Abstract
Title of Document: STRESS RESPONSE OF BOVINE ARTERY AND RAT
BRAIN TISSUE DUE TO COMBINED TRANSLATIONAL
SHEAR AND FIXED UNCONFINED COMPRESSION
DEFORMATION
Lauren Leahy, M.S. Mechanical Engineering, 2015
Directed By: Dr. Henry W. Haslach, Jr., Mechanical Engineering
During trauma resulting from impacts and blast waves, sinusoidal waves
permeate the brain and cranial arterial tissue, both non-homogeneous biological
tissues with high fluid contents. The experimental shear stress response to
sinusoidal translational shear deformation at 1 Hz and 25% strain amplitude
and either 0% or 33% compression is compared for rat brain tissue and bovine
aortic tissue. Both tissues exhibit Mullins effect in shear. Harmonic wavelet
decomposition, a novel application to the mechanical response of these tissues,
shows significant 1 Hz and 3 Hz components. The 3 Hz component magnitude
in brain tissue, which is much larger than in aortic tissue, may correlate to
interstitial fluid induced drag forces that decrease on subsequent cycles perhaps
because of damage resulting in easier fluid movement. The fluid may cause the
quasiperiodic, viscoelastic behavior of brain tissue. The mechanical response
differences under impact may cause shear damage between arterial and brain
connections.
STRESS RESPONSE OF BOVINE ARTERY AND RAT BRAIN TISSUE
DUE TO COMBINED TRANSLATIONAL SHEAR AND FIXED
UNCONFINED COMPRESSION
By
Lauren Leahy
Thesis submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Masters of Science
2015
Advisory Committee:
Dr. Henry W. Haslach, Jr., Chair
Dr. Balakumar Balachandran
Dr. Amr Baz
c© Copyright by
Lauren Leahy
2015
ii
Contents
1 Introduction 1
2 Background 3
2.1 Phase shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Tissue Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Artery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Rat Brain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Mullins Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Methods 19
3.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Specimen Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.1 Aorta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.2 Brain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.1 Combined Sinusoidal Translational Shear and Fixed Initial Un-
confined Compression . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Methods of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.1 Mullins Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.2 Signal Processing Analysis . . . . . . . . . . . . . . . . . . . . 29
3.4.3 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Results 35
4.1 Qualitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1.1 First Peak Amplitude of Sinusoidal Shear Response . . . . . . 38
4.1.2 Mullins Effect in Shear Stress Response to Translational Shear
Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1.3 Symmetry of one Load-Unload Cycle . . . . . . . . . . . . . . 43
4.1.4 Symmetry about the Horizontal Axis . . . . . . . . . . . . . . 44
4.1.5 Time Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1.6 Limit Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Frequency Decomposition of the Stress Response . . . . . . . . . . . . 51
4.2.1 Frequency Filter . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.2 Fourier Series Analysis . . . . . . . . . . . . . . . . . . . . . . 53
4.2.3 Harmonic Wavelet Analysis . . . . . . . . . . . . . . . . . . . 59
4.2.4 2 Hz Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.5 Results Summary . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Discussion 69
5.1 Solid-Fluid Interaction in the Mechanical Response . . . . . . . . . . 73
5.2 Transient Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3 Frequency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
iii
5.3.1 Fluid Influence on Brain Tissue Response during a Load-Unload
Deformation Cycle . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3.2 Aortic Tissue Stress Response . . . . . . . . . . . . . . . . . . 80
5.4 Role of Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.5 Clinical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.6 Shortcomings or Limitations . . . . . . . . . . . . . . . . . . . . . . . 83
6 Conclusion 85
7 Appendix 86
7.1 Matlab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.1.1 Mullins Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.1.2 Time Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.1.3 Filter Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.1.4 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.1.5 Harmonic Wavelet Analysis . . . . . . . . . . . . . . . . . . . 87
7.1.6 Wavelet Examination . . . . . . . . . . . . . . . . . . . . . . . 88
7.2 Planes of Dissection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8 References 90
iv
List of Tables
1 Brain and Aortic Tissue 40 Cycles at 0% and 33% Compression . . . 39
2 Brain Tissue Time Shift at the End of Each of the First 3 Deformation
Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 Fourier Coefficients for Frequencies 0-6 Hz of Shear Stress Response (Pa) 58
4 Frequency Ranges for j Levels . . . . . . . . . . . . . . . . . . . . . . 60
5 Brain Tissue 40 Cycles at 0% Compression (20515a) . . . . . . . . . . 61
6 Brain Tissue 40 Cycles at 33% Compression (12915a) . . . . . . . . . 62
7 Aortic Tissue 40 Cycles at 0% Compression (22515b) . . . . . . . . . 63
8 Aortic Tissue 40 Cycles at 33% Compression (22515d) . . . . . . . . 64
9 L2 Error for Binned Wavelets and All Wavelets . . . . . . . . . . . . 65
10 Fourier Coefficients for Brain Tissue 0% Compression and 2 Hz . . . 66
11 Harmonic Wavelet Coefficients for Brain Tissue at 0% Compression
and 2 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
v
List of Figures
1 Phase Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Circle of Willis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Clark and Glagov Artery Schematic . . . . . . . . . . . . . . . . . . . 10
4 Davis Artery Schematic . . . . . . . . . . . . . . . . . . . . . . . . . 11
5 Dingemans Artery Schematic . . . . . . . . . . . . . . . . . . . . . . 13
6 Rat Brain Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7 Iliff Glymphatic System . . . . . . . . . . . . . . . . . . . . . . . . . 15
8 CSF Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
9 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10 Cleaned Aorta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
11 Skull Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
12 Acoustic Meatus Diagram . . . . . . . . . . . . . . . . . . . . . . . . 25
13 Dissection: Rat Brain - Cut 1 . . . . . . . . . . . . . . . . . . . . . . 26
14 Dissection: Rat Brain - Cut 2 . . . . . . . . . . . . . . . . . . . . . . 26
15 Rat Brain Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
16 Wavelet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
17 Stress Response of Aortic Tissue under 0% Compression . . . . . . . 36
18 Stress Response of Aortic Tissue under 33% Compression . . . . . . . 36
19 Stress Response of Brain Tissue under 0% Compression . . . . . . . . 37
20 Stress Response of Brain Tissue under 33% Compression . . . . . . . 37
21 Exponential Fit Curve of Aortic Tissue under 0% Compression . . . . 40
22 Exponential Fit Curve of Aortic Tissue under 33% Compression . . . 41
23 Exponential Fit Curve of Brain Tissue under 0% Compression . . . . 41
24 Exponential Fit Curve of Brain Tissue under 33% Compression . . . 42
25 Shoulder on Brain Stress Response . . . . . . . . . . . . . . . . . . . 43
26 Time Shift for Aortic Tissue Stress Response . . . . . . . . . . . . . . 45
27 Time Shift for Brain Tissue Stress Response under 0% Compression . 47
28 Time Shift for Brain Tissue Stress Response under 33% Compression 47
29 Stress Versus Strain Curve for Aortic Tissue under 0% Compression . 48
30 Stress Versus Strain Curve for Aortic Tissue under 33% Compression 49
31 Stress Versus Strain Curve for Brain Tissue under 0% Compression . 50
32 Stress Versus Strain Curve for Brain Tissue under 33% Compression . 50
33 Pass Filter Frequency Decomposition for Brain Tissue Stress Response 52
34 Sum of Pass Filter Frequency Decomposition for Brain Tissue Stress
Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
35 Fourier Coefficient Magnitudes: Mover Displacement . . . . . . . . . 54
36 Fourier Series Decomposition: Mover Displacement . . . . . . . . . . 54
37 Fourier Coefficient Magnitudes: Aortic Stress Response . . . . . . . . 56
38 Fourier Series Decomposition: Aortic Stress Response . . . . . . . . . 56
39 Fourier Coefficient Magnitudes: Brain Stress Response . . . . . . . . 57
40 Fourier Series Decomposition: Brain Stress Response . . . . . . . . . 57
41 Harmonic Wavelet Decomposition Goodness of Fit: Brain, 0% com-
pression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
vi
42 Harmonic Wavelet Decomposition: Artery, 0% Compression . . . . . 61
43 Harmonic Wavelet Decomposition: Artery, 33% Compression . . . . . 62
44 Harmonic Wavelet Decomposition: Brain, 0% Compression . . . . . . 63
45 Harmonic Wavelet Decomposition: Brain, 33% Compression . . . . . 64
46 Harmonic Wavelet Decomposition: Brain, 2 Hz, 0% Compression . . . 66
47 Fourier Series Decomposition: Brain Stress Response, 2 Hz . . . . . . 67
48 Axons in Shear Schematic . . . . . . . . . . . . . . . . . . . . . . . . 70
49 Positive Shear Direction . . . . . . . . . . . . . . . . . . . . . . . . . 71
50 Center Shear Direction . . . . . . . . . . . . . . . . . . . . . . . . . . 71
51 Negative Shear Direction . . . . . . . . . . . . . . . . . . . . . . . . . 71
52 Shear Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
53 Damaged Brain Tissue . . . . . . . . . . . . . . . . . . . . . . . . . . 79
54 Planes of Dissection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
1
1 Introduction
Medical information about the mechanical behavior of brain and artery tissues must
extend beyond simple characterization of the tissue response to forces to include its
correlation with damage since damage in these vital organs can cause life changing
effects. Mild traumatic brain injuries result from impacts, high frequency blast waves
and inertial acceleration. In the skull, brain matter and arteries lie within close prox-
imity to each other; the Circle of Willis (on the inferior side of the brain) joins several
arteries together and branches off to the many smaller arteries that supply oxygen to
the cerebrum (Campellone 2015). To better understand brain and aortic tissue’s sus-
ceptibility to damage, the structures of each tissue, both non-homogeneous biological
tissues with high fluid contents, are compared to explore whether the structural dif-
ferences can explain the differing mechanical response of the two tissues and whether
the differing responses may cause damage between the tissues.
In brain tissue research, the mechanics of damage is not well defined. It is hy-
pothesized that the drop in load carrying ability directly correlates to the fluid-solid
interaction in the material; the easier the ability for the fluid in the structure to flow
(and therefore the more damage that has been inflicted) the more drastic the drop
in the load carrying ability. Damage is defined to be a reduction in load carrying
capability which may be due to the breaking of bonds within each structure.
The mechanics of an initial insult and repeated loading is of interest not only
because many other brain researchers neglect to include the initial impact loading
data within their findings, but also because repetitive loadings may cause an increase
in damage. Repeated loading does not reduce the carrying ability of aortic tissue in
vivo because the aorta undergoes life-long cycles of pressure from blood. The aorta
is a natural load bearing material, while brain tissue does not experience regular
mechanical insults. The stress responses of brain and aorta tissues in the two materials
are speculated to be different due to their difference in structures; however some
2
response traits should be similar because both tissues have a high fluid content.
The shear stress response to sinusoidal translational shear deformation is used to
compare these materials because blood pressure for arterial tissue and shock waves
for brain tissue are sinusoidal. Combined compression and shear tests model both
the longitudinal and shear component in a deformation wave due to impact for the
brain tissue as well as model the combined compression and shear applied to an
artery as blood passes through it. The response to repeated loadings in the brain
is unknown; however the tissue might not be able to withstand this type of loading
without damage.
The goal of this research is to determine how two particular load-bearing and non-
load bearing hydrated soft tissues differ in their shear stress response to combined
translational shear deformation and unconfined compression. One hypothesis is that
a mechanical cause of brain tissue damage under an external mechanical insult is
the increased hydrostatic pressure in, and pathological flow of, the extracellular fluid
(ECF). A three parameter model is used to fit the peak stress response of bovine
aortic tissue and rat brain tissue. Fourier series and harmonic wavelet frequency de-
compositions are used to analyze the frequencies that exist within the stress response
and are related to the structure and fluid in each tissue.
3
2 Background
Aortic tissue and brain tissue are both non-linear viscoelastic materials. Viscoelastic
materials are those that exhibit both elastic and viscous properties. These materials
behave elastically while loaded quickly but exhibit a continued increase in strain
as the stress is held constant called creep. Likewise, viscoelastic materials, when
held at a constant deformation after an initial deformation, experience a decrease in
stress called stress relaxation. Stress responses of viscoelastic materials are heavily
rate dependent due to their viscous properties. The stress response to sinusoidal
deformation may be sinusoidal or quasi-periodic, oscillations that may follow a regular
pattern but do not have a fixed period. A sinusoidal shear stress response may exhibit
a phase shift compared to the deformation which may depend on amplitude, frequency
and damping. If the response is quasiperiodic, the time shift may also depend on these
factors and may change with the number of cycles. The dependence for linear elastic
materials can give guidance in assessing the influence of these factors for both brain
and aortic tissue. The damping factor may be related to the role of the interstitial
fluid which may be responsible for the viscoelastic response in both tissues.
2.1 Phase shift
Linear viscoelastic materials subject to an oscillatory strain input respond with a
stress having the same frequency as the strain, however the curves are not synchro-
nized in time. The difference in synchronization can be characterized by the phase
shift. The phase shift is the difference in time between the displacement and stress
sine waves as they cross through the horizontal axis measured with respect to a cor-
responding cycle in radians or degrees. A phase shift can be measured only when
both a sinusoidal strain input versus time curve and a force versus time curve exhibit
the same frequency response. The phase angle, δ, numerically describes the lead or
4Figure 1: Stress and strain curves having the same frequency but different amplitudes, for
which stress leads strain by δ.
lag that one curve has compared to the other; a curve leads if its corresponding cycle
crosses the horizontal axis at an earlier time. The phase angle can also be calculated
as a time shift given as
∆t =
δ
2pif
(1)
where f is the frequency.
The experimentally determined phase shift should give information about the
viscosity of the tissue. The following simple examples give insight into the possible
relation between phase shift and bulk viscosity. The sinusoidal stress and strain for
a strain input are
 = 0 sin(ωt); (2)
σ = σ0 sin(ωt+ δ); (3)
where δ is the phase shift, also called the phase angle. The storage modulus E1 and
the loss modulus E2 are
E1 =
σ0
0
cos(δ); (4)
E2 =
σ0
0
sin(δ). (5)
The tangent of the phase angle δ is calculated by taking the ratio of the loss modulus
5
and the storage modulus,
tan(δ) =
E2
E1
. (6)
In linear viscoelasticity, dynamic mechanical analysis characterizes the phase shift
observed during oscillatory testing by relating the energy stored to the energy lost.
