de-

creases as we move from the centre of the channel (y=0, z=0) upwards or downwards in the z-

direction towards the wall. Therefore, the fluid closer to the wall has a longer residence time

than the fluid at the centre of the channel. This increase in residence time of fluid near the wall

results in an increased formation of Lu-DOTA-TATE as we move upwards or downwards in the

z-direction in Figure 22(b).

Figure 22: Velocity profile (a) and concentration profile of Lu-DOTA-TATE (b) at the outlet of main reaction

channel.

4.1 2D and 3D model Y-shaped model 42

The concentration profile of Lu-DOTA-TATE at the outlet of the channel for the 2D and 3D model

are compared and plotted in Figure 23. This plot shows the concentration as a function of the

position across the channel width (y-direction in Figure 22). Therefore, the concentration profile

for the 3D model was averaged over the channel height in the z-direction. This was done by inte-

grating the velocity weighted concentration over the channel height and dividing this by the

integration of the velocity over the channel height.

(21)

Figure 23 shows that there is a discrepancy in the concentration profile between the two models

resulting from a difference of the velocity profile between the 2D and 3D model.

Figure 23: Concentration profile of Lu-DOTA-TATE at outlet of microfluidic channel for 2D and 3D model

It should be noted, however, that the Lu-DOTA-TATE concentration averaged over the entire

outlet width is quite similar for the 2D and 3D model in Figure 23 (0.3072 mol/m3 for the 2D

model and 0.3171 mol/m3 ). Therefore, it was decided to further investigate the difference of the

average Lu-DOTA-TATE concentration of the 2D and 3D model.

The average concentration at the outlet for the 2D and 3D model has been evaluated for 4 differ-

ent flow rates (1 µL/min, 10 µL/min, 25 µL/min and 100 µL/min). The results are shown in Ta-

ble 4.

Table 4: Comparison 2D and 3D model

Flow rate Residence time (s)

Average concentra-tion 2D model (mol/m3)

Average concentra-tion 3D model (mol/m3)

Difference in reaction yield (%)

1 µL/min 106 0.4906 0.4906 0.0120 10 µL/min 10.6 0.4111 0.4134 0.458 25 µL/min 4.24 0.3072 0.3171 1.97 100 µL/min 1.06 0.1014 0.1225 4.23

4.1 2D and 3D model Y-shaped model 43

The table shows that the difference between the reaction yield reached in the 2D and 3D model

results for low flow rates with reaction yields >80% is below 0.5% and the difference increases

for higher flow rates. The difference between the 2D and 3D model for a flow rate of 100 µL/min

resulting in a reaction yield of ~20% is approximately 4%. The reaction rate constant used in

the comparison in Table 4 equals 1 mol/ m3 s. For comparison, the reaction rate constant used in

this thesis for the formation of Lu-DOTA-TATE at 80 ˚C equals 0.902. The difference between the

2D and 3D model is of similar size for reaction rate constants equal to 0.1 mol/m3 s and 10

mol/m3 s.

The simulation results reveal that the 2D simulation model predicts a lower product formation

than the 3D simulation model for all flow rates considered. This result is in line with the findings

of a simulation study by Sullivan et al. which investigated the diffusion of Mn2+ ( = 2.0E-9 m2 s-

1) and glycerol ( = 1.0E-9 m2 s-1) tracers in a Y-shaped micromixer with rectangular channels

[74]. This study reported that the 2D simulation slightly under estimated the extent of diffusive

dispersion compared to the 3D simulation. The 2D approximation in this study agrees reasona-

bly well with the 3D simulation and found an error of ~ 3%. This finding of Sullivan et al. ex-

plains why the reaction yield in the 2D model in this thesis is lower than the reaction yield of the

3D model.

Based on the similarity of the results of the 2D and 3D simulation models, it was decided to use

2D models in the remainder of this thesis to reduce the computation time of the simulation

models. To illustrate the time saved, the computation time of a 2D model simulation was ap-

proximately 10 minutes while the computation time of the equivalent 3D model simulation was

approximately 8 hours.

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

44

4.2 Influence of channel dimensions on the mixing process and reaction in

a microfluidic channel

This section will discuss the simulation results of two models, a T-shaped and a fully mixed

model, that were applied to investigate the influence the channel dimensions have on the mixing

process and reaction in microchannels. These models correspond to microchannel systems used

for experiments which were performed in conjunction with this thesis [58]. The flow rate ratio

in both models was set according to the value used in the experiments at 2:1 (DOTA-TATE:Lu).

The simulation models in this section use a main channel length of 100 cm and varying values

for the channel diameter (100, 150, 500, 1000, and 1500 μm).

4.2.1 Heating of reaction channel

An important process that was not taken into account in the previous models is heat transfer.

This section will discuss why heat transfer can be neglected for channels with the small diameter

that was used in the previous model (150 μm). In the next parts of this study, however, heat

transfer has to be taken into account, since channels with larger dimensions are investigated.

The reaction of [177Lu]Lu3+ with DOTA-TATE is customarily done at 80 °C [57]. At room tem-

perature, the formation of [177Lu]Lu-DOTA-TATE is very slow. This was validated by an experi-

mental setup described by Liu et al. where a [177Lu]LuCl3 solution (0.260±0.003 mM) and a

DOTA-TATE solution (0.303±0.003 mM) were injected separately through two inlets into the

reaction capillary. The flow rate ratio was set to 2:1 (DOTA-TATE:Lu), resulting in a molar ratio

of 2.33:1. No formation of [177Lu]Lu-DOTA-TATE could be detected when the reaction mixture

was left standing for several hours at room temperature (pH 4.0-4.5) [58]. As explained in sec-

tion 2.4, the reaction rate is increased as the temperature is raised. Therefore, the reaction

channel needs to be heated to the desired reaction temperature (80 ˚C).

4.2.1.1 Theoretical calculation

The temperature profile of laminar flow in a cylindrical tube with a constant wall temperature

can be calculated by setting up a heat balance over a thin slice dx as depicted in Figure 24. The

heat balance is shown in equation (22) [40].

(22)

In equation (22), D represents the diameter of the tube, is the density of the solution, is the

average velocity of the fluid, is the specific heat capacity of the solution and is the heat

transfer coefficient. and represent the temperature of the wall and the average tempera-

ture with respect to the position x along the tube.

