Добавил:
Sekretar
kiopkiopkiop18@yandex.ru
t.me/Prokururor I Вовсе не секретарь, но почту проверяю
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3787_Библиотеки_им_академика_М_И_Перельмана
.pdf
Appendix 14: 3D Microstructure-Based Model of Active Coronary Artery (... 295
https://t.me/med1917
()
WMi¼
Z
π
2
1
ℛiθðÞwieðÞdθ þ
2
0
Z
π
2
1
þ
π
π
2
wILeðÞdθ ð4:207Þ
Z
0
1
ℛiθðÞwieðÞdθ
π
2
2
According to microscopic responses of individual elastin fibers under mechanical
loads (Chap. 3), the elastic properties is assumed to be linear, thus the SEF of elastin
is given by:
1
2
w
k
¼
E
e
E
2
ð4:208Þ
where fiber strain e is larger than zero (fiber is only resistant to tensile load), and k
E
stiffness parameter of elastin fiber. IL elastin has a similar function but with a
different stiffness k
:
IL
1
2
w
k
¼
IL
e
IL
2
ð4:209Þ
Because of the wavy nature of collagen fibers, the SEF is considered to account
for the nonlinear elastic behavior (Hollander et al., 2011a):
¼
1
1 þ M
kCe e
ðÞ
C
w
C
where fiber strain e is larger than collagen straightening strain e
collagen can withstand tension (which also denotes fiber waviness). k
1þM
C
0
, beyond which the
0
ð4:210Þ
and MCare
C
parameters characterizing the nonlinear stress–strain response of collagen. It is found
that collagen straightening strain e
follows a beta distribution in LAD adventitia
0
(Chen, Slipchenko, et al., 2013):
De
where B(α
, α2) is a beta function, and a and b the lower and upper bounds of the
1
straightening strain e
ðÞ¼
0
. A uniform distribution of straightening strain is assumed for
0
1
B α
; α
ðÞ
1
2
α11
e0 aðÞ
ðÞ
α1þα21
b aðÞ
b e
α21
0
ð4:211Þ
media collagen since collagen bundles are thinner and have a lower volume fraction
as compared to adventitia. The material properties of elastin and collagen fibers are
assumed to remain constant throughout the vessel wall.
is
296 4 Constitutive Models of Coronary Vasculature
https://t.me/med1917
Active Stresses of Coronary Artery with SMC Contraction
Since the force of SMC contraction is generated by active energy consuming
processes, a stored elastic energy (i.e., SEF), which is a function of strain state, is
not appropriate. An empirical length–tension relationship is typically employed in
this case. Here, a 3D model is proposed to account for triaxial responses of a single
cell to better predict the overall behavior of blood vessels. A phenomenological
stress–strain law is employed in the cell direction (i.e., major axis of a cell) as
Schmitz and Böl (2011):
hi
σ
SMC
¼ A
ρ
1
λ
2
ρ
2
max
λ
ρ
2
SMC
2
λ
λ
ðÞ
SMC
max
2
þ σ
max
ð4:212Þ
where A is the level of activation (0 is passive state and 1 is fully active, A ¼ 1 for the
present study), λ
SMC
cell stretch), n and n
denotes an optimal stretch ratio at which a SMC generated maximum stress σ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼
0
n FT F
nqis the longitudinal stretch of a SMC (i.e.,
are the longitudinal and transversal vector, respec tively. λ
max
max
. ρ
and ρ2determine the curvature and the skewness of the curve, respectively. When
ρ
> 0, the absolute slope of lower stretch region (λ
2
of high stretch region (λ
ρ
¼ 0 provi des a symmetric curve as employed in previous models.
2
SMC
> λ
), while ρ2< 0 leads an inverse behavior; and
max
SMC
λ
) is smaller than that
max
An analogous relation is used to account for SMC transverse stress in one of
minor axes (there are two minor axes of a single cell: transverse and radial axes) as:
hi
of which λ
σ
0
SMC
0
SMC
ρ
1
¼ τA
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q
¼
n0 FT F
ρ
0
2
λ
λ
max
ρ2 2
SMC
0
n
is SMC stretch in transverse direction, and τ is a
0
λ
λ
SMC
max
2
þ σ
max
ð4:213Þ
dimensionless parameter, that determines the contractile properties in this direction,
which can be regarded as the ratio of active axial stress to circumferential stress.
