A COMPUTATIONAL STUDY ON THE BIPHASIC RESPONSE OF BRAIN TISSUE UNDER INDENTATION

By

RUIZHI WANG

A THESIS PRESENTED TO THE GRADUATE SCHOOL

OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2017

© 2017 Ruizhi Wang

To my mentor, Dr. Malisa Sarntinoranont

4

ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Malisa Sarntinoranont, for leading me into

this fascinating area and guiding me through my thesis study. I am grateful for her

valuable advice, kindly encouragement and care in my academic pursuits. I also thank

Dr. Youping Chen for helpful discussions.

5

TABLE OF CONTENTS page

LIST OF TABLES ............................................................................................................ 6

LIST OF FIGURES .......................................................................................................... 7

ABSTRACT ..................................................................................................................... 8

CHAPTER

1 MOTIVATION AND SPECIFIC AIMS ...................................................................... 10

1.1 Motivation ......................................................................................................... 10 1.2 Specific Aims .................................................................................................... 13

1.2.1 Specific Aim 1: Develop a Systematic Approach for Characterizing Biphasic Mechanical Properties from Creep Indentation ............................... 13

1.2.2 Specific Aim 2: Estimate Biphasic Properties in Rat Brain Slices ............ 14

2 CHARACTERIZING THE BIPAHSIC PROPERITES OF SOFT TISSUES THROUGH CREEP INDENTATION ....................................................................... 16

2.1 Introduction ....................................................................................................... 16 2.2 Theory Framework and Computational Modeling ............................................. 17

2.2.1 Theory of Porous Media .......................................................................... 17 2.2.2 Modeling Biphasic Creep Indentation ...................................................... 18

2.3 Determination of Biphasic Mechanical Properties ............................................. 20 2.4 Discussion and Conclusion ............................................................................... 22

3 ESTIMATION OF BIPHASIC MECHANICAL PROPERTIES OF RAT BRAIN SLICES ................................................................................................................... 29

3.1 Introduction ....................................................................................................... 29

3.2 Methods ............................................................................................................ 31 3.3 Results .............................................................................................................. 33 3.4 Discussion ........................................................................................................ 34 3.5 Conclusion ........................................................................................................ 36

4 CONCLUSION AND FUTURE WORK .................................................................... 44

4.1 Conclusion ........................................................................................................ 44 4.2 Future Work ...................................................................................................... 46

LIST OF REFERENCES ............................................................................................... 48

BIOGRAPHICAL SKETCH ............................................................................................ 54

6

LIST OF TABLES

Table page 3-1 Estimates of biphasic brain tissue properties. .................................................... 38

7

LIST OF FIGURES

Figure page 1-1 Normalized creep curves of multiple anatomical regions in rat brain.. ................ 15

2-1 Schematic of creep indentation setup. ................................................................ 25

2-2 Finite element mesh and boundary conditions used for simulating submerged biphasic indentation. ........................................................................................... 25

2-3 Relation between Poisson’s ratio and the deformation ratio. .............................. 26

2-4 Relation between shear modulus and the instantaneous displacement. ............ 26

2-5 Influence of permeability on the creep time. ....................................................... 27

2-6 Influence of indentation force on the creep time. ................................................ 27

2-7 Steps for characterizing biphasic properties of soft tissues from creep indentation. ......................................................................................................... 28

3-1 Brain structure and multiple anatomical regions. ................................................ 39

3-2 Optical coherence tomography (OCT) image of the interior of a rat brain slice where fibrous white matter regions is next to more uniform gray matter regions. ............................................................................................................... 40

3-3 Continuous spherical fiber distribution. ............................................................... 40

3-4 Distribution of principal stresses at instantaneous and equilibrium states .......... 41

3-5 Sensitivity Analysis of the Tension/Compression Stiffness of the solid matrix with increasing Poisson’s ratio. ........................................................................... 42

3-6 Stress-strain relations for three anatomical regions under uniaxial compression and tension. ................................................................................... 43

8

Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science

A COMPUTATIONAL STUDY ON THE BIPHASIC RESPONSE OF BRAIN TISSUE UNDER INDENTATION

By

Ruizhi Wang

April 2017

Chair: Malisa Sarntinoranont Major: Mechanical Engineering

Biphasic theory can provide a mechanistic description of deformation and

transport phenomena in soft tissues, and has been used to model surgery and drug

delivery in the brain for decades. Knowledge of corresponding mechanical properties of

brain is needed to accurately predict tissue deformation and flow transport for these

applications. Properties measured from previous studies fall in relatively large ranges,

and thus require further validation and refinement. Indentation is a widely used testing

technique to characterize biphasic materials. The purpose of this thesis is to improve

the understanding of biphasic response of brain tissue under creep indentation, and

estimate biphasic properties of brain slices from previous experimental data.

First, a systematic approach for determining biphasic properties through creep

indentation was developed. Poisson’s ratio and shear modulus of the solid matrix, as

well as, the hydraulic permeability were related to features in creep curves based on the

instantaneous volume conservation of biphasic creep and Darcy’s law. Second, a finite

element model of creep indentation was created, and biphasic properties of brain slices

were estimated by comparing simulation results to experimental data. Due to the fibrous

structure of brain tissues, the solid matrix was assumed to be composed of a neo-

9

Hookean ground matrix reinforced by fibers that exhibits tension-compression

nonlinearity during deformation. Estimated modulus and hydraulic permeability fell

within an acceptable range compared with those in previous studies. A sensitivity

analysis points to the necessity of considering tension-compression nonlinearity when

the material undergoes a large creep deformation ratio.

10

CHAPTER 1 MOTIVATION AND SPECIFIC AIMS

1.1 Motivation

Brain is a hydrated structure consisting of a porous solid matrix and interstitial

fluid. Under mechanical stimuli, the long-term redistribution of fluid in the interstitial

space is essential for its rheological behavior. Biphasic theory, originally developed in

soil mechanics (Biot, 1941) and later extensively used to model soft biological tissues, is

therefore employed to investigate various mechanical phenomena affecting the brain.