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

45

Figure 24:Differential volume element used in heat balance [40]

The average temperature, , equals at the beginning of the heated section (x=0). The follow-

ing expression can be derived to calculate the average temperature as a function of x in equation

(23) [40].

(23)

The heat transfer coefficient (h) can be calculated by using equation (24):

(24)

where Nu is the dimensionless Nusselt number and is the thermal conductivity.

For the case of a confined laminar flow where the wall temperature increases from T0 to Tw at

z=0 and remains at this constant temperature for z>0, two regions can be distinguished as

shown in Figure 25: the thermal entrance region and the thermally developed region.

Thermal entrance region

In the thermal entrance region, a temperature gradient exists in the thermal boundary layer ( ).

The thickness of the boundary layer increases further down the stream until it reaches the tube

center at x= Lt. The length of this region is called the thermal entrance length (Lt) and can be

calculated by the expression in equation (25).

(25)

The symbol in equation (25) is the thermal diffusivity of the fluid ( =0.143*10-6 m2 s-1 for wa-

ter). The thermal entrance length in a channel with a diameter of 100 μm for the highest applied

velocity used in this study (0.20 m/s) equals 0.7 mm. Compared to the total length of the reac-

tion channel (100-116 cm), this is negligible.

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

46

Thermally developed region

The region x> Lt is called the thermally developed region. The temperature still increases in this

region, but the dimensionless temperature profile is invariant with axial distance x [75, 76].

Figure 25: Thermal entrance length and developing temperature profile (based on [76])

The Nusselt number is a function of axial position x in the thermal entrance region and Nu be-

comes a constant in the thermally developed region. For laminar flow in a cylindrical pipe the

expressions for Nu in equations (26) and (27) can be used [40].

(26)

(27)

For simplicity, the thermal entrance length is neglected and Nu is regarded to have a constant

value of 3.66 throughout the entire channel.

As the fluid flows through the heated section of the reaction channel, the temperature of the

fluid increases from T0 (20˚C) to Tw (80 ˚C). The length needed to achieve a certain temperature

increases for an increase of the average velocity (see equation (23)). Therefore, the maximum

required length to reach the desired temperature is calculated for this study by using the maxi-

mum applied velocity (0.2 m/s).

Equation (23) is used to find the required length for the solution to reach an average tempera-

ture of 79.5˚C for an applied average velocity of 0.2 m/s. For a channel with a diameter of 100

μm, the required length to reach the desired temperature is approximately 0.5 cm. Similarly, the

required length for a channel with a diameter of 150 μm is 1 cm. In these cases, the required

length to reach the desired temperature is negligible to the total length of the main reaction

channel (100-116cm).

For a channel with a diameter of 1500 μm, however, the required length for the solution to reach

the desired temperature is 103 cm, which can no longer be neglected.

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

47

4.2.1.2 Simulation results

The influence of the channel diameter on the temperature profile can also be observed in the

simulation results of the T-shaped 2D model as shown in Figure 26. The figure shows the tem-

perature profile of the fluid at 0.5 cm downstream in the heated main reaction channel. It is

nearly uniform at 80˚C for a channel diameter of 100 μm. For a channel diameter of 1500 μm, on

the other hand, the temperature profile is still developing and the fluid at the center of the chan-

nel is still at the initial temperature of 20˚C

Figure 26: Simulation results of the temperature profile at 0.5 cm down the main reaction channel for a chan-nel diameter of 100 μm (Figure 24a) and 1500 μm (Figure 24b) with an applied average velocity of 0.2 m/s.

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

48

4.2.2 T-shaped 2-dimensional model

The velocity profile of the T-shaped 2D model is shown in Figure 27. The figure shows that the

inlet flow rate of DOTA-TATE is twice the inlet flow rate of lutetium.

Figure 27: Velocity profile for the T-shaped 2D model

To investigate the influence of the diameter of the main channel on the reaction yield, the same

T-shaped geometry was simulated for different values of the channel diameter. The reaction

yield of Lu-DOTA-TATE as a function of residence time is shown in Figure 28 for channel diame-

ters of 100, 150, 500, 1000, and 1500 μm. The figure shows that the reaction yield of Lu-DOTA-

TATE is reduced if the channel diameter is increased.

As explained before, both diffusion as well as the heating process are dependent on the diameter

of the channel. The diffusion time required for the reactants to diffuse across the channel width

is proportional to the square of the channel diameter. The required time downstream for the

solution to reach the desired temperature is also proportional to the channel diameter squared.

(28)

(29)

Therefore, an increase of the channel diameter will lead to slower mixing. The heating process

occurs more slowly as well, as it takes a longer time for the fluid to reach the desired tempera-

ture. The temperature of the fluid will impact the formation rate of Lu-DOTA-TATE due to the

temperature dependency of the kinetic rate constant (section 2.4) [58]. These heat transfer and

diffusion limitations will reduce the reaction yield of Lu-DOTA-TATE for increasing diameters.

The ratio of the thermal diffusivity and mass diffusion coefficient is captured by the Lewis num-

ber (section 2.3.4). The Lewis number for this system equals

, indicating that

the heat diffuses approximately 20 times faster than the chemical species.

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

49

Figure 28: Influence of channel diameter on reaction yield.

The simulation results have been used to determine the required residence times for each di-

ameter to achieve the desired reaction yield of 99% which are shown in Table 5. The results

again confirm that an increase in diameter leads to a slower reaction. The residence time re-

quired to reach 99% reaction yield for a T-shaped system with a channel diameter of 1500 μm

(123.8 seconds) is almost triple the value of the residence time required for a system with a

channel diameter of 100 μm (45.5 seconds).

For all diameters studied in this thesis, the reaction time is significantly reduced compared to

the conventional vessel based method which requires a reaction time of approximately 20 min-

utes (experiments conducted in small volume reaction tubes 40-75 μl) [13, 77].

Table 5: Residence time needed for 99% conversion for T-shaped model.

