SPATIAL VARIATIONS IN THE MECHANICAL PROPERTIES OF THE THORACIC
AORTA
By
Jungsil Kim

A DISSERTATION
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Mechanical Engineering
2011

ABSTRACT
SPATIAL VARIATIONS IN THE MECHANICAL PROPERTIES OF THE THORACIC
AORTA
By
Jungsil Kim

Cardiovascular disease (CVD) is the leading cause of death in the United States and
a major cause of disability worldwide. A common type of vascular diseases in the U.S. is
aortic aneurysm. Surgical interventions, such as open surgery or endovascular aneurysm
repair (EVAR), are often required to avoid the high risk associated with aneurysm rupture.
To better understand the role of biomechanics in aortic diseases and develop their clinical
interventions, there is a need to understand the mechanical behavior of healthy as well as
diseased aortas and their eﬀects on aneurysm expansion and rupture potential. While it is
well known that the mechanical properties of a blood vessel vary with location and age, little
attention has been paid to its circumferential variations. Therefore, the goal of this study
is to investigate spatial variations in the mechanical behavior of the descending thoracic
aorta. Toward this end, the following has been accomplished: 1) a biaxial experimental
apparatus with a stereo-vision system, which allows us to track the three-dimensional (3D)
motion of the aorta during the inﬂation test, was developed, 2) the inﬂation tests at ﬁxed
longitudinal stretch ratios were performed for two longitudinal portions – the proximal and
distal – and four circumferential regions – the anterior, left lateral, posterior, and right
lateral – of the porcine thoracic aorta, 3) stress-strain analysis were developed based on
the approximation of the aortic wall surface using a set of continuous base functions in
a curvilinear coordinate system, 4) the variations of stretch, stress, stiﬀness deﬁned as a
change in the circumferential stress corresponding to a change in the circumferential stretch,
and pressure-strain elastic modulus were statistically analyzed, and 5) material parameters
were estimated by a parameter estimation method using a constitutive model based on the

constrained mixture approach.
The experimental results showed that the posterior region was much stiﬀer than the
anterior region. However, the physiological stiﬀness represented by the pressure-strain elastic
modulus did not show a signiﬁcant diﬀerence among the circumferential regions for the
proximal and distal portions of the thoracic aorta. In addition, the stress showed a signiﬁcant
diﬀerence among the circumferential regions, and the stretch was relatively uniform. In
the parameter estimation, material parameters of elastin and collagen were dominant in
the mechanical response, but the role of the smooth muscle seemed to be insigniﬁcant. A
signiﬁcant diﬀerence was found in the parameters of the elastin and collagen ﬁber between
the anterior and posterior regions.
In conclusion, this study presents an experimental method and analysis to measure local
deformation of a blood vessel. Furthermore, it shows that there exists consistent spatial
variations in the mechanical properties of the thoracic aorta. These ﬁndings increase our
understanding of vascular mechanics and adaptation, and will eventually help to improve
clinical treatments and interventions of vascular diseases.

Table of Contents
List of Tables

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

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

xii

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Anatomy and histology of the aorta . . . . . . . . . . . . . . . . . .
1.2 Mechanical behavior of the aorta . . . . . . . . . . . . . . . . . . .
1.3 Variation and heterogeneity of the mechanical behavior of the aorta
1.4 Experimental characterization of mechanical behavior of the aorta .
1.5 Objective of this study . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

1
2
6
11
12
14

2 EXPERIMENTAL METHODS . . . . . . . . .
2.1 Experimental apparatus . . . . . . . . . . . . .
2.1.1 Calibration of sensors . . . . . . . . . . .
2.2 Specimen preparation . . . . . . . . . . . . . . .
2.3 Test protocol . . . . . . . . . . . . . . . . . . .
2.4 Marker tracking and 3D position reconstruction
2.4.1 Calibration of cameras . . . . . . . . . .
2.4.2 Image processing . . . . . . . . . . . . .
2.4.3 Marker tracking . . . . . . . . . . . . . .
2.4.4 3D position reconstruction . . . . . . . .
2.5 Histology sample preparation . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

17
17
23
25
26
27
27
29
30
31
32

3 BIOMECHANICAL ANALYSIS . . . . . .
3.1 Kinematics in curvilinear coordinate systems
3.2 Numerical analysis and approximation . . .
3.2.1 Coordinate transformation . . . . . .
3.2.2 Deformation gradient . . . . . . . . .
3.2.3 Approximation of the surface . . . .
3.2.4 Least squares estimation . . . . . . .
3.3 Thickness of the aortic wall . . . . . . . . .
3.4 Curvature of the aortic wall . . . . . . . . .
3.5 Stress and strain analysis . . . . . . . . . . .
3.6 Statistical analysis . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

33
33
36
36
38
39
40
41
41
43
48

. . . . . . . . . .

50

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

4 CONSTITUTIVE RELATIONS AND PARAMETERS
iv

4.1

Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Constitutive models . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . .

50
53
57

5 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Validation of the experimental measurements . . . . . . . . . . . . . . . . .
5.2 Validation of the approximation using continuous functions . . . . . . . . . .
5.3 Variation of the aortic wall thickness . . . . . . . . . . . . . . . . . . . . . .
5.4 Radius of curvature of the aortic wall . . . . . . . . . . . . . . . . . . . . . .
5.5 Stress and strain relationship of the aorta . . . . . . . . . . . . . . . . . . . .
5.6 Circumferential variation in the stiﬀness . . . . . . . . . . . . . . . . . . . .
5.7 Longitudinal variation in the mechanical behavior of the aorta . . . . . . . .
5.8 Material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9.1 Nonlinear, anisotropic behavior of the aorta . . . . . . . . . . . . . .
5.9.2 Variation in stiﬀness . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9.3 Heterogeneous composite of the aorta . . . . . . . . . . . . . . . . . .
5.9.4 Relation of heterogeneous mechanical properties with pathological conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9.5 Material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9.6 Limitation of this study . . . . . . . . . . . . . . . . . . . . . . . . .

58
58
59
60
60
62
67
69
70
72
72
73
73

6 CONCLUSION

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

A Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1 Camera matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91
91

74
75
76

B Analysis . . . . . . . . . . . . . . . . . . . . . . . .
B.1 Derivatives of the parameters for the curvature .
B.1.1 Reference Conﬁguration . . . . . . . . .
B.1.2 Current Conﬁguration . . . . . . . . . .
B.2 Gaussian quadrature . . . . . . . . . . . . . . .

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

. 96
. 96
. 96
. 99
. 103

C Results . . . . . . . . . . . . . . .
C.1 Information of the aorta sample
C.2 Stress-stretch plots . . . . . . .
C.3 Stretch, stress, and stiﬀness . .
C.4 Material parameters . . . . . .

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

106
106
107
118
122

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

v

List of Tables
5.1

The mean and standard deviation of material parameters for all aorta samples 71

C.1 Information of all aorta samples . . . . . . . . . . . . . . . . . . . . . . . . . 106
C.2 The mean and the standard deviation of the circumferential stretch of the
thoracic aorta at the transmural pressure of 8.00 kPa (60 mm Hg), 12.00 kPa
(90 mm Hg), and 16.00 kPa (120 mm Hg) for each proximal circumferential
region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
C.3 The mean and the standard deviation of the circumferential stress of the
thoracic aorta at the transmural pressure of 8.00 kPa, 12.00 kPa, and 16.00
kPa for each proximal circumferential region (unit: kPa) . . . . . . . . . . . 119
C.4 The mean and the standard deviation of the circumferential stiﬀness of the
thoracic aorta at the transmural pressure of 8.00 kPa, 12.00 kPa, and 16.00
kPa for each proximal circumferential region (unit: kPa) . . . . . . . . . . . 120
C.5 The mean and the standard deviation of the pressure-strain elastic modulus
of the proximal and distal portions of the thoracic aorta for each proximal
circumferential region (unit: kPa) . . . . . . . . . . . . . . . . . . . . . . . . 121
C.6 Material parameters for the sample 1 . . . . . . . . . . . . . . . . . . . . . . 122
C.7 Material parameters for the sample 2 . . . . . . . . . . . . . . . . . . . . . . 122
C.8 Material parameters for the sample 3 . . . . . . . . . . . . . . . . . . . . . . 123
C.9 Material parameters for the sample 4 . . . . . . . . . . . . . . . . . . . . . . 123
C.10 Material parameters for the sample 5 . . . . . . . . . . . . . . . . . . . . . . 123
C.11 Material parameters for the sample 6 . . . . . . . . . . . . . . . . . . . . . . 124
C.12 Material parameters for the sample 7 . . . . . . . . . . . . . . . . . . . . . . 124
C.13 Material parameters for the sample 8 . . . . . . . . . . . . . . . . . . . . . . 124

vi

C.14 Material parameters for the sample 9 . . . . . . . . . . . . . . . . . . . . . . 125
C.15 Material parameters for the sample 10 . . . . . . . . . . . . . . . . . . . . . 125
C.16 Material parameters for the sample 11 . . . . . . . . . . . . . . . . . . . . . 125

vii

List of Figures
1.1

Anatomy of the aorta. (For interpretation of the references to color in this
and all other ﬁgures, the reader is referred to the electronic version of this
dissertation). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Schematic cross section of the wall of an elastic artery. Adapted from (Rhodin,
1980). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Histology of the cross-section of the aorta (a) and its small section stained
with hematoxylin and eosin (H & E) (b), Verhoeﬀ-Van Gieson (VVG) (c),
picrosirius red (d), and the polarized image of picrosirius red stain (e). . . .

5

Typical stress-strain curves of a circumferential strip of an artery during the
uniaxial test (Holzapfel et al., 2000) . . . . . . . . . . . . . . . . . . . . . .

6

2.1

Schematic diagram of the experimental set-up for the extension-inﬂation test.

18

2.2

Schematics of the experimental device. . . . . . . . . . . . . . . . . . . . . .

19

2.3

Schematics of the test chamber to show calculation of the transmural pressure 20

2.4

Two camera set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5

Overview of the experimental process using the extension-inﬂation test device. 22

2.6

Schematic drawing of an upper cannula connector. . . . . . . . . . . . . . . .

23

2.7

A representative example for the linear least squares regression of load cell
data (a) and pressure transducer data (b). . . . . . . . . . . . . . . . . . . .

24

(a) The porcine aorta sample from the thoracic to the abdominal, (b) the
thoracic aorta with surrounding tissues, (c) the thoracic aorta after removing
surrounding tissues, and (d) a cannulated aorta specimen with micro spheres
attached on the wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

Schematics of a world point and its corresponding image point . . . . . . . .

27

2.10 Calibration jig and camera set-up . . . . . . . . . . . . . . . . . . . . . . . .

28

1.2
1.3

1.4

2.8

2.9

viii

21

2.11 An example of the processing to extract the position of markers in pixel. (a)
the entire image, (b) the masked image, and (c) thresholded image . . . . . .

30

2.12 Stereo images from left side camera (a) and right side camera (b), and the
reconstructed marker positions in 3D (c) . . . . . . . . . . . . . . . . . . . .

31

3.1

The surfaces in the reference conﬁguration and the deformed conﬁguration are
approximated by continuous functions of two variables, Θ and S, convected
to the surface. The deformation gradient of the surface corresponding to the
mapping is denoted by F. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.2

Free body diagram of membrane tissue. . . . . . . . . . . . . . . . . . . . . .

44

3.3

(a) Free body diagram of a blood vessel sample during the inﬂation test and
(b) the cross-section of the aortic wall. . . . . . . . . . . . . . . . . . . . . .

46

Comparison of stretch-stress plots before and after the rotation of a specimen.
A: anterior region, L: left lateral region, P: posterior region, and R: right
lateral region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Comparison of the distribution of circumferential stretch between two methods: (a) stretch distribution by linear approximation with triangular elements
and (b) stretch distribution by the approximation in the present study. Black
dots represent markers in the deformed conﬁguration. . . . . . . . . . . . . .

59

Mean and standard deviation of the aortic wall thickness along the circumference: (a) the proximal thoracic aorta, (b) the distal thoracic aorta. 0◦ :
the anterior region, 90◦ : left lateral region, 180◦ : posterior region, and 270◦ :
right lateral region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

A representative example of the distributions of the radius of curvature for
the four circumferential regions of one thoracic aorta sample at transmural
pressures of 8.00 kPa (60 mm Hg), 12.00 kPa (90 mm Hg), and 16.00 kPa
(120 mm Hg), respectively, and a longitudinal stretch ratio (ΛZ ) of 1.35. . .

62

A representative example of the ﬁve repeated loading and unloading curves of
the anterior region of the thoracic aorta. . . . . . . . . . . . . . . . . . . . .

63

A representative example of the distributions of circumferential stretch for the
four circumferential regions of one thoracic aorta sample at transmural pressures of 8.00 kPa, 12.00 kPa, and 16.00 kPa, respectively, and a longitudinal
stretch ratio (ΛZ ) of 1.35. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

5.1

5.2

5.3

5.4

5.5
5.6

ix

5.7

5.8

5.9

A representative example of the distributions of longitudinal stretch for the
four circumferential regions of one thoracic aorta sample at transmural pressures of 8.00 kPa, 12.00 kPa, and 16.00 kPa, respectively, and a longitudinal
stretch ratio (ΛZ ) of 1.35. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

A representative example of the circumferential Cauchy stress and circumferential stretch curve (a) and the longitudinal stress and circumferential stretch
curve (b) at a longitudinal stretch ratio (ΛZ ) of 1.35. . . . . . . . . . . . . .

66

A representative example of the circumferential stiﬀness and transmural pressure curve at a longitudinal stretch ratio (ΛZ ) of 1.35. . . . . . . . . . . . .

67

5.10 Comparison of the mean circumferential stretch of the proximal thoracic aorta
at the transmural pressure of 12.00 kPa for each longitudinal stretch ratio (ΛZ ). 68
5.11 Comparison of the mean circumferential Cauchy stress of the proximal thoracic aorta aat the transmural pressure of 12.00 kPa for each longitudinal
stretch ratio (ΛZ ). Asterisk represents the signiﬁcant diﬀerence (*: p<0.05,
**: p<0.01, ***: p<0.005) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

5.12 Comparison of the mean circumferential stiﬀness of the proximal thoracic
aorta among the four circumferential regions at the transmural pressure of
12.00 kPa for each longitudinal stretch ratio (ΛZ ). . . . . . . . . . . . . . . .

79

5.13 Comparison of the mean pressure-strain elastic modulus of the proximal thoracic aorta at the transmural pressure of 12.00 kPa for each longitudinal
stretch ratio (ΛZ ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

5.14 Comparison of the mean stretch at the anterior and posterior regions of the
proximal and distal portions of the thoracic aorta at the transmural pressure
of 12.00 kPa for each longitudinal stretch ratio (ΛZ ). . . . . . . . . . . . . .

81

5.15 Comparison of the mean circumferential stress at the anterior and posterior
regions of the proximal and distal portions of the thoracic aorta at the transmural pressure of 12.00 kPa for each longitudinal stretch ratio (ΛZ ). . . . . .

82

5.16 Comparison of the mean circumferential stiﬀness at the anterior and posterior regions of the proximal and distal portions of the thoracic aorta at the
transmural pressure of 12.00 kPa for each longitudinal stretch ratio (ΛZ ). . .

83

5.17 Comparison of the mean pressure-strain elastic modulus at the anterior and
posterior regions of the proximal and distal portions of the thoracic aorta at
the transmural pressure of 12.00 kPa for each longitudinal stretch ratio (ΛZ ).

84

x

5.18 A representative stress-stretch plot. Stress was calculated by using experimentally measured data and theoretically predicted. M: experimentally measured
data, P: theoretically predicted data . . . . . . . . . . . . . . . . . . . . . .

85

5.19 Comparison of the mean and standard deviation of material parameter c1 (a)
and c2 (b) among the four circumferential regions for the proximal thoracic
aorta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

5.20 Comparison of the mean and standard deviation of material parameter c1 (a)
and c2 (b) between the proximal and distal portions of the thoracic aorta. .

87

A.1 Schematics of a world point and its corresponding image point . . . . . . . .

92

A.2 Image plane and pixel coordinate . . . . . . . . . . . . . . . . . . . . . . . .

93

C.1 Stress-stretch plots for the proximal thoracic aorta at a ﬁxed stretch ratio of
1.35, 1.40, and 1.45 for sample 1 . . . . . . . . . . . . . . . . . . . . . . . . . 107
C.2 Stress-stretch plots for the proximal thoracic aorta at a ﬁxed stretch ratio of
1.35, 1.40, and 1.45 for sample 2 . . . . . . . . . . . . . . . . . . . . . . . . . 108
C.3 Stress-stretch plots for the proximal thoracic aorta at a ﬁxed stretch ratio of
1.35, 1.40, and 1.45 for sample 3 . . . . . . . . . . . . . . . . . . . . . . . . . 109
C.4 Stress-stretch plots for the proximal thoracic aorta at a ﬁxed stretch ratio of
1.35, 1.40, and 1.45 for sample 4 . . . . . . . . . . . . . . . . . . . . . . . . . 110
C.5 Stress-stretch plots for the proximal thoracic aorta at a ﬁxed stretch ratio of
1.35, 1.40, and 1.45 for sample 5 . . . . . . . . . . . . . . . . . . . . . . . . . 111
C.6 Stress-stretch plots for the proximal thoracic aorta at a ﬁxed stretch ratio of
1.35, 1.40, and 1.45 for sample 6 . . . . . . . . . . . . . . . . . . . . . . . . . 112
C.7 Stress-stretch plots for the proximal thoracic aorta at a ﬁxed stretch ratio of
1.35, 1.40, and 1.45 for sample 7 . . . . . . . . . . . . . . . . . . . . . . . . . 113
C.8 Stress-stretch plots for the distal thoracic aorta at a ﬁxed stretch ratio of 1.35,
1.40, and 1.45 sample 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
C.9 Stress-stretch plots for the distal thoracic aorta at a ﬁxed stretch ratio of 1.35,
1.40, and 1.45 sample 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
C.10 Stress-stretch plots for the distal thoracic aorta at a ﬁxed stretch ratio of 1.35,
1.40, and 1.45 sample 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

xi

C.11 Stress-stretch plots for the distal thoracic aorta at a ﬁxed stretch ratio of 1.35,
1.40, and 1.45 sample 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

xii

Chapter 1
INTRODUCTION
Cardiovascular disease (CVD) is the leading cause of death in the United States and a major
cause of disability worldwide. In 2007, CVD accounted for 33.6 percent of all deaths, and
the direct and indirect cost of CVD was estimated at $ 286.6 billion in the U.S. (AmericanHeart-Association, 2011). The high mortality rate of CVD has thus promoted the growth
of research on cardiovascular mechanics. Vascular mechanics that has been developed based
on continuum mechanics helps to understand the mechanical state of vascular tissue in
various physiopathological conditions or in genetic disorders. Although understanding the
biomechanics of the blood vessels has been greatly increased for the last three decades,
the heterogeneity of vascular mechanics is still largely undisclosed and the state-of-the-art
technique is not complete enough to explain regional diﬀerence in vascular behavior of either
healthy and diseased arteries.
One of the most common types of vascular diseases in the U.S. is aortic aneurysm, an abnormal dilation of the aorta more than 3 cm in diameter (American-Heart-Association, 2011),
which occurs mainly among the elderly (Walker et al., 1986). Aortic aneurysms are classiﬁed
by the location where they occur: aortic root, thoracic, and abdominal aortic aneurysms.
Abdominal aortic aneurysms (AAAs) are more common than thoracic aortic aneurysms,
and the mortality rate of ruptured AAAs is up to 90% (American-Heart-Association, 2011).

1

Thus, the majority of studies of aortic aneurysms have been done on the abdominal aorta,
and relatively few have focused on the thoracic aorta. Thoracic aortic aneurysms are also
serious conditions due to the high risk of rupture, and they frequently occur concomitantly
with other diseases such as atherosclerosis and Marfan syndrome (Isselbacher, 2005). The
most severe disease of the aorta is aortic dissections, which are caused by a disruption or
tear of the intimal layer, resulting in a separation in the aortic wall. Aortic dissections have
a higher risk of mortality than thoracic aortic aneurysm and carry a high mortality rate,
which increases 1 percent per hour unless patients ﬁnd treatment within the ﬁrst 24 hours
(Cannon and O’Gara, 2006).
Surgical interventions, such as open surgery or endovascular repair, are often required
to avoid the high risk associated with aneurysm rupture. The decision-making for surgical
repair, however, depends mainly on physicians’ clinical experience. In common practice, a
surgical repair is recommended if the maximum diameter of the aneurysmal aorta exceeded
a certain size (e.g., 5.5 cm for abdominal aortic aneurysms). In reality, however, some small
aortic aneurysms still rupture even before reaching the critical size (Participants, 2002).
Hence, the maximum diameter alone is not a suﬃcient indicator for the risk of rupture.
Abdominal aortic aneurysms grow faster in the anterior side, but their ruptures are found
most in the posterolateral sides (Schwartz et al., 2007). It is rather suggested that mechanical
failure is a local event and occurs when the local mechanical stress exceeds the local strength
(Vorp et al., 1998). Therefore, there is a pressing need for understanding spatial variations
in mechanical properties of healthy as well as diseased aortas and their eﬀects on aneurysm
expansion and rupture potential.

1.1

Anatomy and histology of the aorta

The aorta is the largest artery originating from the left ventricle of the heart and conveys
oxygenated blood to organs and tissues of the body. It is generally classiﬁed into several

2

Brachiocephalic
artery

Left common
carotid artery

Aortic arch

Left subclavian
artery

Ascending
thoracic aorta

Descending
thoracic aorta
Intercostal
arteries

Diaphragm

Abdominal
aorta

Common
iliac arteries

Fig. 1.1: Anatomy of the aorta. (For interpretation of the references to color in this and all
other ﬁgures, the reader is referred to the electronic version of this dissertation).
portions: ascending aorta, aortic arch, and descending aorta, which is again divided to the
thoracic aorta and the abdominal aorta. Fig. 1.1, which was reconstructed from magnetic
resonance imaging (MRI) images, shows more detail anatomy of a human aorta. The descending thoracic aorta begins just beyond the aortic arch as the aorta bends down into
the body and ends at the diaphragm. It is situated on the left side of the vertebral column,
containing several intercostal arteries along its length, which provide blood to the chest area.
In humans, the mean inner radius and wall thickness of the thoracic aorta are approximately
1.25 cm and 2 mm, respectively (Humphrey, 2001). Below the diaphragm, the abdominal

3

Advantitia

Non-myelinated
nerve

Macrophage Fibroblast Bundle of collagen fibrils
Blood vessel
Myelinated
nerve

Collagenous
fibrils
Media

Smooth muscle
cells
Basal lamina
Smooth muscle
cells

Intima

Elastic artery

Elastic lamina
Reticular and fine
collagen fibrils
Endothelial cells

Fig. 1.2: Schematic cross section of the wall of an elastic artery. Adapted from (Rhodin,
1980).
aorta is located in front of the vertebral column, and it bifurcates into the left and right common iliac arteries. The abdominal aorta has several aortic branches, which include the celiac,
superior mesenteric, inferior mesenteric, and renal arteries. The aorta becomes narrower in
diameter as it separates into branches.
The aortic wall is composed of three distinct layers: the tunica intima, tunica media, and
tunica adventitia (Fig. 1.2). The intima is the innermost layer of an aorta and consists of a
layer of endothelial cells and a subendothelial layer of connective tissue. The internal elastic
lamina separates the intima and the media. The media is the middle layer of concentricallyarranged smooth muscle cells, and contains collagen ﬁbers and elastic ﬁbers. The external
elastic lamina separates the media and the adventitia. Finally, the adventitia is the outermost
layer of an aorta and primarily composed of longitudinally arranged collagen ﬁbers with
elastin and ﬁbroblasts. Compared with an elastic artery, an aorta consists of a thick media

4

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

Fig. 1.3: Histology of the cross-section of the aorta (a) and its small section stained with
hematoxylin and eosin (H & E) (b), Verhoeﬀ-Van Gieson (VVG) (c), picrosirius red (d), and
the polarized image of picrosirius red stain (e).
layer and a relatively thin adventitia layer (Fig. 1.3 (a)). The primary structural proteins
of the aortic wall are collagen and elastin.
The cross-sectional histology of the aorta with diﬀerent stains gives us the information
of constituent as shown in Fig. 1.3. The entire aortic ring with surrounding tissue was
stained with picrosirius red (Fig. 1.3 (a)), where collagen is colored red. Nuclei are blue
with hematoxylin and eosin (H & E) stain (Fig. 1.3 (b)) so that cell density can be estimated.
Verhoeﬀ-Van Gieson (VVG) stain is common for examining the elastin because elastic ﬁbers
are colored black, and collagen is red (Fig. 1.3 (c)). In order to examine the orientation
of collagen ﬁbers, polarized light microscopy can be used (Fig. 1.3 (e)). The density of
elastin lamellae varies throughout the aortic wall, and the adventitia layer has denser elastin
lamellae than the media layer.
The microstructure of the aortic wall varies among species. The contents of the collagen
and elastic tissue diﬀer with regard to the portion of the aorta. In this study, we focus on
the descending thoracic aorta and discuss it in detail.