The dynamic modulus of a material is the complex sum of the storage modulus and the
loss modulus. The tangent of the phase shift is the ratio of the elastic and dissipative
energies (or the storage and loss moduli)(Equation 6), therefore, if no energy is lost
or if the material is purely elastic, the phase shift is zero. A material that is purely
viscous, on the other hand, would only have energy lost and would have a phase shift
of 90o.
The Standard Solid model, a spring (R1) connected in parallel with Maxwell’s
model (a spring (R2) and dashpot in series), demonstrates the behavior of a linear
viscoelastic material. This standard linear solid model is
˙ =
R1
η (
η
R2
σ˙ + σ −R1)
R1 +R2
. (7)
For sinusoidal strain control,  = 0 sin(ωt) is substituted into this equation to yield
ω0 cosωt =
R1
η (
η
R2
σ˙ + σ −R10 sinωt)
R1 +R2
. (8)
The differential equation solved to obtain stress is
σ(ωt+ δ) =
1
R1(R22 + η2ω2)
0R2[e
−R2t
η ηω(R21 −R1R2 −R
2
2)
− ηω(R21 −R1R2 −R
2
2) cosωt
+ (R21R2 + η
2R1ω
2 + η2R2ω
2) sinωt] (9)
6
Use the sum of sines to obtain
tan(δ) =
η(−R21 +R1R2 +R
2
2)ω
R21R2 + η2R1ω2 + η2R2ω2
. (10)
If the spring in series with the dashpot (R2) is larger than the stiffness of the spring
in parallel (R1) the tangent will be positive which means that the stress leads the
strain. If the opposite stiffness is true (R1 > R2), the strain leads the stress.
The Zener standard solid model (used by Dennerll et al. (1989) to model an axon
in tension), a spring (R2) connected in series with a Kelvin-Voigt model (a spring
(R1) in parallel with a dashpot) is
σ +
R1
R2
σ +
η
R2
σ˙ = R1+ η˙. (11)
Under sinusoidal displacement control,  = sin(ωt) is substituted into this equation
to produce
(1 +
R1
R2
)σ +
η
R2
σ˙ = R1 sin(ωt) + ηω cos(ωt). (12)
Solve the differential equation to obtain the stress
σ(ωt+ δ) =
R2(−e
−(R1+R2)t
η R2ηω +R2ηω cos(ωt) + (R21 +R1R2 + η
2ω2) sin(ωt))
R21 + 2R1R2 +R
2
2 + η2ω2
.
(13)
Use the sum of sines to obtain
tan(δ) =
R2ηω
R21 +R1R2 + η2ω2
. (14)
The tangent of the phase angle in the Zener model is always positive, meaning
that stress leads strain. Although the Zener and Standard solid models cannot cap-
ture the complexity of heterogeneous biological tissue, these models demonstrate the
dependence of phase shift on ω and the material parameters. The non-linearity of
7
the biological tissue stress response may cause a phase shift that depends on other
factors than the linear model variables. These dependencies guide the investigation
into the cause of any time shift in the brain and aorta shear stress response.
2.2 Tissue Structure
Several arteries lie within close proximity to the brain including the Circle of Willis, a
joining area on the inferior side of the brain that branches into small arteries that carry
oxygen to 80% of the cerebrum (Campellone 2015). Arterial tissue and brain tissue,
although both soft biological tissues with high fluid content, have different physical
requirements. Arteries naturally expand and contract as blood pulses which causes
a compressive stress in the radial direction, a tensile stress in the circumferential
direction, and shear stresses. The effect of shear is of interest because it has not been
studied in much detail. Brain tissue undergoes a combined compressive and shear
stress wave during a blast. A comparison of the response to combined compression
and shear for both the brain and artery should be informed by an analysis of the
structures of each material.
Figure 2: Circle of Willis (Campellone 2015)
8
2.2.1 Artery
Arterial tissue is a natural load bearing, soft biological tissue with high interstitial
fluid content and is comprised of three main layers called the intima (inner), media
(middle) and adventitia (outer). The intima is composed of endothelial cells, con-
nective tissue and axially oriented smooth muscle cells with an underlying layer of
collagen fibers. Ground matter is situated on either side of these fibers parallel to the
intimal and adventitial faces. Histology has shown a dominant direction of collagen
fibers in the circumferential direction with local variation. The internal elastic lamina
which separates the intimal and medial layers is comprised of an elastin sheet that is
responsible for transporting fluid. The media is comprised of many smooth muscle
cells, elastin and collagen (Figure 3). The fluid is located in the smooth muscle cells
and in the extracellular matrix (ECM). The extracellular fluid (ECF) is responsible
for transporting nutrients to the cells. Versican proteoglycans are the main filler sub-
stance in between cells which bond to water and allow for diffusion of nutrients to the
cells. These proteoglycans form large complexes and attach to collagen to regulate
the movement of molecules through the arterial wall.
Three schematic models are proposed in the literature for the structural organi-
zation of the aortic wall. These models eventually may allow an assessment of the
corresponding effects each set of substructures would have on the mechanical response.
Clark and Glagov (1985) proposed from histology images that the arterial wall
is comprised of overlapping, musculo-elastic fascicles which consist of smooth muscle
cells, elastin sheets, and collagen fibers which are similarly oriented. The commonly
oriented smooth muscle cells are encompassed by a matrix mat comprised of a fine
mesh-work of collagen fibrils. This matrix is related to a “Chinese finger trap” in Clark
and Glagov (1979), stating that the matrix holds the cells together when tension
forces exist. These fascicles were observed to be anchored by elastic fibers and in
the media lie mainly circumferentially. Wavy collagen bundles that never interfere
9
with one another are oriented in the same direction as nearby elastin forms sheets.
Clark and Glagov note that there were connections between elastin fibers of each
layer that are “more easily disrupted than the elastin systems bracketing the layers.”
They argue that the medial wall is comprised of a repeating sequence of “elastin-
cells-elastin - collagen bundles - elastin- cells- elastin” (Figure 3). While the fascicles
are predominately oriented in the circumferential direction in the media, Clark and
Glagov found that the fascicles favor an axial orientation in the adventitia and luminal
side. Clark and Glagov conclude that there exists distinctly aligned collagen and
elastin on both sides of a cell layer. This predominately circumferential oriented
fascicle direction theory would create a mechanical response that hardens because
the matrix mat would create an increase in strength as a circumferential tensile force
is exerted. This theory implies a significant difference in shear response with respect
to specimen orientation, when deformed with or across the specimen fiber direction.
Using Clark and Glagov’s schematic model, when the tissue experiences a shear
force the longitudinally oriented specimens would predominantly test the connections
between the wall layers and the connections between the fascicles. The circumferen-
tially oriented specimens would predominantly test the strength of the collagen fibers
in shear. Both oriented specimens could test the strength of the interfiber bonds
between collagen fibers (Haslach et al. 2015b).
Much like the observations of Clark and Glagov, Davis (1993) reported that in
the aortic media there exists circumferentially oriented smooth muscle cell layers
alternating with elastic laminae (Figure 4). Davis observed that smooth muscle cells
are connected to the elastic laminae (which Davis refers to as musculo-elastic units)
within the aortic wall in alternating diagonal patterns. Davis boldly hypothesizes
that the tangential vessel wall stresses are not affected during regular blood pressure
induced deformation. Davis observed that a single elastic lamina separates each
smooth muscle cell layer in mice which differs from the double elastic laminae found
10
Figure 3: Clark and Glagov Artery Schematic (C indicates the circumferential place, L
indicates the longitudinal plane) Each musculo-elastic fascicle contains commonly oriented
cells (Ce) enclosed in a matrix mat (M) surrounded by a system of elastic fibers (E) and
wavy collagen bundles (F). (Clark and Glagov 1985)
in larger mammals’ aortic walls as described by Clark and Glagov. Davis mentions the
findings of Pease and Paule (1960) that describe smooth muscle cell layers of the rat
aortic media as a single layer of obliquely oriented cells with each cell extending from
one elastic lamina to the next, the orientation of the smooth muscle cells changing
direction in successive layers. This herringbone design theory creates a different
mechanical response than the Clark and Glagov model.
Davis’ herringbone schematic design (Figure 4) implies that the tissue would ex-
perience a tensile force within the contractile-elastic units when a longitudinal shear
stress is applied; the angled contractile-elastic units would be stretched out of the
circumferential plane. The response due to pure circumferential tension creates in-
ternal shearing between the elastin layers due to the alternating contractile-elastic
units. The mechanical response due to shear would create a tensile stress within the
contractile-elastic units oriented in one direction and would create a shear stress in
the smooth muscle cells in the layers with opposing contractile-elastic unit direction.
Much like Davis’ discussion of musculo-elastic units, Dingemans et al. (2000)
discusses a similar basic unit of the aortic media, the lamellar unit, which consists
11
Figure 4: Davis Artery Schematic (Davis 1993)
of two elastic lamellae and intervening smooth muscle cells. These lamellar units are
comprised of a single layer of smooth muscle cells with elastic lamellae on both sides
(Figure 5). Dingemans noted that in collapsed specimens there exists wavy elastin
lamellae and that while parts of the lamellae were smooth, there were also irregular
focal protrusions that extended into the interlamellar space. The Dingemans model
agrees with Clark and Glagov (1985) in that the smooth muscle cells are oriented
mainly in the circumferential direction. Dingemans et al. observed that there are
relatively few direct contacts of smooth muscle cells with solid elastin lamellae and
that the smooth muscle cell extensions were found in close contact with irregular
protrusions from the elastin lamellae. Dingemans makes a contrasting argument to
Clark and Glagov’s “Chinese finger trap” theory of collagen fibril net organization
that preferential localization of collagen fibrils immediately adjacent to the smooth
12
muscle cell surface was not observed. Dingemans et al. interpret their images as
showing that the collagen fibrils appear loosened and spiraled. Dingemans et al. state
that they did not observe any organization of the aortic medial in ‘musculo-elastic
fascicles’ as described by Clark and Glagov (1985) in several non-human species.
Dingemans et al. describe the structure of the aortic wall saying the longitudinal
surface ridges of the smooth muscle cells are connected by long elastin protrusions
to the lamellae on either side. The smooth muscle cells are also connected with
the elastin lamellae via oxytalan fibers that contain collagen. The collagen fiber
bundles are described to lie directly against the elastin lamellae in the circumferential
direction. The fibronectin appeared to not only connect smooth muscle cells to the
elastin sheets, but also to each other. Dingemans et al. observe that the smooth
muscle cells were connected by numerous gap junctions. Dingemans et al. conclude
that the collagen fibrils located adjacent to the smooth muscle cell surface as noted
in Clark and Glagov (1985), were absent.
Dingemans structural theory would create a mechanical response that would pri-
marily test the tensile strength of the elastin protrusions and after some displacement,
the smooth muscle cells and the oxytalan fibers that link them to the elastin sheets
would also be in shear and tension respectively. The linking elastin protrusions exhibit
different directions that would correlate to a lack of a significant difference between
the tested specimen orientations within small shear deformation; Dingemans theory
of predominately circumferential collagen fibers would affect the mechanical response
after the fibers become taut. Holzapfel et al. (2000) assume this mechanical response
within their mathematical models.
Dingemans et al. note the proteoglycans that surround all collagen fibrils are
oriented perpendicularly to the fibril direction. The ground substance appears com-
pletely filled with large proteoglycans seen on the very cell surface, but the elastin
lamellae and the oxytalan fibers do not contain proteoglycans. The location of pro-
13
Figure 5: Dingemans Artery Schematic. The schematic shows smooth muscle cells (SMC)
and elastin lamellae (EL) and their connections. The collagen fibers (Coll) lie closely to the
elastin lamellae. Oxytalan fibers (Ox) connect the elastin lamellae to the smooth muscle
cells. Deposits (D) of collagen and proteoglycans are found on the cell surface. (Dingemans
2000)
teoglycans can indicate the primary locations of extra-cellular fluid within the tissue.
The collagen fibers act as a restraint to deformation. When the artery tissue
is relaxed, the collagen fibers are slack and when the tissue expands due to blood
pressure, the elastin sheets stretch and the collagen fibers become taut, restraining
the tissue.
Haskett et al. (2010) discuss probable cross-linking that bond between circum-
ferentially oriented bands of collagen fibers. This theory was drawn from tensile
testing performed perpendicular to the primary collagen fiber direction that results
in a higher stiffness than the tests performed in the direction of the collagen fibers.
2.2.2 Rat Brain
Brain tissue is a heterogeneous soft biological tissue comprised of many subsections
of both white and grey matter and is not a natural load bearing material (Figure
6). Brain tissue solid matter is primarily neurons and glial cells. A neuron is a
14
nerve cell that transmits information through electrical and chemical signals. The
axons of the neuronal cells are responsible for carrying the electrical impulses from
the cell body toward the axon terminal to send information to other axon dendrites
which creates a network of neurons. Brain tissue has both a high intra- and extra-
cellular fluid content of around 80% of its mass. Extracellular fluid (ECF) which
makes up 20% (Verkman 2013) of the brain’s volume is responsible for transporting
nutrients from the capillaries to the cells and for removing waste from the cells as
part of the glymphatic system. The ECF creates an internal pressure in the brain
that is balanced by a linked network of axons in tension. Brain tissue does not
have a structured framework supporting the tissue as does the artery. The balance
between the internal pressure of the ECF and the tension within the axons allows
the tissue to keep its shape, and neurites adjust their length accordingly to maintain
a steady tension (Van Essen 1997). Brain tissue has viscoelastic properties and is
under resting tension which is demonstrated by the tissue’s ability to spring back
to its original position after a transient deformation. Van Essen states that all cells
have an natural pressure differential across their plasma membrane, resulting from a
mechanism to regulate the osmotic balance, but pressure can also arise from extrinsic
sources.