Diameter (μm) Residence time (s) needed to reach a reaction yield of 99 %

100 45.5 150 45.7 500 51.2 1000 77.1 1500 123.8

The temperature profile develops much faster than the diffusion process of the reactants as Le ≈

20. Therefore, it is expected that the dependency of the required residence time on the channel

diameter can largely be contributed to the diffusion of chemical species. This was checked by

removing the heat transfer process from the system by setting both inlet streams to the desired

inlet temperature of 80 ˚C (in the normal T-shaped model, the inlet streams are set to 20 ˚C and

heated after the two inlet streams are combined). Indeed, the required residence time to reach a

reaction yield of 99% in a system with channel diameter of 500 μm that has a uniform tempera-

ture of 80 ˚C equals 51.2 seconds and is not different from the T-shaped system with heat trans-

fer (51.2 seconds). The required residence time is significantly reduced, on the other hand, if the

role of mass transfer is reduced by having the reactants pre-mixed before they enter the main

reaction channel (explained in more detail in section 4.2.3). Therefore, it can be concluded that

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

50

heat transfer limitations do not play a significant role for the T-shaped model and the depend-

ency of the required residence time on channel diameter is mainly governed by the diffusion of

the reactants.

As discussed above, mixing occurs through diffusion and the characteristic mixing time is pro-

portional to the square of the channel diameter. Therefore, it is expected that the required resi-

dence to achieve the desired reaction yield of 99% scales quadratically with the channel diame-

ter. This behaviour is supported by the results obtained in the simulations shown in Table 5.

Figure 29 shows the dependence of the required residence time on the channel diameter and the

data points were fitted to a quadratic equation. The quadratic relation provides a good fit to the

simulation results with a R2-value of 0.9979.

Figure 29: Residence time needed for 99% conversion for T-shaped model as a function of channel diameter.

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

51

4.2.3 Fully mixed model

An additional computational model was built to simulate the reaction behaviour for the situation

when the reactants are fully mixed as they enter the main reaction channel. The simulation re-

sults are compared to experimental data that was obtained in conjunction with this thesis. The

experimental setup used a static mixing tee (IDEX-540) which contains a frit to accomplish near

instantaneous mixing of the reactants at the entrance of the main reaction channel [58].

Despite the fact that the reactants are fully mixed as they enter the main reaction channel, diffu-

sion still needs to be taken into consideration. This can be best understood by considering the

Lu-DOTA-TATE concentration profile at the channel outlet as shown in Figure 30. It can be seen

that most of the product is formed at the walls of the channel as a result of the laminar flow pro-

file that leads to a longer residence time of the fluid near the walls. This means that the a signifi-

cant larger quantity of the reactants are consumed near the channel walls, and need to be con-

tinuously replenished by molecular diffusion.

Figure 30: Concentration Lu-DOTA-TATE at the outlet of the channel for a channel diameter of 1000 μm and a

velocity of 0.05 m/s, which equals a residence time of 20 seconds.

An increasing channel diameter will thus lead to a larger influence of diffusion on the reaction.

slower development of the temperature profile. To isolate the influence of molecular diffusion

on the reaction yield of Lu-DOTA-TATE, a simulation was performed that removed the heat

transfer process and used an uniform temperature of 80 ˚C. A reaction yield of 99% was ob-

tained after 45.8 seconds for a channel diameter of 500 μm while 49.5 seconds were required to

reach a reaction yield of 99% for a channel diameter of 1500 μm. In the absence of heat transfer,

this increase in required residence time as a function of channel diameter proves that molecular

diffusion should still be taken into account.

If heat transfer is also taken into account by setting the inlet stream temperature to 20 ˚C and

the wall of entire reaction channel to 80 ˚C, the required residence time for a channel diameter

of 1500 μm increases further from 49.5 s (system with uniform temperature of 80 ˚C) to 53.9 s.

This means that both heat transfer and mass transfer play a role in the formation of Lu-DOTA-

TATE in microfluidic channel where the reactants are pre-mixed.

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

52

The reaction yield of Lu-DOTA-TATE for the fully mixed model as a function of residence time is

shown in Figure 31 for channel diameters of 100, 150, 500, 1000, and 1500 μm. The figure

shows that again the reaction yield is reduced for increasing diameters. The reduction in reac-

tion yield, however, is smaller compared to the T-shaped model results (see Figure 28) where

the reactants are not mixed. This can be explained by the fact that, in contrast to the T-shaped

model, the reactants are already fully mixed.

The reduction in reaction yield is governed by both heat transfer as well as mass transfer con-

siderations. However, the impact of diffusion limitations in the fully mixed model is significantly

reduced compared to the T-shaped model.

Figure 31: Influence of channel diameter on reaction yield when the reactants are fully mixed.

The simulation results for the reaction yield were compared to the values obtained in the ex-

periments using a mixing tee for a channel diameter of 100 μm [58]. Figure 32 shows the reac-

tion yield of Lu-DOTA-TATE as a function of residence time for the experimental and simulation

results. The results of the simulations fall within the error bars in Figure 32, indicating a good

agreement between simulation and experiment.

Figure 32: Comparison of reaction yield of experimental results vs simulation results in a 100 μm channel. Experimental conditions: concentration lutetium inlet = 0.260±0.003 mM and concentration DOTA-TATE

inlet= 0.303±0.003 mM. Flow rate ratio set to 2:1 (DOTA-TATE:Lu).

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

53

The results in Figure 31 show that the reaction yield for 100, 150, and to a lesser extent 500 μm

channels are very similar. The maximum difference in reaction yield observed between the 100

μm and 150 μm channel is less than 1%. These simulation results coincide with the experimental

results using different channel diameters (ranging from 50 to 254 μm) for which no differences

in reaction yield could be observed [58]. The experimental reaction yields that were obtained for

channel diameters of 50, 100, 150, and 254 μm are shown in Figure 33.

Figure 33: Influence of channel diameter on the reaction yield of [177Lu]Lu-DOTA-TATE [58].

Based on the simulation results of the fully mixed model, the minimum residence time to reach

99% conversion has been determined for the different channel diameters as shown in Table 6.

This is the minimum residence time that can be accomplished for the case when diffusion limita-

tions do not play a role and is reached when the reactants are fully mixed at the entrance of the

main reaction channel. The results in Table 6 show that the required residence time to reach

99% is prolonged from 44.6 second for a 100 μm channel to 53.9 seconds for a 1500 μm channel.