5

Fig. 1.4: Typical stress-strain curves of a circumferential strip of an artery during the uniaxial
test (Holzapfel et al., 2000)

1.2

Mechanical behavior of the aorta

The mechanical properties of an aorta vary with species, age, location, and its pathological
conditions. Although there are many documents to report the characteristics of mechanical
behavior of the aorta, some important ﬁndings are brieﬂy summarized in this section.
Like many other biological soft tissues, an aorta undergoes relatively large deformations
and its behavior is nonlinear, inelastic, and anisotropic over ﬁnite strains. Fig. 1.4 shows
the typical stress-strain response of a circumferential arterial strip during the uniaxial test
in passive condition. An arterial tissue displays stress softening during the ﬁrst few loading
cycles, as shown in Fig. 1.4. Then it exhibits a nearly repeatable cycle behavior (Yin and
Fung, 1971). Due to the repeatable behavior in cyclic loading, the mechanical behavior of
arterial tissues are often described by using pseudoelasticity. For example, Fung et al. (1979)
stated that, “Since stress and strain are uniquely related in each branch of a speciﬁc cyclic
6

process, we can treat the material as one elastic material in loading and another elastic
material in unloading. Thus, we can borrow the method of the theory of elasticity to handle
an inelastic material. To remind us that we are really dealing with an inelastic material,
we call it pseudoelasticity”. However, the hysteresis, the diﬀerence between the loading
and unloading curves, is typically small and a majority of experimental studies of a blood
vessel have used only loading curves for characterizing the pseudoelastic behavior. Fung
et al. (1979) also suggested that it is convenient to simplify a complex property of an artery
although the pseudoelasticity is not an intrinsic property.
Long-term vascular adaptation of a blood vessel occurs under various pathophysiological
conditions, such as aging (Groenink et al., 1999; Spina et al., 1983) and cardiovascular diseases (Angouras et al., 2000; Thubrikar et al., 2001; Vande Geest et al., 2006). It appears
that changes in the mechanical state of a blood vessel wall play important roles in vascular
adaptations (Humphrey, 2008). An important goal of vascular mechanics is to accurately
estimate the mechanical state of blood vessels in vivo, which is the key determinants of vascular homeostasis. Although the time course and detailed mechanisms of vascular homeostasis
remain unclear, many studies have suggested that blood vessels alter their microstructures
to restore a preferred (homeostatic) state (Humphrey, 2001).
In the study of vascular mechanics, the discovery of residual stress in an artery has been
of great importance. When an arterial ring is cut radially, the ring springs open. Chuong
and Fung (1986) reported that the stress resultant and stress moments were zero after the
radial cut from the static equilibrium. They identiﬁed a stress-free state and showed that
circumferential residual strain reduced the transmural gradients of stress and strain (Chuong
and Fung, 1986).
During the last several decades, mathematical descriptions for mechanical behavior of
blood vessels have been extensively developed. Takamizawa and Hayashi (1987) proposed
the uniform transmural circumferential strain hypothesis, but Guo et al. (2005) found that
the uniformity of the transmural strain was limited when the opening angles of arteries

7

were smaller than 180◦ . Zhang et al. (2005) proposed the uniform biaxial strain hypothesis,
including the uniformity of the both circumferential and axial strains in vivo. Later, it was
also shown that the vascular smooth muscle action contributes to more uniform strain and
stress distribution throughout the arterial wall (Rachev and Hayashi, 1999).
Many constitutive models have been developed to describe the mechanical behavior of
a blood vessel under speciﬁc conditions of interest. Strain energy function, commonly in
exponential, polynomial, and logarithmic forms, is used to determine a constitutive model in
hyperelasticity. The simplest strain-energy function W per unit volume is the neo-Hookean
model
W = c1 (Ic − 3)

(1.1)

where c1 is a material parameter having the dimension of stress and Ic = trC. It is appropriate for homogeneous, isothermal, and hyperelastic materials. Another commonly used
strain energy function for rubber-like material is the Mooney-Rivlin model

W = c2 (Ic − 3) + c3 (IIc − 3)

(1.2)

where c2 and c3 are material parameters, and
IIc =

1
(trC)2 − trC2 .
2

(1.3)

Vaishnav et al. (1973) proposed a two dimensional (2D) strain-energy function in polynomial form to describe the three-dimensional (3D) anisotropic behavior of passive vascular
tissue as
2
2
3
2
2
3
W = a1 EΘΘ + a2 EΘΘ EZZ + a3 EZZ + a4 EΘΘ + a5 EΘΘ EZZ + a6 EΘΘ EZZ + a7 EZZ (1.4)

where ai (i = 1, · · · , 7) are material parameters having in the unit of stress.
A well-known strain energy function for soft tissue was proposed by Chuong and Fung
8

(1983). It is a 3D strain energy function W per unit initial volume in the undeformed
conﬁguration such that
1
W = c4 exp (Q)
2

(1.5)

where c4 is a material parameter having the same unit of stress and Q is given by
2
2
2
Q = b1 ERR + b2 EΘΘ + b3 EZZ + 2b4 ERR EΘΘ + 2b5 EΘΘ EZZ + 2b6 EZZ ERR

(1.6)

where bi (i = 1, · · · , 6) are non-dimensional material parameters, and EIJ (I, J = R, Θ, Z)
are Green’s strain components in the cylindrical polar coordinate (R, Θ, Z). Later, a modiﬁed
3D strain-energy function of Fung’s type form was formulated by Humphrey (1995) with
shear terms in the exponential function.
Another type of a 2D strain energy function in logarithmic form was proposed by Takamizawa
and Hayashi (1987) as
W = −c5 Ln(1 − Q )

(1.7)

where c5 is a material parameter and the function Q is given by
1
1
2
2
Q = d1 EΘΘ + d2 EZZ + d3 EΘΘ EZZ .
2
2

(1.8)

Here di (i = 1, 2, 3) are non-dimensional material parameters.
A combined polynomial and exponential form of strain energy function was proposed by
Holzapfel and Weizsacker (1998) as
c
c
W = 6 (Ic − 3) + 7 {exp(Q) − 1} .
2
2

(1.9)

where material parameters c6 and c7 having the unit of stress. Later Holzapfel et al. (2000)
proposed another strain energy function, which modeled an arterial tissue corresponding to
the isotropic deformations and anisotropic deformations, considering two collagen ﬁber fam-

9

ilies. To determine the isotropic response, neo-Hookean model was used and the anisotropic
response was determined by an exponential function such that

Waniso (I4 , I6 ) =

k1
2k2

exp k2 (Ii − 1)2 − 1

(1.10)

i=4,6

where k1 a material parameter in the unit of stress and k2 is a dimensionless parameter. I4
and I6 are invariants of C. Holzapfel and others also developed two-layer model (media and
adventitia as a ﬁber-reinforced composite) with residual strains. Yet another strain energy
function was proposed by Zulliger et al. (2004b)
W = fe ce (Ic − 3)3/2 + fc

1
Wc
2

1
I4 − 1 + Wc
2

I6 − 1

.

(1.11)

where fe and fc are the area fraction of elastin and collagen, respectively, ce the elastic
constant, and Wc the strain-energy function of the collagen. It accounts for wall compositions
and structures, such as the wavy nature of the collagen ﬁbers and their angle.
While the aforementioned constitutive models have been developed for traditional stressstrain analysis, the constitutive model based on a constrained mixture approach has been
developed to describe continuous growth and remodeling of vascular tissue (Baek et al., 2006;
Humphrey and Rajagopal, 2002; Zeinali-Davarani et al.). That is, it deals with not only the
separate contribution of each constituent of a blood vessel but also the continuous production
and removal of each constituent during the vascular adaptation. Thus, a constrained mixture
model can provide the potential role of each constituents, elastin, collagen, and smooth
muscle, in the mechanical behavior of stressed material.

10

1.3

Variation and heterogeneity of the mechanical behavior of the aorta

The mechanical properties of the aorta vary with locations on the aortic tree (Han and Fung,
1995), and, indeed, the aortic wall is not isotropic (Weizsacker, 1988; Zou and Zhang, 2009).
For example, the dog thoracic aorta in situ was stiﬀer in the longitudinal than the circumferential direction (Patel et al., 1969), there is the longitudinal variation in circumferential
strain and elastic modulus of mouse aortas (Huang et al., 2006), and the stretch ratio and
Cauchy stress were lower in the thoracic aorta than the abdominal aorta (Guo and Kassab,
2004).
The arterial wall is a heterogeneous composite consisting mainly of collagen, elastin,
and smooth muscle cells. The relative amount of these constituents is responsible for the
mechanical properties of a blood vessel and varies with locations along the arterial tree
(Fischer and Llaurado, 1966; Halloran et al., 1995; Lillie and Gosline, 2007; Purslow, 1983;
Roach and Song, 1994; Stergiopulos et al., 2001). The histology of the aorta is diﬀerent
between the thoracic and abdominal regions. The thoracic aorta contains relatively more
elastin, whereas the abdominal aorta contains relatively more collagen (Thubrikar, 2007).
Elastin and collagen are the major components of the ascending thoracic aorta, and smooth
muscle cells are the major components of the abdominal aorta.
The inﬂuence of elastin on the mechanical properties of the aortic tissue has been investigated (Gundiah et al., 2007; Lillie and Gosline, 2007; Zou and Zhang, 2009). Elastin ﬁbers
are primarily responsible for the linear behavior of an artery in the low pressure range (Gundiah et al., 2007; Shadwick, 1999; Stergiopulos et al., 2001). On the other hand, collagen is
recruited in the higher pressure range and contributes to the nonlinear behavior of arterial
tissue with elastin (Groenink et al., 1999; Shadwick, 1999). The orientation of collagen and
the amount of cross-linking contribute to the mechanical behaviors of vascular tissue (Haskett et al., 2010; Holzapfel et al., 2002). Stergiopulos et al. (2001) attempted to characterize

11

the elastic properties and composition of the separated inner and outer halves of porcine
aortic media, and they concluded that these properties were similar. The three-dimensional
structure of the rat aortic media was observed using 3D confocal microscopy, the number of
medial lamellae in anterior region were greater than the posterior (O’Connell et al., 2008).

1.4

Experimental characterization of mechanical behavior of the aorta

For the experimental studies, ex vivo including in vitro tests are commonly employed, although in vivo tests would characterize the biomechanical behavior of a blood vessel under
actual physiological conditions. It is due to the limitation of in vivo tests such as the complexity of in vivo physiological conditions, diﬃculty of obtaining accurate strain measurements,
and the uneasiness of controlling experiments.
A variety of ex vivo testing methods have been developed to characterize the mechanical
properties of vascular tissue using diﬀerent shapes of tissue specimens, such as strips, ﬂat
sheets, rings, or cylindrical tubes. The most common biomechanical test is the uniaxial
extension test with a strip for investigating mechanical behavior of the vascular tissue in one
direction under cyclic loading (Gundiah et al., 2007; Iliopoulos et al., 2009b; Lally et al.,
2004; Okamoto et al., 2002; Sokolis, 2007), but it is not suitable to quantify the anisotropic
behavior of vascular tissues. The ring test (Guo and Kassab, 2004; Huang et al., 2006; Lillie
and Gosline, 2007) is another type of the uniaxial test on arterial rings. It is commonly
used to quantify the yield strength or active smooth muscle tone of a blood vessel in the
circumferential direction, but it is insuﬃcient to study the anisotropy of vascular tissues
because it does not measure deformation and force in axial direction. A planar biaxial
test with excised ﬂat sheet specimens has been popular for investigating biaxial mechanical
behavior of aortic tissue (Lally et al., 2004; Okamoto et al., 2002; Tremblay et al., 2010;
Vande Geest et al., 2006; Zhou and Fung, 1997). However, lateral excision of the tissue may
12

alter the spatial distribution of residual stress in the specimen and, thus, it is diﬃcult to
simulate the deformation of the artery experienced during the cardiac cycle. On the other
hand, the inﬂation test with a cylindrical tube segment is the preferred test for obtaining
biaxial mechanical properties of a blood vessel (Blondel et al., 2001; Humphrey et al., 1993;
Langewouters et al., 1984; Schulze-Bauer et al., 2002). This test reﬂects closely the motion
of the aortic wall during the cardiac cycle so that we estimate in vivo stress, although the
surrounding tissue are not usually taken into account in the ex vivo test.
Since vascular tissues experience ﬁnite deformation and stress during the test, a videobased tracking technique is typically applied, with multiple markers embedded or aﬃxed to
the tissue specimen (Everett et al., 2005; Hu et al., 2007; Hsu et al., 1995; Saravanan et al.,
2006; Thubrikar et al., 1990; Zhang et al., 2002). In the traditional ex vivo inﬂation test, a
blood vessel is assumed to be a perfect cylindrical tube and only the outer diameter change
of the vessel is measured during the test (Blondel et al., 2001; Langewouters et al., 1984;
Schulze-Bauer et al., 2003; Valdez-Jasso et al., 2009). Hence, the spatial distribution of local
mechanical properties of the vessel is not commonly characterized in the traditional inﬂation
test.
Recent advances in the 3D medical imaging have allowed one to use a tracking technique
to measure nonuniform deformation during the test. For example, Draney et al. (2004)
quantiﬁed in vivo cyclic strain of porcine aortas using cine phase contrast magnetic resonance
imaging (MRI). In their study, the cyclic strain were not uniform along the circumference of
the aorta, but their results were highly deviated. Danpinid et al. (2010) proposed a method
to investigate the local stress-strain relationship of both normal and pathological murine
abdominal aortas using an ultrasound motion estimation technique, and it allowed one to
identify the vascular disease. Recently, Genovese (2009) introduced a new optical system
with a concave conical mirror, which provides the full-ﬁeld measurement of 3D deformation
of an artery, and Avril et al. (2010) developed an inverse method to determine material
parameters using the experimental data under the traditional assumption. The accuracy of

13

the method and its utility is, however, not yet clearly demonstrated and there is still a need
for developing more analysis methods to make full use of 3D spatial information of the artery.
Development of such strain measurement techniques promotes the study for characterizing
the heterogeneity in the mechanical properties of a blood vessel over a wide range of pressure,
and thus more experimental techniques and novel analysis methods need to be developed.

1.5

Objective of this study

While it is well known that the mechanical properties of a healthy blood vessel vary with
location (Lillie and Gosline, 2007; Sokolis, 2007) and age (Groenink et al., 1999; Spina
et al., 1983), their circumferential variations have received little attention. In the majority
of biomechanical analysis in vascular studies, an artery has been considered as a hollow
conduit with a constant thickness, and uniform stress and strain distribution in the vessel
wall has been assumed in the circumferential direction (Chuong and Fung, 1986; Takamizawa
and Hayashi, 1987; Zhang et al., 2005). Furthermore, in most previous experimental studies
using the inﬂation test, only outer diameter change associated with a pressure change was
measured, and the circumferential variation in the local mechanical properties of the vessel
was not usually characterized. Recent biomedical imaging studies, however, showed the
non-uniform circumferential behavior of the aorta during the cardiac cycle (Draney et al.,
2002, 2004), the asymmetric distension of the aortic wall (van Prehn et al., 2009a,b), and
an increased amount of the anterior aortic wall motion relative to the posterior (Goergen
et al., 2007). Therefore, it is essential to consider nonuniform deformation of the aorta in
the circumferential direction to obtain a physiologically more accurate mechanical model.
Nonetheless, few experimental studies have quantiﬁed the circumferential variation in the
mechanical behavior of the aorta.
The goal of this study is to investigate the spatial variations in the mechanical properties
of the thoracic aorta. In order to investigate the spatial variation, the aortic wall is classiﬁed

14

by the location of the aorta: the proximal and distal portions in the longitudinal direction;
and the anterior (A), left lateral (L), posterior (P), and right lateral (R) regions in the
circumferential direction. The speciﬁc aims of this study are:
• to develop a biaxial experimental apparatus to measure the local deformation of a
blood vessel
• to perform the inﬂation tests at ﬁxed longitudinal stretch ratios for the two longitudinal
portions and the four circumferential regions of the porcine thoracic aorta
• to develop a stress-strain analysis using 3D experimental data
• to investigate variations in stretch, stress, stiﬀness, and pressure-strain elastic modulus
and analyze statistically
• and to estimate material parameters using a constrained mixture model
This dissertation consists of six chapters. Chapter 2 describes the experimental apparatus designed and constructed for this study. It includes features of the extension-inﬂation
test device and a stereo vision system that allows us to track 3D motion of the aorta during the test. After that, it describes sample preparation and the protocol of the inﬂation
test. Camera calibration and image processing is addressed lastly. Chapter 3 focuses on
biomechanical analysis to determine the local mechanical properties of the thoracic aorta. It
introduces the kinematics in the curvilinear coordinate system, approximation of the surface,
and then equations of the circumferential stretch, stress, and stiﬀness in both longitudinal
and circumferential directions.
Chapter 4 presents a constitutive model to describe the mechanical response of the both
proximal and distal portions of the thoracic aorta during the inﬂation test. It includes the
parameter estimation study using a constrained mixture model.
Chapter 5 shows the results of experiments, stress-strain analysis, material parameters,
and the statistical analysis. The major ﬁnding of this study and discussion are ﬁnally
15

presented.
Lastly, Chapter 6 summarizes this dissertation and contribution of this study and then
brieﬂy discusses potential future study.

16

Chapter 2
EXPERIMENTAL METHODS
2.1

Experimental apparatus

An extension-inﬂation test system (Fig. 2.1 and Fig. 2.2) was developed to measure the biaxial deformation of a blood vessel. It consists of three main parts: motion, data acquisition,
and vision systems.
A polycarbonate test chamber is placed on the lower crosshead in the center of the test
device. Two moving crossheads are driven by a stepper motor in opposite directions, while
the center of the device remains ﬁxed. A specimen is connected to a syringe pump (KDS
210, KD Scientiﬁc, MA, USA) by a tube. Bidirectional infusion-withdrawal motion of the
syringe pump inﬂates and deﬂates the specimen.
A 34.47 kPa (5 psi) gauge pressure transducer (FPG2AT, Honeywell-Sensotec, OH, USA)
is installed to the tube, which is connected to the specimen. In this study, the transmural
pressure is deﬁned as the diﬀerence between internal pressure and external pressure of the
region where markers attached to the specimen. In Fig. 2.3, transmural pressure can be

17

DAQ
DAQ

Frame
Frame
Grabber
Grabber

A/D
A/D

PC
PC

Upper Crosshead
Upper Crosshead
Load
Cell
Aorta
Sample

CCD

Test
Chamber

Lower Crosshead
Lower Crosshead

Motor
Motor

Pressure Transducer
Syringe Pump

Fig. 2.1: Schematic diagram of the experimental set-up for the extension-inﬂation test.
calculated via
trans = P inter − P exter
PB
B
B

(2.1)

= PA − γh1 − γh2

(2.2)

= PA − γ (h1 + h2 )

(2.3)

= PA − γh.

(2.4)

where PA and PB are the pressure at the location of A and B, respectively, and γ is the
speciﬁc weight. The axial load applied to the specimen during the test is measured using a
4.54 kg (10 lb) load cell (Model 34, Tension/Compression, Honeywell-Sensotec, OH, USA),
18

Fig. 2.2: Schematics of the experimental device.
which is attached to the upper cannula of the specimen. The measurements are collected in
a computer by a data acquisition board at a 100 Hz sampling rate.
The stereo vision system, consisting of two CCD monochrome cameras (KPM2A, 768H
× 494V, Hitachi, Japan) with ﬁxed focal length lenses (HF25HA-1B, Fujinon, Japan), allows
the tracking of the 3D position of markers attached to the specimen. Two cameras are placed
at diﬀerent positions but at the same height, focusing on the central region of the test device
where the specimen is mounted (Fig. 2.4). The distance of two cameras is approximately
30 cm, and the angle between two cameras and the specimen is approximately 30 degrees.
The frame grabber obtains images of markers aﬃxed to the specimen using the two cameras
alternately at 5 frames per 2 seconds.
A computer program is designed to control the experimental system using Labview 8.2.

19

markers

P
o

h2
cannulas

PB
B
h
h1
specimen
pressure transducer
PA

Fig. 2.3: Schematics of the test chamber to show calculation of the transmural pressure
The speed and direction of the syringe pump are controlled by the automated Labview
program based on the pressure readings.
A few considerations were made when designing the test device. First, this test device
has a multiple-cannula mounting connector for testing several types of blood vessels. This
test device is able to test various sizes of specimens, whereas most of existing pressurization
testers have been used only for small vessels or short segments of blood vessels. Second,
this test device has a vertical mounting system. The vertical set-up is designed to avoid
the bending of a blood vessel caused by gravity, since the target length of an aorta sample
in this study was about 12 cm. In addition, the vertical mounting system allows us to use
multiple cameras around the specimen to record its deformation from multiple directions.
Third, its height is adjustable, while the center of the device is ﬁxed. Due to the irregular
length of blood vessels and diﬀerent longitudinal extension for testing, the adjustable height
of a test device was needed. Two parallel crossheads maintain the center of the specimen in
longitudinal direction, while the crossheads travel in opposite directions along two vertical
threads. Lastly, it has less incidence of and is easy to remove air bubbles. At the end of the

20

(a) Camera set-up with the specimen

specimen

30 cm

30º
c
camera

c
camera

(b) Schematics of camera set-up
Fig. 2.4: Two camera set-up
tube away from the specimen, saline solution is injected into the tube using a three-wayvalve. To remove air bubbles inside the specimen and the tube, upper cannulas (2.6) were

21

Start

Diameter of syringes
Flow rate of syringes
n=1
n = n+1
Syringe pump
infusion

No

Data
acquisition

Image
acquisition

Pg ≥ 160 mm Hg

Yes
Syringe pump
withdrawal

No

Pg ≤ 10 mm Hg

Yes
No

n=5

Yes
Stop

Fig. 2.5: Overview of the experimental process using the extension-inﬂation test device.

22

Fig. 2.6: Schematic drawing of an upper cannula connector.
specially designed. They have a tiny hole, which can be opened while the saline solution is
ﬁlling the specimen and can be closed after the removal of air for the test.

2.1.1

Calibration of sensors

The load cell has been calibrated by using weights with a string. Five diﬀerent weights in turn
are hung, and each voltage output from the data acquisition board is recorded, respectively.
A linear least squares regression is used to determine an equation of best-ﬁt line for the test
data, that is, the known weights and the voltage outputs of load cell corresponding to the
weights. The slope and y-intercept of the ﬁtting equation are used to convert voltage outputs
into axial loads applied to the specimen during the test. Fig. 2.7 (a) shows an example of
the least squares regression for calibrating the load cell.
Likewise, the pressure transducer has been calibrated by using a single-limb manometer
connected to the pressure transducer with a vinyl tube. Five diﬀerent column heights of
water in the manometer and their corresponding voltage outputs of the pressure transducer
are recorded. The room temperature is accounted for calculating the hydrostatic pressure for
23

0.7
Experimental data
Linear fitting line

Weight (kg)

0.5

0.3

y = 214.85 x − 0.0068477
0.1

−0.1
0

0.5

1

1.5

2

2.5

Load cell output (voltage)

3
−3

x 10

(a) Load cell
10

Gage Pressure (kPa)

Experimental data
Linear fitting line
8

6

4
y = 344.24 x + 0.14214
2

0
0

0.005

0.01

0.015

0.02

0.025

Pressure transducer output (voltage)
(b) Pressure transducer
Fig. 2.7: A representative example for the linear least squares regression of load cell data (a)
and pressure transducer data (b).
the case of each water column. A linear least squares regression is used again to determine a
relationship between the pressure and voltage output of the pressure transducer. The slope

24

and y-intercept of the ﬁtting equation are used to calculate the internal gauge pressure of
the specimen. Fig. 2.7 (b) shows an example of the least squares regression for calibrating
the pressure transducer.