For example, the development of hydrocephalus (Kaczmarek et al., 1997; Peña et al.,

1999; Taylor and Miller, 2004; Smillie et al., 2005; Wirth and Sobey, 2009), tissue

deformation during neurosurgery (Paulsen et al., 1999; Miga et al., 2000; Platenik et al.,

2002; Lunn et al., 2006), as well as convection-enhanced drug delivery (Basser, 1992;

Lonser et al., 2002, 2007, 2014; Morrison et al., 2007; Jagannathan et al., 2008; Ding et

al., 2009; Chen and Sarntinoranont, 2007; Astary et al., 2010; Kim et al., 2010, 2012a,

2012b; Dai et al., 2016; Croteau et al., 2005; Vogelbaum et al., 2007). Knowledge of

biphasic mechanical properties of brain is important for analyzing the internal stresses

and flow and mass transport in these processes. Despite this need, only a few

experimental studies that directly measure such properties have been conducted

(Franceschini et al., 2006; Cheng and Bilston, 2007; Wagner and Ehlers, 2008; Weaver

et al., 2012; Tavner et al., 2016). In recent years, elastography has become an effective

technique for in vivo measurement of biphasic properties of soft biological tissues

(Konofagou et al., 2001; Righetti et al., 2004; Berry et al., 2006a, 2006b; Perriñez et al.,

2010; Weaver et al., 2012).

11

Generally, brain tissues have been found to be extremely soft and can sustain

large strain. They also exhibit tension-compression nonlinearity, hysteresis and strain-

rate dependence during deformation. However, because of different sample preparation

and loading conditions used in these experiments, the properties have been found to

vary in a relatively large range. The estimated values for Young’s modulus of the solid

matrix fall within the range of several hundred (Taylor and Miller, 2004; Cheng and

Bilston, 2007) to several thousand (Franceschini et al., 2006; Wagner and Ehlers, 2008;

Weaver et al., 2012; Mehrabian et al., 2015) in Pascal units. In a study on the vasogenic

brain edema, the range of Poisson’s ratio of the solid matrix has been investigated to

range between 0.3 and 0.35 (Drake et al., 1996), which has been adopted by many

researchers in computer simulations of brain as a biphasic material. By comparing the

initial stiffness modulus obtained in the free drainage tests to that obtained in the

uniaxial tension/compression tests, Franceschini et al. concluded that the initial drained

Poisson’s ratio is equal to 0.496 (Franceschini et al., 2006). Hydraulic permeability can

be estimated from the spread of dye through brain following cold-induced edema, and

falls in the range between 1.0×10-13

and 1.0×10-12

m4/N s (Reulen et al., 1977). Higher

values of the permeability were reported in oedometric tests on human tissue excised

during autopsy, which range from 6.15×10-12

to 1.58×10-9

m4/N s and have a mean

value of 2.42×10-10

m4/N s (Franceschini et al., 2006), whereas results from an artificial

cerebrospinal fluid (CSF) permeation test using lamb brains appear to agree with the

lowest value in the above range (Tavner et al., 2016). The permeability in the white

matter of calves identified through unconfined compression experiments is similar to the

value of 1.0×10-11

m4/N s (Cheng and Bilston, 2007). The optimized mode of

12

permeability of cat’s brains obtained from perfusion tests is 2.19×10-12

m4/N (Mehrabian

et al., 2015). Moreover, the appropriate constitutive relation for the solid matrix is also

open to debate. While pure linear elastic and hyperelastic models have been widely

used to explore matrix deformation, there are direct experimental evidences supporting

the existence of viscosity in the solid phase (Franceschini et al., 2006; Cheng and

Bilston, 2007).

Given the large ranges of the Young’s modulus, Poisson’s ratio and hydraulic

permeability obtained from previous studies, it is necessary to test more tissue samples

to further validate the properties and refine the parameters. Meanwhile, many

constitutive models used for brain are phenomenological, with material coefficients

possessing no physical meanings. Because of the heterogeneity and complex

microstructure of brain, to form a more accurate and realistic constitutive relation,

specific anatomical regions should be differentiated, and contributions from different

structural components should be taken into consideration.

The biphasic nature of hydrated tissues affords their rheology, so biphasic

mechanical properties can be extracted from the time-dependent deformation of such

tissues. Among all the experimental approaches for determining the in situ mechanical

properties of soft tissues, indentation is popular due to several advantages. It is

noninvasive and easy to perform, and can probe local properties. Also, a classical

mathematical solution for contacting elastic bodies is available (Mak et al., 1987).

Specifically, a spherical indenter has the advantage of avoiding stress concentrations

and singularities which can develop using flat-ended indenters (Lee et al., 2008). By

combining theoretical analysis and numerical simulations, Hu et al. developed a simple

13

method for characterizing the biphasic properties of gels from stress relaxation

indentation (Hu et al., 2010, 2011). In a previous study of our group, the creep

deformation of the cerebral cortex, hippocampus and caudate/putamen in acute rat

brain tissue slices under micro-indentation was recorded using an optical coherence

tomography (OCT) system (Lee et al., 2014). The optimized normalized displacement-

time curves for each anatomical region are shown in Figure 1-1.

1.2 Specific Aims

The objective of this project was to estimate the biphasic mechanical properties

of rat brain slices from the previous experimental data. In the process, a systematic

approach for extracting the biphasic mechanical properties of soft tissues from creep

indentation was developed as a preliminary. A constitutive relation based on the

microstructure of brain was chosen to represent the solid matrix. Biphasic mechanical

properties were estimated inversely by fitting finite element simulation results to

experimental data.

1.2.1 Specific Aim 1: Develop a Systematic Approach for Characterizing Biphasic Mechanical Properties from Creep Indentation

The distinct role that stiffness, compressibility and hydraulic permeability each

plays in the response of tissues under creep indentation was analyzed using a

homogeneous, isotropic neo-Hookean solid matrix. Results were verified through finite

element simulations. Critical methods in modeling biphasic indentation were also

discussed. Features in creep curves were linked to material properties, and the

summarized approach was presented in a flow chart. Tedious curve-fitting procedures

can be avoided when determining biphasic mechanical parameters.

14

1.2.2 Specific Aim 2: Estimate Biphasic Properties in Rat Brain Slices

Based on the fibrous structure of brain tissue, a fiber-reinforced tension-

compression nonlinear constitutive relation was chosen for the solid matrix. Biphasic

mechanical properties were estimated by fitting computational results to experimental

data. Estimated properties data was compared with those in previous studies. The

influence of tension-compression nonlinearity on tissue response to indentation and

compression was elucidated through a sensitivity study.