In the radial direction of a cell, a similar stress–strain law should be considered.
SMCs, however, are largely compressed in the radial direction and stretched in other
two directions under current distention-extension loading conditions. Moreover,
Eq. (4.212) is a strongly nonlinear and non-monotonic function, of which a large
span in stretch variable may lead to divergence of the solutions. A linear stress–strain
law is thus considered as a simplification of the stiffness of active SMCs in the radial
direction as:
1
of which λ
00
SMC
ratio of the vessel wall as SMCs arrange in θ z plane, and k
radial direction during SMC contraction.
σ
00
SMC
¼ k
SMC
λ
00
SMC
ð4:214Þ
is the radial stretch of SMCs which is equivalent to the radial stretch
is the stiffness in
SMC
Appendix 14: 3D Microstructure-Based Model of Active Coronary Artery (... 297
https://t.me/med1917
The second Piola–Kirchhoff stress of each constituent is thus derived as:
Z
π
ω
ij
0
ℛijθðÞ
∂w
∂e
∂e
i
dθ ð4:215Þ
∂E
S
i
∂W EðÞ
¼
∂E
E, C, SMC
¼
X
i
and the total stress is the volume-weighted sum as given by:
S ¼ f
of which f
f
SMCSSMC
SILþ fESEþ fCSCþ f
IL
+ fCSCpresents passive second Piola–Kirchhoff stress of vessel wall.
ESE
is the active stress generated by SMC contraction, of which the compo -
SMC
S
SMC
ð4:216Þ
nents can be writt en as:
S
SMC ij
¼ σ
SMC
∂λ
∂E
SMC
ij
þ σ
0
SMC
∂λ
∂E
0
SMC
ij
þ σ
00
SMC
∂λ
∂E
00
SMC
ij
ð4:217Þ
The Cauchy stress components (i.e., Eq. 4.202) of the vessel will then be obtained
by substituting the constitutive laws for individual fibers and cells (Eqs. 4.209–
4.214, and 4.217) into Eq. (4.216).
The full microstructural model has 15 geometrical parameters and requires
9 material parameters of individual fibers and cells. The orientation and undulation
distribution parameters of fibers and cells and obtained statistical distributions based
on two groups (one for adventitia and another for media) of coronar y artery
specimens have been measured (Chap. 3). These statistical measured parameters
are directly integrated into the model to predict mechanical responses of additional
porcine LAD arteries (n ¼ 5). The 9 material parameters are determined by optimization with appropriate boundary condition (Eqs. 4.200 and 4.201). Moreover,
microstructural geometrical parameters are refined for each sample by imposing
restrictions to ensure fiber orientation and waviness still follow statistic distributions
measured to obtain a better agreement between model predictions and
experimental data.
Parameter Estimation
Parameters are optimized by least squares fit to the experimental data by minimizing
an objective function based on the sum of squared residuals (SSE) between model
predictions and experimental data. In general, passive and active parameters are
determined separately. The passive material parameters of fibers are first determined
by pressure–radius and pressure–force relations, and subsequently integrated into the
active model to determine active parameters of SMCs. These passive parameters,
however, are determined in a limited loading range, i.e., range of only passive
loading conditions. The passive parameters, determined under axial stretch ratio
298 4 Constitutive Models of Coronary Vasculature
https://t.me/med1917
λz¼ 1.3 and circumferential stretch ratio λθfrom 0.9 to 1.8 (corresponding pressure
varying from 20mmHg to 160 mmHg), are integrated into the active model to
determined active parameters under λ
¼ 1.3 and lower λθfrom 0.6 to 1.7. Thus,
z
the active parameters are under- or overestimated somewhat as passive parameters
are not validated in the range of λ
from 0.6 to 0.9, which is an important region of
θ
vessel active response (i.e., pressure varying from 20 to 60 mmHg). Therefore, an
objective function simultaneously accounting for both passive and active behaviors
is defined as follows:
2
SSE ¼
1
nm
!
n,m
X
,j
i
P
r
br
o
o
ij
4
P
σ
^
r
o
!