2.2

Specimen preparation

(a)

(b)

(c)
thorcic aorta
microspheres
(d)
abdominal aorta
Fig. 2.8: (a) The porcine aorta sample from the thoracic to the abdominal, (b) the thoracic
aorta with surrounding tissues, (c) the thoracic aorta after removing surrounding tissues,
and (d) a cannulated aorta specimen with micro spheres attached on the wall.
Aortas from approximately six-month-old pigs (Fig. 2.8 (a)) were obtained from a local
slaughterhouse. They were stored at 5◦ C until testing within 24 hours after death. For
the inﬂation test, the peripheral, loose connective tissues are carefully removed from the
adventitia of the aorta, while aortic branches are kept intact (Fig. 2.8 (b) and (c)). The
proximal or distal portion of straight descending thoracic aorta, approximately 15 cm long,
is isolated. All aortic branches are ligated with nylon thread, and both ends of the isolated
aorta are connected to cannulas with care to ensure the aorta is not bent or twisted. Approximately 140 black spheres with a diameter of 550 µm, as markers, are attached to the
aorta along its circumference. They are attached to the aortic wall near either the ﬁrst to
third intercostal arteries for the proximal portion of the thoracic aorta or the ﬁfth to seventh
25

intercostal arteries for the distal portion, avoiding the regions of aortic branches (Fig. 2.8
(d)). Additional microspheres are attached to the proximal and distal regions of the specimen with 7 cm distance to obtain the reference measurements of longitudinal stretches. In
order to reduce the amount of glue, a pulled-glass micropipette is used. Then the specimen
is mounted vertically in the test chamber and the inside of the specimen and test chamber
are ﬁlled with 0.9% NaCl solution. The osmolarity is kept for all tests to limit the eﬀect of
osmolarity on the mechanical behavior of a vessel (Guo et al., 2007).

2.3

Test protocol

The unloaded reference length wherein the blood vessel is at zero longitudinal force and at
zero transmural pressure is recorded, ignoring the eﬀect of the weight of specimens. The
specimen is preconditioned longitudinally and circumferentially to obtain repeatable mechanical response of the blood vessel and to reduce possible stress concentration induced
during the preparation. The specimen is then elongated at stretch ratios of 1.35, 1.40, or
1.45 with respect to the reference length. The aorta has a pre-stretch in the longitudinal
direction at the in vivo state, and its in situ stretch ratio was estimated between 1.2 and
1.45 (Han and Fung, 1995). For each ﬁxed longitudinal stretch ratio, the inﬂation test is performed by pressurizing and depressurizing the specimen ﬁve times at approximately 1/110
Hz frequency over a transmural pressure range from 1.33 kPa to 21.33 kPa (10 - 160 mm
Hg) at room temperature. Digital images of markers aﬃxed to the specimen are collected
simultaneously with pressure and axial load measurements.
In this study, the aortic wall is classiﬁed into four regions in the circumferential direction.
The wall between two consecutive intercostal arteries is deﬁned as the posterior region, its
opposite across the lumen is deﬁned as the anterior region, and the two side regions between
the anterior and posterior regions are determined as the left and right lateral regions. For
the proximal thoracic aorta, the inﬂation test is repeated at the four circumferential regions

26

Y

X

X
ycam
x

xcam

C

(px, py)
Camera
center

Z
Principal axis

Image plane
Fig. 2.9: Schematics of a world point and its corresponding image point
(A, L, P, and R) after a 90-degree rotation of the specimen between one region and another.
For the distal thoracic aorta, the test is repeated at two circumferential regions (A, P) after
180-degree rotation. Great care is taken in ensuring no twisting of the specimen after the
rotation.
After the inﬂation test, two cross-sectional cuts of the specimen to obtain an aortic ring
are made at the test region where the markers are aﬃxed. Three repeated wall thickness
measurements per specimen are taken at every 45 degrees along the circumference using a
digital vernier caliper. Then, the aortic ring segments are ﬁxed in 10 % formal saline for
histology examination.
For this study, seven samples of the proximal thoracic aorta and four samples of the distal
thoracic aorta were used, and their experimental information is tabulated in Table C.1 in
Appendix C.

2.4
2.4.1

Marker tracking and 3D position reconstruction
Calibration of cameras

A 3D calibration jig, which displays array of dots with known spacing, was used. A ﬁnite
projective camera model have been employed to determine a camera matrix. Projection

27

Y

Z

Camera 2
X

Calibration jig

Camera 1

Fig. 2.10: Calibration jig and camera set-up
under this model is a linear mapping in the form of x = MX using homogeneous coordinates,
where a camera matrix M describes how a 3D world point X = (X, Y, Z)T projects to a 2D
pixel point x = (x, y)T in the image coordinate. In brief, the camera matrix is determined
via three steps. First, a 3D world point (X, Y, Z) is transformed to a point in the camera
reference frame by rotation and translation. Next, the point is projected onto the image
plane at (xcam , ycam ). Lastly, the point (xcam , ycam ) on the image plane is mapped into the
pixel coordinates in an image. (The details are introduced in Appendix A.) Therefore, the

28

homogeneous linear mapping can be written as


 

 X 




 sy 





M11 M12 M13 M14  
 








 
 Y 
 M

=
sx
M22 M23 M24 
  21


 


 Z 








 s 


M31 M32 M33
1



 1 


(2.5)

where s is a scaling factor and Mij is an element of the calibration matrix.
In equation (2.5), there are 11 unknowns, and more than 6 pairs of 3D world points and
their corresponding pixel points in the image are needed. World points are obtained from
the calibration jig (Fig. 2.10) and the centers of their pixels in images are extracted through
image processing of the images. Using n-number of pixel points corresponding to the world
points of the calibration jig, equation (2.5) can be rewritten in the form of u = Ab. Here,
u represents a vector of an image pixel point of x-axis and y-axis alternately, A represents
n × 11 matrix, and b represents all elements of the calibration matrix in a vector. In order
to determine the optimized parameter b of the calibration matrix, the pseudo inverse (least
squares) calculation has been applied. The parameters b is obtained by b = (AT A)−1 AT u
via minimizing

2.4.2

Ab − u 2 .

Image processing

Two CCD cameras obtain serial digital images of the specimen during the inﬂation test. The
center of each marker is extracted via the following steps. Firstly, the image mask from the
entire image, that is a small region of markers on the specimen, is deﬁned and the masked
region is extracted. Next, the manual threshold type is selected and its range for the masked
image is set up, when thresholded markers are displayed clearly from the background. If
dark objects are being looked for, markers are shown as dark objects and the aortic wall
is shown relatively brighter. After that, border objects of the masked image are removed.
Then, the center of mass for markers in x and y directions are obtained from each images
29

(a)

(b)

(c)

Fig. 2.11: An example of the processing to extract the position of markers in pixel. (a) the
entire image, (b) the masked image, and (c) thresholded image
via the particle analysis, which is a built-in function of NI Vision Assistant.

2.4.3

Marker tracking

Using the 2D position data of markers in pixels, the motion of markers is traced over the
inﬂation test. However, the total number of particles in images is not the same as the
number of markers. Thus, Matlab programming, so-called the marker tracking algorithm,
has been developed to track markers in which we are interested. The algorithm consists of the
followings steps. First, some markers of interest are selected from the ﬁrst image and saved
with labels. The positions of markers ((xi , yi ) in the ﬁrst image, where i denotes the label
of a marker) are compared with the new positions of markers in next image ((xj , yj ), where
j denotes the label of a marker). Second, the distance between markers from two images
(| (xi , yi ) − (xj , yj ) |) are calculated. If a marker of i from the ﬁrst image had a minimum
distance with a marker of j from the next image, we assume that those two markers are the
same ones and their distance and labels are recorded. If more than two markers of j in the
new image have a same marker of i as a distance between two markers from two consecutive
images is minimum, the label of i is repeated. Then a marker of j with a smaller distance
from a marker of i is taken as new position of the marker of i in the previous image. The
other marker of j which has the same marker of i as the minimum distance is discarded.

30

15

Y

10
5
5

0
-45

0

-40
-35
(a)

X

(b)

Z

-5
(c)

(unit: mm)

Fig. 2.12: Stereo images from left side camera (a) and right side camera (b), and the reconstructed marker positions in 3D (c)
Therefore, the number of new markers remains as the same number of markers from the ﬁrst
image. However, some of markers from the ﬁrst image may not be shown later. In this case,
the minimum distance of the markers may be bigger than the averaged minimum distance
between two consecutive images for all markers. If some markers had a much bigger distance
than the averaged minimum distance, they are discarded and the same steps are repeated to
ﬁnd the minimum distance between an image and the next image for the markers. Finally,
the new positions of markers are determined. The same process is repeated for the next
image with the updated positions of markers until the positions of markers in the last image
are determined.

2.4.4

3D position reconstruction

Once camera matrices are determined, 3D positions of markers can be obtained using pairs
of 2D stereo images taken from both cameras. Linear mapping derived from equation (2.5)
for each camera yields four equations with three unknowns (X, Y, Z), which represents a 3D
world point corresponding to a pair of image points (x1 , y 1 ) and (x2 , y 2 ) for both cameras,

31

as




















1 
y 1 − M14 





1 − M1 

x
24



1
1
 M11 − M31 y1

 M 1 − M 1 x1
 21
31

1
1
M12 − M32 y1




1
1
M13 − M33 y1  

 X 




1 − M1 x  


M23
1 
33

1
1
M22 − M32 x1
=
2 
2
2
2
2
2
2
y 2 − M14   M11 − M31 y2 M12 − M32 y2 M13 − M33 y2
 
 


2 
2
2
2
2
2
2
x2 − M24 
M21 − M31 x2 M22 − M32 x2 M23 − M33 x2


Y








 Z 

(2.6)

where superscript 1 and 2 denote two diﬀerent cameras. Likewise the process of determining
a camera matrix, the pseudo inverse (least squares) calculation is employed to determine the
unknown world point (X, Y, Z). Fig. 2.12 shows the stereo images and the reconstructed 3D
positions of markers.

2.5

Histology sample preparation

At the mid-section of the sample where markers are attached, two transverse cuts of the
aorta are made. The aortic ring is ﬁxed in a 10 % formalin solution at room temperature
for 3 days and is embedded in paraﬃn for approximately 24 hours. Then it is sectioned at 5
µm thickness and stained with hematoxylin and eosin (H&E), Verhoeﬀ-Van Gieson (VVG),
and picrosirius red.

32

Chapter 3
BIOMECHANICAL ANALYSIS
3.1

Kinematics in curvilinear coordinate systems

In cardiovascular mechanics, curvilinear coordinates, such as cylindrical polar coordinate and
spherical polar coordinate, are often used to describe the geometrical shape of a blood vessel
better. For a curvilinear coordinate system, an orthogonal natural basis, called covariant base
vectors, are deﬁned as Gi in the reference conﬁguration and gi in the deformed conﬁguration.
Gi and gi need not to be of unit magnitude, and in general they depend on position.
Let us consider db as a diﬀerential vector in space. It can be written as
db = dz i gi = dz 1 g1 + dz 2 g2 + dz 3 g3

(3.1)

with respect to the deformed curvilinear coordinates z i .
A reciprocal (contravariant) basis vectors Gi in the reference conﬁguration and gi in the
deformed conﬁguration are determined to satisfy the orthogonality condition via
j

i
Gi · Gj = δj ,

(3.2)

j

i
gi · gj = δj

(3.3)

Gi · Gj = δi ,
gi · gj = δi ,
33

j

j

where δi = 1 if i = j and δi = 0 if i = j.
A diﬀerential vector db in equation (3.1) can be written using contravariant basis vector
g i as
db = dzi gi = dz1 g1 + dz2 g2 + dz3 g3

(3.4)

with respect to the deformed curvilinear coordinate zi .
In the Cartesian coordinate system, db = dxi ei , where the normal basis ei is written as
ei =

∂b
∂xi

(3.5)

Likewise, in curvilinear coordinate system, the vector db is expressed as db = dz i gi , and
orthogonal basis vector gi is
gi =

∂b
.
∂z i

(3.6)

Using a chain rule, gi can be rewritten as
∂b
∂b ∂xk
∂xk
gi = i =
=
e
∂z
∂z i k
∂xk ∂z i

(3.7)

∂b
∂b ∂z α
∂z α
= α i =
gα .
∂z ∂z
∂z i
∂z i

(3.8)

gi =

When the covariant basis vector gi are orthogonal at each point, the orthonormal vector ˆ
e
is determined by the division of the covariant basis vector by its magnitude, such that

ˆi =
e

gi
(no summation on i)
| gi |

(3.9)

where ˆi does not change with position, and
e
| gi |=

√

gi · gi =

√

gii (no summation).

(3.10)

Basic manipulations of components for the curvilinear coordinate system are analogous
34

to those for the Cartesian coordinate system. However, the diﬀerentiation of a position
vector with respect to the curvilinear coordinate system yields two terms. For example, a
diﬀerential vector dv can be rewritten as
∂v i
∂v
∂
∂gi
= m v i gi = m gi + v i m ,
∂z m
∂z
∂z
∂z

(3.11)

here the second term can be
∂
∂gi
m = ∂z m
∂z

∂xk
e
∂z i k

=

∂ 2 xk
e .
∂z m ∂z i k

(3.12)

After substituting equation (3.8) into the second term in equation (3.11)
∂gi
∂ 2 xk ∂z j
= m i k gj .
∂z m
∂z ∂z ∂x

(3.13)

The equation of the balance of linear momentum is div(t) + ρb = ρa, where t is the
Cauchy stress tensor, b is the body force vector, and a is the acceleration vector. The
divergence of a tensor t in curvilinear coordinate system is given by
div(t) = gm ·

∂gj
∂gi
∂tij
ij
m gi ⊗ gj + tij ∂z m ⊗ gj + t gi ⊗ ∂z m
∂z

.

(3.14)

The deformation gradient tensor F is the gradient of mapping function to describe the
motion of a continuum. When a position vector for a material point in the undeformed
conﬁguration X has displacement, the material point has new position x in the deformed
conﬁguration. For an inﬁnitesimal change of the position vector in the curvilinear coordinate
system, the deformation gradient F is the fundamental measure of the deformation as

F=

∂x
∂xi
i
=
gi ⊗ GJ = FJ gi ⊗ GJ .
∂X
∂X J

35

(3.15)

i
i
where FJ = ∂xJ . Then the determinant of the deformation gradient J is given by
∂X

J = detF =

g1 · (g2 × g3 )
.
G1 · (G2 × G3 )

(3.16)

The covariant component of C is
i
C = FA gi ⊗ GA

T

j

· FB gj ⊗ GB

j

i
= FA FB gij GA ⊗ GB

(3.17)

= CAB GA ⊗ GB

3.2

Numerical analysis and approximation

The cylindrical coordinate is often used in vascular mechanics to describe the positions of
material particles. It is appropriate to solve blood vessel problems since the base vectors of
the curvilinear coordinate need not be independent of position. In this study, the positions
of markers are determined in the Cartesian coordinate from the image processing, and then
they are converted to the cylindrical polar coordinate.

3.2.1

Coordinate transformation

The 3D positions of markers aﬃxed to the specimen are obtained before and during the
inﬂation test. The load-free state is regarded as the reference conﬁguration, and the deformed
state during the test is regarded as the deformed conﬁguration. Because of the cylindrical
shape of the vessel wall, the position of a marker (X, Y, Z) in the Cartesian coordinate system
is converted to variables (Θ, S, R) in the cylindrical polar coordinate system via

X − Xo = R cos Θ,

Y − Yo = R sin Θ,

36

Z=S

(3.18)

x (θ (Θ , S ), s (Θ, S ), r (Θ, S ) )

X (Θ (Θ), S (S ), R (Θ, S ) )

F

x

S

X
S
Θ

Θ
z

Z

y

X

Y

x

Fig. 3.1: The surfaces in the reference conﬁguration and the deformed conﬁguration are
approximated by continuous functions of two variables, Θ and S, convected to the surface.
The deformation gradient of the surface corresponding to the mapping is denoted by F.
where {Xo , Yo } is the center-line approximated by the marker positions. Likewise, the position (x, y, z) in the deformed conﬁguration is converted to variables (θ, s, r) via

x − xo = r cos θ,

y − yo = r sin θ,

z=s

(3.19)

The outer surfaces of the vessel wall are parameterized using two variables Θ and S, which
provide a convected curvilinear coordinate system for the surface during the deformation.
The normalized variables S and Θ can be obtained via

S=

2(S − Smin )
−1
Smax − Smin

(3.20)

Θ=

2(Θ − Θmin )
− 1,
Θmax − Θmin

(3.21)

and then −1 ≤ S ≤ 1 and −1 ≤ Θ ≤ 1.
All variables in the reference and deformed conﬁgurations are expressed as functions of

37

Θ and S, as
Θ = Θ(Θ),

S = S(S),

ˆ
θ = θ(Θ, S),

3.2.2

s = s(Θ, S),
ˆ

ˆ
R = R(Θ, S),
r = r(Θ, S).
ˆ

(3.22)
(3.23)

Deformation gradient

The covariant base vectors of the surface, Gi in the reference conﬁguration and gi in the
deformed conﬁguration, are deﬁned as

Gi =

∂X
,
∂Σi

gi =

∂x
∂Σi

(3.24)

where Σi are two variables convected to the surface Θ and S.
In a convected curvilinear coordinate system, the 2D deformation gradient F∗ of the
surface is given by
F∗ = gi ⊗ Gi

(3.25)

where ⊗ denotes the tensor product. Since the covariant and contravariant base vectors are
ˆ
generally not orthonormal, local orthonormal base vectors Ei in the reference conﬁguration
and ˆi in the current conﬁguration are determined at each point as
e
G1
ˆ
E1 =
,
| G1 |
ˆ1 =
e

g1
,
| g1 |

ˆ ˆ
G2 − (G2 · E1 )E1
ˆ
E2 =
ˆ ˆ
| G2 − (G2 · E1 )E1 |
ˆ2 =
e

g2 − (g2 · ˆ1 )ˆ1
e e
| g2 − (g2 · ˆ1 )ˆ1 |
e e

(3.26)

(3.27)

where the subscripts 1 and 2 represent the circumferential and longitudinal directions, respectively.
The covariant and contravariant base vectors thus can be expressed in terms of the local
ˆ
orthonormal base vectors Ei or ˆi . Finally, the 2D deformation gradient F with respect to
e
ˆ
ˆ
the local orthonormal base vector can be written as Fpq = ep · FEq .
38

3.2.3

Approximation of the surface

The surfaces of the vessel wall are globally parameterized using two variables S and Θ, which
are convected to the two-dimensional curvilinear coordinate system. The surfaces of vessel
wall in both conﬁgurations are approximated using Legendre polynomial functions, which
make two variables orthogonal to each other.
ˆ ˆ ˆ
The four functions (R, θ, s, and r) in equation (3.22) and (3.23) are approximated by
ˆ
the ﬁnite series of continuous base functions of Θ and S

ˆ ˆˆ ˆ
R, θ, s, r =

nU

αj φj (Θ, S)

(3.28)

j=1

where nU is the number of degrees of the base function and αj is a parameter. The base
function φj is deﬁned as a combination of Legendre polynomials of Θ and that of S, that
is, φj (Θ, S) = ψk (Θ)ψl (S) where ψk and ψl represent k th and lth degrees of Legendre
polynomials, respectively, for each variable. For example, ψ0 = 1 and nth degree (n=1, 2, 3,
... ) of Legendre polynomials of a variable x are given by
n
1 dn
2
ψn (x) = n
.
n x −1
2 n! dx

(3.29)

In this study, 3rd and 4th degrees of Legendre polynomials are used in the longitudinal and
circumferential directions, respectively.
In our pilot study, the edge of the surface domain has a large variation in stretch distribution, and so the accuracy of the Legendre polynomial approximation decreased near
the edge of the domain. Once the coordinate was transformed to the cylindrical coordinate
and the positions of markers were approximated, diﬀerent areas of the central region of the
domain were tested for determining the optimal area of the central region, which had a high
accuracy of the approximation. The stretch distributions of a silicon tube, which has relatively uniform stretch distribution, with diﬀerent areas of central region were compared to

39

each other. The central region of domain with -0.7 < Θ < 0.7 and -0.7 < S < 0.7 showed
nearly uniform stretch distribution, so that the reduced domain were used for stress-strain
analysis.

3.2.4

Least squares estimation

Let Φexp (Θn , S n ) be the experimental data of marker positions at a point (Θn , S n ) for
n = 1, 2, ..., nD , where nD is the number of markers. We used 36 ± 6 markers in this study.
Parameters αi for each approximation function can be estimated by the least squares method
with an optimization function E


nD  nU

E=
n=1



j=1

2

exp
αj φj Θn , S n − Φn (Θn , S n )
.


(3.30)

Taking partial derivatives of E in equation (3.30) with respect to αi


∂E
=2
∂αi

nD  nU
n=1



exp

αj φj Θn , S n − Φn



Θn , S n

j=1

φ Θn , S n .
 i

(3.31)

At the minimum, ∂E/∂αi = 0, which leads to the linear equation Kα = f , where
α = {α1 , ..., αnU }T and

nD

Kij =

φi (Θn , S n )φj (Θn , S n )

(3.32)

n=1
nD

fi =

exp

Φn (Θn , S n )φi (Θn , S n ).

(3.33)

n=1

The ﬁtting parameters α for the Legendre polynomials can be obtained by solving the above
linear equations using the experimental data.
α = LT L

−1

40

LT Φexp

(3.34)

where L = φ1 , φ2 , ..., φN U

3.3

T

.

Thickness of the aortic wall

In the load-free state, the aortic wall thickness H0 at a certain position along the circumference is approximated by a curve ﬁt to the measurements of aortic wall thickness at every 45
degrees, using the Fourier series

H0 (θ) = a0 + a1 cos(θ) + a2 sin(θ) + a3 cos(2θ) + a4 sin(2θ).

(3.35)

The aortic wall thickness during the inﬂation test in the deformed conﬁguration can be
calculated by the incompressibility condition (Carew et al., 1968), which means that the
determinant of the deformation gradient is unity (det F=1). In 3D, the deformation gradient
can be written as





 F11 F12 0

F =  F21 F22 0


0
0 λ3







(3.36)

where λ3 = h/H0 , h being the deformed wall thickness. The 1 and 2 directions are in-plane
(i.e., circumferential and longitudinal directions) and the 3 direction out-of-plane (radial
direction). From the incompressibility condition, the wall thickness during the deformation
is given by
h=

3.4

H0
.
F11 F22 − F12 F21

(3.37)

Curvature of the aortic wall

The principal curvature of the aortic wall is obtained by the ﬁrst and second fundamental
forms of a surface in diﬀerential geometry. We use the same parameters in the fundamental
forms of a surface as parameters in our approximation, that is, S and Θ.
41

The ﬁrst fundamental form of a surface is
2

2

I = EdS + 2F dSdΘ + GdΘ = dx · dx

where
E=
F =
G=

∂x 2
+
∂S

∂y 2
+
∂S

∂z 2
∂S

(3.38)

(3.39)

∂y ∂y
∂z ∂z
∂x ∂x
+
+
∂S ∂Θ ∂S ∂Θ ∂S ∂Θ

(3.40)

∂x 2
+
∂Θ

(3.41)

∂y 2
+
∂Θ

∂z 2
.
∂Θ

The second fundamental forms of a surface are
2
2
ˆ
II = LdS + 2M dSdΘ + N dΘ = −dx · dN

where

∂2x
∂S 2
1
∂x
L= √
EG − F 2 ∂S
∂x
∂Θ
∂2x
∂S∂Θ
1
∂x
M=√
2
∂S
EG − F
∂x
∂Θ
∂2x
∂Θ2
1
∂x
N=√
EG − F 2 ∂S
∂x
∂Θ

∂2y
∂S 2
∂y
∂S
∂y
∂Θ
∂2y
∂S∂Θ
∂y
∂S
∂y
∂Θ
∂2y
∂Θ2
∂y
∂S
∂y
∂Θ

∂2z
∂S 2
∂z
∂S
∂z
∂Θ
∂2z
∂S∂Θ
∂z
∂S
∂z
∂Θ
∂2z
∂Θ2
∂z
∂S
∂z
∂Θ

(3.42)

(3.43)

(3.44)

(3.45)

ˆ
and the unit surface normal vector N is given by
g × g2
ˆ
N= 1
.
|g1 × g2 |
42

(3.46)

The detail derivatives of diﬀerentiations are introduced in Appendix B.
Let κ be a normal curvature and it has two extremal values, which are the principal
curvature, κn (Struik, 1988),
2

II
LdΘ + 2M dΘdS + N dS 2
κ=
=
.
2
2
I
EdΘ + 2F dΘdS + GdS

(3.47)

EG − F 2 κ2 − (EN + GL − 2M F ) κ + LN − M 2 = 0.

(3.48)

Therefore, we have

The two extreme values of κn in equation (3.47) are two principal curvatures κ1 and κ2 .
We assumed that the outer radii of curvatures of the aortic wall are

r1 =

1
,
κ1

r2 =

1
.
κ2

(3.49)

Here the sign convention is that the radius of curvature is positive if the center of curvature
is on the r-negative side of the surface. The detail derivatives of the parameters are included
in Appendix B.