15

Figure 1-1. Normalized creep curves of multiple anatomical regions in rat brain.

Black=Cerebral Cortex, Blue=Hippocampus, Red=Putamen (Lee et al., 2014).

16

CHAPTER 2 CHARACTERIZING THE BIPAHSIC PROPERITES OF SOFT TISSUES THROUGH

CREEP INDENTATION

2.1 Introduction

The biphasic nature of hydrated biological tissues affords their rheology. When

external loads are applied on a biphasic material, the solid matrix deforms, as well as

the porous space, and the interstitial fluid migrates, which in turn alters the pressure on

the matrix. The bulk material reaches its equilibrium state asymptotically with the

interstitial fluid. For a biphasic soft tissue with a neo-Hookean solid matrix that

undergoes steady-state deformation, there are three material parameters governing its

mechanical response, Young’s modulus 𝐸 and Poisson’s ratio 𝜈 of the solid matrix, as

well as the hydraulic permeability 𝑘. Since the intrinsic interaction between the solid

phase and the fluid phase provides the mechanism for the explicit viscoelastic behavior

of biphasic tissues, their material parameters can be estimated from the time-dependent

phenomena.

Biological tissues are usually very soft and slippery, which make in situ testing

difficult to set up. Indentation is a commonly used technique for measuring properties of

soft materials due to several advantages (Chen et al., 2007). It is noninvasive and easy

to perform, and can probe local properties. Also, a classical mathematical solution for

contacting elastic bodies is available (Mak et al., 1987). As an added benefit, a

spherical indenter can avoid stress concentrations and singularities which can develop

using flat-ended indenters (Lee et al., 2008).

Biphasic tissues exhibit viscoelasticity under mechanical stimuli. Two main

characteristics associated with viscoelasticity are stress relaxation and creep. Hu et al.

developed a simple method for characterizing the biphasic properties of gels from stress

17

relaxation indentation (Hu et al., 2010, 2011). However, the relation between creep

indentation and biphasic properties has not been fully discussed yet. By combining

theoretical analysis and numerical simulations, the mechanism of the flow-dependent

creep deformation of porous media can be elucidated.

2.2 Theory Framework and Computational Modeling

2.2.1 Theory of Porous Media

Biphasic theory is based on the theory of mixture as each spatial point in the

material is assumed to be occupied simultaneously by a material point of a solid and

fluid phase (Mow et al., 1980). The complicated microstructure can be smeared out, and

principles in continuum mechanics can thus be used in the macroscopic level. It is

assumed that both the solid and fluid phases are incompressible. The constitutive

equation for the bulk material is

𝛔 = −𝑝𝐈 + 𝛔𝐸 (2-1)

where 𝛔 is the Cauchy stress tensor for the mixture; 𝛔𝐸 is the contact stress from

the deformation of the solid matrix; 𝑝 is the interstitial fluid pressure and 𝐈 is the identity

tensor. The constitutive model for the isotropic neo-Hookean solid matrix is given by a

strain energy density function,

𝑊 =𝜇𝑠

2(𝐼1 − 3) − 𝜇𝑠 ln 𝐽 +

𝜆𝑠

2(ln 𝐽)2 (2-2)

where 𝜇𝑠, 𝜆𝑠 are the Lamé’s elastic constants of the solid matrix; 𝐼1(= Tr(𝐛)) is

the first invariant of the left Cauchy-Green deformation tensor 𝐛(= 𝐅𝐅𝑇 where 𝐅 is the

elastic deformation gradient tensor); and 𝐽(= det 𝐅) is the elastic volume ratio or

Jacobian of the deformation. Lamé’s constants are related to Young’s modulus and

Poisson’s ratio (𝐸, 𝜈) of the solid matrix, which were used in this study, by 𝜆𝑠 =

18

𝐸𝜈/[(1 + 𝜈)(1 − 2𝜈)] and 𝜇𝑠 = 𝐸/[2(1 + 𝜈)]. The Cauchy stress tensor from the

deformation of the solid matrix is obtained by differentiating the strain energy density

with respect to the strain tensor

𝝈𝐸 =𝜇

𝐽(𝐛 − 𝐈) +

𝜆

𝐽(ln 𝐽)𝐈 (2-3)

Fluid flow is described by Darcy’s law as

𝑘 ∇𝑝 = 𝐯𝑠 − 𝐯 (2-4)

where 𝐯 = ∅𝑠𝐯𝑠 + ∅𝑓𝐯𝑓 is the volume-averaged bulk velocity; ∅𝑠, ∅𝑓 are the solid

and fluid volume fractions (∅𝑠 + ∅𝑓 = 1); 𝐯𝑠, 𝐯𝑓 are the velocity vectors of solid and fluid;

and 𝑘 is the hydraulic permeability which is assumed constant.

The conservation of mass for the mixture requires

∇ ⋅ 𝐯 = 0 (2-5)

Under quasi-static conditions with negligible body force, the conservation of

momentum results in the equation

∇ ⋅ 𝛔 = 0 (2-6)

Darcy’s law as well as the equations of conservation of mass and momentum

govern the mechanical response of the biphasic material.

2.2.2 Modeling Biphasic Creep Indentation

Creep is an increase of strain in material under constant stress. In our previous

experiments on acute rat brain, cylindrical tissue slices were submerged in solution to

maintain their viability, and to reduce friction and adhesion between the tissue layer and

the indenter. Tissue bottom was fixed to a rigid and impermeable substrate. A constant

indentation force was ensured by carefully placing stainless steel spherical beads on

the center of submerged tissue slices, as shown in Figure 2-1, and the applied force

19

was calculated by subtracting the buoyancy force from the gravitational force of the

beads (F = 37 μN). The time-dependent deformation in each anatomical region was

observed by an optical coherence tomography (OCT) system for over 10 minutes. In

this study, the tissues were assumed to be a biphasic material with a neo-Hookean solid

matrix. The mechanical properties of the solid matrix (𝐸, 𝜈) as well as the hydraulic

permeability 𝑘 can be determined from the experimental creep curves.