This is equivalent to an increase in required residence time of approximately 20% as the channel

diameter is increased from 100 μm to 1500 μm. Based on these findings, the conclusion is made

that only slightly slower reactions are expected for the formation of Lu-DOTA-TATE as the chan-

nel diameter is increased up to 1.5 mm when the reactants are fully mixed at the entrance of

main reaction channel.

Table 6: Residence time needed for 99% conversion for fully mixed model

Diameter (μm) Residence time (s) needed to reach reaction yield of 99 %

100 44.6 150 44.7 500 46.3 1000 49.9 1500 53.9

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

54

4.2.4 Comparison of fully mixed model and T-shaped model

A direct comparison of the reaction yield for the T-shaped model and the fully mixed model for

channel diameters of 100 μm and 1000 μm is provided in Table 7.

Table 7: Reaction yield for T-shaped model and fully mixed model for a channel diameter of 100μm and 1000μm

Residence time (s)

Reaction yield (%) Reaction yield (%) Diameter 100 μm Diameter 1000 μm

T-shaped model Fully mixed model T-shaped model Fully mixed model 5 49.38 50.06 13.23 35.79

10 71.77 72.13 34.82 63.47

20 89.87 90.03 64.13 86.23

50 99.31 99.39 94.20 99.01

100 99.91 99.99 99.75 99.99

As shown in Table 7, the simulation results for the T-shaped model and the fully mixed model

are very similar for a channel diameter of 100 μm. The discrepancy of the reaction yield between

the models with a channel diameter of 100 μm is less than 1% for all residence times. The simi-

larity of the T-shaped model and the fully mixed model is also illustrated by the residence time

needed to reach a reaction yield of 99% shown in Table 5 and Table 6. For the T-shaped micro-

fluidic system, it takes 45.5 seconds while the fully mixed model reaches a reaction yield of 99%

after 44.6 seconds.

This can be explained by consideration of the Damkӧhler number. For a channel diameter of 100

μm, Da equals 0.14. This means that the overall reaction in the system is limited by the reaction

rate. In this region, diffusion does not have a limiting role on the overall reaction yield. In other

words, the reaction in the system occurs on a slower time scale than the mixing process. As a

result, the reaction yields obtained for a channel diameter of 100 μm are very similar irrespec-

tive of whether the reactants are mixed or not at the entrance of the main reaction channel.

For a channel diameter of 1000 μm, on the other hand, a significant difference can be observed

in the reaction yield for the T-shaped model and the fully mixed model. The maximum difference

in reaction yield between the T-shaped model and the fully mixed model in Table 7 is approxi-

mately 29% (for 10 seconds residence time). Furthermore, the residence time needed to reach a

reaction yield of 99% equals 77.1 seconds and 49.9 seconds for the T-shaped system and the

fully mixed system respectively (see Table 5 and Table 6).

For a channel diameter of 1000 μm, the Damkӧhler number equals 14.3. In this case, the system

is in the diffusion limited region. This means that the mixing time is slower than the reaction

time scale. In other words, diffusion becomes a limiting factor for the reaction yield that is ob-

tained in the situation when the reactants are not mixed at the entrance of the main reaction

channel. Therefore, the reaction yields obtained for a channel diameter of 1000 μm depends on

whether the reactants are mixed or not at the entrance of the main reaction channel.

Based on the simulation results in Table 5, Table 6 and Table 7, a comparison can be made be-

tween a microfluidic system in which the reactants are fully mixed and a system where the reac-

tants are added separately. In the fully mixed model, it is assumed that rapid mixing is achieved

to ensure that the reactants are fully mixed when they enter the main reaction channel. Rapid

mixing is not required for channel diameters of 500 μm and lower as the difference between the

two systems in residence time required to reach a reaction yield of 99% does not surpass 10%.

For channel diameters of 1000 μm and higher, rapid mixing is preferred as the difference in

residence time to reach the required reaction yield surpasses 50%.

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

55

4.2.5 Influence of reaction kinetics

This section will explore the impact of reaction kinetics on the simulation results of the T-shaped

model and the fully mixed model.

First, the situation is considered for faster reactions where the value of the reaction rate con-

stant is increased compared to the reaction simulated in the previous sections. This means that

the ratio of the characteristic time scale of diffusion to the characteristic time scale of chemical

reaction, represented by the Damkӧhler number (kcL2/ Ɗ), increases for faster reactions. In

other words, the rate of reaction has increased relative to the rate of diffusion. As a result, the

relative importance of diffusion limitations grows for increasing Damkӧhler numbers. Therefore,

it is expected that the difference between the fully mixed model and the T-shaped model grows

when Da is increased. To support this, simulations were done for reaction kinetics that were

increased by a factor 10 compared to those available for the synthesis of Lu-DOTA-TATE. The

Arrhenius parameters for this model are set as A=1.24E20 M-1 s-1 and Ea=109.5E3 J mol-1. This

means that the resulting reaction rate constant at 80˚C is set at 9.02 mol m-3 s-1 (compared to

0.902 mol m-3 s-1 used previously). The resulting Damkӧhler number for a system with a channel

diameter of 1000 μm used in this model is equal to 143.

The simulation results of the fully mixed model and the T-shaped model for the faster reaction in

a 1000 μm channel are shown in Table 8.

Clearly, the faster reaction will lead to higher reaction yields for the same residence time com-

pared to the results in Table 7. After a residence time of 20 seconds in the T-shaped model, for

example, the simulation results for the faster reaction in Table 8 reports a reaction yield of

88.6% while a yield of 64.1% is observed for the original reaction kinetics in Table 7.

But more importantly, Table 8 shows that for a residence time of 10 seconds, the T-shaped

model reaches a reaction yield of 68% while the fully mixed model reaches already a reaction

yield well over 99%. The previous simulation results in Table 7 (where the reaction rate con-

stant is tenfold lower than in the simulation used for Table 8) indicate that a reaction yield of

approximately 99% is reached only after a residence time of 50 seconds with the fully mixed

model. The corresponding yield for the T-shaped model after 50 s would be 94%. Thus the dif-

ference between the fully mixed model and the normal tee model increases for faster reactions

as expected based on the increase of the Damkӧhler number. This is relevant for the synthesis of

radiopharmaceuticals such as [68Ga]Ga-THP-TATE [78] which have faster reaction kinetics than

[177Lu]Lu-DOTA-TATE.