Figure 6: Rat Brain Schematic (University of Utah Health Sciences 2015)
15
Astrocytes are star-shaped glial cells that provide biochemical support of the
blood-brain barrier, provide nutrients to the nervous tissue, maintain the extracel-
lular ion balance, and repair/scar the brain after traumatic injury. Three types of
astrocytes are involved in the physical structuring of the brain. Fibrous astrocytes
that are located in white matter that are responsible for connecting the cells to the
outside of capillary walls, protoplasmic astrocytes are located in grey matter and ra-
dial astrocytes are present during childhood development. Glial cells provide support
for the neurons to protect the brain. These cells also supply the neurons with nutri-
ents and oxygen because neurons are very active and generate a lot of waste. The
process of fluid redistribution, nutrient transport and waste removal within the brain
is referred to as the glymphatic system. Cerebrospinal fluid (CSF) is generated in the
4 ventricles of the brain and moves through the subaachnoid compartment (Figure
8). Iliff et al. (2012) describe the flow path of CSF and interstitial fluid (ISF) as
driven by the pulsing arteries in the brain that cause a pressure change between the
para-arterial and paravenous pathways (Figure 7).
Figure 7: Fluid flow in glymphtic system (Iliff 2012)
Iliff et al. propose that channels between cells provide low-resistance pathways for
fluid movement between the paravascular spaces. Due to the gaps between the astro-
cytes endfeet in the astroglial sheath that surround brain blood vessels (Mathiisen et
al. 2010), the nutrient filled fluid is able to flow between the dendrites, but that the
16
Figure 8: CSF Circulation in humans (Silverthorn 2011)
exchange of fluid between the blood and brain is controlled by the size of the narrow
clefts between overlapping endfeet of the astrocytes which may restrict free flow of
the fluid. The CSF distributes the nutrients to the cells and picks up the metabolites
in the process and continues flowing toward the veins. The veins reabsorb the waste
and carry it to the cervical lymphatic system.
Damage within the brain can be categorized into two types: cell damage and
structural damage. Structural damage consists of the breaking of axonal connections
and may result in a separation of subregions. Structural damage could also include
the stretching of the axons. A strain of 20% has been suggested as the threshold for
diffuse axonal injuries (Bain and Meaney 2000). An increase in hydrostatic pressure
17
causes stretching or relocation of subregions. Cell damage consists of the rupture of
cells or change in the cells’ permeability due to stretching of the cell walls.
Unconfined compression tests (Haslach et al. 2015) provide information about
the ECF-solid phase interaction in which the ECF-flow is multi-axial. A constant
deformation rate compression causes a disruption in the brain tissue’s equilibrium
balance of hydrostatic pressure. The lateral dimension changes indicates that the
tissue is compressible due to ECF loss. The tissue’s interstitial fluid may move in all
directions within the tissue causing transverse stress due to the drag force between
the fluid and solid phase. The compression may induce axonal stretching. The
solid matter’s resistance of ECF flow could be caused by the further restriction of
extracellular space in the compressed tissue.
2.3 Mullins Effect
In cyclic tensile tests, some biological tissues exhibit stress softening (Horney et al.
2010). This phenomenon is referred to as Mullins effect. Diani et al. (2009) observed
that in rubber most of the softening appears after the first load. This softening
occurs whenever the strain of the current cycle is less than or equal to the maximum
stretch ever applied. The physical explanation of Mullins effect has not been generally
agreed upon, but may be due to chain breakage, slipping of molecules, and cross-link
ruptures. Blanchard and Parkinson (1952) as well as Bueche (1960) attribute the
Mullins effect to bond rupture and crosslinking. Diani et al. describes the Mullins
effect to be due to the rupture of carbon-black structure in rubber. Typically, on the
initially loading, the material exhibits a relatively stiff response. When the material
is unloaded and then reloaded the stress-strain curve softens significantly. Zuniga
and Beatty (2002) describe the material as having a selective memory of only the
maximum of the previous strains ever experienced and represent the Mullins effect
with a 3 parameter model to fit the exponential type stress softening (Equation 15),
18
F (m,M) = exp(−b
√
M −m) (15)
where M is the maximum previous strain, m is the current magnitude of strain, and
b is a softening parameter.
Mars et al. (2004) observes that the steady state of the response due to torsional
applied shear is softer if larger initial magnitudes of strain are experienced. They
discovered that the majority of the softening occurred in the first 10 cycles of loading
and that under constant-amplitude cyclic loading the stress-strain curve exhibits a
logarithmic trend. In some cases, Mars et al. describe without explanation a new
phenomenon called the ‘healing’ effect in which after several cycles, the stress-strain
response remains relatively constant and then starts to slightly stiffen.
19
3 Methods
Arterial and brain tissue can experience internal deformation waves which consist
of both longitudinal and shear components simultaneously (translational shear and
compressive waves). Aortic and brain tissue are tested in an apparatus that can
operate in translational shear with an initial fixed unconfined compression.
3.1 Apparatus
The apparatus (Figure 9) was constructed to test cut rectangular specimens of di-
mensions up to 25 mm x 25 mm with thicknesses up to 8 mm in shear. The specimen
is placed between two parallel test plates. The bottom plate (9 -10) is mounted to
a Bose 250 gram load cell (Figure 9 -3) and rests on top of a linear bearing (Figure
9-1). The linear bearing ensures that the bottom test plate remains flat and parallel
to the top plate with negligible friction detected by the load cell. The other end of
the load cell is secured to a support bracket (Figure 9-8). This support is essential
to the design because one end of the load cell must be fixed at all times in order to
deflect due to the applied forces. The support bracket is connected to a spring-loaded
displacement stage, ThorLabs T12X (Figure 9-9), that enables the bottom plate and
load cell to move up and down as a unit to increase or decrease the compression
on the specimen and to adjust for different specimen thicknesses. This assembly is
fastened to a mounting bracket (Figure 9-4) that is screwed into a stationary stand.
The top test plate (Figure 9-11) is mounted on a Bose Electroforce Test Bench 200
N test machine that uses a linear motor design that is modified from a Bose sub-
woofer. The Bose machine is composed of a permanent magnet that is suspended
in an electromagnetic field. The magnet is able to produce translational motion as
the electromagnetic field changes. This design reduces any frictional energy loss by
avoiding the use of bearings or seals. The oscillating magnet is attached to an ac-
20
tuator to record any error that occurs. The Bose machine allows the top test plate
to move linearly or sinusoidally at a set rate up to 3.2 m/s or 200 Hz and to a set
final displacement (Chan et al. 2008). The Bose proprietary software, WinTest 4.1,
is used to appropriately assign each variable for a given test.
Figure 9: Combined translational shear and unconfined compression apparatus
3.2 Specimen Preparation
Specimens are cut from bovine aortic tissue and from rat brain tissue.
3.2.1 Aorta
The samples are taken from bovine descending aorta purchased at J. W. Treuth &
Sons (328 Oelle Ave. Baltimore, MD 21228). The aorta must first be cleaned of fat
before the samples are taken from it; excess branches from the aorta as well as any
additional portions of the heart are removed with a No. 10 blade scalpel (Figure 10).
Because the curvature of the aorta is very sharp at the proximal end of the aorta,
21
this section is removed.
Figure 10: Cleaned bovine aorta
Once this process is completed, a ring is cut from a location of the tester’s choice
that is approximately 6+ mm in width. The thickness of the sample varies by location
on the aorta. For our tests, the specimen is cut from the middle section of the aorta.
After the ring is cut open by making a longitudinal incision along the convex side, a
tensile pattern fixture is used to cut the open ring along the circumferential direction
to create a strip with a width and length of approximately 6mm. Blue dye is used to
mark one of the two circumferential sides of each rectangular specimen so the tester
can record whether the specimen is loaded in the circumferential or longitudinal
direction.
3.2.2 Brain
The brain must be removed without damage by the carefully executed following pro-
cedure. (See Appendix, Section 7 for Planes of Dissection, Figure 54).
Place the euthanized rat dorsal surface up on a surgical mat with the anterior
side facing up. Using surgical scissors and tissue forceps, pinch the skin on the dorsal
surface by the nape of the neck and make a longitudinal incision at the mid-line.
Insert one end of the scissor blades under the skin surface to further the incision
longitudinally to the right and left side of the upper neck. With a No. 11 blade
22
scalpel make transverse cuts into the muscle tissue just below the base of the skull,
using the forceps to separate the tissue as the cuts become more medial. Once the
muscle is separated on the posterior and lateral sizes of the neck, open the hand
held bone cutters (Fine Science Tools #16107 − 14) slightly so that the opening of
the curved jaws can fit the width of the spine. Holding the bone cutters vertically,
insert the jaws of the cutters on either side of the spine where the muscle tissue was
separated slightly distal to the skull and deep enough to ensure the length of the jaws
go slightly further than the depth of the spine. Slowly grip the bone cutters to cut
through the spine. Once the bone has separated, use the scalpel to cut the remaining
connecting tissue to complete the decapitation.
Figure 11: Mouse Skull Diagram (The Jackson Laboratory 2015)
After decapitation, make a sagittal incision down the center line of the head from
between the eyes back to the furthest posterior position. By pulling and if needed
making swift superficial cuts along the medial of the skin, separate the skin from the
muscle tissue on both the right and left sides to expose the muscles on the sides of
the skull.
Holding the head in the non-dominant hand and using the bone cutters in the
dominant hand, grab, twist slightly and pull the muscle tissue from the posterior
23
occipital in jaw-sized pieces (Figure 11). When most of the muscle tissue is cleared
from the occipital, rest the right side jaw of the bone cutters with the blade proximal
to the skull on the superior edge of the occipital and while grasping the handles of
the bone cutters allow the left side jaw to gently scrape against the bone to remove
the residual muscle tissue. If any cerebral disks are still connected to the skull, gently
insert one jaw of the bone cutters from the posterior side through the hole of the disk
along the dorsal surface without coming in contact with the spinal cord and make the
cut. Grab the side of the cut disk that agrees with the researcher’s dominant side and
twist and pull toward the distal direction. Once all of the disks have been removed,
insert the bone cutters from the posterior side through the occipital hole, being very
careful not to come in contact with the brain tissue with the rounded edge of the
bone cutters facing medially and the sharp edge of the jaw placed inside of the skull
riding the inner surface of the occipital so that the tip of the jaw blade points toward
the dorsal, lateral corner of the occipital. Cut the occipital at this angle on both the
right and left sides (making sure that the rounded edge of the bone cutters is facing
the medial direction for both). Putting the bone cutters in the sagittal plane with the
rounded edge of the bone cutters facing toward the occipital, insert the lower cutter
jaw under the flap of the occipital that was created from the angled cuts. Lean the
upper jaw on the back of the interparietal and while keeping the upper jaw of the
bone cutters stable (like a joint), rotate the occipital flap away from the brain (in the
posterior and dorsal direction). The dorsal surface of the skull has two visible ridges
running in the sagittal plane on either side of the skull (these ridges exist at the most
medial intersection of the muscle tissue on either side of the skull and the bone).
With the curved edge of the bone cutters facing the medial direction, and directed
in the sagittal plane pointing toward the rostral direction (toward the nose) make
small cuts along the ridge on either side of the skull. Once cuts of approximately 1-3
mm have been made on both sides, the same approach can be used to remove this
24
flap as was used to remove the occipital flap. Work down either side of the ridges
until the brain is exposed and the ridges begin to curve in the medial direction.
The dura mater should still be intact at this point, but must be removed so the
brain is unrestrained. Along the interface of the left and right hemisphere of the
cerebrum, a dark red line should be visible. Using the tip of the bone cutters, lightly
grab this line and pull very gently away from the surface of the brain and snip the
connection. The same should be done along the interface between the cerebellum and
the cerebrum.
Now the top of the brain should be fully exposed, however there are still bones
that are holding the brain in place that lie between the interface of the cerebrum and
cerebellum. The external acoustic meatus can be seen pulled from the interface of the
cerebrum and cerebellum (Figure 12). Bring the bone cutters in the coronal plane,
with the rounded edges facing the table toward the posterior side of the skull. By
tilting the bone cutters so the rounded edge faces toward the medial direction slightly,
insert the more medial directed jaw along the intimal surface of the side wall of the
skull at a depth of approximately 4 mm. Clamp the bone cutters slightly enough
to grab, but not cut, and twist along the lateral-medial axis in the lateral direction.
If the bone cutters are clamped too tightly, the bone will snap while twisting. This
process should be repeated on the other side. The twisting action ensures that the
acoustic meatus, which restrains the brain from being lifted from the skull, is removed
and does not damage the brain tissue in the process of its removal.
Remove the brain from the skull by flipping the skull upside down over a small
tray filled with PBS. Insert a surgical elevator between the ventral side of the brain
and the skull and gently assist the falling process of the brain into the tray. Small
cuts will need to be made to the olfactory bulbs and to any other connecting tissues
from the brain to the base of the skull. Allow the brain to fall into the tray being
sure that the brain is not pushed on by the elevator. This concludes the process of
25
Figure 12: External acoustic meatus on human skull (Seward 2015)
harvesting a rat brain without damaging the tissue prior to testing.
After excising the brain from the rat, an incision is made with a No. 11 blade
scalpel parallel to the frontal plane to separate the cerebellum from the cerebrum
(Figure 13). A surgical elevator is moistened in the PBS to create a thin film of fluid
on the tool. The fluid creates a barrier to ensure that the brain tissue does not stick
to the tool. The surgical elevator is used to gently lift the cerebellum to place it into
a tray of PBS to avoid dehydration of the specimen. An incision is then made down
the center line of the cerebrum to separate its two halves (Figure 14). The separation
of the two cerebrum halves requires multiple stabbing cuts directed straight down
rather than a sawing or pivoting blade cut which could damage the tissue.
Each cerebrum half is cut in the sagittal plane into 3 mm thick specimen; creating
two specimens from each half (Figure 15). These specimens are lifted with a surgical
elevator into the PBS as well. The sagittal plane specimens are trimmed to 12 mm x
6 mm to ensure consistency throughout the tests. When it is time to test a specimen,
the surgical elevator is again moistened and then used to scoop each specimen out of
the tray filled with PBS. Using a surgical elevator is ideal, opposed to using tweezers
or other devices, because the specimen is not squeezed, merely lifted. A total of 24
26
(12 rat brain specimens and 12 aortic) were tested.
Figure 13: Post cerebrum removal
Figure 14: Post cerebellum separation
Figure 15: The left half of the cerebellum cut to make 2 specimens. The incision plane is
facing upward for both specimens showing the heterogeneity. (O indicates the outermost
specimen, I indicates the inner specimen and the arrow pointing toward the front side of
the brain.)