2
P
ij
P
F
^F
ij
þ
σ
^
F
P
ij
F
r
br
o
o
ij
þ
F
σ
^
r
o
!
2
P
!
2
F
ij
F
F
^F
ij
þ
σ
^
F
2
F
F
ij
ð4:218Þ
where i and j denote distension and axial loads at which the corresponding outer
P
radius (r
F
, r
) and axial force F
o
o
ij
ij
P
F
; F
ij
are measured. P denotes passive responses
ij
and F denotes full responses (including both passive and active) of coronary arteries,
n is the number of different pressures and m is the number of different axial stretch
ratios used. σ
is the standard deviation of experimental measurement, andbr
^
,^F
o
ij
are corresponding model predicted outer radius and axial force. The objective
function with more restrictions (i.e., typically separated into two object ive functions)
leads to a better identification of material parameters; especially, for the active
parameters. A genetic algorithm method is employed to search optimal parameter
sets (sum marized in Tables 4.31, 4.32, and 4.33 below) using Fortran language
executed in Linux.
Table 4.31 Material parameter estimates of fibers and smooth muscle cells (SMC) determined
from grouped statistical geometrical distributions of fibers and cells into the model, based on both
passive and full distension–extension experimental data of axial stretch ratios λ
Sample no.
Parameters
Passive k
Goodness-of-fit
of passive data
Active ρ
k
(MPa) 0.19 0.05 0.02 0.23 0.17 0.13 0.08
IL
(MPa) 0.25 0.28 0.13 0.13 0.26 0.21 0.07
E
k
(MPa) 10.0 48.2 10.9 100 36.4 41.1 32.9
C
M
C
2
R
for P 0.80 0.96 0.95 0.96 0.96 0.93 0.06
2
R
for F 0.78 0.77 0.82 0.72 0.86 0.79 0.05
2
R
for r
(MPa) 0.47 0.45 0.63 0.20 0.3 0.410.15
1
ρ
2
λ
max
σ
maxx
123 45
5.96 5.71 6.00 5.99 4.06 5.54 0.75
0.91 0.91 0.96 0.93 0.94 0.93 0.02
o
1.22 0.59 0.03 1.66 0.9 0.88 0.55
1.44 1.06 1.39 1.38 1.36 1.33 0.14
0.10 0.11 0.07 0.06 0.12 0.09 0.02
(MPa)
τ 0.14 0.20 0.45 0.87 0.13 0.36 0.28
¼ 1.3 and 1.5
z
Average SD
(continued)
3
5
ij
Appendix 14: 3D Microstructure-Based Model of Active Coronary Artery (... 299
https://t.me/med1917
Sample no.
Parameters
k
SMC
123 45
Average SD
0.03 0.05
(MPa)
Goodness-of-fit of full
data
2
R
for P 0.83 0.69 0.99 0.99 0.83 0.87 0.11
2
R
for F 0.71 0.77 0.72 0.78 0.86 0.77 0.05
2
R
for r
0.90 0.58 0.85 0.91 0.91 0.83 0.13
o
Reproduced with permission by Chen and Kassab (2017)
is stiffness parameter of inter-lamellar (IL) elastin network
k
IL
is stiffness parameter of elastin fiber
k
E
and MCare parameters characterizing the stiffness and nonlinear parameter of collagen
k
C
and ρ2determine the curvature and the skewness of the length–tension relationship during SMC
ρ
1
contraction
denotes an optimal stretch ratio at which a SMC generated maximum stress σ
λ
max
τ is the ratio of active axial to circumferential stress; k
is the stiffness in radial direction
SMC
max
Table 4.32 Material and geometrical parameter estimates of individual elastin, collagen fibers, and
smooth muscle cells (SMC) with refined microstructural geometrical parameters that still follow
statistic distributions
Sample no.