3.5

Stress and strain analysis

The stretch λi is calculated at each point by
λi =

ˆ
ˆ
Ei · (F∗ T F∗ )Ei ( i=1, 2)

(3.50)

where subscript 1 and 2 denote the circumferential and the longitudinal directions, respectively. In the classical method, the relationship between the transmural pressure and the
tension is referred to as Laplace’s law, that is T = P r , where T is the tension in the walls,
h
P is the transmural pressure across the vessel wall, r is the radius of the vessel, and h is the
43

FP
F2

F1

F1

F2

r1

r2

Fig. 3.2: Free body diagram of membrane tissue.
thickness of the wall. The higher pressure diﬀerence the more tension, the thicker wall the
less tension, and the bigger radius the more tension there is. However, previous studies using
this equation do not take into account changes in radius or tension along the circumference.
In this study, the radius of the aorta depends on the circumferential position. Thus, the
membrane hypothesis was applied to the aortic tissue to determine stress. The eﬀect of
bending throughout the vascular wall was so small that it was ignored when the aorta was
loaded.
For a small section of the aortic segment, depicted in Fig. 3.2, the relationship of the
transmural pressure and the tension was derived in the similar way with the Laplace law.
Three forces are acting upon the wall: one due to pressure (FP ) acting radially and the
others due to tensions (F1 and F2 ) acting tangentially in two principal directions (S1 and
S2). From the equilibrium of forces,

FP = F1 + F2 .

44

(3.51)

From the shell membrane equilibrium equation,
T1 T2
+
=P
r1
r2

(3.52)

where T1 and T2 are membrane principal stress resultants per unit length in circumferential
and longitudinal direction, respectively. Here, the local radius of curvature of the membrane
r1 and r2 in circumferential and longitudinal direction, respectively, are calculated by the
classical diﬀerential geometry of the surface as explained in the previous section.
For a cylindrical polar model, the mean radius of the curvature in the longitudinal direction is much larger than that in the circumferential direction (r2

r1 ). Therefore,

equation (3.52) can be rewritten in terms of the mean circumferential Cauchy stress σ1 , that
is T1 = σ1 /h. The mean circumferential stress σ1 for each circumferential region can be
estimated by
σ1 =

P (r1 − h)
.
h

(3.53)

Here, we assume that stress σ1 is the circumferential stress of the midwall.
Let us consider stress in the longitudinal direction. Bending of the aorta in vivo during
the cardiac cycle is negligible (Choi et al., 2009) and we assume that the blood vessel has no
bending moment applied at the end during the inﬂation test. During the inﬂation test, the
blood vessel wall thickness (h) depends on its circumferential location (θ). The estimation of
the aortic wall thickness was addressed in previous section (equation (3.37)). The bending
moment M on the vessel wall
2π

M=

0

(σ2 x )rm hdθ = 0

(3.54)

where σ2 is the longitudinal stress on the vessel wall, and rm is the mean radius of curvature
at the midwall. The distance x = x − x as shown in Fig. 3.3 (b), where x is the centroid.
¯
¯
Because of the asymmetry, the centroid of area is not same of middle of aortic ring. The

45

t2
σzz
ri

ro
Fz

P
cannula
σzz
t2
(a) Free body diagram

h(θ)

dA

rm

ds

θ

x
x

x’

(b) Cross-section
Fig. 3.3: (a) Free body diagram of a blood vessel sample during the inﬂation test and (b)
the cross-section of the aortic wall.
centroid, x, is calculated by
¯

x=
¯

1 2π
1 2π
2
xdA =
cos θrm hdθ
A 0
A 0

(3.55)

where x = r cos θ.
Because of the asymmetry of the aortic wall thickness, the cross-sectional area of the wall
(A) and the centroid of area (¯) can be calculated as functions of θ. The piecewise area of
x
the aortic wall is dA = hds, where s is the arc of circular shape of the aortic ring. If θ is
46

very small, ds = rm dθ. The total cross-sectional area of the aortic ring, A, is given by
2π

A=

0

2π

dA =

0

rm hdθ

(3.56)

. Let the longitudinal stress assume σ2 = ax + b, where a and b are constants. The forces
in longitudinal direction during the inﬂation test reach the equilibrium and thus
2π
0

σ2 hrm dθ = P π(r1 − h)2 + Fz .

(3.57)

where Fz is axial force to be applied the specimen during the inﬂation test. In equation (3.54)
and equation (3.57), σ2 is substituted in terms of a and b by the assumption σ2 = ax + b.
Then these two equations are arranged in matrix form as
 


2π 2
 0 x hrm dθ

2π
0 x hrm dθ






  
2π

  
0
0 x hrm dθ  a
.
=

2π
2

  
hrm dθ  b  P πri + Fz 
0

(3.58)

Two constants a and b can be estimated using Gaussian quadrature (see Appendix B.2), and
the longitudinal stress σ2 is determined.
In order to estimate material stiﬀness, the circumferential stiﬀness k1 (Pi ) is calculated
by ﬁtting a 6th order of polynomial function to the stress-stretch curve. That is, the circumferential stiﬀness in this study is deﬁned as a tangent of the mean stress-stretch curve at the
transmural pressure Pi such that
k1 (Pi ) =

∂σ1
σ (λ +
|λ (P ) = 1 1
1 i
∂λ1

λ1 ) − σ1 (λ1 −
2 λ1

λ1 )

.

(3.59)

To estimate the structural stiﬀness during the cardiac cycle, the pressure-strain elastic
modulus Ep can also be calculated. Ep was introduced by Peterson et al. (1960) and after that
many others adopted this pressure-diameter relation. The pressure-strain elastic modulus
represents distensibility of a blood vessel by ∆P /∆ , where ∆P is the maximum pulse
47

pressure between the diastolic state and the systolic state and ∆ is the change in strain
during the same period. The pressure-strain elastic modulus resulted from both wall stiﬀness
and the geometry of the blood vessel. Originally it is expressed as
Psys − Pdia
Dsys − Ddia /Ddia

Ep =

(3.60)

where subscripts sys and dia denote the systolic and diastolic conditions, respectively, and D
is the diameter of lumen (Feigl et al., 1963). In this study, the denominator of the equation
(3.60) was divided by Do , which is the diameter of the lumen at the load-free state. Thus,
equation (3.60) can be rewritten as
Psys − Pdia
λsys − λdia /λdia

Ep =

(3.61)

The systolic pressure and diastolic pressure have been assumed Psys =15.20 kPa (114 mm
Hg) and Pdia =10.13 kPa (76 mm Hg), respectively. These pressures were determined when
the external pressure by the surrounding organ and tissue was considered as 5% of the
transmural pressure (Zhang et al., 2005).

3.6

Statistical analysis

The mean of the computed values, such as stretch, stress, and stiﬀness, for the four circumferential regions of the aorta are compared statistically. The mean of a data set xi (i = 1, 2, · · · , N )
is deﬁned as
x=
¯

1
N

N

xi

(3.62)

i=1

and the standard deviation (SD) is computed via

SD =

N
¯2
i=1 (xi − x)

N −1

48

(3.63)

All aorta specimens are assumed to be independent statistical samples. Due to an increased chance of an error in doing multiple two-sample t-tests, ANOVA is suitable in comparing more than two. Hence, the signiﬁcant diﬀerence among the four regions and each
pairs of regions are evaluated by using one-way repeated measures ANOVA, which generalizes t-test to more than two groups, with Sidak adjustment. Also, the signiﬁcant diﬀerence
between two longitudinal portions for each region is evaluated by two-sample t-test. The
diﬀerence is considered signiﬁcant when p < 0.05.

49

Chapter 4
CONSTITUTIVE RELATIONS AND
PARAMETERS
4.1

Constitutive relations

The energy equation is
ρo
where ρo is mass density,

d
dF
= PT :
− 0 · q0 + ρo g
dt
dt

(4.1)

the internal energy density deﬁned per unit mass, P the ﬁrst

Piola-Kirchhoﬀ stress tensor,

∂()
0 the del operator ( 0 = ∂X ), q0 the heat ﬂux vector, and

g a heat addition deﬁned per unit mass. The internal energy is related to the Helmholtz
potential ψ and entropy η, that is,

= ψ + ηT , where T is the temperature.

For an isothermal process with no heat transfer (i.e. T =costant, q0 = 0), the equation
(4.1) yields
−ρ0

dψ
+ PT
dF

:

dF
=0
dt

(4.2)

and the ﬁrst Piola-Kirchhoﬀ stress tensor P can be
PT = ρo

∂ψ
∂ψ
←→ P = ρo T .
∂F
∂F

50

(4.3)

The Helmholtz potential ψ and the strain-energy W are related in an isothermal process,
that is, ρo ψ (F) = W (F). From the material frame indiﬀerence
∂ψ
∂ψ
=2
· FT
T
∂C
∂F

(4.4)

where C = CT and C = FT · F. From equation (4.3) and (4.4), the ﬁrst Piola-Kirchhoﬀ
stress tensor
P=2

∂W
· FT .
∂C

(4.5)

The alternate constitutive relations for the Cauchy stress can be written as (using ˆ =
t
1
J F · P)

ˆ = 2 F · ∂W · FT
t
J
∂C

(4.6)

which is deﬁned as the actual acting force on a body over the current area. However,
equation (4.6) does not take into account the incompressibility constraint (J = ρ0 /ρ = 1).
After applying the constraint, equation (4.6) is rewritten as

t = −pI + 2F ·

∂W
· FT
∂C

(4.7)

where p is a Lagrangian multiplier, which depends on position and time. Each component
of equation (4.7) is
ˆ
t11 = −p + t11
ˆ
t22 = −p + t22

(4.8)

ˆ
t33 = −p + t33 .
The strain energy function W of vascular tissue is a function of the right Cauchy-Green
deformation tensor as
W = W (Cij ),

wherei, j = 1, 2, 3.
51

(4.9)

Here 1, 2, and 3 denote the circumferential, axial, and radial directions, respectively.
For a general hyperelastic behavior, equation (4.6) can be written with an index notation
as
2
∂W
ˆ
tij = FiA
F ,
J
∂CAB jB

where i, j, A, B = 1, 2, 3.

(4.10)

During the inﬂation test, the deformation gradient is assumed to be




 F11 F12 0

F =  F21 F22 0


0
0 λ3





  λ1 0 0
 
= 0 λ
2 0
 
 
0 0 λ3







(4.11)

where λ3 = h/H0 based on the incompressibility assumption. The right Cauchy green tensor
C becomes


2
 λ1


C= 0


0


0

0 

λ2 0  .

2

2
0 λ3

(4.12)

Based on the thin-wall analysis (Holzapfel et al., 2000), let the stress in the radial direction
ˆ
t33 be zero. Then the Lagrangian multiplier p = t33 . Consequently, equation (4.8) is
rewritten as
2
∂W
2
∂W
ˆ
ˆ
t11 = t11 − t33 = F1A
F1B − F3A
F
J
∂CAB
J
∂CAB 3B
2
∂W
2
∂W
ˆ
ˆ
t22 = t22 − t33 = F2A
F2B − F3A
F
J
∂CAB
J
∂CAB 3B
t33 = 0

52

(4.13)

4.1.1

Constitutive models

Mixture model
In this study, the constitutive model based on a constrained mixture approach (Baek et al.,
2006; Humphrey and Rajagopal, 2002) has been adopted to describe the mechanical properties of the aorta. Let the natural conﬁguration represent the stress-free state of a constituent
of aortic wall. Deformation is considered in two steps: ﬁrst, mapping from the natural conﬁguration (stress-free state) of each constituent to the reference conﬁguration (in vivo state),
second mapping from the reference conﬁguration to the deformed conﬁguration.
The arterial wall is assumed to be a mixture of the constituent of i, such as elastin (e),
multiple collagen families (1, · · · , k, · · · , 4), and smooth muscle (m). The strain energy W
of the mixture per unit volume is
W = Σ i ν i W i = ν e W e + Σk ν k W k + ν m W m

(4.14)

where ν i are the unit reference volume for each constituent of i. It has been assumed that
there is no active tone presented during the inﬂation test in this study.
For the elastin, a mapping from the natural conﬁguration to the reference conﬁguration
ˆ
(in vivo state) Ge is deﬁned as
ˆ ˆ ˆ
ˆ
Ge = diag Ge , Ge , Ge
1
2
3

(4.15)

such that the mapping of the elastin from the natural conﬁguration to the deformed conﬁguration
ˆ
Fe = F · Ge
n

(4.16)

and therefore
ˆ
Ce = (Fe )T · Fe = Ge
n
n
n

53

T

ˆ
· C · Ge .

(4.17)

The Cauchy stress ˆ of elastin can be rewritten in terms of Fe and Ce as
t
n
n
2F

∂W e T
∂W e T
F = 2FiA e FBj
∂Ce
∂CAB
∂W e ˆ eT
ˆ
= 2FiA Ge
GlB
Ak ∂C e
n[kl]
e
= 2Fn[ik]

= 2Fe
n

FBj T

∂W e eT
F
e
∂Cn[kl] n[lj]

(4.18)

∂W e eT
F
∂Ce n
n

Strain energy of elastin W e per unit reference volume are given by
c
e
e
e
W e (Ce ) = 1 Cn [11] + Cn [22] + Cn [33] − 3
n
2

(4.19)

e
e
e
e
where Cn [11] ,Cn [22] , and Cn [33] are components of Cn ,

and
e
e
2Fn[ik] Fn[jl]

∂W e
ˆ
= c1 Ge λi
i
e
∂Cn[kl]

ˆ
Ge λj
j

∂W e
.
e
∂Cn[kl]

(4.20)

Each component of equation (4.20) is
e
2Fn[i1]

2
∂W e eT
∂W e eT
e
ˆ
Fn[1j] = 2Fn[11] e
Fn[11] = c1 Ge λ1
1
e
∂Cn [11]
∂Cn [11]

e
2Fn[i2]

2
∂W e eT
∂W e eT
e
ˆ
Fn[2j] = 2Fn[22] e
Fn[22] = c1 Ge λ2
2
e
∂Cn[22]
∂Cn[22]

e
2Fn[i3]

2
∂W e eT
∂W e eT
e
ˆ
Fn[3j] = 2Fn[33] e
Fn[33] = c1 Ge λ3 .
3
e
∂Cn [33]
∂Cn [33]

(4.21)

ˆ
In the radial direction, the homeostatic tensor Ge and stretch of elastin λ3 can be substituted
3
by the circumferential and axial components from the incompressibility constraint, such that
ˆ
Ge λ3 =
3

1
ˆ ˆ
Ge Ge λ1 λ2
1 2

54

.

(4.22)

For the collagen ﬁber, the stretch of k th collagen ﬁber family in the natural conﬁguration
(stress-free state) from the deformed conﬁguration is deﬁned as
λk = Gc λk
n
h

(4.23)

where Gc is homeostatic stretch. The stretch of the k th collagen ﬁber is
h
λk =

λ1 sin αk

2

2

+ λ2 cos αk ,

(4.24)

where αk is the orientation of the k th collagen ﬁber family. The Cauchy stress of the collagen
ﬁber family

2

FiA FjB

∂W k
2 ∂W k
2 ∂W k ∂λk
n
= λk
= λk
,
i
i
k
k2
k 2 ∂λk 2
∂CAB
∂λn
∂λi
i

(4.25)

where the strain energy of collagen W k per unit reference volume is given by
2
c
2
W k λk = 2 exp c3 λk − 1
n
n
4c3

−1

(4.26)

and therefore
∂W k
2
∂λk
n

2
c
2
= 2 exp c3 λk − 1
n
2

∂λk
n

2

= Gc 2 sin αk
h
k2

2

λk − 1
n

(4.27)

2

∂λ1
∂λk
n

2

= Gc 2 cos αk
h
2
∂λk
2

2

.

(4.28)

For the smooth muscle, the stretch in the natural conﬁguration from the deformed conﬁguration is deﬁned as
λm = Gm λ1
n
h

55

(4.29)

where Gm is homeostatic stretch. The Cauchy stress of the smooth muscle
h
FiA FjB

∂W m
∂W m
∂W m ∂λm 2
n
= λm 2 m 2 =
,
i
m
m 2 ∂λm 2
∂CAB
∂λn
∂λi
i

(4.30)

where the strain energy of smooth muscle W m per unit reference volume is given by
W m (λm ) =
n

2
c4
exp c5 (λm )2 − 1
n
4c5

−1

(4.31)

λm 2 − 1
n

(4.32)

and therefore
2
c
∂W m
= 4 exp c5 λm 2 − 1
n
2
2
∂λm
n

∂λm 2
n
= Gm 2
h
∂λm 2
1
∂λm 2
n
= 0.
∂λm 2
2

(4.33)

The volume fraction of the constituent of i is prescribed as following (He and Roach,
1994; Holzapfel et al., 2002; Menashi et al., 1987):
ν e = 0.2,

ν m = 0.2,

ν k = [0.1, 0.1, 0.4, 0.4] (1 − ν e − ν m ) .

(4.34)

In this study, a four-ﬁber family has been considered. It is assumed that a one-ﬁber
family is aligned to the circumferential direction, another the longitudinal direction. The
other a two-ﬁber family is disposed helically with respect to the longitudinal direction.
αk = [0◦ , 90◦ , α◦ , (180 − α)◦ ]
Parameters c1 , c2 , c3 , c4 , c5 , Ge , Ge , Gc , Gm , α to describe the mechanical behavior of
1
2
h
h
the aorta in this study are assumed to be unknown and determined by the parameter estimation.

56

4.1.2

Parameter estimation

The constitutive parameters are estimated by ﬁtting the inﬂation test data of a healthy
porcine thoracic aorta at three diﬀerent longitudinal stretches of 1.35, 1.40. and 1.45. The
stretch and stress are normalized with homeostatic stress and stretch, which are obtained at
95 mm Hg at the longitudinal stretch of 1.35. The magnitude of normalized stretch and stress
depend on the homeostatic ones, but it does not eﬀect the shape of curve in the stress-stretch
plot. Parameters are determined by minimizing the objective function e which represents
˜
diﬀerences between the theoretically predicted results and experimentally calculated results
based on measurements in the two stretching directions
m

e=
˜
k=1

(t11 |t − t11 |e )2 + (t22 |t − t22 |e )2
k
k
2
σh

(4.35)

where the subscripts t and e denote theoretical and experimental values, respectively, and
σh is the homeostatic stress of the artery.

57

Chapter 5
RESULTS AND DISCUSSION
5.1

Validation of the experimental measurements

Cauchy Stress (KPa)

250

200

150

A1
L1
P1
R1
A2
L2
P2
R2

100

50

0
0.9

1

1.1

1.2

1.3

Circumferential Stretch
Fig. 5.1: Comparison of stretch-stress plots before and after the rotation of a specimen. A:
anterior region, L: left lateral region, P: posterior region, and R: right lateral region.
In our pilot study, the inﬂation test for the same region of an aorta was repeated after
rotating the specimen under the same experimental conditions in order to ensure the reproducibility of our experimental results. Fig. 5.1 shows an example of stress-stretch response of
58

(a)

(b)

(cm)

1.6

1.2

1.2

0.8

0.8

Z

1.6

1.07

1.05

1.03
-1.6 -1.4 -1.2
X

-1.6 -1.4 -1.2
X

(cm)

Fig. 5.2: Comparison of the distribution of circumferential stretch between two methods:
(a) stretch distribution by linear approximation with triangular elements and (b) stretch
distribution by the approximation in the present study. Black dots represent markers in the
deformed conﬁguration.
an aorta before and after rotating. There was little diﬀerence in the stress-stretch responses
after the rotation.

5.2

Validation of the approximation using continuous
functions

A ﬂexible silicone tube, which has nearly uniform radius and stretch distributions, was used
to compare the circumferential stretch distributions obtained by two approximations: one
was derived from the continuous functions explained in the previous chapter; and the other
was based on linear polynomial approximation with triangular elements, which is a typical
method used in ﬁnite element analysis and previous experimental studies (Hu et al., 2007;
Saravanan et al., 2006). Although both stretch distributions looked similar, the stretch
distribution obtained by the approximation using continuous functions showed smoother
and slightly higher variation than that obtained by linear approximation (Fig 5.2).
59

In addition to the circumferential stretch, the outer radii of small cylindrical rods were
investigated, assuming each rod had a constant radius. Microspheres (i.e. markers) with
a mean diameter of 196, 390, and 550 µm were attached onto the surfaces of two rods
with diﬀerent diameters of 1.27 and 1.91 cm using adhesive and a pulled glass micropipette.
Pairs of digital images of markers on the specimen were obtained and processed to extract
the positions of markers. The radii of the specimens with diﬀerent sizes of markers were
estimated by the method described in the previous chapter. The radii of the specimens were
also estimated by ﬁtting the equation of a perfect cylinder with the marker positions using
the least squares method. The entire procedure was repeated three times for each specimen
and the average values of the radii were obtained. Overall, the radius diﬀerence between the
two methods was less than 2%.

5.3

Variation of the aortic wall thickness

Aortic wall thickness varied gradually along the circumference (Fig. 5.3). The anterior region
was thickest (2.2 ± 0.3 mm for the proximal thoracic aorta and 1.8 ± 0.3 mm for the distal
thoracic aorta) and the posterior region was thinnest ( 1.5 ± 0.2 mm for the proximal and
1.2 ± 0.1 mm for the distal). The ratio of the mean wall thickness of the anterior region to
the posterior was 1.5 ± 0.1 for the proximal and 1.6 ± 0.2 for the distal. Although the wall
thickness decreased as the distance from the heart further, the ratio remained similar.

5.4

Radius of curvature of the aortic wall

The outer radius of the aorta responded to the change of pressure as shown in Fig. 5.4. At
the load-free state, the mean and standard deviation of the radius of the proximal aorta for
each region were 8.5 ± 0.8 (A), 8.8 ± 1.0 (L), 8.1 ± 1.0 (P), and 8.7 ± 0.9 mm (R). It showed
that the mean radius of the posterior region was slightly smaller than other circumferential
regions. On the other hand, the radii of the four regions became nearly uniform as the
60

3

Thickness (mm)

2.5
2
1.5
1
0.5
0

0

90

180
270
360
Angle (degree)
(a) Proximal portion of the thoracic aorta

0

90

3

Thickness (mm)

2.5
2
1.5
1
0.5
0

180
270
360
Angle (degree)
(b) Distal portion of the thoracic aorta

Fig. 5.3: Mean and standard deviation of the aortic wall thickness along the circumference:
(a) the proximal thoracic aorta, (b) the distal thoracic aorta. 0◦ : the anterior region, 90◦ :
left lateral region, 180◦ : posterior region, and 270◦ : right lateral region
pressure was applied during the inﬂation test. At the transmural pressure of 12.00 kPa, the
mean and standard deviation of the radius for each region were 10.0 ± 0.7 mm (A), 10.1 ±
61

(cm)
2.2

A

L

P

R

2.2

2.2

1.8

1.8

1.8

1.8

1.4

1.4

1.4

1.4

60 mmHg
S

2.2

1.2

1.6

2.0 (rad)

2.2

1.0

1.4

1.8

1.0

1.4

1.8

1.0

1.8

1.8

1.8

1.4

1.4

1.8

1.0

1.4
Θ

1.8

1.8

1.4

1.4

2.2

1.4

90 mmHg
S

2.2

1.8

1.0

2.2

1.4

1.2

1.6

2.0

2.2

1.0

1.4

1.8

1.0

1.4

1.8

2.2

2.2

1.8

1.8

1.8

1.8

1.4

1.4

1.4

1.4

120 mmHg
S

2.2

1.2

1.6
Θ

2.0

1.0

1.4
Θ

1.8

1.0

1.4 1.8
Θ
(unit: cm) 1.0

1.2

1.4

Fig. 5.4: A representative example of the distributions of the radius of curvature for the
four circumferential regions of one thoracic aorta sample at transmural pressures of 8.00
kPa (60 mm Hg), 12.00 kPa (90 mm Hg), and 16.00 kPa (120 mm Hg), respectively, and a
longitudinal stretch ratio (ΛZ ) of 1.35.
0.7 mm (L), 10.2 ± 0.7 mm (P), and 10.1 ± 0.8 mm (R).