27
3.3 Protocol
To begin testing, the Bose stationary stand is loosened from the base and moved
away from the top grip. By doing this, the researcher has more room to carefully
place a drop of 3M VetBond No. 1469SB glue on the bottom plate to secure the
specimen. The specimen can easily be slid off the moistened surgical elevator and
must be precisely placed in its desired alignment on the bottom grip (with the 6 mm
side facing the Bose and the 12 mm side for the brain tissue facing perpendicular to
the Bose and with the circumferential direction parallel to the mover’s direction for
the aortic tissue). The specimen cannot be moved once placed without damaging the
tissue because the glue dries quickly. Another drop of glue is placed on the top of
the specimen. While one hand holds the spring loaded T12X in its lowest position,
the stationary stand is readjusted to a position in which the top and bottom plate
realign. Holding the plate down is essential to avoid additional normal loading on the
specimen. The stationary stand is refastened and the bottom plate is released slowly
upwards. The T12X is adjusted to raise the bottom plate so that the specimen is
squeezed to the desired thickness.
Throughout the test, WinTest 4.1 records the elapsed time, displacement, and the
reactive load induced in the load cell. The scan time is set to the fastest programmable
time of 0.0002 seconds to maximize the number of points during a test. Translational
shear stress is calculated from the detected load from the bottom grip of the apparatus
(Figure 9) by
σ = 10−3
gF
A
(16)
where g is the gravitational constant, F is the detected load (in grams), and A is the
cross sectional area parallel to the test plate (in m2). The translational shear strain
28
is calculated from the detected displacement of the mover by
 =
d
L
(17)
where d is the detected displacement (in mm) and L is the length of the specimen (in
mm) in the direction parallel to the displacement direction.
3.3.1 Combined Sinusoidal Translational Shear and Fixed Initial Uncon-
fined Compression
Aortic tissue experiences sinusoidal shear stresses due to blood pressure oscillations;
likewise, brain tissue can experience sinusoidal shear due to a blast wave. An uncon-
fined compression of either 0% or 33% of the thickness of the specimen is applied prior
to the sinusoidal translational shear test. The amplitude of the shear deformation is
25% of the length. The sinusoidal translational shear frequency is 1 Hz and 40 cycles
which is chosen to emulate the stresses and heart rate induced naturally within aortic
tissue. Aortic tissue, which is naturally load bearing, is loaded within the normal
range of stresses of an aorta. The graphs’ traits are compared with the same loads
exerted on brain tissue, which is not naturally load bearing, to hypothesize what on
the stress-strain graphs could correspond with damage. Brain tissue and aortic tissue
should experience differently not only due to structural differences, but possibly due
to damage as well. A specimen is tested at 2 Hz and 40 cycles to verify whether any
phenomena found in the 1 Hz stress response are characteristic of other frequencies
or if these phenomena only occur with a 1 Hz applied shear deformation.
3.4 Methods of Analysis
The shape and magnitude of the stress-strain data is compared for each compression
and material.
29
3.4.1 Mullins Effect
Zuniga and Beatty (2002) propose that tensile stress softening in rubber can be mod-
eled with an exponential function with three parameters (Equation 15). Taking a
similar approach to their enriched form of the linear vibrations damping envelope
equation, for the brain and artery tests the decreasing trend of the peaks on subse-
quent cycles is fit to a three parameter model. The equation for Y , the peak stress
values with respect to time, is
Y = A exp(−bt) + c. (18)
The first term, A, controls the magnitude of the overall drop in the stress, the second
term, b, influences the quickness of the drop, and the third term, c, allows the model
to approximate the steady state value of the stress over time. A Matlab program is
used to fit each curve and computes the equation variables (Section 7.1).
3.4.2 Signal Processing Analysis
One technique to determine the dominant frequencies within the stress versus time
response is to subject the data to frequency filtering. Frequency filtering is a process
that takes a current signal and applies a pass filter of a set range of frequencies.
By expanding and restraining the band of frequencies allowed, the new signal shows
whether the frequencies chosen are present in the original signal. A Matlab program
is used to filter each stress curve (Section 7.1).
Fourier Series Decomposition is a method to represent a given periodic signal as
a sum of multiple sine and cosine functions. In terms of exp(2piiωt) = cos(2piωt) +
i sin(2piωt) , the Fourier series equation is given by,
h(t) =
n=∞∑
n=−∞
an exp(2piint). (19)
30
The coefficient an is dertmined from orthogonality of the basis functions as
an =
1
T
∫ T
0
h(t)e−2piintdt (20)
where T is the period.
The Fourier transform generalizes the expression to obtain the coefficients of the
Fourier series
H(ω) =
∫ ∞
−∞
h(t)e−iωt dt or
H(f) =
∫ ∞
−∞
h(t)e−i2pift dt (21)
where ω is angular frequency and f is frequency given in Hz.
The signal can be described as a function in the time domain or the frequency
domain. In the frequency domain, the function is complex which gives amplitude
information as well as phase shift information at a given frequency. Using the Fourier
transform, one can pass between the two domains. The inverse Fourier transform is
(from the frequency to the time domain)
h(t) =
∫ ∞
−∞
H(f)e2piift df (22)
where H(f) is the signal in the frequency domain and h(t) is the signal in the time
domain.
One technique to compute the Fourier transform is Fast Fourier transform. Fast
Fourier transform (FFT) is a technique to estimate the Fourier transform of a dis-
crete periodic function. A magnitude spectrum graph displays the real and imaginary
magnitudes of the function in the frequency domain and easily shows which frequen-
cies in the signal are the most significant. A Matlab program is used to decompose
each stress curve using fast Fourier transform (Section 7.1). A Fourier decomposition
31
analysis is applied to the displacement curve to verify that no artifacts exist.
The stress versus time and strain versus time data for each material is studied for
the first three cycles to determine the lag-lead relationship between stress and strain
for the initial response and how it changes with subsequent cycles. Since Fourier
series can only accurately fit periodic functions, the Fourier series may not able to fit
the data; in that case Newland harmonic wavelet analysis is used. Harmonic wavelet
analysis is different from the Fast Fourier Transform because instead of breaking the
signal into harmonic sine functions that do not decay with time, the harmonic wavelet
transform (HWT) breaks the signal into local wavelets that are defined over infinite
time but have significant amplitudes in a finite time. Harmonic wavelets specifically
involve orthogonal wavelets which creates easier mathematics and allows for a unique
set of wavelets in a given signal (Newland 1994b). Harmonic wavelets take the form
of
w(x) =
exp(i4pix)− exp(i2pix)
i2pix
. (23)
Harmonic wavelet transform is used to capture dominant frequencies (j-level) and the
times (k-level translation) that occur within the data. The harmonic wavelet written
in terms of j and k is
w(2jx− k) =
exp(i4pi(2jx− k))− exp(i2pi(2jx− k))
i2pi(2jx− k)
. (24)
The translations in the time domain within a frequency level is induced by k from
the wavelet’s Fourier transform in the frequency domain (Newland 1993). An increase
in k level corresponds with a further shift in time. A wavelet with j = 1, k = 0 and
that is purely real is shown in Figure 16.
The harmonic characteristic of the wavelets allow for the interpretation of the
levels and translations are confined to an octave band of frequencies. The harmonic
wavelet transform uses a FFT and inverse FFT process to analyze discrete experi-
32
Figure 16: A real wavelet graphed versus time for j = 1 and k = 0.
mental data, which does not need to be periodic or sinusoidal. This method looks at
frequency levels that correspond to frequency ranges starting at j = 0 and a frequency
range from 2piT to
4pi
T where T is the total time of the data being evaluated. Each fre-
quency range increases in width as the frequency level increases. The frequency range
doubles with each sequential j level increase and the equation for the frequency range
at any j level is
(2pi)2j
T
≤ f <
(2pi)2j+1
T
. (25)
The total time of the experiment must be rounded down to a new evaluated time,
T = 2n for the FFT computations with the highest integer value of n so that 2n is
less than the original total time. Therefore, the analysis is applied to the number of
cycles that occur in time T . The component frequencies that can be identified are
limited and not exact (since they are given in ranges) however this method finds the
most prominent frequency ranges and the time intervals in which they occur. This
method is very dependent on the number of data points; the more data points over
a given time frame, the more frequencies that can be analyzed and the more tightly
defined time intervals can be examined at a given j level (Equation 25). A faster
programmed sampling time gives more points in a test, but the sampling rate on
33
the Bose test machine does have a limit (0.0002 s). This method can give us the
transient information as well as information on how the frequency components may
change with time.
Both the FFT and the harmonic wavelet method give information about the aver-
age value of the curve or measure of symmetry around the time-axis, given as a zero
frequency level amplitude. They also both provide information about the importance
or dominance of certain frequencies.
To determine the goodness of fit of the decomposition as compared to the original
curve, the L2 error is calculated as the ratio of the L2 norm of the difference between
the curves to the L2 norm of the original curve
|G− g|
|g|
=
√∑
|G− g|2
√∑
|g|2
(26)
where G is the decomposed function and g is the original stress signal. A Matlab
program is used to organize the wavelets into 20 bins in order of their magnitudes. The
bins magnitude ranges are determined by a step size of max−min20 where the maximum
and minimum wavelet magnitudes are calculated (Section 7.1). An increase in bin
number corresponds to a decrease in wavelet amplitude. The bin number captures
the highest magnitude wavelet in bin number 1 with decreasing magnitudes as the
bin number increases.
3.4.3 Statistical Analysis
Two-factor ANOVA (Analysis of Variance) tests are used to examine statistical de-
pendence of the stress response on compression and material. Parts of the stress
versus time response are analyzed including the first stress peak magnitude, the mag-
nitude of stress drop over the cycles (A), the quickness of stress drop over the cycles
(b), and the steady state response (c) (Equation 18). Two-factor ANOVA tests are
34
computed to determine the dependence of the Fourier series coefficients for 0 Hz, 1
Hz, and 3 Hz on material and compression.
Reproducibility, a relative standard deviation, is calculated for the numerical val-
ues used as
R =
SD
AV G
(27)
where SD is the standard deviation and AV G is the average of a set of values.
Reproducibility gives relative meaning to the standard deviation. For example, if the
standard deviation is 10 and the average is 1000, the reproducibility is 0.01, but if the
average is 5 the reproducibility is 2. A reproducibility factor of less than 1 shows a
relatively small standard deviation and therefore a likely reproducible set of ANOVA
statistics. Reproducibility loses its meaning as the average approaches zero.
35
4 Results
The cerebral arteries and brain lie in close proximity in the head so that during
a trauma, the brain tissue and the arteries are likely affected at the same time.
The bovine aorta is a model for the cerebral arteries since human cerebral arteries
are not available for testing. Brain and aortic tissue were tested under sinusoidal
translational shear and fixed unconfined compression. Particularly of interest is the
difference between brain tissue, a naturally non-load bearing material, and aortic
tissue, a naturally load bearing material in vivo. The results are presented in two
sections, qualitative and quantitative. First, the qualitative section provides graphs
for each type of test and analysis using values that can be pulled from the graphs:
first peak stress amplitude, change in stress amplitude, symmetry of response, and the
time shift between the stress response and the applied deformation response. Next,
the quantitative section provides signal analysis of each type of test to determine the
frequencies present in the stress response.
4.1 Qualitative Analysis
Throughout this study, repetitions of the same tests were performed not only to
gather statistical evidence, but also because of the amount of specimen variability.
Specimen variation in soft biological tissue stress responses exists based on animal
to animal differences (e.g. size differences). With a lack of exact reproducibility,
the tests were performed several times to gather information about the average and
standard deviation of the data to better characterize and interpret the results.
From the 0% compression graphs (Figure 17, 19), it is clear that the mechanical
behavior of the two tissues is not the same. Within this section, the magnitude of
the stress amplitudes is compared for each tissue at both 0% compression and 33%
compression. The first stress peak amplitude is specifically of interest because this
36
Figure 17: Stress versus time response of aortic tissue under 0% compression and 1 Hz, 25%
translational shear strain (22515b).
Figure 18: Stress versus time response of aortic tissue under 33% compression and 1 Hz,
25% translational shear strain (22515d).
is the tissues’ initial mechanical response to deformation. The change in amplitude
over time is of interest because this is the tissues’ mechanical response to a repetitive
deformation. The time shift between the stress versus time and strain versus time
graphs is analyzed for the first 3 cycles and calculated as the time between each curve
crossing through the horizontal axis. The shape of the stress graphs is checked for
37
Figure 19: Stress versus time response of brain tissue under 0% compression and 1 Hz, 25%
translational shear strain (20515a).
Figure 20: Stress versus time response of brain tissue under 33% compression and 1 Hz,
25% translational shear strain (12915a).
symmetry with respect to a vertical line through the center of a half cycle as well as
with respect to the horizontal axis over all cycles. These graphical comparisons are
meant to help understand the mechanical response of brain tissue as compared to a
natural load carrying tissue.
38
4.1.1 First Peak Amplitude of Sinusoidal Shear Response
The shear stress magnitude is largest on the first deformation cycle (Figure 17, 18, 19,
20). The brain and aortic tissue shear peak stress magnitudes under a deformation
strain of 25% are nearly the same on the first deformation cycles and have average
amplitudes of 2856 Pa and 2578 Pa respectively. It should be noted that the first
peak stress magnitude response does vary by specimen.
Table 1 reports the peak shear stress which occurs on the first deformation cycle
for 24 total specimens (6 of each type of test) of which 12 are under no compression
and 12 are under 33% compression. A two-factor ANOVA with replication determines
the statistical difference between the first peak stress of brain tissue and aortic tissue
as well as the dependence on compression. The brain and aortic tissues do not show a
statistically significant difference in the first stress peak magnitude (P=0.2723). For
both tissues, compression causes a significant increase in stress magnitude (P=0.014).
The standard deviation has little meaning when viewed alone. The reproducibility
ratios for the peak stress magnitudes are less than 1 so these tests satisfy the criterion
for repeatability used in biomechanics.