Corresponding biphasic contact problems were solved using the FEBio software

suite (version 2.5.2, Musculoskeletal Research Laboratories, University of Utah, Salt

Lake City, UT). A wedge with an angle of 90 degrees was created to represent the

tissue slice, and the indenter was modeled as the corresponding part of a sphere, as

shown in Figure 2-3. According to the experimental setup described above, the

boundary conditions set on the models are: (1) fixed displacement and zero fluid

velocity at the bottom of the tissue (rigid and impermeable substrate boundary); (2) zero

interstitial fluid pressure at the surface of the tissue; (3) rigid and impermeable indenter,

i.e., zero normal flow flux in the contact region (this is a time-varying condition changing

with time steps due to expanding contact region); (4) free normal displacement of the

indenter and rigid confinement for the other 5 degrees of freedom; (5) frictionless

biphasic contact between the indenter (master body) and the tissue (slave body). The

rat brain tissue slices were approximately 330 μm thick and had a 1.2 mm radius, and

the radius of the spherical indenter were 500 μm. Both tissue slice and indenter were

meshed using 8-node trilinear hexahedral elements. The final mesh consisted of 8448

elements for the tissue wedge, and 6912 elements for the indenter. Denser meshes

were employed around the contact region for more accurate numerical results. A

20

quarter of the actual force (=9.25 μm) was prescribed downward on the indenter. The

maximum step length allowed is 20s. To capture the instantaneous response, smaller

time steps with minimum length of 0.01s were adopted during the very first simulation

stage (0~1s).

𝛿𝐯𝑠 (virtual velocity of the solid) and 𝛿𝑝 (virtual pressure of the fluid) were used in

zero virtual work assumption to get the weak formulation of biphasic materials, which

was linearized by increments ∆𝐮 and ∆𝑝 (displacement and pressure), and solved as a

transient problem by full Newton iterations. Computational creep curves were obtained

by monitoring the normal displacement of the node in the center of the upper surface of

tissue slices in each time step.

2.3 Determination of Biphasic Mechanical Properties

To determine the three mechanical parameters (𝐸, 𝜈, 𝑘), consider the situation of

creep indentation where the indentation force can be treated as a step function.

Instantaneously after the spherical indenter is released onto the tissue layer, the

interstitial fluid has no time to migrate. Since the fluid is incompressible, and the solid

phase always occupies the same volume as the fluid phase, the bulk material behaves

like an incompressible solid. Poisson’s ratio of the solid matrix at this instant is 0.5, and

the indentation depth ℎ(0) is solely governed by shear modulus 𝐺. After the tissue

reaches equilibrium, the Poisson’s ratio reduces to its true value. So, the ratio of the

displacement at infinite ℎ(∞) to the instantaneous displacement ℎ(0) is a measure of

Poisson’s ratio.

According to Hu et al., the analytical solution to the instantaneous relation

between the force and the indentation depth is expressed as

21

𝐹 =16

3𝐺 ⋅ ℎ(0) ⋅ 𝑎(0) (2-7)

where 𝑎 represents the radius of the contact area. When the contact radius is small

compared to the thickness of the tissue, the Hertzian contact is reached, and 𝑎 takes

the form

𝑎 = √𝑅ℎ (2-8)

where 𝑅 represents the radius of the indenter. As the indentation depth becomes large

enough, the contact portion approaches a spherical cap of height ℎ, and 𝑎 is given by

𝑎 = √2𝑅ℎ (2-9)

After the tissue reaches its equilibrium state, the solid matrix is compressible. So

the force and indentation depth relation should be modified as

𝐹 =16

3

𝐺

2(1−𝜈)⋅ ℎ(∞) ⋅ 𝑎(∞) (2-10)

Dividing equation (2-10) by equation (2-7) we have

ℎ(∞)⋅𝑎(∞)

ℎ(0)⋅𝑎(0)= 2(1 − 𝜈) (2-11)

By substituting equations (2-8) and (2-9) into equation (2-11) we have

[ℎ(∞)

ℎ(0)]

3

2≤ 2(1 − 𝜈) ≤ √2 [

ℎ(∞)

ℎ(0)]

3

2 (2-12)

The right hand side would be reached only when 𝜈 is small, and there is a

significant difference between the initial and infinite contact condition. In our study, the

indentation depth is relatively small compared to the tissue thickness, so the actual

condition should be close to the left boundary in equation (2-12). The real dependence

of Poisson’s ratio 𝜈 on the deformation ratio ℎ(∞)/ℎ(0) is calculated numerically and

presented in Figure 2-3. Also plotted is the relation obtained by enforcing the left

22

boundary in Equation (2-12). The left boundary only deviates a little from the numerical

solution, which convinces us that [ℎ(∞)

ℎ(0)]

3

2= 2(1 − 𝜈) is a fair estimate for Poisson’s ratio.

Equation (2-7) also indicates that in a certain creep indentation system, where 𝐹

and 𝑅 are constant, ℎ(0) can be taken as a measure of 𝐺. Due to the nonlinear relation

between the indentation depth and the contact radius, the 𝐺 − ℎ(0) curve obtained from

computational simulation is also nonlinear, as shown in Figure 2-4.

Permeability 𝑘 determines the velocity of fluid flow traveling through the porous

media under certain pressure gradient. The higher 𝑘 is, the faster interstitial flow

migrates, and the less time it requires to reach equilibrium. 𝑘 does not affect the

instantaneous and equilibrium states of the material, but plays a role in shifting the

creep curve in the horizontal direction, as shown in Figure 2-5.

2.4 Discussion and Conclusion

A systematic approach for characterizing the biphasic mechanical properties of

soft tissues through creep indentation was developed in this chapter. A neo-Hooean

solid matrix was adopted, because the parameters in its strain energy density function

are simple and possess realistic physical meanings, which make it easy to elucidate the

approach with this model. In general, there are five parameters in the indentation

system except the mechanical properties: indentation force 𝐹, indenter radius 𝑅, tissue

thickness 𝑡𝑖, tissue radius 𝑟 and indentation depth ℎ.When the contact radius is much

smaller than the tissue radius, the influence from tissue boundary is negligible and the

system can be simplified to indentation of an infinite elastic layer bonded to a rigid half

space. After applying boundary conditions a complicated boundary value problem is

established, to which the analytical solution is difficult to find.