12345
Average SD
5.68 5.65 5.91 5.58 3.32 5.23 0.96
0.39 0.38 0.37 0.38 0.25 0.35 0.05
8.30 4.97 4.90 5.22 6.97 6.07 1.35
71.9 42.0 47.7 67.2 71.0 60.0 12.6
0.32 0.20 0.20 0.24 0.29 0.25 0.05
0.26 0.10 0.14 0.12 0.24 0.17 0.07
1.74 1.97 1.80 1.96 1.71 1.84 0.11
0.56 0.46 0.26 0.33 0.30 0.38 0.11
0.20 0.37 0.38 0.38 0.26 0.32 0.07
0.30 0.15 0.27 0.16 0.27 0.23 0.06
2.09 2.08 1.58 1.61 1.71 1.81 0.23
0.22 0.44 0.28 0.24 0.47 0.33 0.10
0.29 0.18 0.28 0.13 0.26 0.23 0.06
0.30 0.11 0.13 0.16 0.27 0.19 0.08
1.04 1.83 1.66 1.83 0.79 1.43 0.43
1.45 1.28 1.48 1.35 1.16 1.34 0.12
Fiber passive k
Fiber geometrical e
SMC active ρ
Parameters
(MPa) 0.15 0.04 0.04 0.28 0.39 0.18 0.14
IL
k
(MPa) 0.30 0.28 0.14 0.28 0.35 0.27 0.07
E
k
(MPa) 11.8 55.1 10.9 57.9 11.8 29.5 22.1
C
M
C
0
α
1
α
2
μ
E1
σ
E1
μ
E2
σ
E2
μ
C1
σ
C1
μ
C2
σ
C2
μ
M
σ
M
(MPa) 0.27 0.30 0.47 0.25 0.27 0.31 0.08
1
ρ
2
λ
max
σ
(MPa) 0.08 0.09 0.08 0.07 0.11 0.09 0.01
max
τ 0.32 0.11 0.15 0.26 0.30 0.23 0.08
k
(MPa) 0.03 0.00 0.00 0.03 0.01 0.01 0.01
SMC
SMC geometrical μ
SMC
σ
SMC
0.19 0.27 0.22 0.25 0.29 0.24 0.04
0.15 0.29 0.29 0.12 0.22 0.21 0.07
(continued)
300 4 Constitutive Models of Coronary Vasculature
https://t.me/med1917
Sample no.
12345
0.96 0.93 0.87 0.95 0.98 0.94 0.04
o
Average SD
Goodness-of-fit of full data R
Parameters
2
for P 0.95 0.85 0.99 0.99 0.94 0.94 0.05
2
R
for F 0.83 0.83 0.77 0.93 0.79 0.83 0.06
2
R
for r
Reproduced with permission by Chen and Kassab (2017)
Material parameters definitions are identical to those in Table 4.31, and (μ
, σij) are the mean and
ij
standard deviation of the jth normal distribution of orientation angle of adventitia fiber i (i ¼ C,
E and j ¼ 1,2) where C, E denote collagen and elastin fiber, respectively; (α
the beta distribution of collagen straightening strain e
ening strain in media; (μ
fiber orientation, and (μ
, σM) are the mean and standard deviation of normal distribution of media
M
, σ
SMC
) are the mean and standard deviation of media SMC orientation
SMC
in adventitia; e02denotes collagen straight-
01
, α2) are parameters of
1
Table 4.33 Material and geometrical parameter estimates of fi bers and smooth muscle cells (SMC)
determined by a mean-value approach to eliminate continuous distribution of microstructure
Sample no.