4.1.2 Mullins Effect in Shear Stress Response to Translational Shear De-
formation
The Mullins effect (Section 3.4.1) under sinusoidal deformation describes stress soft-
ening in which the stress magnitude of subsequent reloading cycles significantly de-
creases from the stress of the initial loading cycle. Most of the Mullins effect reports
are for tensile deformation; however, in the present reported data for brain and aortic
tissue a similar effect is seen in translational shear.
The stress response of both materials displays a transient part of about 6-10
cycles during which the peak stress amplitudes decrease over time (Figure 17, 18, 19,
39
Table 1: Brain and Aortic Tissue 40 Cycles at 0% and 33% Compression
Test Name First Peak Stress (Pa) A b c
Aorta, 0%
22515b 2275.38 972.2 0.3715 1440.4
22515c 3338.13 1011.8 0.1734 1995.2
40915a 1640.45 147.7 0.1286 1588.9
40915b 1752.18 208.4 0.153 1642.8
40915c 2100.98 270.1 0.1465 1945.6
40915d 2430.7 213.5 0.1093 2325.2
AVG 2256.303 470.6167 0.180383 1823.017
SD 609.6996 405.9098 0.096132 325.7441
R 0.2702 0.8625 0.5329 0.1786
Aorta, 33%
22515d 2294.4 1063.2 0.4276 2372.6
22515e 2741.3 587.1 0.7343 3344.5
22515f 2043.7 405.5 0.5862 1661.9
40915e 3719.6 496.6 0.1335 3273.4
40915f 3259.3 863.1 0.1612 5542.2
40915g 3346.3 853.4 0.5633 2833.9
AVG 2900.8 711.48 0.43435 3171.4
SD 651.9 253.8 0.2428 1318.4
R 0.2247 0.3567 0.5590 0.4157
Brain, 0%
20515a 2695.0 1822.2 0.7917 1095.7
20515b 2485.2 1666.4 0.5695 904.2
21915a 3284.9 2376 0.2705 810.7
21915b 2252.2 1233.8 0.0758 572.7
42315b 3302.7 2250.5 0.6012 1196.6
42315c 2714.1 1997.3 0.7096 933.07
AVG 2789.0 1891.0 0.5030 918.8
SD 425.38 415.41 0.2744 219.2
R 0.1525 0.2196 0.5456 0.2385
Brain, 33%
12915a 3116.0 1794.1 0.6526 1491
42315d 2839.4 2063.1 0.6929 951.6
42315e 4696.5 4732.8 1.1969 1001.7
42315f 4058.8 2828.6 0.6562 1446.6
42315g 4805.5 4235.4 1.0737 1388
42315h 3001.5 2556.7 0.6678 656.0
AVG 3753.0 3035.1 0.8233 1155.8
SD 882.7 1189.8 0.2451 336.4
R 0.2352 0.3920 0.2977 0.2910
40
20). Table 1 lists the parameter values A, b, and c for the 24 specimens found from
Equation 18. Brain tissue experiences a much larger drop in peak stress magnitude
in the first few cycles than aortic tissue, while aortic tissue has a more gradual decay
in peak stress magnitude.
For both tissues, after the stress drop over the first few cycles, the stress peaks
appear to level out with a more gradual change in stress. However, in some cases
for aortic tissue, the stress peak amplitude gradually decreases as the number of
cycles increases and in other cases there is a slight stiffening past about 20 cycles, or
increase in stress peak amplitude. The brain tissue stress response does not exhibit
any stiffening, only a stress drop in the initial deformation cycles.
Figure 21: Stress versus time response and exponential fit curve of aortic tissue under 0%
compression and 1 Hz, 25% translational shear strain.
The exponential fit is not exact (Figure 21, 22, 23, 24), but the fit does provide
insight on whether the stress response is similar to Mullins effect. A two-factor
ANOVA with replication is used to evaluate the effect of tissue and compression
on each coefficient in the decay equation. The first term, A, from the exponential
fit for brain tissue is significantly larger than that for the aortic tissue (P=1.3E-
41
Figure 22: Stress versus time response and exponential fit curve of aortic tissue under 33%
compression and 1 Hz, 25% translational shear strain.
Figure 23: Stress versus time response and exponential fit curve of brain tissue under 0%
compression and 1 Hz, 25% translational shear strain.
06). The compression causes a significant increase in the first exponential fit term
for both tissues (P=0.020). The second term, b, for the fit, which corresponds to
42
Figure 24: Stress versus time response and exponential fit curve of brain tissue under 33%
compression and 1 Hz, 25% translational shear strain.
the quickness of the drop in stress is also larger (Table 1) for brain tissue than for
aortic tissue (P=0.005). Compression causes an increase in the second term, b, for
both tissues (P=0.0009). The third term, c, corresponds to the steady state stress
magnitude which is higher for the compressed tests for both specimens as compared
to each materials uncompressed test (P=6.13E-05). The third term is lower for brain
tissue than aortic tissue (P=0.0125). The reproducibility factors for A, b, and c are
all less than 1.
43
4.1.3 Symmetry of one Load-Unload Cycle
About the vertical axis through the peak of a half cycle, the aortic tissue is nearly
symmetric and the brain tissue is not symmetric. The load response of brain tissue
displays ‘shoulders’, an asymmetric deformity causing opposing concavity on each
cycle (adding more inflection points on the curve) that differs from a pure sine wave
signal (Figure 25). The shoulders break symmetry about the vertical line through
the peak because they are at different locations on the loading side of a half cycle as
compared to the unloading side. No visible shoulder appears on the first deformation
of the brain (Figure 19, 20). The aortic tissue stress response in a single half cycle
is nearly symmetric with respect to the vertical axis that crosses through the peak
values. No shoulders are visibly present in the aortic tissue response (Figure 17,
18); this difference indicates that some component of one of the tissues significantly
influences its stress response.
Figure 25: Shoulder shown on a stress versus time curve for brain tissue under 0% com-
pression and 1 Hz, 25% translational shear strain.
44
4.1.4 Symmetry about the Horizontal Axis
The vertical symmetry with respect to the horizontal axis in a single cycle varies with
specimen and is not always symmetric because the maximum positive and negative
stresses are unequal in magnitude. The symmetry with respect to the time axis can
be seen more clearly toward the last set of cycles when the stress peaks are mostly
constant (Figure 17, 18, 19, 20). For the compression tests for both specimens,
the symmetry about the horizontal axis is more offset (when the average magnitude
of the entire stress signal is nonzero) than the non-compressed tests so it is not
symmetrical about the horizontal time axis. The stress response of the compressed
tissue tests shows a shift about the horizontal axis (sometimes in the positive and
sometimes negative direction) causing a difference in the positive and negative stress
magnitudes. For the compressed brain tissue tests, the ‘shoulders’ shift with respect
to the horizontal axis and no longer occur on the horizontal axis, but rather on the
negative side of the stress response (Figure 20).
4.1.5 Time Shift
A time shift on each cycle is defined as the time between the stress versus time graph
and the strain versus time graph as they cross through the horizontal time axis on the
corresponding cycle (Section 2.1). The time shift that is defined here is more general
than the phase shift that is commonly known for sine curves because the phase shift
involving sine curves is constant over many cycles. Aortic tissue stress versus time
response with respect to the strain versus time data has a time shift that is smaller
in magnitude than the time shift in brain tissue. The aortic tissue stress time shift
creates a lag in the stress (the stress curve crosses the horizontal time axis after the
deformation reaches zero) for most tests while some have small amplitude time shifts
(near zero). Conversely, the brain tissue stress time shift creates lead in the stress
(the stress curve crosses the horizontal axis before the deformation reaches zero).
45
For aortic tissue, the average time shift is approximately 0.04 s with displacement
leading the stress curve (Figure 26). For brain tissue, the average time shift is approx-
imately 0.1 s with the stress leading the displacement curve. The brain tissue stress
versus time response is quasiperiodic; the shift changes over time within the brain
stress data and typically increases and eventually stabilizes (Figure 27, 28) (Table 2).
All reproducibility factors are less than 1 from Table 2.
Figure 26: Strain leads stress for the aortic tissue stress versus time response under 0%
compression and 1 Hz, 25% translational shear strain.
46
Table 2: Brain Tissue Time Shift at the End of Each of the First 3 Deformation Cycles
Test Name Shift 1 Shift 2 Shift3
Brain, 0%
20515a 0.0914 0.1004 0.1029
20515b 0.0776 0.0938 0.0956
21915a 0.0674 0.1053 0.109
21915b 0.0626 0.063 0.0638
42315b 0.0742 0.096 0.1119
42315c 0.0864 0.077 0.1182
AVG 0.0766 0.0892 0.1002
SD 0.0109 0.0160 0.0194
R 0.1433 0.1797 0.1940
Brain, 33%
12915a 0.1376 0.1524 0.1529
42315d 0.1012 0.112 0.114
42315e 0.0835 0.1282 0.1348
42315f 0.0879 0.0943 0.1009
42315g 0.1064 0.1156 0.1187
42315h 0.0883 0.1022 0.108
AVG 0.1008 0.1174 0.1215
SD 0.02003 0.02068 0.01914
R 0.1987 0.1760 0.1575
47
Figure 27: Stress leads strain for the brain tissue stress versus time response under 0%
compression and 1 Hz, 25% translational shear strain.
Figure 28: Stress leads strain for the brain tissue stress versus time response under 33%
compression and 1 Hz, 25% translational shear strain. The time shift changes on subsequent
cycles and is different for the front and back side of a half cycle.
48
4.1.6 Limit Cycle
Stress versus strain graphs over 40 cycles help to visualize whether a limit cycles exists
and the presence of hysteresis (Figure 29, 30, 31, 32). The stress versus strain graphs
are compared to the stress versus time analysis of the Mullins effect to help visualize
the transient response. The number of cycles prior to the limit cycle is related to the
quickness of stress drop variable, b, from the exponential fit equation (18); the higher
the b value in the exponential fit equation, the quicker the stress drops and the fewer
number of cycles before steady state. The magnitude of the drop in the stress-strain
curves over the first few cycles is related to the magnitude of stress drop variable, A,
from the exponential fit equation.
Figure 29: Stress versus strain curve for aortic tissue under 0% compression and 1 Hz, 25%
translational shear strain.
The aortic tissue stress-strain graphs are a much different shape than the brain
tissue stress-strain graphs; the brain tissue cycles have more of a pronounced ‘S’
shape and are much thinner in the sense that the area within the loop is smaller
than that of the aortic tissue stress-strain cycles (Figure 29, 30, 31, 32). The stress-
strain graphs for brain tissue exhibit a flat section at low strains and then harden;
this flattening/hardening does not occur in the aortic tissue and does not exist on
49
Figure 30: Stress versus strain curve for aortic tissue under 33% compression and 1 Hz,
25% translational shear strain.
the first cycle of the brain tissue response. The flat region of the brain tissue stress-
strain response correlates to the shoulder in the brain tissue stress-time response.
Compression causes an increase in the area within the stress-strain cycles for both
materials. If the material were elastic, then there would be no area within the cycles.
The area in the loops of the stress-strain curves are often used to measure to amount
of energy absorbed and dissipated, however no attempt was made to measure the
area enclosed because it is not clear that in this case the area enclosed is the energy
lost; many other factors may cause the appearance of energy dissipation (eg. fluid
redistribution and solid phase damage). The compression in brain tissue causes a
negative shift of the center of the loop (Figure 32) which correlates to the shift in
the stress from the stress-time graph (Figure 20). The loops are graphed clockwise
through time so the gap between the horizontal axis and where the stress-strain curve
crosses through the zero-strain axis verifies that stress leads strain for the brain tissue
because stress is zero before strain is zero.
50
Figure 31: Stress versus strain curve for brain tissue under 0% compression and 1 Hz, 25%
translational shear strain.
Figure 32: Stress versus strain curve for brain tissue under 33% compression and 1 Hz, 25%
translational shear strain.
51
4.2 Frequency Decomposition of the Stress Response
One long-term goal is to correlate the frequency components of the stress response to
the behavior of the fluid and solid portions of the tissues that might cause differences
in the response such as the shoulders in the brain tissue. A frequency filter is used to
determine the main frequencies responsible for the shoulders in the brain tissue stress
response. The frequency of the stress response and the magnitude of each frequency
component from each tissue are further examined in a Fourier series decomposition,
and in a harmonic wavelet decomposition as described in the Methods section (3.4.2).
4.2.1 Frequency Filter
Matlab is used to apply a pass filter on the uncompressed brain stress response data to
graph only the part of the stress versus time signal with a frequency response within
the frequency range given. Expanding and constricting the pass filter from 0-1 Hz,
2-3 Hz and 0-3 Hz, shows the shear stress response of brain tissue to an applied 1 Hz
sinusoidal deformation has a 1 Hz component which is the same as the applied strain
frequency and a 3 Hz component which captures the shoulders in the stress response
(Figure 33). The fit is not perfect with only the sum of 1 and 3 Hz components,
however the sum captures the majority of the response (Figure 34). Other higher
frequencies may also exist to capture the rest of the signal response.
52
Figure 33: Frequency decomposition using two pass filters for 1 Hz and 3 Hz, of a original
stress versus time curve for brain tissue under 0% compression and 1 Hz, 25% translational
shear strain.
Figure 34: Sum of 1 Hz and 3 Hz curves detected from pass filters and original stress versus
time curve for brain tissue under 0% compression and 1 Hz, 25% translational shear strain.
53
4.2.2 Fourier Series Analysis
To verify the shape and frequency of the mover displacement amplitude and to ensure
that no artifacts in the controlled displacement can account for some characteristics
of the shear stress response, Fourier series decomposition is used to analyze the dis-
placement versus time data.
The frequencies extracted from the BOSE displacement versus time data for a
programmed displacement amplitude of 3 mm at 1 Hz are 0, 1, and 3 Hz with Fourier
coefficient amplitudes respectively 0.1547, 2.9916, and 0.0129 (Figure 35, 36). The
most significant frequency is 1 Hz with small amplitude frequencies of 0 and 3 Hz so
the applied displacement is close to a 1 Hz signal. Figure 35 is a frequency spectrum
graph which displays the magnitude, Y (t) (in mm), of each sine function with respect
to each frequency. Figure 36 is the Fourier decomposition displacement versus time
curve plotted on top of the original displacement data.
