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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
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
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
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
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.