12345
Average SD
5.66 5.66 5.71 5.31 5.13 5.50 0.23
0.23 0.30 0.29 0.22 0.24 0.26 0.03
0.39 0.21 0.38 0.36 0.26 0.32 0.07
0.23 0.31 0.36 0.23 0.20 0.27 0.06
1.70 1.62 1.81 1.63 1.90 1.73 0.11
0.38 0.30 0.33 0.40 0.32 0.36 0.04
1.76 1.85 1.74 1.70 1.77 1.76 0.05
0.27 0.21 0.26 0.15 0.19 0.22 0.04
1.80 1.57 1.73 1.47 1.27 1.57 0.19
1.48 1.22 1.48 1.47 1.48 1.43 0.10
Fiber passive k
Fiber geometrical e
SMC active ρ
Parameters
(MPa) 0.25 0.0 0.03 0.29 0.22 0.16 0.12
IL
k
(MPa) 0.23 0.28 0.21 0.21 0.28 0.24 0.03
E
k
(MPa) 10.5 16.9 10.7 42.7 36.1 23.4 13.5
C
M
C
01
e
02
μ
E1
μ
E2
μ
C1
μ
C2
μ
M
(MPa) 0.48 0.36 0.53 0.27 0.24 0.38 0.11
1
ρ
2
λ
max
σ
(MPa) 0.08 0.10 0.08 0.08 0.12 0.09 0.02
max
τ 0.13 0.14 0.24 0.17 0.38 0.21 0.09
k
(MPa) 0.01 0.00 0.00 0.03 0.01 0.01 0.01
SMC
SMC geometrical μ
Goodness-of-fit of full data R
SMC
2
for P 0.93 0.9 0.86 0.99 0.99 0.93 0.05
2
R
for F 0.84 0.63 0.70 0.96 0.85 0.80 0.12
2
R
for r
0.26 0.23 0.27 0.20 0.12 0.22 0.05
0.96 0.86 0.79 0.96 0.89 0.89 0.06
o
Reproduced with permission by Chen and Kassab (2017)
and μi2are the two mean orientation angles of adventitia fiber i (i ¼ C, E), and μMand μ
μ
i1
orientation angles of media fiber and SMC; e
and e02denotes collagen straightening strain in
01
SMC
are
adventitia and media, respectively
References 301
https://t.me/med1917
References
Achille, P. D., Celi, S., Puccio, F. D., & Forte, P. (2011). Anisotropic AAA: Computational
comparison between four and two fiber family material models. Journal of Biomechanics, 44,
2418–2426. https://doi.org/10.1016/j.jbiomech.2011.06.029
Agoras, M., Lopez-Pamies, O., & Ponte Castañeda, P. (2009). A general hyperelastic model for
incompressible fiber-reinforced elastomers. Journal of the Mechanics and Physics of Solids, 57,
268–286. https://doi.org/10.1016/j.jmps.2008.10.014
Azuma, T., & Hasegawa, M. (1971). A rheological approach to the architecture of arterial walls.
The Japanese Journal of Physiology, 21,37–47. https://doi.org/10.2170/jjphysiol.21.27
Azuma, T., & Oka, S. (1971). Mechanical equilibrium of blood vessel walls. American Journal of
Physiology, 221, 1310–1318. https://doi.org/10.1152/ajplegacy.1971.221.5.1310
Baek, S., Gleason, R. L., Rajagopal, K. R., & Humphrey, J. D. (2007). Theory of small on large:
Potential utility in computations of fluid–solid interactions in arteries. Computer Methods in
Applied Mechanics and Engineering, 196, 3070–3078. https://doi.org/10.1016/j.cma.2006.06.
018
Brown, I. A. (1973). A scanning electron microscope study of the effect of uniaxial tension on
human skin. The British Journal of Dermatology 89, 383–393.
Burton, A. C., & Yamada, S. (1951). Relation between blood pressure and flow in the human
forearm. Journal of Applied Physiology, 4, 329–339. https://doi.org/10.1152/jappl.1951.4.5.329
Carew, T. E., Vaishnav, R. N., & Patel, D. J. (1968). Compressibility of the arterial wall.
Circulation Research, 23,61–68. https://doi.org/10.1161/01.RES.23.1.61
Chen, H., Guo, X., Luo, T., & Kassab, G. S. (2016). A validated 3D microstructure-based
constitutive model of coronary artery adventitia. Journal of Applied Physiology, 121(1),
333–342. https://doi.org/10.1152/japplphysiol.00937.2015
Chen, H., & Kassab, G. S. (2017). Microstructure-based constitutive model of coronary artery with
active smooth muscle contraction. Scientific Reports, 7(1), 9339. https://doi.org/10.1038/
s41598-017-08748-7
Chen, H., Liu, Y., Slipchenko, M. N., Cheng, J.-X., & Kassab, G. S. (2011). The layered structure
of coronary adventitia under mechanical load. Biophysical Journal, 101, 2555–2562. https://doi.
org/10.1016/j.bpj.2011.10.043
Chen, H., Liu, Y., Zhao, X., Lanir, Y., & Kassab, G. S. (2011). A micromechanics finite-strain
constitutive model of fibrous tissue. Journal of the Mechanics and Physics of Solids, 59,
1837. https://doi.org/10.1016/j.jmps.2011.05.012
1823–
Chen, H., Luo, T., Zhao, X., Lu, X., Huo, Y., & Kassab, G. S. (2013). Microstructural constitutive
model of active coronary artery media. Biomaterials, 34(31), 7575–7583. https://doi.org/10.