The Bose displacement mover is electromagnetically controlled which allows the
mover to move nearly friction-free as compared to hydraulic testing machines. Further
sometimes during testing with hydraulic machines, spurious waves are induced in the
hydraulic fluid that can add components to the controlled displacement of other
hydraulic testing machines.
The frequency components of the shear stress response for each material at 0%
and 33% compression are analyzed by a Fourier series decomposition. In particular,
Fourier series decomposition is used to further examine the 3 Hz frequency component
for the brain tissue response determined from the frequency pass filter analysis. For
both aortic tissue and brain tissue, the significant frequencies from the Fourier series
decomposition are 0, 1, and 3 Hz (Figure 37, 39). The reproducibility factor was less
than 1 for all of the 1 Hz and 3 Hz components which are most significant due to their
large magnitudes (Table 3). Small 2 Hz and 4 Hz components appear in some tests.
A two-factor ANOVA with replication is used to determine the significance of tissue
54
Figure 35: Fourier coefficient magnitudes versus frequency for BOSE displacement mover
at 1 Hz and 3 mm amplitude.
Figure 36: Fourier series decomposition curve (blue) and original curve (red) for BOSE
displacement mover at 1 Hz and 3 mm amplitude. The two curves lie on top of each other.
and compression on the Fourier coefficients for each dominant frequency (0, 1, and 3
Hz). The magnitude of the Fourier coefficient for the 0 Hz component corresponds
55
to a magnitude shift or the average stress value and did not significantly depend on
tissue type or compression. The 0 Hz Fourier coefficient is not statistically different
with respect to tissue type or compression. The 1 Hz Fourier coefficient is higher in
aortic tissue (P=1.2E-05) and under 33% compression is higher than 0% compression
in both materials (P=0.051). The 3 Hz Fourier coefficient is higher in brain tissue
(P=8.87E-07) and is not statistically affected by compression, however visually the
shoulders are influenced by compression as a shift toward the negative stress side with
respect to the horizontal axis (Figure 20).
The Fourier Series Decomposition cannot model the transient portion of either
set of data (Figure 38, 40). The brain has a quasiperiodic stress response (Table 2)
that makes the Fourier series not able to model it correctly since the Fourier series
summands are periodic sine functions each of which has a fixed phase shift.
56
Figure 37: Fourier coefficient magnitudes versus frequency for aortic tissue under 1 Hz, 0%
compression, 25% translational shear strain (22515b).
Figure 38: Fourier decomposition curve versus time (blue) and original stress versus time
(red) curve for aortic tissue under 1 Hz, 25% translational strain (22515b).
57
Figure 39: Fourier coefficient magnitudes versus frequency for brain tissue under 1 Hz, 0%
compression, 25% translational shear strain (21915a).
Figure 40: Fourier decomposition curve versus time (blue) and original stress versus time
(red) curve for brain tissue under 1 Hz, 25% translational strain (21915a).
58
Table 3: Fourier Coefficients for Frequencies 0-6 Hz of Shear Stress Response (Pa)
0 Hz 1 Hz 2 Hz 3 Hz 4 Hz 5 Hz 6 Hz
Artery, 0%
22515b 755.8 1749.5 32.4 144.4 15.1 11.8 2.2
22515c 1092 2613.6 46.4 187.4 33.4 12.5 3.9
40915a 86.8 1621.3 0 18.6 0.7 0.9 0.3
40915b 89.8 1639.8 0 32.8 0.9 1.1 0
40915c 102.6 1885.3 21.4 39.9 0.8 0.5 0
40915d 121.7 2274.7 31.4 32.9 0 0 0
AVG 374.7833 1964.0333 21.9333 76 8.4833 4.4666 1.0666
SD 438.6019 398.3133 18.7628 71.2886 13.5181 5.9674 1.6317
R 1.1702 0.2028 0.8554 0.9380 1.5935 1.3360 1.5297
Artery, 33%
22515d 121.5 2110.5 52.7 278.9 14.7 16.5 4
22515e 138.2 2969.4 132.6 175 13.4 18.2 0
22515f 76.5 1638.6 51 75.5 9.5 11.5 2.6
40915e 178.2 3229.8 0 41.2 0 1.4 0
40915f 299.1 5561.9 88.6 80.7 3.1 2.4 0
40915g 257.3 2600.4 25 68.7 2.2 1.3 0
AVG 178.4666 3018.4333 58.3166 120 7.15 8.55 1.1
SD 84.8893 1372.5405 46.9702 90.1555 6.2227 7.8298 1.7606
R 0.4756 0.4547 0.8054 0.7512 0.8703 0.9157 1.6006
Brain, 0%
20515a 59.8 717.4 0 324 0 92 5.1
20515b 53.4 629.6 0 265.5 0 68.2 2.6
21915a 67.8 757.8 25 310.8 0 93.2 4.2
21915b 68.1 690.9 25 280.1 16.7 73.1 4.5
42315b 66.8 810.1 49.7 352.5 16.4 104.8 3.4
42315c 56.6 642.2 41 282.4 19.2 84.9 4.8
AVG 62.0833 708 23.45 302.55 8.7166 84.6 4.1
SD 6.35308 68.9368 20.5020 32.5757 9.5980 14.6977 0.9380
R 0.1023 0.0973 0.8742 0.1076 1.101 0.1737 0.2288
Brain, 33%
12915a 523.6 1041.5 270 388.9 32.2 106.2 13.9
42315d 58.9 713.5 41.3 265.2 9.7 78.2 0
42315e 132.1 811.1 0 320.7 0 98.3 0
42315f 86.5 1092.8 33.8 391.5 0 101.8 0
42315g 115 966.6 75.1 350 17.7 103.5 0
42315h 56.8 625.7 0 166 0 48.7 2.1
AVG 162.15 875.2 70.0333 313.7166 9.9333 89.45 2.6666
SD 179.5852 187.5448 101.9412 86.3042 13.0552 22.3424 5.5669
R 1.1075 0.2142 1.4556 0.2751 1.3142 0.2497 2.0875
59
4.2.3 Harmonic Wavelet Analysis
Fourier series decomposition provides information about the dominant frequencies
present in the stress versus time response, but does not fit the data with precision since
the shear stress response for brain tissue is not periodic. Harmonic wavelet analysis
provides information about the change in amplitude and the location of the frequency
components with respect to time. Figure 41 displays the same 0% compression brain
test from Figure 40; it can be seen that the harmonic wavelet decomposition analysis
has a better fit than the Fourier series decomposition analysis.
Figure 41: Harmonic wavelet decomposition curve and the original stress curve for brain
tissue under 0% compression and 1 Hz, 25% translational shear strain. The curves lie on
top of one another. (20515a)
Recall from Section 3.4.2, the harmonic wavelet analysis with a time step of 0.0002
s between sample points over 40 cycles requires a total evaluated time of T = 26.2142.
The harmonic wavelet decomposition level j = 4 corresponds to the frequency range
of 0.6 − 1.2 Hz (Equation 25, Table 4) which is interpreted as the 1 Hz frequency
response because the Fourier series decomposition contained dominant 1Hz and 3 Hz
components. A j = 6 level is interpreted as the 3 Hz frequency response by the same
argument.
60
Table 4: Frequency Ranges for j Levels
j level Frequency Lower Bound (Hz) Frequency Upper Bound (Hz)
1 0.1 0.2
2 0.2 0.3
3 0.3 0.6
4 0.6 1.2
5 1.2 2.4
6 2.4 4.9
7 4.9 9.8
8 9.8 19.5
9 19.5 39.1
10 39.1 78.1
The coefficient magnitude for the j = −1 level corresponds to a vertical shift in the
signal at all points (much like the Fourier series coefficient for the 0 Hz component).
Table 5, 6, 7, and 8 show the j and k levels for the most significant wavelets computed
from the harmonic wavelet decomposition. Figures 42, 43, 44, 45 show the harmonic
wavelet decomposition of the dominant wavelets (only the wavelets in Tables 5 to 8)
along with the original stress data.
The dominant frequency consistently present within the harmonic wavelet analysis
for brain tissue and aortic tissue is 1 Hz, which is the applied deformation frequency.
The brain tissue analysis detects a dominant 3 Hz component throughout the signal.
The 3 Hz component does not appear at every value of k, however more 3 Hz compo-
nents may exist within the many more small magnitude wavelets in Bin 20. Wavelets
also interact and may overlap to create a 3 Hz signal in the areas in time where the
k values are not dominant. The lowest k values (wavelets at the beginning of the
signal in time) have the largest magnitude of all of the wavelets in the signal. The 3
Hz component only exists at low values of k, or in the first few cycles for the aortic
tissue.
61
Table 5: Brain Tissue 40 Cycles at 0% Compression (20515a)
Bin number j level k level
1 4, 4 0, 1
7 4 2
11 4 4
12 4, 4 3, 5
13 4 ,4, 4, 6 7, 8, 14, 1
14 4, 4, 4, 4 6, 10, 11, 13
15 4, 4, 5, 6 9, 12, 0, 2
16 6 0
17 4, 6, 6, 6 15, 3, 4, 7
18 6, 6, 6, 6, 6, 6, 6, 6, 6 5, 6, 9, 10, 11, 12, 14, 16, 19
Figure 42: Harmonic wavelet decomposition curve (including only the binned wavelets from
Table 5) and the original stress curve for aortic tissue under 0% compression and 1 Hz, 25%
translational shear strain. (20515a)
An increase in bin number corresponds to a decrease in wavelet amplitude. An
increase in k level corresponds with a further shift to a later time. The bin number
captures the highest magnitude wavelet in bin number 1 with decreasing magnitudes
as the bin number increases. In general at a fixed j level, the k values trend from
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Table 6: Brain Tissue 40 Cycles at 33% Compression (12915a)
Bin number j level k level
1 4 0
3 4 1
7 4 2
11 4, 4, 4 4, 5, 14
12 4, 4, 4, 4, 4 3, 7, 8, 10, 11
13 4, 4, 4 6, 9, 13
14 4 12
16 4, 5, 6 15, 0, 1
17 6 0
18 6, 6 2, 3
Figure 43: Harmonic wavelet decomposition curve (including only the binned wavelets from
Table 6) and the original stress curve for aortic tissue under 33% compression and 1 Hz,
25% translational shear strain. (12915a)
low to high with respect to the bin number which correlates to the drop in amplitude
with respect to time. The first 5 largest amplitudes are consistently j = 4, and low
level k. Small k values correspond to the stress at the beginning of the response (first
few cycles).
As described in Section 3.4.2, the bins are organized by magnitude. For all tests,
Bin 20 contains a large amount of wavelets with the bottom 5% magnitudes. If a bin
63
Table 7: Aortic Tissue 40 Cycles at 0% Compression (22515b)
Bin number j level k level
1 4 2
2 4 1
3 4, 4 0, 14
4 4 5
5 4, 4, 4 4, 8, 11
6 4, 4, 4, 4, 4, 4 3, 6, 7, 9, 10, 13
7 4 12
8 4 15
17 5 0
18 -1 0
19 6 0
Figure 44: Harmonic wavelet decomposition curve (including only the binned wavelets from
Table 7) and the original stress curve for brain tissue under 0% compression and 1 Hz, 25%
translational shear strain. (22515b)
contains many wavelets (greater than 50), then the bin is not considered for analysis
(This only happens for these tissue responses in the smallest magnitude bins, 19 and
20). Any other bin excluded from the tables above is due to the lack of any wavelets
in that particular magnitude range.
The harmonic wavelet analysis successfully fits both tissue stress response curves
as shown by the very low L2 Error for All Wavelets in Table 8 and Figure 41. The L2
64
Table 8: Aortic Tissue 40 Cycles at 33% Compression (22515d)
Bin number j level k level
1 4, 4 1,2
4 4, 4 0, 14
5 4, 4, 4 4, 5, 8
6 4, 4, 4 3, 7, 11
7 4, 4, 4, 4, 4 6, 9, 10, 12, 13
9 4 15
16 5 0
19 6, 7, 15 0, 0, 0
Figure 45: Harmonic wavelet decomposition curve (including only the binned wavelets from
Table 8) and the original stress curve for brain tissue under 33% compression and 1 Hz,
25% translational shear strain. (22515d)
norm error (Equation 26) depicts the standard deviation as a relative fraction that
contains a more obvious meaning. The L2 error for the binned wavelets (top 90%
in magnitude) are low and show a good fit without using all wavelets (wavelets from
Bin 20 are excluded). The harmonic wavelet decomposed artery stress response has
a lower L2 error value than the brain stress response L2 error.
65
Table 9: L2 Error for Binned Wavelets and All Wavelets
Test L2 Error Binned Wavelets L2 Error All Wavelets
Artery, 0%
22515b 0.1284 6.73E-08
22515c 0.1146 1.03E-08
40915a 0.0725 6.67E-09
40915b 0.0667 4.02E-08
40915c 0.0675 1.15E-08
40915d 0.0652 1.73E-09
AVG 0.0858 2.29E-08
SD 0.0280 2.55E-08
Artery, 33%
22515d 0.1545 8.94E-09
22515e 0.106 1.42E-08
22515f 0.0913 1.53E-08
40915e 0.0707 5.51E-09
40915f 0.0701 8.73E-09
40915g 0.0933 1.62E-07
AVG 0.0976 3.58E-08
SD 0.0311 6.20E-08
Brain, 0%
20515a 0.3517 8.73E-06
20515b 0.3567 2.89E-06
21915a 0.3723 3.28E-07
21915b 0.3563 4.37E-08
42315b 0.3605 2.16E-08
42315c 0.3626 5.62E-06
AVG 0.3600 2.93E-06
SD 0.007 3.58E-06
Brain, 33%
12915a 0.4743 2.74E-06
42315d 0.3504 6.54E-09
42315e 0.318 1.06E-08
42315f 0.3343 3.29E-08
42315g 0.311 1.44E-08
42315h 0.237 7.02E-07
AVG 0.3375 5.83E-07
SD 0.0775 1.08E-06
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4.2.4 2 Hz Analysis
To verify that the shoulder response was not a phenomena of only a 1 Hz applied de-
formation, a test was completed at 2 Hz under 0% compression and 25% translational
strain; the shoulders are still present (Figure 46). The Fourier series decomposition
coefficients are given in Table 9.
Figure 46: Stress versus time for brain tissue under 0% compression and 2 Hz, 25% trans-
lational shear strain.