5.5

Stress and strain relationship of the aorta

The circumferential stretch and the circumferential Cauchy stress during the inﬂation test
at three ﬁxed longitudinal stretch ratios of 1.35, 1.40, and 1.45 were computed for each
circumferential region of each specimen. In a repeated cyclic loading condition, loading and
62

Transmural Pressure (kPa)

25
20
15
10
5
0
−5

0.9

1

1.1

1.2

1.3

1.4

Circumferential Stretch
Fig. 5.5: A representative example of the ﬁve repeated loading and unloading curves of the
anterior region of the thoracic aorta.
unloading curves exhibited a small hysteresis (i.e., pseudo-elastic behavior) like other vascular
tissues (Fig. 5.5). Therefore, the last loading curve was used for stress-strain analysis in this
study.
The circumferential stretch (Fig. 5.6) and the longitudinal stretch (Fig. 5.7) were distributed nonuniformly within the domain and they varied according to the circumferential
region. As the pressure increased, the circumferential stretch was increased gradually, but
the longitudinal stretch was relatively uniform compared to the circumferential stretch.
Likewise, the stress also changed with respect to the change of pressure and varied with
the circumferential region. The averaged stretch and stress within the domain for each region
were calculated for comparison among the four circumferential regions of the aorta. Fig. 5.8
displays the circumferential and longitudinal stress-stretch curves for the four regions. All
aorta samples exhibited similar nonlinear stress-stretch behavior. The stress-stretch curves
deviated from each other progressively as the pressure increased. The extent of deviation
for the circumferential stress-stretch curve was more than that of the longitudinal one, and

63

A

L

(cm)

P

2.2

2.2

1.8

1.8

1.4

R

1.4

2.2

60 mmHg
S

2.2

1.8

1.8

1.0

1.4

1.4
1.2 1.81.6

2.0

(rad)

1.0

1.4

1.8

1.4
1.0

2.2

1.8

1.4

1.8

2.2

1.8

1.4

1.4

90 mmHg
S

2.0

1.0

1.4

1.8

1.0

2.2

1.8

1.8

1.4

1.4

2.2

1.8

1.4

1.4

120 mmHg
S

1.4
1.0

1.4
Θ

1.8

1.8

1.8

1.8

1.6 2.0
Θ

1.4

2.2

2.2

1.2

1.0

1.4

1.4
1.8
1.6

1.8

1.8

1.8

1.4
1.2

1.4

2.2

2.2

1.0

1.0

1.0

1.4 1.8
Θ

1.0

1.4 1.8
Θ
1.1 1.2 1.3 1.4

Fig. 5.6: A representative example of the distributions of circumferential stretch for the four
circumferential regions of one thoracic aorta sample at transmural pressures of 8.00 kPa,
12.00 kPa, and 16.00 kPa, respectively, and a longitudinal stretch ratio (ΛZ ) of 1.35.
the overall change of the circumferential stress was larger than the longitudinal stress. In
the low pressure range, the stress-stretch response for each region was nearly linear, but it
became nonlinear after the transmural pressure exceeded a transition point, which ranged
from 10.93 to 14.53 kPa (82 - 109 mm Hg).
The circumferential stiﬀness of the four circumferential regions is plotted with respect
to the increase of pressure in Fig. 5.9. In the low pressure range, the stiﬀness was nearly
constant, which corresponded to the linear stress-stretch response, but increased markedly
after the transition point.
64

A

L

P

R

2.2

2.2

2.2

1.8

1.8

1.8

1.8

1.4

1.4

1.4

1.4

60 mmHg
S

2.2

1.2

1.6

2.0

1.0

1.4

1.8

1.0

1.4

1.8

1.0

2.2

1.8

1.8

1.4

1.4

1.8

1.0

1.4
Θ

1.8

1.8

1.4

1.4

2.2

1.8

1.8

1.0

2.2

1.4

1.4

90 mmHg
S

2.2

1.2

1.6

2.0

1.0

1.4

1.8

1.0

1.4

1.8

2.2

2.2

2.2

1.8

1.8

1.8

1.8

1.4

1.4

1.4

1.4

120 mmHg
S

2.2

1.2

1.6 2.0
Θ

1.0

1.4
Θ

1.8

1.0

1.4 1.8
Θ

1.2

1.3

1.4

Fig. 5.7: A representative example of the distributions of longitudinal stretch for the four
circumferential regions of one thoracic aorta sample at transmural pressures of 8.00 kPa,
12.00 kPa, and 16.00 kPa, respectively, and a longitudinal stretch ratio (ΛZ ) of 1.35.
The mean of the circumferential stretch, which was averaged within the domain, was
computed for each region of all aorta samples. There was no signiﬁcant diﬀerence in the
mean stretch among the four circumferential regions at the transmural pressure of 12.00 kPa
(90 mm Hg) (Fig. 5.10). The mean and standard deviation of the stretch at the transmural
pressure of 12.00 kPa and at longitudinal stretch of 1.35 were 1.20 ± 0.06 (A), 1.22 ± 0.03
(L), 1.22 ± 0.05 (P), and 1.21 ± 0.04 (R). The mean and standard deviations of the stretch
at diﬀerent pressure and diﬀerent longitudinal stretch are tabulated in Appendix C.

65

Circumferential Cauchy Stress (kPa)

(a)

250

200

A
L
P
R

100

0

1

1.1

1.2

1.3

1.4

Circumferential Stretch

Longitudinal Stress (kPa)

(b)

120
A
L
P
R
80

40

0

1

1.1

1.2

1.3

1.4

Circumferential Stretch

Fig. 5.8: A representative example of the circumferential Cauchy stress and circumferential stretch curve (a) and the longitudinal stress and circumferential stretch curve (b) at a
longitudinal stretch ratio (ΛZ ) of 1.35.
In the same manner, the mean of circumferential stress averaged within the domain was
computed for each region, and a signiﬁcant diﬀerence was found among the four circumferential region. For the proximal portion of the thoracic aorta, in the post-hoc analysis, there

66

Circumferential Stiffness (kPa)

2500

2000

A
L
P
R

1000

0

5

10

15

20

25

Transmural Pressure (kPa)
Fig. 5.9: A representative example of the circumferential stiﬀness and transmural pressure
curve at a longitudinal stretch ratio (ΛZ ) of 1.35.
was a signiﬁcant diﬀerence between the anterior and posterior regions, the anterior and left
lateral regions, the anterior and right lateral regions, the left lateral and posterior regions,
and the posterior and right lateral regions at the transmural pressures above 8.00 kPa (60
mm Hg). For the distal portion of the thoracic aorta, a signiﬁcant diﬀerence was also found
between the anterior and posterior regions. The mean and standard deviations of the stress
at diﬀerent pressure and diﬀerent longitudinal stretch are tabulated in Appendix C.

5.6

Circumferential variation in the stiﬀness

The circumferential stiﬀness in this study is deﬁned as a change in stress with respect to a
change in stretch in the circumferential direction. Hence, it presents the intrinsic stiﬀness
of the vascular tissue. There was a signiﬁcant diﬀerence in the stiﬀness among the four
circumferential regions. The mean stiﬀness of the posterior region had the highest value,
and the anterior region had the lowest. The diﬀerence in the stiﬀness among the four
regions became more prominent as pressure increased. Fig. 5.12 shows the mean stiﬀness
67

Circumferential Stretch

1.4
1.3
1.2
1.1
1

A
L
P
R
Circumferential Region

(a) Λ=1.35

Circumferential Stretch

1.4
1.3
1.2
1.1
1

A
L
P
R
Circumferential Region

(b) Λ=1.40

Circumferential Stretch

1.4
1.3
1.2
1.1
1

A
L
P
R
Circumferential Region

(c) Λ=1.45
Fig. 5.10: Comparison of the mean circumferential stretch of the proximal thoracic aorta at
the transmural pressure of 12.00 kPa for each longitudinal stretch ratio (ΛZ ).

68

and the statistical evaluation of their signiﬁcance at the transmural pressure of 12.00 kPa,
which is within the transition region for the thoracic aorta at each ﬁxed longitudinal stretch
ratio. The stiﬀness at the anterior region diﬀered signiﬁcantly from the posterior. However,
the mean pressure-strain elastic modulus, which represents the structural stiﬀness, had no
signiﬁcant diﬀerence among the four regions (Fig. 5.13). The mean and standard deviation
of the circumferential stiﬀness for each region at the investigated transmural pressure of 8.00,
12.00, and 16.00 kPa are listed in Table C.5 in Appendix C.

5.7

Longitudinal variation in the mechanical behavior
of the aorta

The aorta at all tested regions exhibited typical nonlinear pseudoelastic behavior. For the
mean circumferential stretch of the distal thoracic aorta, there was no signiﬁcant diﬀerence
between the anterior and posterior regions, which is the same trend as the proximal thoracic
aorta (Fig. 5.14). The values of mean stretch of the anterior and posterior regions of the
distal thoracic aorta had a higher standard deviation than that of the proximal. The stretch
of the anterior region in the distal portion was slightly higher than the proximal, although
the diﬀerence was not statistically signiﬁcant. The stretch of the posterior region had almost
the same amount of the stretch in both proximal and distal portions.
The mean stress of the distal thoracic aorta was higher in the both anterior and posterior
regions compared to that of the proximal portion at the same transmural pressure 12.00 kPa
(Fig. 5.15). Similar to the proximal portion, the mean stress of the posterior region at the
distal portion was signiﬁcantly higher than that of the anterior region. The speciﬁc values
of the stretch and stress are included in Appendix C.
The mean circumferential stiﬀness was also changed with the change in pressure (Fig.
5.16). The diﬀerence in the mean stiﬀness of the posterior region from the proximal to
the distal portions of the thoracic aorta was signiﬁcantly higher than that of the anterior
69

region, and their diﬀerence became larger as pressure increased. The posterior region remains
signiﬁcantly stiﬀer than the anterior regions for both longitudinal portions. However, the
pressure-strain elastic modulus did not show the signiﬁcant diﬀerence between the proximal
and distal portions for both anterior and posterior regions, although the mean pressurestrain elastic modulus was higher at the distal portion (Fig. 5.17). It may be due to the
large standard deviation. The speciﬁc values of the circumferential stiﬀness and pressurestrain elastic modulus are included in Appendix C.

5.8

Material parameters

Material parameters are one of common indicators to describe the mechanical response of the
vascular tissue using a constitutive model. Parameter estimation was performed to determine
the material parameters of the aorta for each region. A representative best-ﬁt curve of the
theoretical prediction to the experimentally measured data is shown in Fig. 5.18. For most
of the test samples, the regression curves ﬁt better to the circumferential stress-stretch curve
than the longitudinal curve.
The mean and standard deviation of all parameters are shown in Table 5.1. Parameter
c1 presents the isotropic behavior of the elastin of the aorta. The c2 is a parameter of the
exponential part of the constraint mixture model, which represents the mechanical behavior
of the four collagen ﬁber families (See Chapter 4.4.1). The anterior region of the thoracic
aorta has the biggest mean value of c1 in the proximal portion, but there was no signiﬁcant
diﬀerence among the four circumferential regions (Fig. 5.19). On the other hand, the parameter c2 in the anterior region has the smallest mean value among the four circumferential
regions (Fig. 5.19), and a signiﬁcant diﬀerence was found between the anterior and right
lateral regions. For the statistical analysis of c1 and c2 , data from all 11 aortic sample were
used regardless of the longitudinal portion. In a statistical analysis, there were signiﬁcant
diﬀerences of c1 and c2 between the anterior and posterior regions (Fig. 5.19 and 5.20).

70

Table 5.1: The mean and standard deviation of material parameters for all aorta samples
Parameter \ Region
c1 (kPa)
c2 (kPa)
c3
c4 (Pa)
c5
Ge
1
Ge
2
Gc
h
Gm
h
α(◦)

Anterior
406 ± 96
155 ± 63
2.73 ± 2.87
(31.2 ± 83.7) × 10−9
(1.93 ± 4.74) × 10−6
1.12 ± 0.06
1.19 ± 0.04
1.09 ± 0.01
1.10 ± 0.00
26.5 ± 6.4

Posterior
318 ± 93
318 ± 209
6.33 ± 7.91
(1.48 ± 4.54) × 10−6
(7.7 ± 17.1) × 10−3
1.23 ± 0.07
1.15 ± 0.06
1.09 ± 0.01
1.10 ± 0.00
31.9 ± 12.1

(a) The anterior and posterior regions
Parameter \ Region
c1 (kPa)
c2 (kPa)
c3
c4 (Pa)
c5
Ge
1
Ge
2
Gc
h
Gm
h
α(◦)

Left lateral
355 ± 90
226 ± 104
4.10 ± 3.47
(0.685 ± 0.108) ×10−9
0.202 ± 0.522
1.19 ± 0.10
1.14 ± 0.06
1.09 ± 0.01
1.10 ± 0.00
24.5 ± 11.7

Right lateral
345 ± 76
197 ± 76
5.83 ± 5.57
(23.6 ± 37.1) ×10−9
2.28 ± 5.36
1.20 ± 0.07
1.17 ± 0.04
1.09 ± 0.01
1.10 ± 0.00
30.0 ± 4.7

(b) The left and right lateral regions

71

While the material parameters in the unit of stress corresponding to the behavior of the
elastin (c1 ) and collagen (c2 ) have large values, the parameters for smooth muscle (c4 ) very
small. All homeostatic stretch parameters, Ge , Ge , Gc , and Gm are less than 1.2 for each
1
2
h
h
constituent. The averaged angle of collagen ﬁber family for each region is about 28◦ .

5.9
5.9.1

DISCUSSION
Nonlinear, anisotropic behavior of the aorta

The stress-stretch curves for all local regions of the thoracic aorta in this study exhibited
nonlinear behavior similar to other studies (Groenink et al., 1999; Stergiopulos et al., 2001;
Sokolis, 2007). The circumferential stress increased linearly up to a transition point beyond
which the stress changed rapidly with the increase of the stretch. This nonlinear trend of the
circumferential stress with respect to the stretch (or strain) is consistent with other works
for the human thoracic aorta (Spina et al., 1983), the canine thoracic aorta (Zhou and Fung,
1997), and ovine thoracic aorta (Wells et al., 1999). The range of linear circumferential
stress-stretch response in this study is at a relatively higher pressure than other porcine
arteries such as porcine basilar arteries and coronary arteries (Hu et al., 2007; Pandit et al.,
2005; Wang et al., 2006).
The transition from linear to nonlinear behavior of the aorta occurred at 10.93 to 14.53
kPa (82 - 109 mm Hg), which falls within the in vivo pressure range from 10.13 to 15.20
kPa (76 - 114 mm Hg) when the radial compression of the aorta by the surrounding tissue is
assumed to be around 5% of internal pressure (Zhang et al., 2005). Other studies (Danpinid
et al., 2010; Shadwick, 1999) have also suggested that the transition corresponded to normal
physiological conditions.
The circumferential stress-strain relationship diﬀers from the longitudinal stress-strain
relationship. It infers that the mechanical properties of the aorta is anisotropic, which is a
well-known property of the soft tissue reported in many other studies (Fung, 1993; Humphrey,
72

2001).

5.9.2

Variation in stiﬀness

This study supports that the mechanical properties of the aorta depend on location. One
major ﬁnding in this study is the circumferential variation in the stiﬀness of the healthy
thoracic aorta. The posterior region of the aorta was signiﬁcantly stiﬀer than the anterior. It
may be related to the bigger motion of the anterior region than the posterior (Goergen et al.,
2007). However, the posterior region was signiﬁcantly thinner than the anterior, so that the
structural stiﬀness, which is represented by the pressure-strain elastic modulus, remained
non-signiﬁcant. This ﬁnding suggests that the circumferential distention of the proximal
thoracic aorta may be uniform in vivo even with the signiﬁcant variations in stiﬀness of
the aortic tissue in the circumferential direction. However, the validation of this suggestion
requires a better understanding of the eﬀect of the surrounding tissues and the spine on
biomechanics of the aortic wall in vivo.
This study also reveals the longitudinal variation in stiﬀness of the healthy thoracic aorta.
The stiﬀness in the distal portion is higher than the proximal portion for the both anterior
and posterior regions. This trend is consistent with previous studies (Purslow, 1983; Zou
and Zhang, 2009).

5.9.3

Heterogeneous composite of the aorta

The arterial wall is a heterogeneous composite consisting mainly of collagen, elastin, and
smooth muscle cells. The amount of these contents varies with locations along the arterial
tree (Fischer and Llaurado, 1966; Halloran et al., 1995; Purslow, 1983) and are responsible
for the mechanical properties of a blood vessel (Lillie and Gosline, 2007; Roach and Song,
1994; Stergiopulos et al., 2001). Many studies have reported that elastin ﬁbers are primarily
responsible for the linear behavior of an artery in the low pressure range (Gundiah et al.,
2007; Shadwick, 1999; Stergiopulos et al., 2001). The inﬂuence of elastin on the mechanical
73

properties of aortic tissue has been investigated (Gundiah et al., 2007; Lillie and Gosline,
2007; Zou and Zhang, 2009). These previous studies imply that the stiﬀness of the aortic wall
may be aﬀected by the amount of elastin, the orientation of elastin ﬁbers, and the density
of cross-linking. On the other hand, collagen is recruited in the higher pressure range and
contributes to the nonlinear behavior of arterial tissue (Groenink et al., 1999; Shadwick,
1999). The orientation of collagen ﬁbers and the amount of cross-linking inﬂuence the
mechanical response of vascular tissue (Haskett et al., 2010; Holzapfel et al., 2002). Many
previous studies suggest that the microstructure and relative content of each component may
cause the variation in stiﬀness among the diﬀerent circumferential regions shown in the this
study.

5.9.4

Relation of heterogeneous mechanical properties with pathological conditions

The mechanical properties of a diseased aorta vary spatially with its pathological condition.
Previous studies (Iliopoulos et al., 2009a; Raghavan et al., 2006; Thubrikar et al., 2001)
suggested nonuniform changes of the aortic wall during the development of aortic diseases,
such as aortic aneurysms. Iliopoulos et al. (2009b) reported a preferential bulging in the
anterior side of the human ascending thoracic aneurysms. Choudhury et al. (2009) showed
that the elastic modulus of the ascending thoracic aorta depended on the circumferential
region as well as the pathological tissue type.
On the other hand, the variation in mechanical properties of the aorta may have an
aﬀection on vascular remodeling and the pathological condition. For example, the progression
of abdominal aortic aneurysms is associated with an increase in circumferential stiﬀness
(Vande Geest et al., 2006).
Therefore, the mechanical properties of vascular tissue may aﬀect the pathological condition or vice versus. Previous studies were, however, not enough to make a decisive conclusion
about the relations of heterogeneous biomechanical properties with the development of aortic
74

diseases. It promotes further comparative studies of local mechanical properties of aortic
walls between healthy and diseased samples.

5.9.5

Material parameters

The mechanical behavior of a blood vessel has been characterized with a constitutive model
based on a constrained mixture approach using experimental data. Previous studies have
used many diﬀerent constitutive models and diﬀerent measures of strain. Recently, many
adopted constitutive models motivated from arterial structure (Holzapfel et al., 2000; Zulliger
et al., 2004b). One of the structurally motivated models to describe arterial mechanics is
the constitutive model based on a constrained mixture theory. This model suggests that
the mechanical behavior of a blood vessel can be explained by major constituents, that is,
elastin, collagen ﬁbers, and smooth muscles. Parameters for elastin (c1 ) and collagen (c2 )
that we found in this study have larger values compared to those for smooth muscles as
shown in Table 5.1. These results suggest that the elastin and collagen ﬁbers are major
components contributing to the mechanical response of the aorta.
In our study, the mean values of c1 and c2 for the distal thoracic aorta are smaller than
the proximal, respectively. It is in a good agreement with the previous study of Halloran
et al. (1995), which showed that the content of collagen and elastin, respectively, decreases
from proximal to distal portion of the aorta.
For each circumferential region, the mean value of c1 is higher at the proximal portion
than the distal and c2 is higher in the distal portion than the proximal. It implies that the
elastin is more dominant in the proximal portion and the collagen is in the distal. Indeed,
the content of elastin is higher in the proximal portion than the distal, and that of collagen
is higher in the distal (Davidson et al., 1986).
In the comparison of the material parameters between the anterior and posterior regions,
c1 of the anterior is higher than that of the posterior for both proximal and distal portions
of the thoracic aorta, although there was no signiﬁcant diﬀerence in the proximal portion. It
75

may be because of the diﬀerent number of elastin lamella between two circumferential regions.
O’Connell et al. (2008) examined the number of medial lamellae for the rat abdominal aorta,
and a signiﬁcant diﬀerence was found in the anterior and posterior regions .
The smooth muscle contraction aﬀects the stress and strain distribution in arteries
(Rachev and Hayashi, 1999; Zulliger et al., 2004a), and the elastin lamina is much stiﬀer
than the smooth muscle layer (Matsumoto et al., 2004). In this study, the active tone was
not accounted for and the parameters of smooth muscle, c4 and c5 , were found to be very
small. Therefore, the role of smooth muscle in the passive mechanical behavior of the thoracic
aorta was insigniﬁcant.
Most of previous studies have assumed that the elastin has isotropic properties. However,
Rezakhaniha et al. (2011) reported recently that the anisotropic properties of elastin provide
better curve ﬁt to characterize the mechanical response of a blood vessel. Hence, there is a
need to study the mechanical properties and behavior of elastin.

5.9.6

Limitation of this study

Although our validation study showed that the errors were negligible, there might be measurement errors caused by the rotation of the specimen when measuring the strain at four
local circumferential regions of the thoracic aorta. The accuracy of the strain measurement
may be enhanced by using multiple cameras simultaneously around the specimen or using a
concave conical mirror (Genovese, 2009).
We found that the values of stress and strain at the distal portion have larger variances
compared to those at the proximal portion, which might attribute to the small number of
specimen from the distal portion. The number of samples also aﬀect the statistical analysis
on signiﬁcance test. Therefore, we suggest that more experimental test with specimens from
the distal portion can strengthen the statistical analysis presented in this study.
Another shortcoming of this study is that the histology of the aortic wall was not fully
examined to validate the prediction based on the mixture theory-based constitutive model
76

for the spatial variation in the mechanical properties of the thoracic aorta. The histological and microstructural features, such as the number of elastin lamellae, may correlate to
the circumferential variation, and further investigation will be needed to understand the
heterogeneity of the thoracic aorta in this study.

77

Circumferential Cauchy Stress (kPa)

170
150

***

***

**
*

***

100

50

0

A
L
P
R
Circumferential Region

Circumferential Cauchy Stress (kPa)

(a) Λ=1.35

140

***

***

**
*

***

100

50

0

A
L
P
R
Circumferential Region

Circumferential Cauchy Stress (kPa)

(b) Λ=1.40
170
150

***

***

**
*

***

100

50

0

A
L
P
R
Circumferential Region

(c) Λ=1.45
Fig. 5.11: Comparison of the mean circumferential Cauchy stress of the proximal thoracic
aorta aat the transmural pressure of 12.00 kPa for each longitudinal stretch ratio (ΛZ ).
Asterisk represents the signiﬁcant diﬀerence (*: p<0.05, **: p<0.01, ***: p<0.005)
78

Circumferential Stiffness (kPa)

1000

**

**

750
500
250
0

A
L
P
R
Circumferential Region

Circumferential Stiffness (kPa)

(a) Λ=1.35
1000

**

**

750
500
250
0

A
L
P
R
Circumferential Region

Circumferential Stiffness (kPa)

(b) Λ=1.40
1000

**

**

750
500
250
0

A
L
P
R
Circumferential Region

(c) Λ=1.45
Fig. 5.12: Comparison of the mean circumferential stiﬀness of the proximal thoracic aorta
among the four circumferential regions at the transmural pressure of 12.00 kPa for each
longitudinal stretch ratio (ΛZ ).

79

Pressure−Strain Elastic Modulus (kPa)

100
75
50
25
0

A
L
P
R
Circumferential Region

Pressure−Strain Elastic Modulus (kPa)

(a) Λ=1.35
100
75
50
25
0

A
L
P
R
Circumferential Region

Pressure−Strain Elastic Modulus (kPa)

(b) Λ=1.40
100
75
50
25
0

A
L
P
R
Circumferential Region

(c) Λ=1.45
Fig. 5.13: Comparison of the mean pressure-strain elastic modulus of the proximal thoracic
aorta at the transmural pressure of 12.00 kPa for each longitudinal stretch ratio (ΛZ ).
80

Circumferential Stretch

1.4
1.3
1.2
1.1
1

AP PP
AD PD
Circumferential Region

(a) Λz =1.35

Circumferential Stretch

1.4
1.3
1.2
1.1
1

AP PP
AD PD
Circumferential Region

(b) Λz =1.40
Circumferential Stretch

1.4
1.3
1.2
1.1
1

AP PP
AD PD
Circumferential Region

(c) Λz =1.45
Fig. 5.14: Comparison of the mean stretch at the anterior and posterior regions of the
proximal and distal portions of the thoracic aorta at the transmural pressure of 12.00 kPa
for each longitudinal stretch ratio (ΛZ ).