23

By making use of the instantaneous volume conservation, 𝜈 and 𝐺 are directly

related to the instantaneous and equilibrium displacement. The curve shown in Figure

2-3 is valid for characterizing 𝜈 from any indentation system with different indentation

force and dimensions, as these factors are cancelled out in calculating the deformation

ratio. However, the relation between 𝐺 and ℎ(0) is under the influence of 𝐹 and 𝑅, so

the curve in Figure 2-4 is only suitable for the specific case in this chapter. Although so

far there is not a quantitative link between permeability 𝑘 and the time required to reach

equilibrium, once 𝜈 and 𝐺 are determined and the initial and infinite states are satisfied,

it is easy to identify 𝑘 through binary search. It should be noted that unlike stress

relaxation indentation, where the time required to reach equilibrium is quadratic to

indentation depth, the creep time only changes slightly with different indentation force in

our case, as shown in Figure 2-6. So, the influence of 𝐹 might be negligible on the

quantitative relation between 𝑘 and creep time 𝑡.

Based on the analysis above, the systematic approach for characterizing the

biphasic mechanical properties is concluded as follows

(1) Identify Poisson’s ration 𝜈 from the deformation ratio ℎ(∞)/ℎ(0);

(2) Determine shear modulus 𝐺 from the instantaneous displacement ℎ(0);

(3) Estimate permeability 𝑘 through binary search by the time required to reach

equilibrium.

The steps are also shown in the flow chart (Figure 2-7).

Tedious curve-fitting procedures can be replaced by this simple and systematic

approach. The approach can be further extended to any biphasic material with a

hyperelastic solid phase that can be expressed using a strain energy density function.

24

The instantaneous displacement depends on the parameters in the deviatoric part of the

function, while the deformation ratio is dominated by the coefficients in the dilatational

part.

25

Figure 2-1. Schematic of creep indentation setup.

Figure 2-2. Finite element mesh and boundary conditions used for simulating submerged biphasic indentation. Dimensions were taken from rat brain tissue

tests (Lee et al., 2014). Tissue radial boundary, l = 1.2mm, initial tissue thickness, ti = 330μm, and spherical indenter radius, R = 500μm.

26

Figure 2-3. Relation between Poisson’s ratio and the deformation ratio.

Figure 2-4. Relation between shear modulus and the instantaneous displacement.

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0 0.1 0.2 0.3 0.4 0.5

ℎ(∞

)/ℎ

(0)

𝜈

Numerical Solution

Left Boundary

0

10

20

30

40

50

0 500 1000 1500 2000 2500 3000 3500

𝒉(𝟎

)/𝝁𝒎

G/Pa

27

Figure 2-5. Influence of permeability on the creep time.

Figure 2-6. Influence of indentation force on the creep time.

0

20

40

60

80

0 100 200 300 400 500 600

𝒉/𝝁𝒎

Time/s

k=5e-12

k=1e-12

k=3e-13

k=1e-13

0

20

40

60

80

100

120

0 100 200 300 400 500 600

𝒉/𝝁𝒎

Time/s

F=57e-6N

F=47e-6N

F=37e-6N

F=27e-6N

F=17e-6N

28

Figure 2-7. Steps for characterizing biphasic properties of soft tissues from creep indentation.

29

CHAPTER 3 ESTIMATION OF BIPHASIC MECHANICAL PROPERTIES OF RAT BRAIN SLICES

3.1 Introduction

Previous studies have shown that brain tissue can be described as a biphasic

continuum, where a porous solid matrix is fully saturated with interstitial fluid

(Franceschini et al., 2006; Cheng and Bilston, 2007). Biphasic models couple

interaction between fluid pressurization through pores and solid matrix deformation, and

have been widely used in computational investigations of clinical applications such as

surgical planning (Paulsen et al., 1999; Miga et al., 2000; Platenik et al., 2002; Lunn et

al., 2006) and convention-enhanced drug delivery (Basser, 1992; Lonser et al., 2002,

2007, 2014; Morrison et al., 2007; Jagannathan et al., 2008; Ding et al., 2009; Chen and

Sarntinoranont, 2007; Astary et al., 2010; Kim et al., 2010, 2012a, 2012b; Dai et al.,

2016; Croteau et al., 2005; Vogelbaum et al., 2007). Accuracy of corresponding

mechanical properties, i.e., stiffness and compressibility of the solid matrix as well as

hydraulic permeability, is essential for the reliability of simulations. To this end, direct

mechanical testing (Franceschini et al., 2006; Cheng and Bilston, 2007; Wagner and

Ehlers, 2008; Tavner et al., 2016) and imaging methods (Weaver et al., 2012) have

been employed to measure these properties in the brain. Brain tissues are found to be

extremely soft and capable of undergoing large strains. They also exhibit strain-rate

dependence, tension-compression nonlinearity and hysteresis during deformation

(Franceschini et al., 2006). In part because of different sample preparation techniques

and loading conditions used in experiments, properties such as the Young’s modulus

have been found to vary in a relatively large range (300~5000Pa) (Taylor and Miller,

2004; Franceschini et al., 2006; Cheng and Bilston, 2007; Wagner and Ehlers, 2008;

30

Weaver et al., 2012; Mehrabian et al., 2015). The appropriate choice of constitutive

relation for the solid matrix phase is also open to debate and depends on the application

of the model.

Biphasic theory which accounts for the intrinsic interaction between the porous

solid matrix and the interstitial fluid can explain the explicit viscoelastic behavior of soft

tissues (Mow et al., 1980). Features in the time-dependent deformation of tissues can

be related to the biphasic mechanical properties (Hu et al., 2010, 2011; Chen et al.,

2016). For a better understanding of the viscoelastic properties in brain, in a previous

study of our group, creep deformation of the cerebral cortex, hippocampus and

caudate/putamen in acute rat brain tissue slices under micro-indentation was recorded

using an optical coherence tomography (OCT) system (Lee et al., 2014). Use of tissue

slices allowed us to indent within various regions and care was taken to maintain

neuronal viability over the test period. The specific locations of these regions in brain

are shown in Figure 3-1.