Table 10: Fourier Coefficients for Brain Tissue 0% Compression and 2 Hz
Frequency Fourier Coefficient Magnitude
0 Hz 30.8052
2 Hz 817.4784
4 Hz 43.5473
6 Hz 337.1983
10 Hz 96.2478
The dominant frequency values from the 2 Hz test are twice those obtained from
the 1 Hz tests due to the doubling in applied deformation frequency. The 2, 6, and
10 Hz coefficients (Table 9, Figure 47) correlate to dominant 1, 3, 5 Hz components
in the 1 Hz tests. Harmonic wavelet analysis has a fractional error of 0.2965 for the
wavelets in bin 1-19. Table 11 shows the harmonic wavelet coefficients for the brain
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tissue under an applied 2 Hz deformation frequency and 0% compression. A j level
of 4 corresponds to 2 Hz and a j level of 6 corresponds to 6 Hz (Equation 25).
Figure 47: Fourier series decomposition coefficients versus frequency for brain tissue under
0% compression and 2 Hz, 25% translational shear strain.
Table 11: Harmonic Wavelet Coefficients for Brain Tissue at 0% Compression and 2 Hz
Bin number j level k level
1 4 0
4 4 1
11 4 2
14 4, 4, 5, 6 4, 5, 0, 1
15 4, 4, 4 3, 7, 8
16 4, 4, 4, 4, 4 6, 10, 11, 13, 14
17 4, 4 9, 12
18 5, 6, 6, 6, 6 1, 0, 2, 3, 4
19 4, 5, 6, 6, 6, 6, 6, 6, 0, 15, 5, 6, 7, 8, 9, 10,
6, 6, 6, 6, 6, 6, 6 11, 12, 14, 15, 16, 19, 24
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4.2.5 Results Summary
The stress response of bovine aortic tissue and rat brain tissue under combined un-
confined compression and translational sinusoidal shear deformation exhibits many
similarities. The first peak stress amplitudes (the largest peak in the signal) of both
materials are statistically similar. The stress response for both materials displays a
transient response and drop in stress magnitude after subsequent cycles. The brain
tissue stress drops more in magnitude and more rapidly over the cycles than the aortic
tissue stress response causing the brain tissue to have a lower steady state stress value.
The aortic tissue stress response for some specimens slightly stiffens after 20 cycles;
the brain tissue stress response does not display stiffening. Compression causes an
increase in the stress response and an increase in the magnitude of the stress drop.
Neither stress response is in phase with the displacement and the brain tissue
response exhibits a variable time shift (quasiperiodic).The brain tissue stress response
leads the displacement response which differs from the lagging stress response of the
aortic tissue. The aortic tissue stress response is nearly symmetric with respect to
the vertical axis of one half cycle but the brain tissue stress response is not symmetric
due to shoulders in the curve.
A Fourier series decomposition of the stress response of both materials detects a 1
Hz and 3 Hz component. Likewise, a harmonic wavelet decomposition captures how
each frequency component changes with respect to time. The shoulders correspond to
a 3 Hz component that exists throughout the signal of the brain tissue response; the
aortic tissue stress response 3 Hz component does not persist throughout the signal.
The Fourier 1 Hz component has a larger magnitude in the stress response of aortic
tissue and the Fourier 3 Hz component has a larger magnitude in the stress response
of brain tissue.
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5 Discussion
The mechanisms underlying the mechanical response of either aortic or brain tissue
are not yet definitively determined. Because both tissues are hydrated, the interstitial
fluid probably plays a significant role in the mechanical response. The influence of the
fluid-solid interaction appears in the transient stress response, the stress frequencies
detected from frequency analyses and the shape of each stress versus time response,
each of which is interpreted using the structural properties.
Aortic tissue can hold a translational shear stress due to the highly structured,
layered form of the tissue (Section 2.2.1). The structure of brain tissue is much
different than aortic tissue and is maintained by a balance between axonal tension
and hydrostatic pressure (Van Essen, 1997). A possible explanation for how brain
tissue holds a translational shear stress is that the hydrostatic pressure within the
tissue increases with an applied unconfined compression or with an applied shear that
causes distortion and that the axons tension increases to compensate for the pressure
change. Since axons exist in all directions (Figure 48), an applied shear stress would
cause some of those axons to stretch in tension. The increased tension in the axons
could resist further stretch of the axons which may resist shear deformation.
A positive and negative mover direction is defined by the Wintest programming
and refers to a relative center, or zero displacement (B), position (Figure 50, 52) of
the Bose mover. The positive and negative displacement indicates the location of the
mover, away (Figure 49 and point A in Figure 52) or toward (Figure 51 and point
C in Figure 52), the Bose machine with respect to the center position. The center
position is where the mover is located at the beginning of every test. A positive and
negative stress is directly calculated from the detected load cell reading; a negative
load indicates a compressive load from the bottom plate pushing on the load cell and
a positive load indicates a tensile load. The displacement reading corresponds to the
top surface of the specimen attached to the Bose mover and the load detected is the
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specimen action on the bottom plate.
Figure 48: (a) Axons are oriented in all directions; each line in this schematic represents a
possible direction of axons in undeformed brain tissue. (b) Schematic of a brain specimen
in translational shear. Red lines represent directions that are in tension and blue lines
represent compression (axons would be ‘slack’).
71
Figure 49: Brain tissue specimen at positive shear position corresponding to point A in
Figure 52.
Figure 50: Brain tissue specimen at center shear position corresponding to point B in Figure
52.
Figure 51: Brain tissue specimen at negative shear position corresponding to point C in
Figure 52.
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Figure 52: A is the positive shear position, B is the center, or zero, shear position, and C is
the negative shear position. A, B, and C correspond to the position of the mover on each
displacement cycle.
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5.1 Solid-Fluid Interaction in the Mechanical Response
Brain tissue contains both solid and fluid phases that may affect the shear stress
response of the tissue separately as well as interfere with each other. The extracellular
space (ECS) of the brain contains and allows the extracellular fluid (ECF) to move.
Normally, in vivo the ECF is known to transport nutrients and remove cell waste.
Pulsing arteries cause a pressure differential that drive the fluid from the para-arterial
to the paravenous pathways (Section 2.2.2). The volume and construction of the ECS
controls the tortuosity, the measure of path curvature that the fluid must travel in a
given physical length, of the tissue. To better explain the tissue’s stress response, the
behavior of the solid and fluid phases and their interaction will be interpreted during
the applied deformation.
The solid-fluid interaction is defined by the influence that the fluid phase has on
the solid phase of the material or vice versa. The solid phase can affect the fluid
because an applied stretch of the solid phase may cause pressure differentials in the
ECF and fluid redistribution. The fluid can affect the solid phase due to drag. A drag
force is equal and opposite on the fluid and solid parts, but the fluid phase cannot
hold a shear stress. Only the solid phase is connected to the load cell and therefore
is the only phase that can create a stress from the load cell attached to the bottom
grip. Drag can result from fluid friction on a surface of the solid matter wall or from
fluid hitting the solid matter surface normal to the fluid flow.
Stress relaxation is largely due to fluid redistribution in hydrated viscoelastic ma-
terials such as brain tissue (Haslach et al. 2015a) or cartilage (Mow et al. 1980). A
redistribution of fluid causes a decrease in pressure and therefore a drop in axonal
tension and a drop in stress. Stress relaxation occurs after a deformation is applied
and then fixed over time while the stress drops through time often to a steady state
stress value. Since stress relaxation occurs at a fixed strain as fluid redistribution,
fluid redistribution is likely to also occur in cyclic loadings. If the fluid within the
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brain tissue structure redistributes from high pressure locations toward low pressure
locations, the total stress supported would decrease over time. The stress relaxation
may also be dependent on the moisture content in the aorta because hydration lowers
the glass transition of elastin and causes the elastin to be rubbery at body tempera-
ture. Lillie and Gosline (1990) state that elastin, when dry, exhibits little elasticity
and hydrated elastin exhibits significant elasticity.
5.2 Transient Response
Over the 40 cycles of applied deformation, the stress response of the brain and aortic
tissue drops over the first few cycles and levels off to an apparent limit cycle. The
transient stress response may be due to several different causes in aortic tissue than in
brain tissue. The aortic tissue has residual stresses within the tissue in vivo that are
lost once the specimen is cut. Rings that are slit gradually open over time probably
because the residual stresses relax in the elastic sheets (Vaishnav and Vossoughi 1987,
Rahal et al. 2006). The relaxation of residual stresses prior to testing may cause the
transient response of aortic tissue possibly due to relaxed collagen fibers and elastin
sheets prior to testing. In vivo, the aortic tissue would not exhibit a transient response
because it is always undergoing cyclic loadings. If the aortic tissue has relaxed and
is not in its natural, original in vivo stress carrying state, the stress response in vitro
may be affected.
The brain tissue transient stress response may be due to a buildup of pressure
within the tissue causing axonal stretch and possible axonal breakage which would
change the tortuosity of the material. Damage of the solid matter could create ad-
ditional pathways for the fluid to flow. The redistribution of the fluid could cause a
decrease in pressure on subsequent loadings and therefore less axonal stretch which
leads to a lower stress carrying ability. If the fluid carries hydrostatic pressures high
enough to damage the solid state of the tissue, additional pathways would be cre-
75
ated rapidly which correlates with the stress peak drop that occurs quickly and in
large magnitudes in the first few cycles. A theorized principle used throughout this
discussion is the easier the fluid can flow, the lower the shear stress measured.
The transient stress response of both tissues is similar to the Mullins effect in
rubber. The Mullins effect is speculated to be an effect due to bond breaking, and it
is likely that aortic and brain tissue also experience bond breaking. The exponential
fit equation (Equation 18) provides insight on how quickly the stress is dropping and
by how much. The brain tissue drops quickly and in large magnitudes; this could be
due to large quantities of axonal breakage and the opening of pathways in the tissue.
The stress-strain curves help visualize the transient response as the loops change
after each cycle (Section 4.1.6). The brain tissue stress-strain curves include just a
few initial, transient curves that lie outside of an apparent ‘limit cycle’, but have a
large difference in stress magnitude between those cycles whereas the aortic tissue
does not have as large of a stress magnitude change on subsequent cycles after the
transient response. This difference reflects the explanation given about the transient
response and possible damage in the brain tissue during testing.
The limit cycle would be expected for the artery but not the brain; the aortic tissue
is naturally cyclically loaded and brain tissue is not significantly loaded naturally.
(The brain does experience small changes in pressure due to the pulsing arteries.)
The limit cycle does not have a physiological explanation in the brain stress response.
A steady state, or limit cycle in brain tissue, possibly occurs when the pressure
buildup no longer causes axonal breakage and the ECF redistributes more freely.
5.3 Frequency Analysis
The harmonic wavelet decomposition analysis here is a novel application of the tech-
nique to the mechanical response of soft, hydrated, biological tissue and is successful
in giving information about the material stress response. Harmonic wavelets have
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previously been used in music analysis and signal analysis (Newland 1993, 1994a).
Harmonic wavelets successfully fit the quasiperiodic, changing magnitude stress re-
sponse, unlike Fourier series analysis. The harmonic wavelets have a significant non-
zero magnitude during a finite time interval that differs from Fourier series analysis
that uses infinite homogeneous magnitude sine waves. Harmonic wavelet decomposi-
tion analysis allows for the capture of change in magnitude and variable time shifts
of each cycle.
Frequencies other than the expected 1 Hz and the harmonic 3 Hz components
exist in the harmonic wavelet decomposition at the beginning of the signal (k = 0)
for aortic and brain tissue. These other frequencies could be due to the starting up
of the Bose displacement mover or may be due to the breaking of bonds in the tissue.
The 3 Hz component in both the Fourier series and harmonic wavelet decomposition
corresponds to the appearance of a shoulder in the brain tissue stress response. The
aortic tissue stress response does not exhibit a distinct shoulder like the brain tissue
stress response, but does exhibit a smaller magnitude 3 Hz component. The shoulder
does not occur on the first quarter cycle of initial loading, but otherwise the shoulder
appears to maintain a similar shape throughout the cycles in the brain tissue tests.
How the fluid may induce the shoulder in the brain tissue shear stress response is
further examined below.
5.3.1 Fluid Influence on Brain Tissue Response during a Load-Unload
Deformation Cycle
The shear stress leads translational deformation in the brain tissue response. An
increase in hydrostatic pressure and resistance to shear deformation could make the
stress lead the displacement curves during the initial applied half sine wave. Axons
have viscoelastic properties that when in tension exhibit a stress leading response,
that is effectively modeled by Dennerll et al. (1989) using Zener’s model, the Kelvin-
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Voight model in series with a spring (R1 =4 microdynes/µm, R2=9 microdynes/µm,
η= 6,000 microdynes*s/µm). The phase shift, δ, is calculated from the parameters
found by Dennerell et al. (1989) by solving the differential equations for the Zener
model. The phase shift is converted to time by Equation 1. Using a fixed frequency
of 1 Hz in Mathematica, the time shift calculated for an axon is 0.0000169s. This
time shift due to axonal stretch is much smaller than the observed time shift in the
brain tissue stress-time curve. The axonal material properties may cause some of the
time shift within the data, but the remainder of the time shift is speculated to be due
to the fluid. Time shift may be related to the viscoelastic response.
As the mover displacement starts to slow down to change directions at the peak
of the first half sine (Figure 49, point A on Figure 52), the difference in hydrostatic
pressure within the tissue due to a distortion of the extracellular space forces the
fluid to redistribute and reduce hydrostatic pressure which causes a stress relaxation
to occur because the zero velocity position is temporarily a fixed deformation. This
fluid redistribution that is initiated by a slowing in the mover until the displacement
approaches zero velocity could cause a drop in stress prior to the peak deformation
which would cause stress to further lead prior to the peak of the deformation sine
waves. If the fluid is driven to redistribute due to the pressure differential in the
tissue, the axonal tension would decrease as the fluid moves to equalize the pressure
difference and the stress would drop. As the mover stops (Figure 49, point A on
Figure 52) and changes direction, the stress continues to relax due to further fluid
redistribution while the solid state is at its most distorted configuration. As the mover
begins to return toward its initial position (Figure 50, point B on Figure 52), axonal
tension and tissue distortion decrease.