As an increase in temperature will lead to faster reaction kinetics (see section 2.4), this finding is

also relevant for the synthesis of Lu-DOTA-TATE at higher temperatures than 80˚C. It should be

noted, however, that the impact of the heat transfer process needs to be reconsidered when

higher temperatures are applied.

Table 8: Reaction yield for T-shaped model and fully mixed model for a channel diameter of 1000 μm with reaction 10 times faster than the reaction of Lu-DOTA-TATE.

Residence time (s)

Reaction yield (%) Diameter 1000 μm

T-shaped model Fully mixed model 2.5 18.44 60.91 5 45.09 94.43

10 68.28 99.86 20 88.61 99.99 50 99.78 100.00

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

56

The same line of reasoning can be applied for the case of a slower reaction. As the reaction rate

constant is reduced, the Damkӧhler number decreases and thus the impact of diffusion limita-

tions are reduced. As a result, the difference between the fully mixed model and the T-shaped

model is reduced.

Simulations have been done for reaction kinetics that were reduced by a factor 10 compared to

those available for the synthesis of Lu-DOTA-TATE. The Arrhenius parameters for this model are

set as A=1.24E18 M-1 s-1 and Ea=109.5E3 J mol-1. This means that the resulting reaction rate con-

stant at 80˚C is set at 0.0902 mol m-3 s-1. The resulting Damkӧhler number for this model is equal

to 1.43.

The simulation results of the fully mixed model and the T-shaped model for the slower reaction

in a 1000 μm channel are shown in Table 9. The maximum difference in reaction yield between

the T-shaped model and the fully mixed model in Table 9 is approximately 8.5% for a residence

time of 25 seconds. This difference is much smaller compared to the difference in reaction yield

observed for the Lu-DOTA-TATE reaction in Table 7 which shows a difference of 29%. This find-

ing confirms that a decrease of the Damkӧhler number will reduce the difference between the T-

shaped and fully mixed model as the relative importance of diffusion limitations are diminished

for decreasing Damkӧhler numbers.

Table 9: Reaction yield for T-shaped model and fully mixed model for a channel diameter of 1000 μm with reaction 10 times slower than the reaction of Lu-DOTA-TATE

Residence time (s)

Reaction yield (%) Diameter 1000 μm

T-shaped model Fully mixed model 25 20.24 28.78 50 41.24 48.39

100 66.86 70.93 200 88.08 89.49 500 99.20 99.35

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

57

To further illustrate the influence of the reaction rate on the simulation results of the T-shaped

model and the fully mixed model, the residence time required to reach a reaction yield of 99%

has been calculated for the 'slow' (kLu-DOTA-TATE*10-1, reaction 10 times slower than the reaction of

Lu-DOTA-TATE) and 'fast' reaction (kLu-DOTA-TATE*10, reaction 10 times faster than the reaction of

Lu-DOTA-TATE). Furthermore, the ratio of the required residence time of the T-shaped model

(tT-shaped model) to the required residence time of the fully mixed model (tfully mixed model) has been

calculated.

The results in Table 10 show that for the 'fast' reaction, the required residence time for the T-

shaped model is more than fivefold the required residence time for the fully mixed model. For

the ‘slow’ reaction, on the other hand, the required residence time for the T-shaped model has

increased approximately 5% compared to the fully mixed model. In other words, the relative

difference between the simulation results of the T-shaped model and the fully mixed model in-

creases for increasing Damkӧhler numbers.

These findings can be explained by the fact that the relative importance of diffusion limitations

grows for increasing Damkӧhler numbers. Therefore, increasing Damkӧhler numbers will lead

to an increasing difference in reaction yield between the case where the reactants are not mixed

(T-shaped model) and the case where the reactants are already fully mixed before they enter the

main reaction channel (fully mixed model).

Table 10: Residence time needed for 99% conversion for fully mixed model and T-shaped model in a 1000 μm channel for different reaction rates.

Residence time (s) needed to reach reaction yield of 99 % Fast reaction

(kLu-DOTA-TATE*10) Lu-DOTA-TATE reaction

(kLu-DOTA-TATE) Slow reaction

(kLu-DOTA-TATE*10-1) T-shaped model 39.93 77.1 473.72

Fully mixed model 7.25 49.9 450.70

5.51 1.55 1.05

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

58

4.2.6 Difference in residence time between the fully mixed and T-shaped model as a

function of the Damkӧhler number

The simulation results presented in section 4.2.2 - 4.2.5 are summarized in this section by com-

paring the required residence time to achieve a reaction yield of 99% for the T-shaped model (tT-

shaped model) and the fully mixed model (tfully mixed model) for the varying diameters and reaction rate

constants used in the previous sections.

The ratio of tT-shaped model and tfully mixed model is used to evaluate the relative difference of the T-

shaped model and the fully mixed model. The different values of the diameters and reaction rate

constants used in the simulations in the previous sections are captured by the Damkӧhler num-

ber

.

As explained in section 2.3.2, the Damkӧhler number captures the importance of molecular dif-

fusion relative to chemical reaction. As a result, it is expected that different T-shaped systems

with the same value of Da will result in a similar value for the ratio of

. This will

be checked by simulating and comparing models with different diameters but the same

Damkӧhler number.

Table 11 presents three data points obtained the formation of Lu-DOTA-TATE (k=0.902 mol m-2)

in a microfluidic channels with diameters of 500, 1000, and 1500 μm and the corresponding

ratios

.

Table 11: Damkӧhler number for three different channel diameters and the corresponding ratio of required residence times of the T-shaped model and fully mixed model obtained for the formation of Lu-DOTA-TATE

( = 7.3E-9 m2/s)

Diameter (μm) Da

500 3.57 1.11 1000 14.3 1.54 1500 32.1 2.30

The table shows that as the value of the channel diameter is doubled from 500 μm to 1000 μm,

Da increases by a factor 22=4 from 3.57 to 14.3. Similarly, Da increases by a factor 32=9 from

3.57 to 32.1 as the channel diameter is tripled from 500 μm to 1500 μm.