1016/j.biomaterials.2013.06.035
Chen, H., Slipchenko, M. N., Liu, Y., Zhao, X., Cheng, J.-X., Lanir, Y., & Kassab, G. S. (2013).
Biaxial deformation of collagen and elastin fibers in coronary adventitia. Journal of Applied
Physiology (1985), 115(11), 1683–1693. https://doi.org/10.1152/japplphysiol.00601.2013
Chen, H., Zhao, X., Berwick, Z. C., Krieger, J. F., Chambers, S., & Kassab, G. S. (2016).
Microstructure and mechanical property of glutaraldehyde-treated porcine pulmonary ligament.
Journal of Biomechanical Engineering, 138(6), 061009–061003. https://doi.org/10.1115/1.
4033300
Chen, H., Zhao, X., Lu, X., & Kassab, G. S. (2013). Nonlinear micromechanics of soft tissue.
International Joural of Non-linear Mechanics, 56,79–85. https://doi.org/10.1016/j.
ijnonlinmec.2013.03.002
Chuong, C. J., & Fung, Y. C. (1983). Three-dimensional stress distribution in arteries. Journal of
Biomechanical Engineering, 105(3), 268–274. https://doi.org/10.1115/1.3138417
Chuong, C. J., & Fung, Y. C. (1984). Compressibility and constitutive equation of arterial wall in
radial compression experiments. Journal of Biomechanics, 17,35–40. https://doi.org/10.1016/
0021-9290(84)90077-0
302 4 Constitutive Models of Coronary Vasculature
https://t.me/med1917
Chuong, C. J., & Fung, Y. C. (1986). On residual stresses in arteries. Journal of Biomechanical
Engineering, 108, 189–192. https://doi.org/10.1115/1.3138600
Coley, D. A. (1999). An introduction to genetic algorithms for scientists and engineers. New York:
World Scientific Publishing Company.
Criscione, J. C., Humphrey, J. D., Douglas, A. S., & Hunter, W. C. (2000). An invariant basis for
natural strain which yields orthogonal stress response terms in isotropic hyperelasticity. Journal
of the Mechanics and Physics of Solids, 48, 2445–2465. https://doi.org/10.1016/S0022-5096
(00)00023-5
Dahl, S. L. M., Vaughn, M. E., Hu, J.-J., Driessen, N. J. B., Baaijens, F. P. T., Humphrey, J. D., &
Niklason, L. E. (2008). A microstructurally motivated model of the mechanical behavior of
tissue engineered blood vessels. Annals of Biomedical Engineering, 36, 1782–1792. https://doi.
org/10.1007/s10439-008-9554-4
Decraemer, W. F., Maes, M. A., & Vanhuyse, V. J. (1980). An elastic stress-strain relation for soft
biological tissues based on a structural model. Journal of Biomechanics, 13, 463–468. https://
doi.org/10.1016/0021-9290(80)90338-3
Deng, S. X., Tomioka, J., Debes, J. C., & Fung, Y. C. (1994). New experiments on shear modulus
of elasticity of arteries. American Journal of Physiology-Heart and Circulatory Physiology,
266,H1–H10. https://doi.org/10.1152/ajpheart.1994.266.1.H1
Dobrin, P. B. (1978). Mechanical properties of arteries. Physiological Reviews, 58, 397–460.
https://doi.org/10.1152/physrev.1978.58.2.397
Farahani, K., & Naghdabadi, R. (2000). Conjugate stresses of the Seth–Hill strain tensors. Inter-
national Journal of Solids and Structures, 37, 5247–5255. https://doi.org/10.1016/S0020-7683
(99)00209-7
Findley, W. N., Lai, J. S., & Onaran, K. (1989). Creep and relaxation of nonlinear viscoelastic
materials. New York: Dover.