81

Circumferential Cauchy Stress (kPa)

200

*
***

*

150
100
50
0

AP PP
AD PD
Circumferential Region

Circumferential Cauchy Stress (kPa)

(a) Λz =1.35
200

*
***

*

150
100
50
0

AP PP
AD PD
Circumferential Region

Circumferential Cauchy Stress (kPa)

(b) Λz =1.40
200

***

*

150
100
50
0

AP PP
AD PD
Circumferential Region

(c) Λz =1.45
Fig. 5.15: Comparison of the mean circumferential stress at the anterior and posterior regions
of the proximal and distal portions of the thoracic aorta at the transmural pressure of 12.00
kPa for each longitudinal stretch ratio (ΛZ ).
82

Circumferential Stiffness (kPa)

1500

*

1000

**
500

0

AP PP
AD PD
Circumferential Region

Circumferential Stiffness (kPa)

(a) Λz =1.35
1500

1000

*

**

500

0

AP PP
AD PD
Circumferential Region

Circumferential Stiffness (kPa)

(b) Λz =1.40
1500

*
*

1000

**
500

0

AP PP
AD PD
Circumferential Region

(c) Λz =1.45
Fig. 5.16: Comparison of the mean circumferential stiﬀness at the anterior and posterior
regions of the proximal and distal portions of the thoracic aorta at the transmural pressure
of 12.00 kPa for each longitudinal stretch ratio (ΛZ ).

83

Pressure−Strain Elastic Modulus (kPa)

120

80

40

0

AP PP
AD PD
Circumferential Region

Pressure−Strain Elastic Modulus (kPa)

(a) Λz =1.35
120

**

80

40

0

AP PP
AD PD
Circumferential Region

Pressure−Strain Elastic Modulus (kPa)

(b) Λz =1.40
120

**

80

40

0

AP PP
AD PD
Circumferential Region

(c) Λz =1.45
Fig. 5.17: Comparison of the mean pressure-strain elastic modulus at the anterior and posterior regions of the proximal and distal portions of the thoracic aorta at the transmural
pressure of 12.00 kPa for each longitudinal stretch ratio (ΛZ ).
84

Circumferential Stress (kPa)

160
1.35 M
1.40 M
1.45 M
1.35 P
1.40 P
1.45 P

120

80

40

0
0.8

0.9
1
Circumferential Stretch

1.1

(a) Circumferential stress-circumferential stretch

1.35 M
1.40 M
1.45 M
1.35 P
1.40 P
1.45 P

Longitudial Stress (kPa)

160

120

80

40
0.98

1.02
1.06
Longitudial Stretch

1.1

(b) Longitudinal stress-circumferential stretch
Fig. 5.18: A representative stress-stretch plot. Stress was calculated by using experimentally
measured data and theoretically predicted. M: experimentally measured data, P: theoretically predicted data

85

Material Parameter c1

5

x 10 5

4
3
2
1
0

4

A

L
P
R
Circumferential Portion
(a) Material parameter c1

x 10

5

Material Parameter c

2

*
3

2

1

0

A

L
P
R
Circumferential Portion
(b) Material parameter c2

Fig. 5.19: Comparison of the mean and standard deviation of material parameter c1 (a) and
c2 (b) among the four circumferential regions for the proximal thoracic aorta.

86

Material Parameter c1

6

x 10

5

*

5
4
3
2
1
0

Material Parameter c 2

6

A
P
Circumferential Portion
(a) Material parameter c1
x 105

*

5
4
3
2
1
0

A
P
Circumferential Portion
(b) Material parameter c2

Fig. 5.20: Comparison of the mean and standard deviation of material parameter c1 (a) and
c2 (b) between the proximal and distal portions of the thoracic aorta.

87

Chapter 6
CONCLUSION
The majority of computational studies of vascular adaptation have been performed based
on the assumption of the homogeneous mechanical properties of a blood vessel. Although
a few studies reported the regional heterogeneity in the mechanical properties of the aorta
along the aortic tree, the circumferential variation has not been taken into account (Guo
and Kassab, 2004; Sokolis, 2007). Recently the local mechanical properties of healthy and
diseased ascending thoracic aortas were studied using uniaxial or biaxial tests, but there
is little experimental data, based on author’s knowledge, to quantify the circumferential
variation in mechanical properties of a healthy thoracic aorta. Hence, in this study, the
circumferential variation in stiﬀness of the porcine descending thoracic aorta was experimentally investigated. In order to characterize the local mechanical properties of the aorta,
an ex vivo extension-inﬂation apparatus with a stereo vision system was developed, which
allowed us to measure the 3D deformation of a blood vessel, and a stress-strain analysis
method using 3D experimental data was developed as well. Then, the material parameters
were estimated by the parameter estimation method using a constitutive model based on
the constraint mixture theory.
The experimental results of this study indicated a nonuniform distribution of stretch
and stress in a local region and nonuniform thickness of aortic wall along the circumference.

88

The stress-stretch response during the inﬂation test depends on the local region. These
results showed that there was the variation in the stiﬀness, deﬁned as the tangent of stressstretch response, among the four circumferential regions and two longitudinal portions. In
addition, material parameters showed that there was no signiﬁcant diﬀerence between two
circumferential regions (the anterior and posterior regions) for the both proximal and distal
portions of the aorta, but that there was a signiﬁcant diﬀerence between two longitudinal
portions (the proximal and the distal portions) for the anterior and posterior region of the
aorta.
One major ﬁnding of this study was that the posterior region of the porcine thoracic
aorta was signiﬁcantly stiﬀer than the anterior, that is, the circumferential variation in the
mechanical properties of the aorta. Interestingly, however, there was no signiﬁcant diﬀerence
in the pressure-strain elastic modulus for the proximal thoracic aorta. Another ﬁnding was
that the distal portion of the porcine thoracic aorta was signiﬁcantly stiﬀer than the proximal
portions. It shows the longitudinal variation in the mechanical behavior of the thoracic aorta.
The parameter estimation based on the mixture theory-based constitutive model suggests
that the proximal thoracic aorta exhibited elastin-dominant mechanical behavior and the
distal thoracic aorta exhibits the collagen-dominant mechanical behavior. It may cause the
longitudinal variation in the mechanical behavior.
Finally, it is suggested that the experimental method and analysis presented in this study
can be used in the study of vascular biomechanics. The consistent spatial variations in the
mechanical properties of the thoracic aorta increase our understanding of vascular remodeling
and adaptation during the progression of vascular diseases or vice versa, and will eventually
help to improve clinical treatments and interventions of vascular disease.

89

APPENDICES

90

Appendix A
Methods
A.1

Camera matrix

In general, points in world is expressed in terms of the world coordinate frame. Two coordinate frames between world and camera one may be related via rotation and translation.
Under the pinhole camera model (See Fig. 2.4), a point in the world, X = (X, Y, Z)T , is
mapped to the point on the image plane, xcam = (x, y)T in terms of the camera coordinate
frame. For the central projective camera, it is a linear mapping between homogenous vectors
of world and image points. In the image coordinate (f X/Z, f Y /Z, f )T , where f is the focal
length, the last coordinate of the image point can be ignored. Then,


 fX



fY




 Z






f 0 0 0 





 0 f 0 0
=

 
 






0 0 1 0 




















Y 

X








1 

(A.1)

Z

In practical, the origin in image plane is not at the principal point, (px , py )T which is the
point where the line from the camera center perpendicular to the image plane meets the
image plane. Then a point Xcam is mapped to the point (f X c /Z c + px , f Y c /Z c + py )T on
91

Fig. A.1: Schematics of a world point and its corresponding image point
the image plane.


 f X + Zpx



f Y + Zpy





Z







f 0 px 0 





 0 f p 0
=
y

 
 







0 0 1 0 

















Y 

X








1 

(A.2)

Z

In general, the camera matrix is determined via three steps. In ﬁrst step, points (x, y, z)
in the camera reference can be obtained the rotation and translation of 3D world points
(X, Y, Z) via









T t

R


=

 Z 



0T 1 
















 1 


 X 












 Y 



X 





Y 

Z 





1 

(A.3)

where, RT and t represent the rotation matrix and the translation vector, respectively, from
the world reference coordinate to the camera reference coordinate.
Next, projection of the points onto the image plane at (xcam , ycam ) is written as

92

x

ycam
cam
xcam
cam

y

(x0 0
(x0 , y0)
Principal point
Principal point

Fig. A.2: Image plane and pixel coordinate



 xcam



y
 cam



 f








 1 0 0 0 






 0 1 0 0 








 0 0 1 0 






X 





Y 

(A.4)


Z 





1 

where, f is the focal length.
Lastly, the mapping from points (xcam , ycam ) on the image plane to the pixel coordinates
in an image is
 


 

 x 
αx 0 x0   xcam
 

 

  1



= 
 y  f  0 αy y0   ycam
 


 

 

 1 
 f
0 0 1













(A.5)

where, αx and αy are scaling in x and y directions, respectively, (x0 , y0 ) is the principal point

93

in the image plane (Fig. A.2). This 3 × 3 matrix is called as camera calibration matrix,


 
 X 






 




 x 


αx 0 x0   1 0 0 0 
 


 


T t
  1
 Y 


 R

=  0 αy y0   0 1 0 0  

y


  f
 




 0T 1  Z 
 


 


 1 


0 0 1
0 0 1 0



 1 


(A.6)

1
In homogeneous coordinates, scaling factors f are irrelevant so that it can be discard. A

single camera is characterized by 4 internal camera parameters and 6 external camera parameters (3 translations and 3 rotations).
Therefore a ﬁnite projective pinhole camera is expressed in terms of a linear mapping of
homogeneous coordinates as xcam = P X, where P = KR[I| t] is the camera matrix with
11 degrees of freedom. That is, P can be as


 

 X 













 sx  P11 P12 P13 P14  




 Y 
 


P

=
P22 P23 P24 
sy
  21



 

 Z 









 s 


P31 P32 P33 1 


 1 

(A.7)

where a scaling factor s = P31 X + P32 Y + P33 Z. Then, all image points yield 2n × 11
matrix in the form of u = Ab and pseudo-inverse method can be applied to ﬁnd a vector
b. The camera matrix is determined by the known measurement of calibration jig. At least
6 pairs of image points and world points of the jig are required. Then equation (5) can be
rewritten as

94




 x1 












 y1 










 x2 

















y2
.
.
.
yn



0
0
 0

X Y Z
1
1
 1


0
0
 0
=

 

  X2 Y2 Z2

 

  .
.
.

.
.
  .
.
.
.

 




Xn Yn Zn

0 X1 Y1 Z1 1 −x1 X1 −x1 Y1
1

0

0

0

0 −y1 X1

−y1 Y1

0 X2 Y2 Z2 1 −x2 X2 −x2 Y2
1
.
.
.

0
.
.
.

0
.
.
.

0
.
.
.

0 −y2 X2
.
.
.
.
.
.

−y2 Y2
.
.
.

1

0

0

0

0 −yn Xn −yn Yn



















−x1 Z1  






−y1 Z1  





−x2 Z2  


−y2 Z2  



. 
. 

. 





−yn Zn 

















P11
P12
P13
P14
P21
P22
P23
P24
P31
P32
P33
































































(A.8)

where n is the number of points. Pseudo Inverse Method (Least Squares Estimation) is
employed to ﬁnd the optimized unknown aﬃne camera matrix parameters Pij .

95

Appendix B
Analysis
B.1
B.1.1

Derivatives of the parameters for the curvature
Reference Conﬁguration

ˆ
ˆ
In the reference conﬁguration, S = S(S), Θ = Θ(Θ) and R ≈ R(S, Θ) = R(S(S), Θ(Θ)) =
ˆ
R(S, Θ), where
ˆ
R=

n

δk Pk (S, Θ)

(B.1)

k=1

The ﬁrst derivatives of X and Y with respect to S and Θ are
ˆ
∂R
∂X
≈
cos Θ
∂S
∂S

(B.2)

ˆ
∂X
∂R
ˆ ∂cos Θ
≈
cos Θ + R
∂Θ
∂Θ
∂Θ

(B.3)

ˆ
∂Y
∂R
≈
sin Θ
∂S
∂S

(B.4)

ˆ
∂Y
∂R
ˆ ∂sin Θ
≈
sin Θ + R
∂Θ
∂Θ
∂Θ

(B.5)

∂S
∂Z
=
∂S
∂S

(B.6)

96

∂Z
∂S
=
∂Θ
∂Θ
where,
ˆ
∂R
=
∂S
ˆ
∂R
=
∂Θ

n

(B.7)

δk

∂Pk (S, Θ)
∂S

(B.8)

δk

∂Pk (S, Θ)
∂Θ

(B.9)

k=1
n
k=1

∂cos Θ ∂Θ
∂cos Θ
=
∂Θ ∂Θ
∂Θ

(B.10)

∂sin Θ
∂sin Θ ∂Θ
=
∂Θ ∂Θ
∂Θ

(B.11)

∂cos Θ
= − sin Θ
∂Θ

(B.12)

∂sin Θ
= cos Θ
∂Θ

(B.13)

Θmax − Θmin
∂Θ
=
2
∂Θ

(B.14)

∂S
Smax − Smin
=
2
∂S

(B.15)

∂S
=0
∂Θ

(B.16)

The second derivatives of X and Y with respect to S and Θ are
∂2X
∂S
∂ 2X
∂Θ

=
2

ˆ
∂ 2R
∂S

=
2

ˆ
∂ 2R

cos Θ + 2
2
∂Θ

2

cos Θ

ˆ
∂ R ∂cos Θ
∂Θ ∂Θ

(B.17)

2
ˆ ∂ cos Θ
+R
2
∂Θ

∂ 2X
∂ 2R
∂R ∂cos Θ
=
cos Θ +
∂S∂Θ
∂S∂Θ
∂S ∂Θ
∂ 2Y
∂S

2

=

ˆ
∂2R
∂S

97

2

sin Θ

(B.18)

(B.19)
(B.20)

ˆ
∂ 2R
=
sin Θ + 2
2
2
∂Θ
∂Θ

ˆ
∂ R ∂sin Θ
∂Θ ∂Θ

∂ 2Y

2
ˆ ∂ sin Θ
+R
2
∂Θ

∂ 2Y
∂ 2R
∂R ∂sin Θ
=
sin Θ +
∂S∂Θ
∂S∂Θ
∂S ∂Θ
∂ 2Z

∂ 2S

∂S

∂S

=
2

(B.21)

(B.22)
(B.23)

2

∂ 2Z

∂2S
=
2
2
∂Θ
∂Θ

(B.24)

∂ 2S
∂2Z
=
∂S∂Θ
∂S∂Θ

(B.25)

where,

n

∂ 2R
∂S

=
2

δk
n

δk
=
2
∂Θ
k=1
∂ 2R
=
∂S∂Θ

2

∂Θ

∂ 2 sin Θ
2

∂Θ

∂S

k=1

∂ 2R

∂ 2 cos Θ

∂ 2 Pk (S, Θ)

∂ 2 Pk (S, Θ)

n

δk
k=1

2

2

∂Θ

∂ 2 Pk (S, Θ)
∂S∂Θ

(B.26)

(B.27)

(B.28)

=

∂cos Θ ∂Θ 2 ∂cos Θ ∂ 2 Θ
+
2
∂Θ
∂Θ
∂Θ
∂Θ

(B.29)

=

∂sin Θ ∂Θ 2 ∂sin Θ ∂ 2 Θ
+
2
∂Θ
∂Θ
∂Θ
∂Θ

(B.30)

∂Θ
=0
∂Θ
∂ 2S
∂S

2

∂ 2S
2

∂Θ

(B.31)

=0

(B.32)

=0

(B.33)

∂ 2S
=0
∂S∂Θ

98

(B.34)

B.1.2

Current Conﬁguration

The polar coordinate (r, θ, s) will be converted to the cartesian coordinate (x, y, z)
ˆ
x = r cos θ ≈ r s, θ cos θ S, Θ
ˆ

(B.35)

ˆ
y = r sin θ ≈ r s, θ sin θ S, Θ
ˆ

(B.36)

z = s ≈ s S, Θ
ˆ

(B.37)

where,

n

s=
ˆ

αk Pk S, Θ

(B.38)

βk Pk S, Θ

(B.39)

γk Pk s, θ

(B.40)

k=1

ˆ
θ=

n
k=1
n

r=
ˆ
k=1

The ﬁrst forms of the surface are
ˆ
∂x
∂ˆ
r
ˆ ˆ ∂cos θ
=
cos θ + r
∂S
∂S
∂S

(B.41)

ˆ
∂x
∂ˆ
r
ˆ ˆ ∂cos θ
=
cos θ + r
∂Θ
∂Θ
∂Θ

(B.42)

ˆ
∂y
∂ˆ
r
ˆ ˆ ∂sin θ
=
sin θ + r
∂S
∂S
∂S

(B.43)

ˆ
∂ˆ
r
∂y
ˆ ˆ ∂sin θ
=
sin θ + r
∂Θ
∂Θ
∂Θ

(B.44)

∂z
∂ˆ
s
≈
∂S
∂S

(B.45)

∂ˆ
s
∂z
≈
∂Θ
∂Θ

(B.46)

99

where,
ˆ
s
∂ˆ
r
∂ˆ ∂s ∂ˆ ∂ˆ ∂θ ∂ θ
r
r
=
+
ˆ
∂s ∂ˆ ∂S ∂θ ∂ θ ∂S
s
∂S

(B.47)

ˆ
∂ˆ
r
∂ˆ ∂s ∂ˆ ∂ˆ ∂θ ∂ θ
r
s
r
=
+
ˆ
∂s ∂ˆ ∂Θ ∂θ ∂ θ ∂Θ
s
∂Θ

(B.48)

ˆ ∂sin θ ∂ θ
ˆ ˆ
∂sin θ
=
ˆ
∂S
∂ θ ∂S

(B.49)

ˆ ˆ
ˆ ∂sin θ ∂ θ
∂sin θ
=
ˆ
∂Θ
∂ θ ∂Θ

(B.50)

ˆ ∂cos θ ∂ θ
ˆ ˆ
∂cos θ
=
ˆ
∂S
∂ θ ∂S

(B.51)

ˆ ∂cos θ ∂ θ
ˆ ˆ
∂cos θ
=
ˆ
∂Θ
∂ θ ∂Θ

(B.52)

∂ˆ
s
=
∂S
∂ˆ
s
=
∂Θ
ˆ
∂θ
=
∂S
ˆ
∂θ
=
∂Θ

n

αk

∂Pk S, Θ
∂S

k=1
n

αk

∂Pk S, Θ
∂Θ

k=1
n

βk

∂Pk S, Θ
∂S

k=1
n

βk

∂Pk S, Θ
∂Θ

k=1

(B.53)

(B.54)

(B.55)

(B.56)

and where,
∂s
2
=
∂ˆ smax − smin
s
ˆ
ˆ

(B.57)

∂θ
2
=
ˆ θmax − θmin
ˆ
ˆ
∂θ

(B.58)

∂ˆ
r
=
∂s
∂ˆ
r
=
∂θ

n

γk
k=1
n

γk

∂Pk s, θ
∂s
∂Pk s, θ

k=1

100

∂θ

(B.59)

(B.60)

ˆ
∂cos θ
ˆ
= − sin θ
ˆ
∂θ

(B.61)

ˆ
∂sin θ
ˆ
= cos θ
ˆ
∂θ

(B.62)

The second forms of the surface are
∂ 2x
∂S

∂S

ˆ
∂ˆ ∂cos θ
r
∂S ∂S

ˆ
cos θ + 2
2

∂ 2x

∂ 2r
ˆ

∂Θ

∂Θ

=
2

ˆ
cos θ + 2
2

+r
ˆ

ˆ
∂ˆ ∂cos θ
r
∂Θ ∂Θ

∂ 2r
ˆ

=
2

ˆ
∂ 2 cos θ

+r
ˆ

∂S

2

ˆ
∂ 2 cos θ
2

∂Θ

2
ˆ
ˆ
ˆ
∂ 2x
∂ 2r
ˆ
r
r
ˆ ∂ˆ ∂cos θ + ∂ˆ ∂cos θ + r ∂ cos θ
=
cos θ +
ˆ
∂S∂Θ
∂S∂Θ
∂Θ ∂S
∂S ∂Θ
∂S∂Θ

∂ 2y
∂S

∂S

ˆ
∂ˆ ∂sin θ
r
∂S ∂S

ˆ
sin θ + 2
2

∂ 2y

∂ 2r
ˆ

∂Θ

∂Θ

=
2

ˆ
sin θ + 2
2

+r
ˆ

ˆ
∂ˆ ∂sin θ
r
∂Θ ∂Θ

∂ 2r
ˆ

=
2

ˆ
∂ 2 sin θ

+r
ˆ

∂S

2

ˆ
∂ 2 sin θ
2

∂Θ

2
ˆ
ˆ
ˆ
∂ 2r
ˆ
r
r
∂ 2y
ˆ ∂ˆ ∂sin θ + ∂ˆ ∂sin θ + r ∂ sin θ
=
sin θ +
ˆ
∂S∂Θ
∂S∂Θ
∂Θ ∂S
∂S ∂Θ
∂S∂Θ

∂2z

∂ 2s
ˆ

∂S

∂S

≈
2

(B.63)

(B.64)

(B.65)
(B.66)

(B.67)

(B.68)
(B.69)

2

∂ 2z

∂ 2s
ˆ
≈
2
2
∂Θ
∂Θ

(B.70)

∂ 2z
∂ 2s
ˆ
≈
∂S∂Θ
∂S∂Θ

(B.71)

Equations in the second forms are
∂ 2 r ∂s 2
ˆ
=
2
2 ∂ˆ
s
∂s
∂S
∂ 2r
ˆ

∂ 2 r ∂θ
ˆ
+
2
ˆ
∂θ ∂ θ

2

s
r
r
∂ˆ 2 ∂ˆ ∂ 2 s ∂ˆ 2 ∂ˆ ∂s ∂ 2 s
s
ˆ
+
+
2 ∂S
∂s ∂ˆ
∂s ∂ˆ ∂S 2
s
s
∂S
ˆ
∂θ
∂S

2

∂ˆ ∂ 2 θ
r
+
ˆ
∂θ ∂ θ2

101

ˆ
∂θ
∂S

2

+

ˆ
∂ˆ ∂θ ∂ 2 θ
r
ˆ
∂θ ∂ θ ∂S 2

(B.72)

∂ 2 r ∂s 2
ˆ
=
2
s
∂s2 ∂ˆ
∂Θ
∂ 2r
ˆ

∂ 2 r ∂θ
ˆ
+
2
ˆ
∂θ ∂ θ

2

∂ˆ 2 ∂ˆ ∂ 2 s ∂ˆ 2 ∂ˆ ∂s ∂ 2 s
s
r
s
r
ˆ
+
+
∂s ∂ˆ2 ∂Θ
∂s ∂ˆ ∂Θ2
s
s
∂Θ
2

ˆ
∂θ
∂Θ

ˆ
∂θ
∂Θ

∂ˆ ∂ 2 θ
r
+
ˆ
∂θ ∂ θ2

2

+

ˆ
∂ˆ ∂θ ∂ 2 θ
r
ˆ
∂θ ∂ θ ∂Θ2

ˆ
∂ 2r
ˆ
∂ 2 r ∂s 2 ∂ˆ ∂s
ˆ
s
∂ˆ ∂ 2 s ∂ˆ ∂s
r
∂ˆ ∂s ∂ 2 s
r
s
=
+
+
2 ∂ˆ
2 ∂S ∂Θ
s ∂S ∂Θ ∂s ∂ˆ
∂s ∂ˆ ∂S∂Θ
s
s
∂s
∂S∂Θ
2 ˆ
ˆ
ˆ
∂ 2 r ∂θ
ˆ
∂ θ ∂θ
∂ˆ ∂ 2 θ ∂ θ ∂θ
r
∂ˆ ∂θ ∂ 2 θ
r
+
+
+
2
ˆ
ˆ
ˆ
∂θ ∂ θ ∂S ∂Θ ∂θ ∂ θ2 ∂S ∂Θ ∂θ ∂ θ ∂S∂Θ
2

(B.73)

(B.74)

ˆ ˆ
∂cos θ ∂ 2 θ
ˆ
∂ θ ∂S 2

(B.75)

ˆ
ˆ 2 ∂cos θ ∂ 2 θ
ˆ ˆ
∂ 2 cos θ ∂ θ
=
+
ˆ
ˆ
∂Θ
∂ θ2
∂ θ ∂Θ2

(B.76)

ˆ ∂ 2 cos θ ∂ θ ∂ θ
ˆ ˆ ˆ ∂cos θ ∂ 2 θ
ˆ
ˆ
∂ 2 cos θ
=
+
ˆ
ˆ
∂S∂Θ
∂ θ2 ∂S ∂Θ
∂ θ ∂S∂Θ

(B.77)

ˆ
∂ 2 cos θ
2

∂S

ˆ
∂ 2 cos θ
2

∂Θ

ˆ
∂ 2 sin θ

ˆ
ˆ
∂ 2 cos θ ∂ θ
=
ˆ
∂S
∂ θ2

+

ˆ
ˆ ˆ
ˆ 2 ∂sin θ ∂ 2 θ
∂ 2 sin θ ∂ θ
=
+
ˆ
ˆ
∂S
∂ θ2
∂ θ ∂S 2

ˆ
ˆ 2 ∂sin θ ∂ 2 θ
ˆ ˆ
∂ 2 sin θ ∂ θ
=
+
ˆ
ˆ
∂Θ
∂ θ2
∂ θ ∂Θ2

(B.79)

ˆ ˆ ˆ ∂sin θ ∂ 2 θ
ˆ
ˆ
ˆ ∂ 2 sin θ ∂ θ ∂ θ
∂ 2 sin θ
=
+
ˆ
ˆ
∂S∂Θ
∂ θ2 ∂S ∂Θ
∂ θ ∂S∂Θ

2

ˆ
∂sin θ
∂S

ˆ
∂ 2 sin θ

∂S

∂
=
∂S

(B.80)

2

∂Θ

where,
∂2s
ˆ
∂S

2

∂2s
ˆ
2

∂Θ

n

=

αk

∂ 2 Pk S, Θ
∂S

k=1
n

=

αk

∂ 2 Pk S, Θ

k=1

∂2s
ˆ
=
∂S∂Θ

2

n

αk
k=1

102

2

∂Θ

∂ 2 Pk S, Θ
∂S∂Θ

(B.78)

(B.81)

(B.82)

(B.83)

n

ˆ
∂ 2θ
∂S

=

2

∂ 2 Pk S, Θ

βk

∂S

k=1
n

ˆ
∂ 2θ

∂ 2 Pk S, Θ

=
βk
2
∂Θ
k=1

∂ 2r
ˆ
=
∂s2

βk
n

γk
k=1
n

=
2

∂ 2 Pk S, Θ
∂S∂Θ

k=1

∂ 2r
ˆ
∂θ

2

∂Θ

n

ˆ
∂ 2θ
=
∂S∂Θ

γk

∂ 2 Pk s, θ
∂s2
∂ 2 Pk s, θ
∂θ

k=1

∂ 2r
ˆ
=
∂s∂θ

2

n

γk

2

∂ 2 Pk s, θ
∂s∂θ

k=1

(B.84)

(B.85)

(B.86)

(B.87)

(B.88)

(B.89)

∂ 2s
=0
∂ˆ2
s
∂2θ
=0
ˆ
∂ θ2
ˆ
∂ 2 cos θ
∂
=
ˆ
ˆ
∂ θ2
∂θ
ˆ
∂ 2 sin θ
∂
=
2
ˆ
ˆ
∂θ
∂θ

B.2

(B.90)
(B.91)

ˆ
∂cos θ
ˆ
∂θ

=

ˆ
∂sin θ
ˆ
∂θ

=

∂
ˆ
ˆ
− sin θ = − cos θ
ˆ
∂θ

(B.92)

∂
ˆ
ˆ
cos θ = − sin θ
ˆ
∂θ

(B.93)

Gaussian quadrature

Gaussian quadrature rule is an approximation of the integral of a function and it is usually
represented by a sum of a weight and a function at speciﬁed points within the domain of
integration. It yields an exact result for polynomials by a suitable choice of the points xi
and weights wi for i = 1, ..., n. The integration domain for the rule is conventionally taken

103

as [-1, 1], so the rule is stated as
n

1
−1

f (x)dx ≈

wi f (xi ).