A Damkӧhler number of 14.3 or 32.1 can also be accomplished in a 500 μm channel by changing

one of the other variables present in the definition of the Damkӧhler number (recall Da=

kcL2/ ). If the diffusion coefficients of the reactants are, for instance, reduced by a factor of 4

and 9 for the T-shaped microfluidic system with a channel diameter of 500 μm, this will result in

Da=14.3 and Da=32.1, respectively. The results of the systems where the diffusion coefficient is

reduced by factor 4 and 9 are shown in Table 12. The ratios of required residence times in Table

12 do not perfectly align with the ratios in Table 11. A Damkӧhler number of 14.3 for a 1000 μm

channel and a diffusion coefficient of 7.3 E-9 m2/s resulted in a ratio of the required residence

times of 1.54 (see Table 11) while the same Damkӧhler number of 14.3 for a 500 μm channel

and a diffusion coefficient of 1.8 E-9 m2/s resulted in a ratio of the required residence times of

1.59 (see Table 12). Similarly, for a Damkӧhler number of 32.1 for a 1000 μm channel and =

7.3E-9 m2/s a ratio of the required residence times of 2.30 was found while a Damkӧhler num-

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

59

ber of 32.1 for a 500 μm channel and = 0.81E-9 m2/s resulted in a ratio of 2.49. This misalign-

ment is caused by the influence of heat transfer as the temperature profile develops faster in a

500 μm compared to a 1000 and 1500 μm channel. This influence of heat transfer was checked

by performing a simulation in the absence of heat transfer where the temperature was set uni-

formly to 80˚C. In this case, an identical Damkӧhler number for a channel diameter of 500 μm

and 1500 μm resulted in an identical ratio of the required residence times.

Table 12: Different Damkӧhler numbers for a 500 μm channel and the corresponding ratio of required resi-dence times of the T-shaped model and fully mixed model

Diameter (μm)

Diffusion coefficient

(m2/s) Da

500 7.3 E-9 14.3 1.59 500 0.81E-9 32.1 2.49

In summary, the presence of heat transfer will cause, to a limited extent, deviations in the ratio

of the required residence times for systems with the same value of the Damkӧhler number but

different reaction rate constants, diameters or diffusion coefficients. However, the complete data

obtained in this thesis, as shown in Figure 34 which plots the ratio of tT-shaped model and tfully mixed

model versus the Damkӧhler number, can be used as a rough approximation to determine the dif-

ference between a T-shaped model and a fully mixed model for the synthesis of Lu-DOTA-TATE

and other products which follow second-order reaction kinetics. The curve in Figure 34 can

serve as a quick estimate to determine if enhanced mixing methods are required to ensure that

the reactants are pre-mixed when entering the main reaction channel.

Figure 34: Ratio of the required residence time to achieve a reaction yield of 99% for the T-shaped model (tT-

shaped model) and the fully mixed model (tfully mixed model) as a function of the Damkӧhler number.

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

60

Figure 34 shows that difference in the residence time required to achieve 99% between the T-

shaped model and the fully mixed model is relatively small (differences less than 11%) for Da<4.

For Da~14, this difference grows to larger than 50%, and the required residence time of the T-

shaped model is 5.5 times the required residence time of the fully mixed model for Da~143.

Based on this finding, it can be concluded that enhanced mixing methods are not required for

Da<4 as the T-shaped model and the fully mixed model result in similar required residence

times to reach the desired reaction yield.

For Da>4, however, the difference between the models increases and diffusion limitations in the

T-shaped model start to have an effect, resulting in longer reaction times. In this situation, it is

preferred to achieve rapid mixing which ensures that the reactants are fully mixed as they enter

the main reaction channel. Therefore, enhanced mixing methods (not solely based on diffusion

as is the case for the T-shaped model) are required to reduce the reaction time.

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

61

4.2.7 Four inlet model

As discussed in the previous chapter, systems with faster reaction kinetics and larger channel

diameters resulting in Da>4, require efficient mixing at the beginning of the main reaction chan-

nel to minimize the required reaction times. In the experiments with [177Lu]Lu-DOTA-TATE, this

was done by use of a state mixing tee (IDEX-540) which contains a frit that enforces mixing of

reactants [58]. However, it is difficult to integrate this kind of system into a microfluidic chip.

Therefore, alternative methods are investigated to accelerate mixing.

One method to accelerate the mixing process, is to reduce the diffusion distance. This can be

accomplished by the application of parallel lamination where the reactants are fed through mul-

tiple inlets as explained in section 3.4 [62, 63]. The 2-dimensional four inlet model described in

section 3.4 was used to investigate the impact of the parallel lamination technique on the reac-

tion yield of Lu-DOTA-TATE. Figure 35(a) shows the DOTA-TATE concentration and Figure

35(b) shows the lutetium concentration at the start of the main reaction channel. In contrast to

the 2 laminae formed in the Y-shaped and T-shaped models (Figure 18), it can be clearly ob-

served in Figure 35 that 4 laminae are formed that consist of lutetium and DOTA-TATE in an

alternating order. This setup decreases the required diffusion distances significantly.

The simulation result for the reaction yield of the four inlet model with a channel diameter of

1000 μm is shown in Table 13 for different residence times. The table shows that the reaction

yield achieved in the four inlet model is higher than the reaction yield that was reached in the T-

shaped model. This increase in reaction yield can be attributed to reduction of the diffusion dis-

tance that was accomplished due to the increase of the number of laminae compared to the T-

shaped model. As a result, the mixing process occurs faster and more product is formed. This is

in line with the findings of a simulation study by Fu et al. which showed that at a distance of 2.5

mm downstream, the mixing efficiency increased from 40% for the T-shaped geometry to 58%

for a double T-shaped geometry [79]. A numerical study by Wang et al. investigated the mixing

effects with chemical reaction using the the iodide–iodate method in a microfluidic system with

6 inlets (instead of 4 inlets used in this thesis). They report that the flow lamination in the 6 inlet

geometry accelerates the whole whole mixing and reaction process tenfold [80].

The reaction yield in the four inlet model is, however, still lower than the reaction yield achieved

in the fully mixed model. This is caused by the fact that the mixing process is not optimized and

diffusion limitations still play a role. The potential of the parallel lamination technique used is

exemplified by the development of the SuperFocus that consists of 138 inlet microchannels that

are combined into the mixer resulting in 4 μm thin laminae. Experiments with methylene blue

have revealed that a mixing time of 4 milliseconds can be reached [81] [82].