In the current study, a biphasic finite element model of creep indentation was

created to compare to these data and extract mechanical properties. The deformation

ratio for the three regions ranged from 2.5~3.0, whereas the greatest deformation ratio

in Figure 2-3 is approximately 1.58. Consequently, a pure uniform neo-Hookean model

is inadequate for describing the mechanical response of the solid matrix under

indentation. Since the three brain regions are intricately structured and heterogeneous,

it is necessary to take the contributions of different microstructural components into

consideration. Cerebral gray matter is mainly composed of nerve cell bodies (soma),

neuropil and interconnected glial cells, while white matter mainly consists of myelinated

31

axons which provides a fibrous structure, as shown in Figure 3-2. Given this tissue

structure, the solid matrix in our model was defined as a tension-compression nonlinear

solid mixture where a compressible, isotropic, neo-Hookean ground matrix was

reinforced by fibers. Fibers in brain have received enormous attention in recent years,

due to their connections with traumatic brain injury (TBI), brain development, growth

and folding, as well as, improved mechanics models for brain deformation. Based on

dynamic shear testing and indentation, a fiber-reinforced single phase hyperelastic,

transversely isotropic model was developed for white matter (Feng et al., 2013, 2016).

In this study, the Young’s modulus (𝐸), fiber modulus (𝜉) and permeability (𝑘) in each

anatomical region were estimated for Poisson’s ratio (𝜈) ranging from 0.35 to 0.49. A

sensitivity analysis of the creep indentation response was also performed.

3.2 Methods

Numerical simulations were performed using a biphasic continuum framework

(Mow et al., 1980). A constant, homogeneous and isotropic hydraulic permeability was

assumed. The solid matrix of brain tissue was treated as a compressible, isotropic, neo-

Hookean ground reinforced by continuously distributed fibers. Fibers were considered to

only sustain tension and have zero response to compression. It is assumed that at any

spatial point in the material, fiber bundles are evenly distributed to form a spherical

surface around the point. The fiber strain energy density function was defined as

(Ateshian et al., 2010)

𝜓 =𝜉

𝛼𝛽(𝑒𝑥𝑝[𝛼(𝐼𝑛 − 1)𝛽] − 1) (3-1)

where fiber modulus (𝜉) represents the stiffness of fiber bundles. Material coefficients 𝛼

and 𝛽 were set to 0 and 2. 𝐼𝑛 is the square of the fiber stretch given by

32

𝐼𝑛 = 𝜆𝑛2 = 𝐍 ⋅ 𝐂 ⋅ 𝐍 (3-2)

where 𝐍 is the unit vector along the fiber direction, and 𝐂 is the right Cauchy-Green

deformation tensor. The Cauchy stress of the fiber bundles is given by

𝝈(𝒏) = 𝐻(𝐼𝑛 − 1)2𝐼𝑛

𝐽

𝜕𝜓

𝜕𝐼𝑛𝒏 ⊗ 𝒏 (3-3)

where 𝐻 is a unit step function to ensure a tensile-stress-only response. 𝐽 is the

Jacobian of the deformation gradient 𝐅, and 𝒏 = 𝐅 ⋅ 𝐍/𝐼𝑛. Under the assumption of

spherical fiber angular distribution, the Cauchy stress tensor was obtained by

integrating 𝝈(𝒏) over all possible fiber directions, as shown in Figure 3-3.

𝝈 = ∫ ∫ 𝝈(𝒏)𝜋

0

2𝜋

0𝑠𝑖𝑛𝜑𝑑𝜑𝑑𝜃 (3-4)

The stress tensor for the solid matrix was the sum of stresses for all solid ground

and fiber components. Only creep indentation of submerged brain tissue slices was

considered in this study. This biphasic contact problem was solved using the FEBio

software suite (version 2.5.2, Musculoskeletal Research Laboratories, University of

Utah, Salt Lake City, UT) for the geometry shown in Figure 2-2. A wedge with an angle

of 90 degrees and height of 330 microns was created to represent the tissue slice, and

the indenter was modeled as the corresponding part of a rigid, impermeable and

frictionless sphere. A quarter of the actual force (=9.25 μN) was prescribed downward

on the indenter. To capture the instantaneous response, small time steps were

prescribed during the initial loading stage (0 to 1 s). Each indentation simulation was for

10 minutes, and the vertical displacement of the tissue surface was monitored to

generate creep curves. To be consistent with literature (Drake et al., 1996; Franceschini

et al., 2006; Cheng and Bilston, 2007), the range of Poisson’s ratio ranged from 0.35-

0.49. To back out the pure tension/compression stiffness of the solid matrix in our

33

model, single phase cylindrical bars defined using the estimated properties for the solid

matrix were created. Poisson’s effect was eliminated by setting 𝜈 as 0 for each data set,

and steady-state uniaxial tension/compression simulations were performed to generate

the stress-strain curves.

3.3 Results

The distinct role that 𝜉, 𝐸 and 𝑘 plays in shaping the creep curve reported in an

articular cartilage indentation test using a permeable cylindrical indenter (Chen et al.,

2016) is also observed in this study with an impermeable spherical indenter. 𝜉

dominates the instantaneous displacement while the infinite deformation is determined

by 𝐸. The time required to reach equilibrium is governed by 𝑘. The optimization scheme

involves two levels of binary search. At the lower level, given a 𝜉, 𝐸 and 𝑘 are optimized

to fit the equilibrium deformation and creep time of the curve. At the higher level, 𝜉 is

searched to match the deformation ratio. Adjustment of 𝜉 requires a new search of 𝐸

and 𝑘. The tension/compression distribution at instantaneous and equilibrium states can

be observed in plots of principal stresses, as shown in Figure 3-4. Estimated biphasic

properties data in each anatomical region are listed in Table 3-1.

With each of the three anatomical regions considered, 𝐸 was found to decrease

~87% as 𝜈 increased from 0.35 to 0.49, see Figure 3-5 (A). As 𝜈 was increased, the

initial increase of 𝜉 was followed by a dramatic drop as 𝜈 reached 0.49, as shown in

Figure 3-5 (B). The sensitivity of both 𝐸 and 𝜉 to choice of 𝜈 increased for a nearly

incompressible ground matrix. Creep response was more sensitive to Young’s modulus

than fiber modulus. Furthermore, taking 𝜉/𝐸 as an approximate measure of tension-

compression nonlinearity of the solid matrix, it was observed that the contrast between

34

tensile and compressive modulus is enhanced by a larger Poisson’s ratio, see Figure 3-

5 (C). It was also noted that within each brain region, 𝑘 changed only slightly with

different properties of the solid matrix.