(B.94)

i=1

An integral over an arbitrary interval [a, b] should be changed into an integral over [-1, 1] to
apply the Gaussian quadrature rule. This change of interval can be done as the following:
b
a

f (x)dx =

b−a 1 b−a
b+a
f(
x+
)dx
2
2
2
−1

(B.95)

After applying the Gaussian quadrature rule, the following approximation can be
n

b

b−a
f (x)dx ≈
wi f
2
a
i=1

b−a
b+a
xi +
2
2

.

(B.96)

For example, the total cross-sectional area of the vessel ring can be divided by constant
ﬁnite domains.

Ne

2π
0

h(θ)r(θ)dθ =

b

i=1 a

h(θ)r(θ)dθ

(B.97)

where N e is total number of ﬁnite intervals over the domain and a and b are the starting
and ending position of each ﬁnite interval. In this case, total domain length is L0 = 2π,
each ﬁnite domain length is L = 2π/N e , then a = (j − 1)L and b = jL for j th ﬁnite domain.
In order to use Gaussian quadrants, the ﬁnite domain length can be converted to [-1 1].
Therefore,
b
a

f (x)dx =

b−a 1
f
2
−1

b+a b−a
+
t dt
2
2

(B.98)

Using the Gaussian quadrants, the integral can be expressed by a summation of function of
Gaussian points and weights. The integration part can be
Ng

1
−1

f (t)dt =

wj f (tj )dt
j=1

104

(B.99)

where N g is total number of Gaussian points. equation (B.97) can be rewritten as
Ne Ng

2π
0

h(θ)r(θ)dθ =

b−a
2

wj
i=1 j=1

h

b+a b−a
+
θ r
2
2 j

b+a b−a
+
θ
2
2 j

dθ
(B.100)

Some low-order Gaussian points and weights are listed below.
Number of points, n

Points, xi

Weights, wi

1

0
√
±1/ 3

2

0
√
± 15/5

8/9

2
3

105

1

5/9

Appendix C
Results
C.1

Information of the aorta sample

In this study, we tested 7 proximal thoracic aortas and 4 distal thoracic aortas were tested.
For the convenience of displaying the results, we name each sample and tabulate their information in Table C.1.

Sample
No.
1
2
3
4
5
6
7
8
9
10
11

Table C.1: Information of all aorta samples
Longitudinal portion
Circumferential region
(Prox/Dis)
(A/L/P/R)
Prox
A, L, P, R
Prox
A, L, P, R
Prox
A, L, P, R
Prox
A, L, P, R
Prox
A, L, P, R
Prox
A, L, P, R
Prox
A, L, P, R
Dis
A, L, P, R
Dis
A, P (L, R for Λ=1.35, 1.40)
Dis
A, P
Dis
A, P

106

Histology
sample
No
Yes
Yes
Yes
Yes
Yes
Yes
No
No
Yes
Yes

Longitudinal Stress (kPa)

Stress-stretch plots

300

200

100

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

140

60

1.1
1.3
Circumferential Stretch

140

100

60

0.9

1.1
1.3
Circumferential Stretch

A
L
P
R

100

0.9

Longitudinal Stress (kPa)

0
0.9

Longitudinal Stress (kPa)

Circumferential Stress (kPa)

Circumferential Stress (kPa)

Circumferential Stress (kPa)

C.2

1.1
1.3
Circumferential Stretch

140

100

60

0.9

1.1
1.3
Circumferential Stretch

Fig. C.1: Stress-stretch plots for the proximal thoracic aorta at a ﬁxed stretch ratio of 1.35,
1.40, and 1.45 for sample 1

107

Longitudinal Stress (kPa)

200

100

300

200

100

0
0.9

300

200

100

0
0.9

100

100

1.1
1.3
Circumferential Stretch

150

100

50
0.9

1.1
1.3
Circumferential Stretch

1.1
1.3
Circumferential Stretch

150

50
0.9

1.1
1.3
Circumferential Stretch

A
L
P
R

150

50
0.9

1.1
1.3
Circumferential Stretch
Longitudinal Stress (kPa)

0
0.9

Longitudinal Stress (kPa)

Circumferential Stress (kPa)
Circumferential Stress (kPa)
Circumferential Stress (kPa)

300

1.1
1.3
Circumferential Stretch

Fig. C.2: Stress-stretch plots for the proximal thoracic aorta at a ﬁxed stretch ratio of 1.35,
1.40, and 1.45 for sample 2

108

Longitudinal Stress (kPa)

200

100

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

180

80
1.1
1.3
Circumferential Stretch

180

130

80
0.9

1.1
1.3
Circumferential Stretch

A
L
P
R

130

0.9

Longitudinal Stress (kPa)

0
0.9

Longitudinal Stress (kPa)

Circumferential Stress (kPa)
Circumferential Stress (kPa)
Circumferential Stress (kPa)

300

1.1
1.3
Circumferential Stretch

180

130

80
0.9

1.1
1.3
Circumferential Stretch

Fig. C.3: Stress-stretch plots for the proximal thoracic aorta at a ﬁxed stretch ratio of 1.35,
1.40, and 1.45 for sample 3

109

Longitudinal Stress (kPa)

200

100

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

80

1.1
1.3
Circumferential Stretch

180

130

80

0.9

1.1
1.3
Circumferential Stretch

A
L
P
R

130

0.9

Longitudinal Stress (kPa)

0
0.9

Longitudinal Stress (kPa)

Circumferential Stress (kPa)
Circumferential Stress (kPa)
Circumferential Stress (kPa)

300

180

1.1
1.3
Circumferential Stretch

180

130

80

0.9

1.1
1.3
Circumferential Stretch

Fig. C.4: Stress-stretch plots for the proximal thoracic aorta at a ﬁxed stretch ratio of 1.35,
1.40, and 1.45 for sample 4

110

Longitudinal Stress (kPa)

200

100

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

130

80
1.1
1.3
Circumferential Stretch

180

130

80
0.9

1.1
1.3
Circumferential Stretch

A
L
P
R

180

0.9

Longitudinal Stress (kPa)

0
0.9

Longitudinal Stress (kPa)

Circumferential Stress (kPa)
Circumferential Stress (kPa)
Circumferential Stress (kPa)

300

1.1
1.3
Circumferential Stretch

180

130

80
0.9

1.1
1.3
Circumferential Stretch

Fig. C.5: Stress-stretch plots for the proximal thoracic aorta at a ﬁxed stretch ratio of 1.35,
1.40, and 1.45 for sample 5

111

Longitudinal Stress (kPa)

200

100

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

130

80
1.1
1.3
Circumferential Stretch

180

130

80
0.9

1.1
1.3
Circumferential Stretch

A
L
P
R

180

0.9

Longitudinal Stress (kPa)

0
0.9

Longitudinal Stress (kPa)

Circumferential Stress (kPa)
Circumferential Stress (kPa)
Circumferential Stress (kPa)

300

1.1
1.3
Circumferential Stretch

180

130

80
0.9

1.1
1.3
Circumferential Stretch

Fig. C.6: Stress-stretch plots for the proximal thoracic aorta at a ﬁxed stretch ratio of 1.35,
1.40, and 1.45 for sample 6

112

Longitudinal Stress (kPa)

200

100

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

130

80
1.1
1.3
Circumferential Stretch

180

130

80
0.9

1.1
1.3
Circumferential Stretch

180

130

80
0.9

1.1
1.3
Circumferential Stretch

A
L
P
R

180

0.9

Longitudinal Stress (kPa)

0
0.9

Longitudinal Stress (kPa)

Circumferential Stress (kPa)
Circumferential Stress (kPa)
Circumferential Stress (kPa)

300

1.1
1.3
Circumferential Stretch

Fig. C.7: Stress-stretch plots for the proximal thoracic aorta at a ﬁxed stretch ratio of 1.35,
1.40, and 1.45 for sample 7

113

Longitudinal Stress (kPa)

200

100

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

100

1.1
1.3
Circumferential Stretch

130

100

70

0.9

1.1
1.3
Circumferential Stretch

A
L
P
R

70

0.9

Longitudinal Stress (kPa)

0
0.9

Longitudinal Stress (kPa)

Circumferential Stress (kPa)
Circumferential Stress (kPa)
Circumferential Stress (kPa)

300

130

1.1
1.3
Circumferential Stretch

130

100

70

0.9

1.1
1.3
Circumferential Stretch

Fig. C.8: Stress-stretch plots for the distal thoracic aorta at a ﬁxed stretch ratio of 1.35,
1.40, and 1.45 sample 8

114

Longitudinal Stress (kPa)

200

100

1.1
1.3
Circumferential Stretch
Longitudinal Stress (kPa)

0
0.9

300

200

100

0
0.9

1.1
1.3
Circumferential Stretch
Longitudinal Stress (kPa)

Circumferential Stress (kPa)

Circumferential Stress (kPa)
Circumferential Stress (kPa)

300

300

200

100

0
0.9

1.1
1.3
Circumferential Stretch

A
L
P
R

230
180
130
80
0.9

1.1
1.3
Circumferential Stretch

230
180
130
80
0.9

1.1
1.3
Circumferential Stretch

230
180
130
80
0.9

1.1
1.3
Circumferential Stretch

Fig. C.9: Stress-stretch plots for the distal thoracic aorta at a ﬁxed stretch ratio of 1.35,
1.40, and 1.45 sample 9

115

Longitudinal Stress (kPa)

200

100

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

300

200

100

0
0.9

100
70

1.1
1.3
Circumferential Stretch

1.1
1.3
Circumferential Stretch

160
130
100
70
0.9

1.1
1.3
Circumferential Stretch

A
P

130

0.9

Longitudinal Stress (kPa)

0
0.9

Longitudinal Stress (kPa)

Circumferential Stress (kPa)
Circumferential Stress (kPa)

Circumferential Stress (kPa)

300

160

1.1
1.3
Circumferential Stretch

160
130
100
70
0.9

1.1
1.3
Circumferential Stretch

Fig. C.10: Stress-stretch plots for the distal thoracic aorta at a ﬁxed stretch ratio of 1.35,
1.40, and 1.45 sample 10

116

Longitudinal Stress (kPa)

200

100

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

1.1
1.3
Circumferential Stretch

300

200

100

0
0.9

140

100
1.1
1.3
Circumferential Stretch

180

140

100
0.9

1.1
1.3
Circumferential Stretch

A
P

180

0.9

Longitudinal Stress (kPa)

0
0.9

Longitudinal Stress (kPa)

Circumferential Stress (kPa)
Circumferential Stress (kPa)
Circumferential Stress (kPa)

300

1.1
1.3
Circumferential Stretch

180

140

100
0.9

1.1
1.3
Circumferential Stretch

Fig. C.11: Stress-stretch plots for the distal thoracic aorta at a ﬁxed stretch ratio of 1.35,
1.40, and 1.45 sample 11

117

C.3

Stretch, stress, and stiﬀness

Table C.2: The mean and the standard deviation of the circumferential stretch of the thoracic
aorta at the transmural pressure of 8.00 kPa (60 mm Hg), 12.00 kPa (90 mm Hg), and 16.00
kPa (120 mm Hg) for each proximal circumferential region
Λ=1.35
Anterior
Left lateral
Posterior
Right lateral
Λ=1.40
Anterior
Left lateral
Posterior
Right lateral
Λ=1.45
Anterior
Left lateral
Posterior
Right lateral

8.00 kPa
1.10 ± 0.04
1.11 ± 0.02
1.12 ± 0.03
1.10 ± 0.02
8.00 kPa
1.09 ± 0.04
1.10 ± 0.02
1.11 ± 0.03
1.10 ± 0.02
8.00 kPa
1.07 ± 0.05
1.09 ± 0.02
1.11 ± 0.04
1.09 ± 0.02

12.00 kPa
1.20 ± 0.06
1.22 ± 0.03
1.22 ± 0.05
1.21 ± 0.04
12.00 kPa
1.19 ± 0.07
1.21 ± 0.03
1.22 ± 0.05
1.20 ± 0.04
12.00 kPa
1.17 ± 0.07
1.20 ± 0.03
1.21 ± 0.05
1.19 ± 0.04

16.00 kPa
1.29 ± 0.08
1.30 ± 0.03
1.29 ± 0.06
1.29 ± 0.05
16.00 kPa
1.28 ± 0.08
1.28 ± 0.03
1.28 ± 0.06
1.28 ± 0.05
16.00 kPa
1.25 ± 0.08
1.27 ± 0.03
1.28 ± 0.06
1.27 ± 0.05

(a) The proximal thoracic aorta
Λ=1.35
Anterior
Posterior
Λ=1.40
Anterior
Posterior
Λ=1.45
Anterior
Posterior

8.00 kPa
1.12 ± 0.07
1.13 ± 0.06
8.00 kPa
1.11 ± 0.08
1.12 ± 0.06
8.00 kPa
1.09 ± 0.09
1.11 ± 0.06

12.00 kPa
1.23 ± 0.10
1.22 ± 0.09
12.00 kPa
1.21 ± 0.11
1.20 ± 0.08
12.00 kPa
1.20 ± 0.11
1.20 ± 0.09

16.00 kPa
1.31 ± 0.11
1.27 ± 0.10
16.00 kPa
1.28 ± 0.11
1.25 ± 0.09
16.00 kPa
1.27 ± 0.12
1.25 ± 0.10

(b) The left and right lateral regions

118

Table C.3: The mean and the standard deviation of the circumferential stress of the thoracic
aorta at the transmural pressure of 8.00 kPa, 12.00 kPa, and 16.00 kPa for each proximal
circumferential region (unit: kPa)
Λ=1.35
Anterior
Left lateral
Posterior
Right lateral
Λ=1.40
Anterior
Left lateral
Posterior
Right lateral
Λ=1.45
Anterior
Left lateral
Posterior
Right lateral

8.00 kPa
45 ± 8
55 ± 8
63 ± 9
52 ± 9
8.00 kPa
46 ± 8
56 ± 9
64 ± 9
53 ± 8
8.00 kPa
46 ± 9
53 ± 8
67 ± 10
54 ± 8

12.00 kPa 16.00 kPa
82 ± 16
126 ± 25
100 ± 16 154 ± 22
113 ± 17 172 ± 29
96 ± 17
146 ± 29
12.00 kPa 16.00 kPa
83 ± 16
128 ± 25
102 ± 15 155 ± 24
116 ± 19 176 ± 30
97 ± 16
150 ± 26
12.00 kPa 16.00 kPa
84 ± 17
130 ± 27
104 ± 17 158 ± 25
120 ± 21 181 ± 34
99 ± 18
152 ± 27

(a) The proximal thoracic aorta
Λ=1.35
Anterior
Posterior
Λ=1.40
Anterior
Posterior
Λ=1.45
Anterior
Posterior

8.00 kPa 12.00 kPa
52 ± 11
95 ± 20
79 ± 9
142 ± 15
8.00 kPa 12.00 kPa
54 ± 10
97 ± 19
80 ± 9
141 ± 15
8.00 kPa 12.00 kPa
54 ± 10
97 ± 19
82 ± 10 148 ± 24

16.00 kPa
146 ± 31
204 ± 26
16.00 kPa
146 ± 29
203 ± 21
16.00 kPa
148 ± 30
215 ± 32

(b) The distal thoracic aorta

119

Table C.4: The mean and the standard deviation of the circumferential stiﬀness of the
thoracic aorta at the transmural pressure of 8.00 kPa, 12.00 kPa, and 16.00 kPa for each
proximal circumferential region (unit: kPa)
Λ=1.35
Anterior
Left lateral
Posterior
Right lateral
Λ=1.40
Anterior
Left lateral
Posterior
Right lateral
Λ=1.45
Anterior
Left lateral
Posterior
Right lateral

8.00 kPa 12.00 kPa
310 ± 47 416 ± 104
359 ± 33 523 ± 99
420 ± 58 634 ± 91
347 ± 50 489 ± 82
8.00 kPa 12.00 kPa
310 ± 47 416 ± 104
360 ± 33 523 ± 99
420 ± 58 634 ± 91
347 ± 51 487 ± 82
8.00 kPa 12.00 kPa
321 ± 51 441 ± 84
370 ± 42 552 ± 98
436 ± 57 674 ± 67
362 ± 54 535 ± 87

16.00 kPa
671 ± 185
1065 ± 255
1246 ± 136
964 ± 117
16.00 kPa
724 ± 175
1118 ± 230
1304 ± 188
1043 ± 124
16.00 kPa
765 ± 177
1140 ± 195
1394 ± 344
1109 ± 153

(a) The proximal thoracic aorta
Λ=1.35
Anterior
Posterior
Λ=1.40
Anterior
Posterior
Λ=1.45
Anterior
Posterior

8.00 kPa
348 ± 61
525 ± 84
8.00 kPa
332 ± 38
592 ± 100
8.00 kPa
345 ± 54
563 ± 95

12.00 kPa
468 ± 70
1012 ± 346
12.00 kPa
513 ± 23
1018 ± 245
12.00 kPa
516 ± 69
1085 ± 244

16.00 kPa
989 ± 418
2151 ± 444
16.00 kPa
945 ± 416
1842 ± 188
16.00 kPa
1039 ± 394
1925 ± 768

(b) The distal thoracic aorta

120

Table C.5: The mean and the standard deviation of the pressure-strain elastic modulus of the
proximal and distal portions of the thoracic aorta for each proximal circumferential region
(unit: kPa)
Λ=1.35
Proximal Distal
Anterior
52 ± 19 56 ± 8
Left lateral
56 ± 10
Posterior
62 ± 14 89 ± 28
Right lateral 54 ± 9
Λ=1.40
Proximal Distal
Anterior
52 ± 19 56 ± 8
Left lateral
56 ± 10
Posterior
62 ± 14 89 ± 28
Right lateral 54 ± 9
Λ=1.45
Proximal Distal
Anterior
52 ± 19 56 ± 8
Left lateral
56 ± 10
Posterior
62 ± 14 89 ± 28
Right lateral 54 ± 9
-

121

C.4

Material parameters

Table C.6: Material parameters for the sample 1
Parameters \ Regions
Anterior
Left lateral
Posterior
2
2
c1 (kPa)
3.49 ×10
2.73 ×10
3.79 ×102
c2 (kPa)
9.11 ×101
1.64 ×102
2.62 ×101
c3
5.79
9.57
28.56
−8 1.11 ×10−15 8.29 ×10−8
c4 (Pa)
2.37 ×10
c5
1.67 ×10−7 2.00 ×10−16 1.54 ×10−6
e
G1
1.19
1.35
1.30
Ge
1.12
1.07
1.11
2
c
Gh
1.11
1.09
1.09
Gm
1.10
1.10
1.10
h
◦)
α(
20.14
10.00
28.80

Right lateral
2.49 ×102
1.01 ×102
17.64
7.69 ×10−8
1.47
1.30
1.17
1.08
1.10
35.83

Table C.7: Material parameters for the sample 2
Parameters \ Regions
Anterior
Left lateral
Posterior
2
2
c1 (kPa)
2.85 ×10
2.79 ×10
2.96 ×102
c2 (kPa)
2.04 ×102
3.02 ×102
2.72 ×102
c3
9.89
8.02
10.68
−7 9.54 ×10−13 2.47 ×10−8
c4 (Pa)
2.81 ×10
c5
4.78 ×10−6 1.80 ×10−3 4.79 ×10−5
Ge
1.17
1.28
1.22
1
e
G2
1.21
1.15
1.20
Gc
1.09
1.09
1.09
h
m
Gh
1.10
1.10
1.10
◦)
α(
38.99
26.23
39.73

Right lateral
3.23 ×102
2.39 ×102
7.56
5.42 ×10−9
14.38
1.24
1.14
1.10
1.10
25.42

122

Table C.8: Material parameters for the sample 3
Parameters \ Regions
Anterior
Left lateral
Posterior
2
2
c1 (kPa)
4.97 ×10
5.34 ×10
5.54 ×102
c2 (kPa)
1.13 ×102
5.89 ×101
1.42 ×102
c3
3.44
3.074
2.93
−10 2.62 ×10−10 2.98 ×10−10
c4 (Pa)
1.74 ×10
c5
2.92 ×10−10 5.24 ×10−8
0.045
e
G1
1.09
1.06
1.13
e
G2
1.24
1.23
1.20
c
Gh
1.08
1.09
1.09
Gm
1.10
1.10
1.10
h
◦)
α(
30.44
43.88
40.12

Right lateral
4.65 ×102
1.78 ×102
1.75
1.09 ×10−9
9.31 ×10−5
1.14
1.24
1.09
1.10
33.75

Table C.9: Material parameters for the sample 4
Parameters \ Regions
Anterior
Left lateral
Posterior
2
2
c1 (kPa)
4.24 ×10
3.97 ×10
3.02 ×102
1
2
c2 (kPa)
5.10 ×10
1.85 ×10
3.06 ×102
c3
0.72
1.13
1.19
−11 9.68 ×10−11 1.59 ×10−10
c4 (Pa)
1.44 ×10
c5
1.12 ×10−9
1.39
0.04
e
G1
1.05
1.07
1.13
e
G2
1.20
1.16
1.17
c
Gh
1.08
1.09
1.09
Gm
1.10
1.10
1.10
h
◦)
α(
27.52
27.18
26.90

Right lateral
4.32 ×102
9.60 ×101
2.09
7.86 ×10−10
0.15
1.09
1.18
1.09
1.10
34.57