As the mover accelerates to the zero position (Figure 50, point B on Figure 52),
the deformation rate reaches its fastest speed as the mover crosses through the zero
position; this speed could be related to the slope of the stress response or to the
78
shoulder phenomenon (Figure 25). The shoulder includes the positions at which the
displacement crosses through the horizontal axis (Figure 50, point B on Figure 52), i.e.
where the mover is accelerating to its fastest speed. Due to a reduction of distortion
in the extracellular space and fluid redistribution during the unloading initial half
sine, the stress reaches zero before the mover reaches the zero position which creates
a leading stress response. The stress therefore may have a new zero stress position,
that may be related to permanent distortions in the tissue.
The mover speed and acceleration could cause an inertial force in the fluid or drag
from the neurons as the fluid passes through the tortuous ECS of the material. The
stress may lead the displacement due to the inertial effects caused by the mover‘s
acceleration when the stress is zero. It may also be possible that, as the mover slows
after passing the zero position, tissue inertia keeps the tissue force on the load cell from
the bottom grip and contributes to an overshoot of negative stress after the half cycle
is completed. When the deformation continues to the zero position, after the stress
has already reached zero, the repositioned fluid and possibly permanently distorted
tissue may now resist the shear deformation in the opposing direction causing the
negative shear stress detected.
When the displacement is at the zero position (Figure 50, point B on Figure
52), the distortion of the ECS may not have fully recovered to its original condition
(Figure 53). The distortion of the ECS may vary during the sinusoidal deformation
and after repetitive loadings. A change in time shift after repeated loadings may
indicate a change in tortuosity of the tissue creating an easier available pathway for
the fluid to flow, which could be related to possible damage. The hydrostatic pressure
may increase in the direction of initial loading and result in a higher stress in that
direction possibly due to the fluid’s initial redistribution and possible initial damage.
The change in stress amplitude after repeated loadings may be caused by the breaking
of some connections within the structure of the tissue.
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Figure 53: The arrow shows the rupture surface of brain tissue.
The shoulder is of great interest in the brain tissue stress response because it does
not appear in aortic sinusoidal deformation and because it may indicate the role of
the ECF in the mechanical response. The shoulder is closely related to the 3 Hz
component detected from the harmonic wavelet decomposition analysis. The 3 Hz
component may be due to a drag force of the fluid exerted on the solid phase. If the 3
Hz component is the stress component required to overcome drag, the bulk direction
of fluid flow would be the same as the sign of the 3 Hz component at any given time
because the force exerted from the ECF on the solid phase is in the direction of the
ECF flow. The 3 Hz component magnitude decreases over the subsequent cycles in
the harmonic wavelet decomposition, suggesting that the drag force may be smaller
as the cycles continue which may be due to the fluid moving more easily. It is not
definite that the drag force is responsible for all of the 3 Hz component because
there are many other possible influences on the stress response of brain tissue such
as stress relaxation, pressure differentials in the tissue, solid frictional forces between
subsections, axonal damage and subsection shearing.
The shoulder does not occur on the initial loading in the first cycle. A possible
explanation of the lack of a shoulder may be due to lack of damage during the initial
cycle. Initially, the ECS has not been distorted by the load, but at each shoulder the
ECS has been previously distorted. Only during the initial loading cycle does the
displacement mover need to “start up” or accelerate up to speed since the mover can-
not instantaneously move at the desired displacement rate. During the first loading,
80
the mover accelerates from the zero position because it is starting from rest, but on
subsequent cycles the maximum velocity occurs at the zero displacement position and
the mover decelerates during the loading. This start up may cause a slightly different
displacement curve because of this acceleration period which may cause a different
stress response.
5.3.2 Aortic Tissue Stress Response
Unlike the brain tissue stress response in which stress leads displacement, the aortic
tissue stress response stress lags the displacement. When the aortic tissue specimen
is cut, the residual stresses from the tissue holding a nearly circular shape are able to
relax and the collagen fibers may crumple (no longer be stretched straight). At the
beginning of the test, the mover displaces in the positive direction (Figure 49, point
A on Figure 52) and for a small initial displacement the stress is very small which
may indicate the small force and stretch required to elongate the collagen fibers from
their crumpled state. Since the top surface of the tissue is controlled by the mover
displacement and the bottom surface of the tissue is attached to the load cell to
detect the load reading, the change in displacement direction of the top plate is not
immediately reflected as a change in direction of the force on the load cell. The stress
lag is due to the viscoelastic properties of the tissue which is comprised of elastin
sheets, smooth muscle cells, and ECF. Recall, a purely elastic material would have
stress and displacement in synchronization and a purely viscous material would have
stress lagging by 90o; Viscoelastic materials have a stress response that lags by a phase
angle between 0o and 90o. The aortic tissue has a nearly constant time shift which
differs from the brain tissue stress response which is quasiperiodic (non-constant time
shift). The change in time shift is likely due to the damage in the brain tissue that
does not occur in the aortic tissue.
As the displacement returns to the zero displacement position (Figure 50, point
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B on Figure 52), the aortic tissue stress response does not have a distinct shoulder
like the brain tissue stress response possibly because the material is more structured
which may prevent the interstitial fluid from moving as easily. Aortic tissue in shear
may prominently test the strength of the collagen, elastin and fibronectin that con-
nects the smooth muscle cells and extracellular matrix (Section 2.2.1). This lack of an
apparent shoulder in the aortic stress response correlates with the much smaller mag-
nitude 3 Hz component from the Fourier series decomposition and harmonic wavelet
decomposition analysis than the brain tissue stress response. If the 3 Hz component
is the stress to overcome drag due to fluid movement, the small magnitude 3 Hz
component from the aortic tissue stress response may correlate to the restriction of
fluid causing less fluid motion throughout the tissue and a smaller drag force. Water
is bonded to proteoglycans in aortic tissue near the smooth muscle cells which may
prevent some fluid from moving. In aortic tissue, the fluid does not have a significant
role in the structure of the tissue, unlike brain in which the structure is due to the
balance of hydrostatic pressure and axonal tension.
An additional possible explanation for the shape of the aortic tissue response is
tissue stiffening (Section 2.2.1). In the stress strain loops, the initial less steep slope
at smaller strains correlates to the higher frequency components in the stress versus
time graphs. This change in slope as the strain increases may be due to material
properties of the aortic tissue; Clark and Glagov (1985) state that the aortic stress
response hardens because of the matrix mat. Both the restricted fluid flow and the
material stiffening could contribute to the small 3 Hz component.
The aortic peak stress response decreases with time in the in vitro tests but is
not expected to occur in vivo, however a transient response may occur because prior
to testing the residual stresses have relaxed which puts the specimen in a slightly
different state. Fluid loss has been seen in aortic tissue ring inflation tests (Haslach
2011, 2015b). Some aortic tissue tests exhibit slight stiffening after many cycles which
82
may be due to the fluid being driven out of the specimen causing tissue dehydration;
this is not likely to occur in vivo where the body is able to rehydrate the tissue.
5.4 Role of Compression
Fixed compression normal to the shear deformation increases the magnitude of the
shear stress response of both tissues. In the brain, this may be due to an increase in
ECF pressure when compressed. In the aortic tissue, compression may cause initial
elongation of collagen fibers and smooth muscle cells which would create a higher
stress under the same applied shear because the fibers may have a higher stretch
magnitude. The stress drops more quickly and in larger magnitudes in the brain
tissue stress response when compressed possibly due to a larger amount of damage
due to the higher internal pressures. The aortic tissue stress response drops more
quickly and in larger magnitudes when compressed possibly due to the fluid driven
out of the specimen more rapidly due to internal pressures. The amount of stress
drop and the quickness of drop is larger in the brain tissue stress response than in the
aortic tissue stress response probably due to the damage that occurs in brain tissue
and fluid redistribution which would occur rapidly whereas the aortic tissue response
is very structured and may not be damaging.
The steady state stress value under compression for aortic tissue is larger than
that of the steady state stress value for brain tissue perhaps because the aortic tissue
holds its structural integrity whereas the brain is likely damaged under similar applied
conditions. Brain tissue in vivo is not expected to hold a stress through time and
meet an apparent limit cycle, however there may be a balance between the amount of
damage and the fluid flow creating a drag force after several cycles. The brain tissue
steady state stress is larger under compression which may be due to an increase in
frictional force between subsections that may slide over one another or a tightening
of pathways for the fluid to flow. Under compression, a higher force is required to
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maintain the 1 Hz deformation with an amplitude of 25% strain.
5.5 Clinical Applications
Fibrous astrocytes connect white matter in the brain to the capillary walls at the
blood brain barrier. The differing responses of cranial artery and brain tissue response
under the same applied conditions could cause an immediate shearing between the
tissues during an external insult and may cause disruptions between the interfaces of
these tissues including the blood-brain barrier (Section 2.2). Because of a difference
in matter density, interfaces between grey and white matter usually damage easier
causing “hemorrhages [to be] characteristically located at the gray-white matter inter-
face” (Wasserman 2014). The density and mechanical response differences may cause
damage between arterial and brain connections. The loss in load carrying ability of
brain tissue may be due to the breaking of neuron-neuron or neuron-glial connections
due to excessive ECF pressure, leading to symptoms of mild traumatic brain injury
(mTBI), especially in the hippocampus. After some of these connections are broken,
the fluid is able to move easier resulting in a lower overall ECF pressure.
The results suggest that the fluid content has a large influence on the stress re-
sponse of the tissue and should lead to future testing to further examine how the fluid
moves during mechanical deformation of the brain tissue. Tests that could track fluid
movement would provide further information on the solid-fluid interaction during an
applied force. Future work should be directed at a mathematical explanation of the
3 Hz response in terms of the drag force in the brain tissue.
5.6 Shortcomings or Limitations
A few limitations are noted for the test protocol. The unconfined compression was
applied before the shear was applied due to the fixture design limitations. The Vet-
bond had to be placed in a thin enough layer that it did not ride up the sides of the
84
specimen, but a thick enough layer for the specimen not to shear off of the plates.
The tuning for the feedback loop of the Wintest 4.1 program for each tissue is not
exact because Wintest approximates tuning based on linear elastic properties. Spec-
imen variability exists for the thickness of aortic tissue because the natural thickness
changes around a ring.
85
6 Conclusion
During trauma resulting from impacts and blast waves, sinusoidal waves permeate the
brain and cranial arterial tissue, both non-homogeneous biological tissues with high
fluid contents. Aortic tissue, used to model cranial arteries, is much more structured
by the bonds between substructures, but the brain tissue structure is maintained by
a balance of pressure and axonal tension. Both tissues exhibit Mullins effect in shear
and have an apparent limit cycle. Harmonic wavelet decomposition shows significant
1 Hz and 3 Hz components. The 3 Hz component magnitude in brain tissue, which is
much larger than in aortic tissue, may correlate to interstitial fluid induced drag forces
that decrease on subsequent cycles perhaps because of damage resulting in easier
fluid movement. Compression increases the stress response of both tissue and the
hydrostatic pressure in brain tissue. A shear deformation causes the pressure inside
the tissue to grow possibly because the fluid is unable to redistribute at a comparable
rate. This pressure buildup is probably highly deformation rate dependent and may
cause disruptions in the connections between substructures of the brain tissue. The
difference between and mismatch of the responses of the arterial and brain tissues may
cause damage between the connections of these two cranial tissues during external
insults.
86
7 Appendix
7.1 Matlab Code
7.1.1 Mullins Effect
The initial step for the Mullins effect analysis is a matlab code written to determine
the positive peak stress peak values for each test and the time corresponding to each
peak stress. Starting guesses for the A, b, and c values are needed to use “lsqcurvefit”
to fit the equation y = A exp bt + c to the peak values. Different results are possible
if different initial guess points are taken so it is essential to choose the guess point to
fit the curve correctly.
7.1.2 Time Shift
The time shift is determined from a matlab code that defines a ratio for all points as
stress(i+1)/stress(i), for i = 1, ..., n for n number of total points, where stress is
σ = 10−3
gF
A
(28)
where F is the detected load in grams, A is the cross-sectional area of the specimen
parallel to the test plate (in m2), and g is the gravitational constant. When this
ratio is negative, this indicates where the stress curve crosses the horizontal axis.
Linear interpolation approximation determines the time that each curve crosses the
horizontal axis. The matlab output is the difference between the time that the stress
crosses the horizontal axis and the time that the strain crosses zero (time is known
and fixed from the programmed frequency), defined as ∆t.
87
7.1.3 Filter Analysis
The pass filter analysis matlab code uses the command idealfilter(f,t,interval, ‘pass’)
which is analyzed at a defined interval on the data. The stress and time data as well
as the interval for the pass filter defined in Hz are input into the program, and the
code outputs a graph of original data and the filtered data. The ideal filter command
uses a range for the filter interval, which was set from 1-3 Hz.
7.1.4 Fourier Series
The Fourier series decomposition matlab code was written to determine the Fourier
series frequencies, coefficients, and phase shifts. The input values for the code are
the time and stress data, and the number of points per second. The code outputs
several graphs including the magnitude frequency spectrum, the real and imaginary
frequency spectrum, the reconstructed Fourier series and the original signal. This
code uses the fast Fourier transform command to analyze the discrete data.
7.1.5 Harmonic Wavelet Analysis
The Harmonic wavelet analysis code has an input of time and stress from the original
data, and outputs of the original signal graph, the reconstructed 20 bins organized by
magnitude of wavelets, the final reconstructed signal aligned with the original signal,
the total time analyzed, the L2 norm for the reconstructed fit, the maximum and
minimum load values of sine signal, and the 20 bin wavelet coefficients for each j and
k level. This code uses the fast Fourier transform command as well as the inverse fast
Fourier transform to perform the harmonic wavelet analysis, as outlined in Newland
(1994b).
88
7.1.6 Wavelet Examination
An additional matlab code was created to better examine a particular wavelet with
given a-values (where a is the complex magnitude of the wavelet). The code inputs
the values determined from the harmonic wavelet analysis into equation
Y = 2 real(a)
sin(4pix)− sin(2pix)
2pix
− 2 imag(a)
−[cos(4pix)− cos(2pix)]
2pix
(29)
to graph the single wavelet with respect to time.
89
7.2 Planes of Dissection
Figure 54: Planes of Dissection (Zimmerman 2015)
90
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