3.4 Discussion

The objective of this study was to estimate the biphasic mechanical properties of

brain slices. Because of the fibrous structure of brain tissue and large deformation

observed during indentation, a fiber-reinforced neo-Hookean constitutive relation was

chosen for the solid matrix. Simulated Young’s modulus of the neo-Hookean ground

and the hydraulic permeability fell within an acceptable range compared with those in

literature (Kyriacou et al., 2002; Franceschini et al., 2006; Cheng and Bilston, 2007;

Wagner and Ehlers, 2008; Weaver et al., 2012; Goriely et al., 2015; Tavner et al.,

2016). The relatively low stiffness might be reflective of collapsed vasculature that is no

longer patent due to tissue slicing. In this study, inclusion of fibers greatly increased the

effective tensile modulus. Stress-strain curves of the solid matrix with a zero Poisson’s

ration obtained from uniaxial tension/compression simulations are shown in Figure 3-6.

Since Poisson’s ratio at the moment the indenter is applied (𝑡 = 0+) is effectively 0.5

before any fluid redistribution occurs, tissue compression under the indenter needs to

be accompanied by tensile radial expansion to satisfy instantaneous volume

conservation. Addition of fibers effectively reduced the instantaneous indentation depth

because of the additional tensile modulus. Consequently, a larger 𝜈 resulted in a

smaller deformation ratio (i.e., the ratio of the equilibrium displacement to the

instantaneous displacement), as well as a larger ratio of tensile modulus to compressive

modulus required to fit the creep curve. Moreover, when 𝜈 is fixed, greater tension-

35

compression nonlinearity could introduce a larger deformation ratio under indentation,

as exhibited by comparison between different anatomical regions in Figure 3-5 (C). For

detailed experimental observations on creep, the reader is referred to Lee et al. (2014).

Actual fiber distribution pattern is likely more organized than in our model, and fiber

orientation could lead to anisotropy in micro-scale anatomical regions. Nonetheless, the

estimated biphasic properties indicate the potential tension-compression of brain

tissues, and the differences in properties that are estimated when fibers are not

included. Such considerations are needed for understanding of tissue behavior since

most tissues deformations incur both tensile and compressive stresses.

Several experiments have attempted to extract mechanical properties of

microstructural components in brain under attention. The elastic modulus of glial cells in

adult wild-type mice CNS measured through transmission electron microscopy and

scanning force microscopy is no more than 300 Pa (Lu et al., 2017), which is similar to

the ground stiffness in our results. The stiffness of axons in brain has been measured

using direct mechanical testing and Atomic force microscopy (AFM) (Xu et al., 2009;

Rajagopalan et al., 2010; Koser et al., 2015). The stiffness of fibers in lamb white matter

varies between 230 to 830 Pa (Feng et al., 2013). Elastic modulus of single axon in rat

embryos measured through AFM indentation ranges between 300 to 1300 Pa

(Magdesian et al., 2012). Axons in chick embryos exhibit relaxation and their stiffness

ranges between 1470 to 9500 Pa (Ouyang et al., 2013), which is also comparable to the

fiber modulus estimated in this study.

In tissues, creep deformation might be a combination of solid matrix

viscoelasticity and fluid-matrix interactions. In this study, fluid viscosity of the solid

36

matrix phase was not taken into consideration, and consolidation was assumed to be

the single mechanism of the time-dependent behavior. A viscoelastic solid matrix could

speed up the creep process and thus a lower 𝑘 is required. Also, a higher instantaneous

stiffness of the ground, as well as, a lower fiber modulus at equilibrium, should be

expected. In fact, the leading mechanism of the delayed volumetric deformation of brain

is believed to be consolidation (Franceschini et al., 2006), so neglecting the viscosity

would not introduce large errors in estimated properties. However, ex vivo tissue

degeneration may lead to cell swelling and thus a lower porosity, and local changes of

pore size due to compression can hinder interstitial flow. As a result, the in vivo

hydraulic permeability at a zero-strain state may be underestimated, and a strain-

dependent 𝑘 can be used in future studies instead of the constant permeability assumed

in the current model. In addition, given that permeability was nearly independent of the

properties of the solid matrix, this might provide a simpler way to determine permeability

from creep indentation, as the matrix properties can be excluded from the quantitative

relation between creep time and permeability.

3.5 Conclusion

In this study, a biphasic finite element model of creep indentation was developed

to compare with previous experimental data. Based on the microstructure of brain

tissues, the solid matrix was assumed to be composed of a neo-Hookean ground matrix

reinforced by continuously distributed fibers that exhibits tension-compression

nonlinearity during deformation. By fixing Poisson’s ratio of the ground matrix, Young’s

modulus, fiber modulus and hydraulic permeability were estimated. Hydraulic

permeability was found to be nearly independent of the properties of the solid matrix.

Estimated modulus (45~1080Pa for the ground matrix, 3430~18200Pa for fibers) and

37

hydraulic permeability (1.36×10-13

~2.88×10-13

m4/N s) fell within an acceptable range

compared with those in previous studies. Results of the sensitivity analysis point to the

necessity of considering tension-compression nonlinearity when the material undergoes

a large creep deformation ratio.

38

Table 3-1. Estimates of biphasic brain tissue properties.

𝜈 𝐸 (Pa) 𝜉 (Pa) 𝑘 (m4/N s)

Cerebral Cortex

0.35 1080 16000 1.36×10-13

0.4 820 17100 1.36×10-13

0.45 500 18200 1.37×10-13

0.49 141 12500 1.38×10-13

Hippocampus

0.35 545 5900 2.03×10-13

0.4 432 6120 2.05×10-13

0.45 265 6210 2.04×10-13

0.49 71 5297 2.04×10-13

Putamen

0.35 323 3670 2.85×10-13

0.4 255 3820 2.87×10-13

0.45 157 3910 2.87×10-13

0.49 45 3430 2.88×10-13

39

Figure 3-1. Brain structure and multiple anatomical regions.

40

Figure 3-2. Optical coherence tomography (OCT) image of the interior of a rat brain slice where fibrous white matter regions is next to more uniform gray matter regions.

Figure 3-3. Continuous spherical fiber distribution.

41

A B

C

D E

F Figure 3-4. Distribution of principal stresses at instantaneous and equilibrium states. A)

instantaneous 1st principal stress distribution, B) instantaneous 2nd principal stress distribution, C) instantaneous 3rd principal stress distribution; D) equilibrium 1st principal stress distribution, E) equilibrium 2nd principal stress distribution, F) equilibrium 3rd principal stress distribution. (Unit: Pa)

42

Figure 3-5. Sensitivity Analysis of the Tension/Compression Stiffness of the solid matrix

with increasing Poisson’s ratio. A) Young’s modulus of the ground matrix, B) Fiber modulus, C) The ratio of Fiber modulus to Young’s modulus, which can be taken as a measure of the tension-compression nonlinearity.