Table C.10: Material parameters for the sample 5
Parameters \ Regions
Anterior
Left lateral
Posterior
2
2
c1 (kPa)
4.87 ×10
3.34 ×10
2.60 ×102
c2 (kPa)
1.80 ×102
3.64 ×102
2.89 ×102
c3
0.12
0.49
4.50
−10 9.34 ×10−10 1.01 ×10−6
c4 (Pa)
2.69 ×10
c5
4.99 ×10−7 2.16 ×10−10 2.80 ×10−7
Ge
1.12
1.18
1.23
1
e
G2
1.15
1.14
1.21
Gc
1.08
1.09
1.08
h
m
Gh
1.10
1.10
1.10
◦)
α(
20.89
15.65
42.12

Right lateral
2.98 ×102
2.88 ×102
4.79
1.31 ×10−9
1.48 ×10−7
1.19
1.17
1.09
1.10
28.77

123

Table C.11: Material parameters for the sample 6
Parameters \ Regions
Anterior
Left lateral
Posterior
2
2
c1 (kPa)
3.96 ×10
3.49 ×10
2.83 ×102
c2 (kPa)
1.98 ×102
3.06 ×102
3.96 ×102
c3
2.57
2.12
1.80
−10 4.81 ×10−10 3.39 ×10−10
c4 (Pa)
6.06 ×10
c5
1.53×10−8
4.92×10−12
9.12×10−8
e
G1
1.11
1.18
1.28
e
G2
1.14
1.06
1.03
c
Gh
1.10
1.11
1.09
Gm
1.10
1.10
1.10
h
◦)
α(
19.58
15.60
10.34

Right lateral
3.36 ×102
2.63 ×102
4.20
7.88 ×10−8
3.18 ×10−7
1.22
1.10
1.08
1.10
24.48

Table C.12: Material parameters for the sample 7
Parameters \ Regions
Anterior
Left lateral
Posterior
Right lateral
2
2
2
c1 (kPa)
3.73 ×10
3.18 ×10
3.32 ×10
3.15 ×102
2
2
2
c2 (kPa)
1.98 ×10
2.04 ×10
2.03 ×10
2.15 ×102
c3
0.84
4.28
4.68
2.82
−11 3.02 ×10−9 1.51 ×10−5 6.10 ×10−10
c4 (Pa)
9.57 ×10
c5
1.53 ×10−10
0.027883
1.98 ×10−6 6.21 ×10−6
e
G1
1.17
1.23
1.23
1.23
e
G2
1.21
1.18
1.18
1.18
c
Gh
1.09
1.11
1.09
1.12
Gm
1.10
1.10
1.10
1.10
h
◦)
α(
31.19
32.74
42.98
27.18

Table C.13: Material parameters for the sample 8
Parameters \ Regions
Anterior
Posterior
2
c1 (kPa)
2.98 ×10
3.22 ×102
c2 (kPa)
1.31 ×102
1.99 ×102
c3
3.01
2.20
−9 1.17 ×10−9
c4 (Pa)
1.76 ×10
c5
1.43 ×10−7 1.80 ×10−4
Ge
1.1
1.24
1
e
G2
1.20
1.17
Gc
1.07
1.11
h
m
Gh
1.10
1.10
◦)
α(
30.09
50.00

124

Table C.14: Material parameters for the sample 9
Parameters \ Regions
Anterior
Posterior
2
c1 (kPa)
5.99 ×10
1.79 ×102
c2 (kPa)
8.42 ×101
8.33 ×102
c3
1.21
1.58
−8 4.70 ×10−10
c4 (Pa)
3.55 ×10
c5
1.56 ×10−5 8.56 ×10−10
e
G1
1.02
1.18
e
G2
1.25
1.20
c
Gh
1.10
1.11
Gm
1.10
1.10
h
◦)
α(
23.62
23.81

Table C.15: Material parameters for the sample 10
Parameters \ Regions
Anterior
Posterior
2
c1 (kPa)
3.13 ×10
3.06 ×102
2
c2 (kPa)
2.41 ×10
4.40 ×102
c3
1.32
3.40
−14 3.93 ×10−9
c4 (Pa)
2.45 ×10
c5
9.29 ×10−11 1.11 ×10−16
e
G1
1.20
1.37
e
G2
1.20
1.05
c
Gh
1.10
1.08
Gm
1.10
1.10
h
◦)
α(
29.97
17.51

Table C.16: Material parameters for the sample 11
Parameters \ Regions
Anterior
Posterior
2
c1 (kPa)
4.45 ×10
2.82 ×102
c2 (kPa)
2.14 ×102
3.94 ×102
c3
1.12
7.64
−10 4.05 ×10−10
c4 (Pa)
4.03 ×10
c5
6.50 ×10−9 2.00 ×10−5
Ge
1.06
1.25
1
e
G2
1.20
1.17
Gc
1.09
1.10
h
m
Gh
1.10
1.10
◦)
α(
18.47
28.71

125

BIBLIOGRAPHY

126

Bibliography
American-Heart-Association, 2011. Heart disease and stroke statistics - 2011 update: A
report from the American heart association. Circulation Journal of the American Heart
Association 123, e18–e209.
Angouras, D., Sokolis, D.P., Dosios, T., Kostomitsopoulos, N., Boudoulas, H., Skalkeas, G.,
Karayannacos, P.E., 2000. Eﬀect of impaired vasa vasorum ﬂow on the structure and
mechanics of the thoracic aorta: Implications for the pathogenesis of aortic dissection.
European Journal of Cardio-Thoracic Surgery 17, 468–473.
Avril, S., Badel, P., Duprey, A., 2010. Anisotropic and hyperelastic identiﬁcation of in
vitro human arteries from full-ﬁeld optical measurements. Journal of Biomechanics 43,
2978–2985.
Baek, S., Rajagopal, K., Humphrey, J., 2006. A theoretical model of enlarging intracranial
fusiform aneurysms. Journal of Biomechanical Engineering 128, 142–149.
Blondel, W.C.P.M., Didelon, J., Maurice, G., Carteaux, J.P., Wang, X., Stoltz, J.F., 2001.
Investigation of 3-D mechanical properties of blood vessels using a new in vitro tests
system: Results on sheep common carotid arteries. IEEE Transactions on Biomedical
Engineering 48, 442–451.
Cannon, C.P., O’Gara, P.T., 2006. Critical pathways in cardiovascular medicine. Lippincott
Williams & Wilkins. second edition.
Carew, T.E., Vaishnav, R.N., Patel, D.J., 1968. Compressibility of the arterial wall. Circulation Research 23, 61–68.
Choi, G., Cheng, C., Wilson, N., Taylor, C., 2009. Methods for quantifying three-dimensional
deformation of arteries due to pulsatile and nonpulsatile forces: implications for the design
of stents and stent grafts. Annals of biomedical engineering 37, 14–33.
Choudhury, N., Bouchot, O., Rouleau, L., Tremblay, D., Cartier, R., Butany, J., Mongrain,
R., Leask, R.L., 2009. Local mechanical and structural properties of healthy and diseased
human ascending aorta tissue. Cardiovascular Pathology 18, 83–91.
Chuong, C., Fung, Y., 1983. Three-dimensional stress distribution in arteries. Journal of
biomechanical engineering 105, 268–274.

127

Chuong, C.J., Fung, Y.C., 1986. On residual stresses in arteries. Journal of Biomechanical
Engineering 108, 189–192.
Danpinid, A., Luo, J., Vappou, J., Terdtoon, P., Konofagou, E.E., 2010. In vivo characterization of the aortic wall stress-strain relationship. Ultrasonics 50, 654–665.
Davidson, J., Hill, K., Alford, J., 1986. Developmental changes in collagen and elastin
biosynthesis in the porcine aorta* 1. Developmental Biology 118, 103–111.
Draney, M.T., Arko, F.R., Alley, M.T., Markl, M., Herfkens, R.J., Pelc, N.J., Zarins, C.K.,
Taylor, C.A., 2004. Quantiﬁcation of vessel wall motion and cyclic strain using cine phase
contrast MRI: In vivo validation in the porcine aorta. Magnetic Resonance in Medicine
52, 286–295.
Draney, M.T., Herfkens, R.J., Hughes, T.J.R., Pelc, N.J., Wedding, K.L., Zarins, C.K.,
Taylor, C.A., 2002. Quantiﬁcation of vessel wall cyclic strain using cine phase contrast
magnetic resonance imaging. Annals of Biomedical Engineering 30, 1033–1045.
Everett, W.N., Shih, P., Humphrey, J.D., 2005. A bi-plane video-based system for studying
the mechanics of arterial bifurcations. Experimental Mechanics 45, 377–382.
Feigl, E.O., Peterson, L.H., Jones, A.W., 1963. Mechanical and chemical properties of
arteries in experimental hypertension. Journal of Clinical Investigation 42, 1640–1647.
Fischer, G., Llaurado, J., 1966. Collagen and elastin content in canine arteries selected from
functionally diﬀerent vascular beds. Circulation Research 19, 394–399.
Fung, Y., 1993. Biomechanics: Mechanical Properties of Living Tissues. Springer. second
edition.
Fung, Y., Fronek, K., Patitucci, P., 1979. Pseudoelasticity of arteries and the choice of
its mathematical expression. American Journal of Physiology - Heart and Circulatory
Physiology 237, H620–H631.
Genovese, K., 2009. A video-optical system for time-resolved whole-body measurement on
vascular segments. Optics and Lasers in Engineering 47, 995–1008.
Goergen, C., Johnson, B., Greve, J., Taylor, C., Zarins, C., 2007. Increased anterior abdominal aortic wall motion: possible role in aneurysm pathogenesis and design of endovascular
devices. Journal of Endovascular Therapy 14, 574–584.
Groenink, M., Langerak, S.E., Vanbavel, E., van der Wall, E.E., Mulder, B.J.M., van der
Wal, A.C., Spaan, J.A.E., 1999. The inﬂuence of aging and aortic stiﬀness on permanent
dilation and breaking stress of the thoracic descending aorta. Cardiovascular Research 43,
471–480.
Gundiah, N., Ratcliﬀe, M.B., Pruitt, L.A., 2007. Determination of strain energy function for
arterial elastin: experiments using histology and mechanical tests. Journal of Biomechanics
40, 586–594.
128

Guo, X., Kassab, G.S., 2004. Distribution of stress and strain along the porcine aorta and
coronary arterial tree. American Journal of Physiology - Heart and Circulatory Physiology
286, H2361–H2368.
Guo, X., Lanir, Y., Kassab, G.S., 2007. Eﬀect of osmolarity on the zero-stress state and
mechanical properties of aorta. American Journal of Physiology - Heart and Circulatory
Physiology 293, H2328–H2334.
Guo, X., Lu, X., Kassab, G.S., 2005. Transmural strain distribution in the blood vessel wall.
American Journal of Physiology- Heart and Circulatory Physiology 288, H881–H886.
Halloran, B., Davis, V., McManus, B., Lynch, T., Baxter, B., 1995. Localization of aortic
disease is associated with intrinsic diﬀerences in aortic structure. Journal of Surgical
Research 59, 17–22.
Han, H.C., Fung, Y.C., 1995. Longitudinal strain of canine and porcine aortas. Journal of
Biomechanics 28, 637–641.
Haskett, D., Johnson, G., Zhou, A., Utzinger, U., Vande Geest, J., 2010. Microstructural and
biomechanical alterations of the human aorta as a function of age and location. Biomechanics and Modeling in Mechanobiology 9, 725–736.
He, C.M., Roach, M.R., 1994. The composition and mechanical properties of abdominal
aortic aneurysms. Journal of vascular surgery 20, 6–13.
Holzapfel, G., Weizsacker, H., 1998. Biomechanical behavior of the arterial wall and its
numerical characterization. Computers in Biology and Medicine 28, 377–392.
Holzapfel, G.A., Gasser, T.C., Ogden, R.W., 2000. A new constitutive framework for arterial
wall mechanics and a comparative study of material models. Journal of elasticity 61, 1–48.
Holzapfel, G.A., Gasser, T.C., Stadler, M., 2002. A structural model for the viscoelastic
behavior of arterial walls: Continuum formulation and ﬁnite element analysis. European
Journal of Mechanics-A/Solids 21, 441–463.
Hsu, F.P.K., Liu, A.M.C., Downs, J., Rigamonti, D., Humphrey, J.D., 1995. A triplane
video-based experimental system for studying axisymmetrically inﬂated biomembranes.
IEEE Transactions on Biomedical Engineering 42, 442–450.
Hu, J.J., Fossum, T.W., Miller, M.W., Xu, H., Liu, J.C., Humphrey, J.D., 2007. Biomechanics of the porcine basilar artery in hypertension. Annals of Biomedical Engineering
35, 19–29.
Huang, Y., Guo, X., Kassab, G.S., 2006. Axial nonuniformity of geometric and mechanical
properties of mouse aorta is increased during postnatal growth. American Journal of
Physiology- Heart and Circulatory Physiology 290, H657–H664.
Humphrey, J., 1995. Mechanics of the arterial wall: review and directions. Critical Reviews
in Biomedical Engineering 23, 1–162.
129

Humphrey, J., Rajagopal, K., 2002. A constrained mixture model for growth and remodeling
of soft tissues. Mathematical Models and Methods in Applied Sciences 12, 407–430.
Humphrey, J.D., 2001. Cardiovascular Solid Mechanics: Cells, Tissues, and Organs. Springer,
New York.
Humphrey, J.D., 2008. Vascular adaptation and mechanical homeostasis at tissue, cellular,
and sub-cellular levels. Cell Biochemistry and Biophysics 50, 53–78.
Humphrey, J.D., Kang, T., Sakarda, P., Anjanappa, M., 1993. Computer-aided vascular
experimentation: A new electromechanical test system. Annals of Biomedical Engineering
21, 33–43.
Iliopoulos, D.C., Deveja, R.P., Kritharis, E.P., Perrea, D., Sionis, G.D., Toutouzas, K.,
Stefanadis, C., Sokolis, D.P., 2009a. Regional and directional variations in the mechanical
properties of ascending thoracic aortic aneurysms. Medical Engineering & Physics 31, 1–9.
Iliopoulos, D.C., Kritharis, E.P., Giagini, A.T., Papadodima, S.A., Sokolis, D.P., 2009b.
Ascending thoracic aortic aneurysms are associated with compositional remodeling and
vessel stiﬀening but not weakening in age-matched subjects. The Journal of Thoracic and
Cardiovascular Surgery 137, 101–109.
Isselbacher, E., 2005. Thoracic and abdominal aortic aneurysms. Circulation 111, 816–828.
Lally, C., Reid, A.J., Prendergast, P.J., 2004. Elastic behavior of porcine coronary artery
tissue under uniaxial and equibiaxial tension. Annals of Biomedical Engineering 32, 1355–
1364.
Langewouters, G.J., Wesseling, K.H., Goedhard, W.J.A., 1984. The static elastic properties
of 45 human thoracic and 20 abdominal aortas in vitro and the parameters of a new model.
Journal of Biomechanics 17, 425–435.
Lillie, M.A., Gosline, J.M., 2007. Mechanical properties of elastin along the thoracic aorta
in the pig. Journal of Biomechanics 40, 2214–2221.
Matsumoto, T., Goto, T., Furukawa, T., Sato, M., 2004. Residual stress and strain in the
lamellar unit of the porcine aorta: experiment and analysis. Journal of biomechanics 37,
807–815.
Menashi, S., Campa, J., Greenhalgh, R., Powell, J., 1987. Collagen in abdominal aortic
aneurysm: typing, content, and degradation. Journal of vascular surgery: oﬃcial publication, the Society for Vascular Surgery [and] International Society for Cardiovascular
Surgery, North American Chapter 6, 578–582.
O’Connell, M.K., Murthy, S., Phan, S., Xu, C., Buchanan, J.A., Spilker, R., Dalman, R.L.,
Zarins, C.K., Denk, W., Taylor, C.A., 2008. The three-dimensional micro-and nanostructure of the aortic medial lamellar unit measured using 3D confocal and electron microscopy
imaging. Matrix Biology 27, 171–181.
130

Okamoto, R.J., Wagenseil, J.E., DeLong, W.R., Peterson, S.J., Kouchoukos, N.T., Sundt III,
T.M., 2002. Mechanical properties of dilated human ascending aorta. Annals of Biomedical
Engineering 30, 624–635.
Pandit, A., Lu, X., Wang, C., Kassab, G.S., 2005. Biaxial elastic material properties of
porcine coronary media and adventitia. American Journal of Physiology - Heart and
Circulatory Physiology 288, H2581–H2587.
Participants, T.U.K.S.A.T., 2002. Long-term outcomes of immediate repair compared with
surveillance of small abdominal aortic aneurysms. New England Journal of Medicine 346,
1445–1452.
Patel, D., Janicki, J., Carew, T., 1969. Static anisotropic elastic properties of the aorta in
living dogs. Circulation Research 25, 765–779.
Peterson, L.H., Jensen, R.E., Parnell, J., 1960. Mechanical properties of arteries in vivo.
Circulation Research 8, 622–639.
van Prehn, J., van Herwaarden, J., Vincken, K., Verhagen, H., Moll, F., Bartels, L., 2009a.
Asymmetric aortic expansion of the aneurysm neck: Analysis and visualization of shape
changes with electrocardiogram-gated magnetic resonance imaging. Journal of Vascular
Surgery 49, 1395–1402.
van Prehn, J., Vincken, K.L., Sprinkhuizen, S.M., Viergever, M.A., van Keulen, J.W., van
Herwaarden, J.A., Moll, F.L., Bartels, L.W., 2009b. Aortic pulsatile distention in young
healthy volunteers is asymmetric: Analysis with ECG-gated MRI. European Journal of
Vascular and Endovascular Surgery 37, 168–174.
Purslow, P., 1983. Positional variations in fracture toughness, stiﬀness and strength of
descending thoracic pig aorta. Journal of Biomechanics 16, 947–953.
Rachev, A., Hayashi, K., 1999. Theoretical study of the eﬀects of vascular smooth muscle
contraction on strain and stress distributions in arteries. Annals of biomedical engineering
27, 459–468.
Raghavan, M.L., Kratzberg, J., Castro de Tolosa, E.M., Hanaoka, M.M., Walker, P., da Silva,
E.S., 2006. Regional distribution of wall thickness and failure properties of human abdominal aortic aneurysm. Journal of biomechanics 39, 3010–3016.
Rezakhaniha, R., Fonck, E., Genoud, C., Stergiopulos, N., 2011. Role of elastin anisotropy
in structural strain energy functions of arterial tissue. Biomechanics and Modeling in
Mechanobiology 10, 599–611.
Rhodin, J.A.G., 1980. Architecture of the vessel wall. In: Comprehensive Physiology .
Roach, M.R., Song, S.H., 1994. Variations in strength of the porcine aorta as a function
of location. Clinical and Investigative Medicine. M´decine clinique et experimentale 17,
e
308–318.
131

Saravanan, U., Baek, S., Rajagopal, K.R., Humphrey, J.D., 2006. On the deformation of
the circumﬂex coronary artery during inﬂation tests at constant length. Experimental
Mechanics 46, 647–656.
Schulze-Bauer, C.A.J., Morth, C., Holzapfel, G.A., 2003. Passive biaxial mechanical response
of aged human iliac arteries. ASME Journal of Biomechanical Engineering 125, 395–406.
Schulze-Bauer, C.A.J., Regitnig, P., Holzapfel, G.A., 2002. Mechanics of the human femoral
adventitia including the high-pressure response. American Journal of Physiology - Heart
and Circulatory Physiology 282, H2427–H2440.
Schwartz, S., Taljanovic, M., Smyth, S., O’Brien, M., Rogers, L., 2007. CT ﬁndings of rupture, impending rupture, and contained rupture of abdominal aortic aneurysms. American
Journal of Roentgenology 188, W57–W62.
Shadwick, R.E., 1999. Mechanical design in arteries. Journal of Experimental Biology 202,
3305–3313.
Sokolis, D.P., 2007. Passive mechanical properties and structure of the aorta: segmental
analysis. Acta Physiologica 190, 277–289.
Spina, M., Garbisa, S., Hinnie, J., Hunter, J.C., Seraﬁni-Fracassini, A., 1983. Age-related
changes in composition and mechanical properties of the tunica media of the upper thoracic
human aorta. Arteriosclerosis, Thrombosis, and Vascular Biology 3, 64–76.
Stergiopulos, N., Vulliemoz, S., Rachev, A., Meister, J.J., Greenwald, S.E., 2001. Assessing
the homogeneity of the elastic properties and composition of the pig aortic media. Journal
of Vascular Research 38, 237–246.
Struik, D.J., 1988. Lectures on Classical Diﬀerential Geometry. Dover Publications, Inc.,
New York. second edition.
Takamizawa, K., Hayashi, K., 1987. Strain energy density function and uniform strain
hypothesis for arterial mechanics. Journal of Biomechanics 20, 7–17.
Thubrikar, M.J., 2007. Vascular Mechanics and Pathology. Springer.
Thubrikar, M.J., Labrosse, M., Robicsek, F., Al-Soudi, J., Fowler, B., 2001. Mechanical properties of abdominal aortic aneurysm wall. Journal of Medical Engineering & Technology
25, 133–142.
Thubrikar, M.J., Roskelley, S., Eppink, R., 1990. Study of stress concentration in the walls
of the bovine coronary arterial branch. Journal of Biomechanics 23, 15–17.
Tremblay, D., Cartier, R., Mongrain, R., Leask, R.L., 2010. Regional dependency of the
vascular smooth muscle cell contribution to the mechanical properties of the pig ascending
aortic tissue. Journal of Biomechanics 43, 2448–2451.
Vaishnav, R., Young, J., Patel, D., 1973. Distribution of stresses and of strain-energy density
through the wall thickness in a canine aortic segment. Circulation Research 32, 577–583.
132

Valdez-Jasso, D., Haider, M.A., Banks, H.T., Bia Santana, D., German, Y.Z., Armentano,
R.L., Olufsen, M.S., 2009. Analysis of viscoelastic wall properties in ovine arteries. IEEE
Transaction on Biomedical Engineering 56, 210–219.
Vande Geest, J.P., Sacks, M.S., Vorp, D.A., 2006. The eﬀects of aneurysm on the biaxial
mechanical behavior of human abdominal aorta. Journal of Biomechanics 39, 1324–1334.
Vorp, D., Raghavan, M., Webster, M., 1998. Mechanical wall stress in abdominal aortic
aneurysm: inﬂuence of diameter and asymmetry. Journal of Vascular Surgery 27, 632–
639.
Walker, J., Johnson, B., Fitz, W., Vaccaro, P., Smead, W., Cooperman, M., 1986. Abdominal
aortic aneurysms in elderly patients. Vascular and Endovascular Surgery 20, 79–83.
Wang, C., Garcia, M., Lu, X., Lanir, Y., Kassab, G.S., 2006. Three-dimensional mechanical
properties of porcine coronary arteries: A validated two-layer model. American Journal
of Physiology - Heart and Circulatory Physiology 291, H1200–H1209.
Weizsacker, H.W., 1988. Isotropy and anisotropy of the arterial wall. Journal of Biomechanics
21, 447–487.
Wells, S.M., Langille, B.L., Lee, J.M., Adamson, S.L., 1999. Determinants of mechanical
properties in the developing ovine thoracic aorta. American Journal of Physiology - Heart
and Circulatory Physiology 277, H1385–H1391.
Yin, F., Fung, Y., 1971. Mechanical properties of isolated mammalian ureteral segments.
American Journal of Physiology 221, 1484–1493.
Zeinali-Davarani, S., Raguin, L.G., Vorp, D.A., Baek, S., . Identiﬁcation of in vivo material and geometric parameters of a human aorta: toward patient-speciﬁc modeling of
abdominal aortic aneurysm. Biomechanics and Modeling in Mechanobiology , 1–11.
Zhang, D., Eggleton, C.D., Arola, D.D., 2002. Evaluating the mechanical behavior of arterial
tissue using digital image correlation. Experimental Mechanics 42, 409–416.
Zhang, W., Herrera, C., Atluri, S.N., Kassab, G.S., 2005. The eﬀect of longitudinal prestretch and radial constraint on the stress distribution in the vessel wall: A new hypothesis.
Mechanics and Chemistry of Biosystems 2, 41–52.
Zhou, J., Fung, Y.C., 1997. The degree of nonlinearity and anisotropy of blood vessel
elasticity. Proceedings of the National Academy of Sciences 94, 14255–14260.
Zou, Y., Zhang, Y., 2009. An experimental and theoretical study on the anisotropy of elastin
network. Annals of Biomedical Engineering 37, 1572–1583.
Zulliger, M., Rachev, A., Stergiopulos, N., 2004a. A constitutive formulation of arterial
mechanics including vascular smooth muscle tone. American Journal of Physiology-Heart
and Circulatory Physiology 287, H1335–H1343.

133

Zulliger, M.A., Fridez, P., Hayashi, K., Stergiopulos, N., 2004b. A strain energy function
for arteries accounting for wall composition and structure. Journal of Biomechanics 37,
989–1000.

134