Table 13: Reaction yield for the four inlet model compared to the T-shaped model and the fully mixed model for a channel diameter of 1000 μm

Residence time (s)

Reaction yield (%) Diameter 1000 μm

Four inlet model T-shaped model Fully mixed model 5 26.53 13.23 35.79

10 54.28 34.82 63.47

20 80.35 64.13 86.23

50 98.06 94.20 99.01

100 99.90 99.75 99.99

4.2 Influence of channel dimensions on the mixing process and reaction in a microfluidic channel

62

Figure 35: Concentration profile of lutetium (a) and DOTA-TATE (b) at the start of main reaction channel of the four inlet model. The lutetium and DOTA-TATE solution are introduced into the main channel in an alter-

nating order. Similarly to the T-shaped model, the flow rate ratio was set to 2:1 (DOTA-TATE:Lu)

4.3 Up-scaling to clinical relevant quantities 63

4.3 Up-scaling to clinical relevant quantities

This section will use the findings of the previous simulation models to make the connection to

requirements of a microfluidic system for [177Lu]Lu-DOTA-TATE synthesis intended for clinical

use.

As explained earlier in this thesis, the final microfluidic system for the synthesis of [177Lu]Lu-

DOTA-TATE should meet the following requirements to be suitable for clinical use in hospitals:

Capacity to process 3 mL in less than 30 mins

Minimized reaction time

Yield of [177Lu]Lu-DOTA-TATE approaching 100%

Methods to enhance mixing in the microfluidic channel are not required for channel diameters of

500 μm and lower as the simulation results have revealed that the required residence time to

achieve a reaction yield of 99% are very similar for the T-shaped model where the reactants

enter separately and the fully mixed model. For channel diameters above 500 μm, diffusion limi-

tations will cause significantly longer reaction times in the T-shaped model compared to the

fully mixed model. Therefore, application of methods to enhance mixing (e.g. parallel lamination

technique discussed in section 4.2.7) are preferred for channel diameters above 500 μm to en-

sure that the reactants are fully mixed as they enter the main reaction channel.

The simulation results of the T-shaped and fully mixed models have revealed that the shortest

reaction times can be accomplished for a channel diameter of 100 μm. As the channel diameter is

increased, the heating process occurs slower, since the required time for the solution to reach

the desired temperature is proportional to the channel diameter squared. However, applying a

microfluidic system with a channel diameter of 100 μm for the clinical synthesis of [177Lu]Lu-

DOTA-TATE is not feasible due to the low flow rates (~10 μl/min) that are required to achieve

the desired residence time.

The simulation results obtained for the fully mixed model and the T-shaped model in Table 5

and Table 6 were used to calculate the maximum flow rate possible in microfluidic systems

(100-1500 μm diameter, 1m length) to still achieve the required residence time for a reaction

yield of 99%. Subsequently, the required process time was determined to produce 3mL of reac-

tion solution that is currently processed in one single batch at Erasmus MC during clinical label-

ing. The results in Figure 36 show that for channel diameters of 500 μm and higher, process

times below 13 minutes can be achieved for both a fully mixed as well as a T-shaped microfluidic

system. The T-shaped geometry is a relative simple geometry that is fairly straightforward to

incorporate in a microfluidic chip compared to the more complex geometries that are required

to enhance mixing. Therefore, the simple T-shaped geometry is most desirable from a manufac-

turing point of view. As shown in Figure 36, the process time for the fully mixed microfluidic

system is significantly reduced compared to the T-shaped system for channel diameters above

500 μm. This means a tradeoff should be made between the increase in cost of manufacturing a

microfluidic system that enhances mixing and the time saved to process the required reaction

volume.

It should be noted that the influence of radiolysis, which will have a larger effect for increasing

diameters, is not included in this thesis.

4.3 Up-scaling to clinical relevant quantities 64

Figure 36: Dependence of highest possible flow rate in the fully mixed and T-shaped geometry to achieve 99% yield on channel diameter, and time required for the production of 3 mL reaction solution.

The calculated process times above are all based on a system containing one single reaction

channel with a length of 100 cm.

An alternative approach would be to design a parallel system consisting of many separate

streams flowing through parallel channels. As indicated above, the shortest reaction time is

reached for a 100 μm channel. Therefore, a system with many parallel 100 μm channels would

result in the shortest process time. Furthermore, the effects of radiolysis are minimized by using

microfluidic channels with small diameters. This becomes highly impracticable, however, due to

high number of parallel streams that are required to reach the desired volume. As a comparison,

it would require 24 parallel streams flowing through channels with a diameter of 100 μm to

achieve the same process time of 12 minutes that is reached in a single channel of 500 μm. The

process time can be further reduced by increasing the number of parallel streams. For example,

the process time required to produce 3mL with 99% yield is less than 3 minutes in a microfluidic

system containing 100 parallel streams with a channel diameter of 100 μm and length of 100 cm.

However, this approach poses practical problems in fabricating and designing such a parallel

system.

Conclusion 65

Chapter 5. Conclusion The main goal of this project was to study the potential of microfluidics for the synthesis of the

radiopharmaceutical [177Lu]Lu-DOTA-TATE by the use of computational methods. To the best of

my knowledge, this is the first computational study that not only combines flow dynamics, mass

transfer and reaction kinetics, but also heat transfer using COMSOL Multiphysics for the purpose

of radiopharmaceuticals synthesis investigation.

The first step in this thesis was to make a comparison of a 2-dimensional and 3-dimensional

computational model for reactions in a cylindrical microfluidic channel. This assessment re-

vealed that the 2D approximation in this study agrees reasonably well with the 3D simulation for

reaction yields above 80% with an error of less than 0.5%. Taking into account the drastic in-

crease in computation time of the 3D simulation compared to the 2D simulation (8 hours for the

3D simulation and 10 minutes for the 2D simulation), it was decided to use 2D models in this

thesis to simulate the synthesis of [177Lu]Lu-DOTA-TATE in a microfluidic channel.