0

400

800

1200

0.35 0.4 0.45 0.49

E (

Pa

)

𝜈

A

0

5

10

15

20

0.35 0.4 0.45 0.49

𝜉(k

Pa

)

𝜈

B

0

20

40

60

80

100

0.35 0.4 0.45 0.49

𝜉/E

𝜈

C

Cortex Hippocampus Putamen

43

Figure 3-6. Stress-strain relations for three anatomical regions under uniaxial

compression and tension. Different dash types represent different Poisson’s ratios assumed. Dash=0.35, dash dot=0.4, solid line=0.45 and round dot=0.49. A) Cerebral Cortex, B) Hippocampus, C) Putamen.

-2000

0

2000

4000

6000

8000

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

𝜎 (Pa)

𝜀

A

-500

0

500

1000

1500

2000

2500

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

𝜎 (Pa)

𝜀

B

-400

0

400

800

1200

1600

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

𝜎 (Pa)

𝜀

C

44

CHAPTER 4 CONCLUSION AND FUTURE WORK

4.1 Conclusion

The work presented in this thesis was focused on the biphasic response of brain

tissue under indentation. Interstitial flow in brain is fundamental to convection-enhanced

delivery, and fluid-matrix interaction provides a mechanistic description of the time-

dependent deformation in brain tissues. So, knowledge of biphasic properties is

essential for a better understanding of brain biomechanics, as well as making medical

surgical treatments safer and more predictable. Brain tissues are intricate and

heterogeneous, and biphasic properties measured from previous experiments fall in a

large range. Indentation is a popular mechanical testing technique due to several

advantages, and has been extensively applied to characterize hydrated soft materials.

Previously in our group, creep indentation was used to measure explicit viscoelastic

properties in cerebral cortex, hippocampus and putamen of acute rat brain tissue slices.

In the current study, tissue slices were treated as a biphasic material, and a biphasic

finite element model of creep indentation was created. Mechanical properties of multiple

anatomical regions were estimated by comparing computational results to experimental

data.

In chapter 2, a systematic approach for characterizing the biphasic mechanical

properties in submerged tissue slices through creep indentation was developed.

Instantaneous volume conservation of the material is key to extract the elastic

properties of the solid matrix. Poisson’s ratio was quantitatively linked to the ratio

between equilibrium and initial indentation depth. Because Poisson’s ratio is effectively

0.5 at the moment the indenter is applied, the instantaneous response of the material is

45

a measure of shear modulus. Flow transport in porous is governed by Darcy’s law. The

hydraulic permeability determines how fast interstitial flow can redistribute under creep

indentation, as well as the time required to reach equilibrium. The critical techniques for

modeling biphasic creep indentation was discussed. Given an experimental creep

curve, once a corresponding computational model is created, biphasic properties can be

estimated by applying binary searches for 𝜈, 𝐺 and 𝑘. Also, simulation results supported

that creep time is nearly independent of the indentation force.

In chapter 3, biphasic mechanical properties of brain slices were estimated by

fitting simulation results to experimental data. Because of the existence of axons in

white matter, anatomical regions in brain were considered as a fibrous structure, and

the solid matrix was assumed to be composed of a neo-Hookean ground matrix

reinforced by continuously distributed fibers that exhibits tension-compression

nonlinearity during deformation. By fixing Poisson’s ratio of the ground matrix, Young’s

modulus, fiber modulus and hydraulic permeability were estimated. Hydraulic

permeability was found to be nearly independent of the properties of the solid matrix.

Estimated modulus (45~1080Pa for the ground matrix, 3430~18200Pa for fibers) and

hydraulic permeability (1.36×10-13

~2.88×10-13

m4/N s) fell within an acceptable range

compared with those in previous studies. Fiber modulus data was also similar to axonal

stiffness measured previously. Pure tensile/compressive modulus were extracted and

plotted using stress-strain curves. The sensitivity analysis shown that for the fiber-

reinforced solid phase, with the compressibility of the ground matrix decreasing, the

ratio of the tensile stiffness to the compressive stiffness should be increased to satisfy

the creep curve. Results of the sensitivity analysis also point to the necessity of

46

considering tension-compression nonlinearity when the material undergoes a large

creep deformation ratio.

4.2 Future Work

Brain tissue is a complicated structure, and its mechanical properties need

continuing exploration before a reliable global model can be obtained (Goriely et al.,

2015). As for the work presented in this thesis, there are several limitations that should

be pointed out, on which future work may be worthwhile.

First, for the creep indentation method to determine biphasic properties, shear

modulus and permeability were not quantitatively linked to the features in a creep curve.

Theoretical or empirical formulations can either provide a simpler way to estimate data

or narrow binary search intervals. To achieve this goal, more theoretical analysis on a

sphere contacting a nonlinear material is required for extracting shear modulus. From

chapters 2 and 3, we reached the conclusion that permeability is independent of the

indentation force and the properties of the solid matrix. Thus we guess that creep time

depends only on permeability and the indenter radius. The following equation can be

obtained from computational simulation and should be helpful to improve our method:

ℎ(𝑡)−ℎ(0)

ℎ(∞)−ℎ(0)= 𝐾(𝑘, 𝑅, 𝑡) (4-1)

where 𝐾 is a function of permeability (𝑘), indenter radius (𝑅) and time (𝑡).

Second, some assumptions made in the fiber model could affect the accuracy of

the estimated data. Given the large strain occurred during indentation, a strain-

dependent permeability can be used in future work instead of the constant permeability

assumed in this study. Also, the influence of fiber orientation on the biphasic mechanical

response of brain tissue under indentation should be investigated. Mechanical

47

properties of anisotropic biphasic models with more organized fiber bundles can be

compared.

48

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BIOGRAPHICAL SKETCH

Ruizhi Wang was born in Huaian, Jiangsu, China in 1993. He received his

bachelor’s degree in mechanical engineering from Huazhong University of Science and

Technology in 2015. In August 2015, he started his graduate studies at the University of

Florida. He joined the Soft Tissue Mechanics and Drug Delivery Laboratory in March

2016.