A comparison has been made between a microfluidic system in which the reactants are fully

mixed and a T-shaped microfluidic system where the reactants are added through two separate

inlets. The evaluation of the simulation results of the T-shaped model and the fully mixed model

has revealed that the reaction times can be greatly reduced compared to reaction times required

in conventional reaction vessels of approximately 20 minutes [13, 77]. In the fully mixed model,

reaction times less than 55 seconds are achieved. Diffusion limitations lead to significantly

slower reactions for larger diameters in the case of the T-shaped model, but nevertheless the

reaction times will be less than 125 seconds for a diameter of 1.5 mm.

This thesis concludes that rapid mixing at the entrance of the main reaction channel is not re-

quired for channel diameters of 500 μm and lower, as the difference between the two systems in

residence time to reach the required reaction yield does not surpass 10%. For channel diameters

of 1000 μm and higher, rapid mixing at the entrance of the main reaction channel is desired as

diffusion limitations will cause significantly longer reaction times in the T-shaped model com-

pared to the fully mixed model. This is illustrated by the fact that the difference in residence time

between the T-shaped model and fully mixed model to reach the required reaction yield sur-

passes 50%.

One of the methods to enhance mixing is by the application of the parallel lamination technique.

A comparison of the four inlet model and the (2 inlet) T-shaped model has shown that higher

reaction yields are achieved in the four inlet model compared to the T-shaped model as a result

of a reduction in diffusion distance that was accomplished by an increase of the number of inlets.

In addition, this thesis attempts to provide a more general picture of the need for enhanced mix-

ing structures in a microfluidic channel for the synthesis of other radiopharmaceuticals besides

[177Lu]Lu-DOTA-TATE. This was accomplished by investigating the difference between the T-

shaped model and the fully mixed model as a function of the Damkӧhler number which captures

the relative rate of chemical reaction compared to the rate of diffusion. The difference between

these two models grows for increasing Damkӧhler numbers as the relative importance of the

diffusion of reactants increases. The difference in the residence time required to achieve 99%

between the T-shaped model and the fully mixed model is relatively small (differences less than

11%) for Da<4. For Da~14, this difference grows to larger than 50%. Based on this finding, it can

Conclusion 66

be concluded that enhanced mixing methods are not required for Da<4 as the T-shaped model

and fully mixed model result in similar required residence times to reach the desired reaction

yield. For Da>4, however, the difference between the models increases as diffusion limitations in

the T-shaped model will cause longer reaction times. Therefore, enhanced mixing methods are

desired for Da>4.

Furthermore, the simulation results were used to determine the required process time of a mi-

crofluidic system for [177Lu]Lu-DOTA-TATE synthesis intended for clinical use. This has revealed

that a simple T-shaped geometry with channel diameters of 500 μm and higher will require

process times below 13 minutes to produce a reaction solution of 3mL [177Lu]Lu-DOTA-TATE

with 99% reaction yield. This is a significant improvement compared to the conventional reac-

tion vessel approach used during clinical labelling that resulted in a process time of 30 minutes.

These findings show that a microfluidic system could be a promising alternative to the conven-

tional, batch-wise technique that is currently used during clinical radiolabelling.

Recommendations

This thesis provides the foundation for the design of a microfluidic system for the synthesis of

[177Lu]Lu-DOTA-TATE. One important factor that has not been included in this thesis, is the im-

pact of radiolytic effects. Therefore, future research should extent on the existing models pre-

sented in this thesis by including the phenomenon of radiolysis.

Another important assumption in the computational models in this thesis is that the channel

walls are considered to be chemically inert. In reality, surface forces can play an important role

due to the high surface to volume ratio of microfluidic systems and can lead to the adsorption of

chemical species [83]. Non-specific adsorption of biomolecules such as DOTA-TATE to the chan-

nel walls can influence the formation of [177Lu]Lu-DOTA-TATE. Therefore, future projects

should study the impact of adsorption in microfluidic channels on product formation.

Furthermore, the results of the T-shaped geometry are solely based on this computational study

and have not been validated by experimental work. Future projects could check the validity of

the T-shaped computational model by conducting experiments based on the settings (channel

dimensions, flow rates, concentrations, temperature etc.) that were used in this thesis.

Appendix I 67

Appendix I The figure below shows a more elaborate reaction scheme depicting the association and disso-

ciation of the various chemical species. L, HL, H2L and H3L represent the different protonated

forms of DOTA-TATE. Furthermore, ML=Lu-DOTA-TATE, MHL= Lu-HDOTA-TATE and MH2L= Lu-

H2DOTA-TATE [84].

Appendix II 68

Appendix II The computational model that was built in COMSOL to simulate the behaviour of the H Chip that

was used in the experimental set-up by Laura Ballemans is shown in Figure 37.

Figure 37: COMSOL model of H chip

The diffusion coefficient of [177Lu]Lu-DOTA-TATE was determined by using a parametric optimi-

zation that fitted the simulation results to the experiment results. In the parametric optimization

technique, the value of the diffusion coefficient was varied with the objective to minimize the

squared difference between the experimental data and the simulation results. The determined

diffusion coefficients of [177Lu]Lu-DOTA-TATE are shown in Figure 38 for three different flow

rates. At a flow rate of 5 ul/min, the diffusion coefficient of [177Lu]Lu-DOTA-TATE has been de-

termined at 1.9 E-10 ±0.36 E-10 m2/s.

This value is in line with a study by Bogdan et al. that found diffusion coefficients between 4.0

and 2.8 m2/s for macrocyclic drug-like molecules in water with a molecular weight ranging from

300 to 730 Dalton [85]. Bogdan et al. report a negative logarithmic relationship between the

diffusion coefficient and molecular weight. This empirical fit can extrapolated to the molecular

weight of DOTA-TATE (1463 Dalton), to give a diffusion coefficient of approximately 2 m2/s and

agrees well with the results that were obtained as part of this thesis resulting in a diffusion coef-

ficient of 1.9 E-10 ±0.36 E-10 m2/s.

Appendix II 69

Figure 38: Diffusion coefficient [177Lu]Lu-DOTA-TATE for different flow rates determined using COMSOL.

0 5 1 0 1 5

0

11 0 -1 0

21 0 -1 0

31 0 -1 0

41 0 -1 0

51 0 -1 0

F lo w ra te (u l/m in )

Dif

fus

ion

co

efi

cie

nt

(m2

/s)

Appendix III

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