Fratzl, P., Misof, K., Zizak, I., Rapp, G., Amenitsch, H., & Bernstorff, S. (1998). Fibrillar structure
and mechanical properties of collagen. Journal of Structural Biology, 122, 119–122. https://doi.
org/10.1006/jsbi.1998.3966
Fung, Y. C. (1993). Biomechanics: Mechanical properties of living tissues (2nd ed.). New York:
Springer.
Fung, Y. C., 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. https://doi.org/10.1152/ajpheart.1979.237.5.H620
Fung, Y. C., & Liu, S. Q. (1995). Determination of the mechanical properties of the different layers
of blood vessels in vivo. Proceedings of the National Academy of Science of the United States of
America, 92, 2169–2173. https://doi.org/10.1073/pnas.92.6.2169
Gaballa, M. A., Jacob, C. T., Raya, T. E., Liu, J., Simon, B., & Goldman, S. (1998). Large artery
remodeling during aging: Biaxial passive and active stiffness. Hypertension, 32(3), 437–443.
https://doi.org/10.1161/01.HYP.32.3.437
Gentleman, E., Lay, A. N., Dickerson, D. A., Nauman, E. A., Livesay, G. A., & Dee, K. C. (2003).
Mechanical characterization of collagen fibers and scaffolds for tissue engineering. Biomate-
rials, 24(21), 3805–3813. https://doi.org/10.1016/S0142-9612(03)00206-0
Gestrelius, S., & Borgström, P. (1986). A dynamic model of smooth muscle contraction. Biophys-
ical Journal, 50, 157–169. https://doi.org/10.1016/S0006-3495(86)83448-8
Ghazanfari, S., Driessen-Mol, A., Strijkers, G. J., Kanters, F. M. W., Baaijens, F. P. T., & Bouten,
C. V. C. (2012). A comparative analysis of the collagen architecture in the carotid artery: Second
harmonic generation versus diffusion tensor imaging. Biochemical and Biophysical Research
Communications, 426(1), 54–58. https://doi.org/10.1016/j.bbrc.2012.08.031
Goldberg, D. E. (1989). Genetic algorithms in search, optimization and machine learning. Boston:
Addison-Wesley.
Green, A. E., & Adkins, J. E. (1960). Large deformations and nonlinear continuum mechanics.
Oxford: Oxford University Press.
References 303
https://t.me/med1917
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
(3), 586–594. https://doi.org/10.1016/j.jbiomech.2006.02.004
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, 283,
H2361–H2368. https://doi.org/10.1152/ajpheart.01079.2003
Guo, X., Xiao, L., & Kassab, G. S. (2005). Transmural strain distribution in the blood vessel wall.
American Journal of Physiology-Heart and Circulatory Physiology, 288(2), H881–H886.
https://doi.org/10.1152/ajpheart.00607.2004
Gurtin, M. E. (1982). An introduction to continuum mechanics. New York: Academic Press.
Hansen, L., Wan, W., & Gleason, R. L. (2009). Microstructurally motivated constitutive modeling
of mouse arteries cultured under altered axial stretch. Journal of Biomechanical Engineering,
131, 101015. https://doi.org/10.1115/1.3207013
Hashin, Z., & Shtrikman, S. (1962). A variational approach to the theory of the elastic behaviour of
polycrystals. Journal of the Mechanics and Physics of Solids, 10, 343–352. https://doi.org/10.
1016/0022-5096(62)90005-4
Hashin, Z., & Shtrikman, S. (1963). A variational approach to the theory of the elastic behaviour of
multiphase materials. Journal of the Mechanics and Physics of Solids, 11, 127–140. https://doi.
org/10.1016/0022-5096(63)90060-7
Hayman, D. M., Zhang, J., Liu, Q., Xiao, Y., & Han, H.-C. (2013). Smooth muscle cell contraction
increases the critical buckling pressure of arteries. Journal of Biomechanics, 46, 841–844.
https://doi.org/10.1016/j.jbiomech.2012.11.040
Herlihy, J. T., & Murphy, R. A. (1973). Length-tension relationship of smooth muscle of the hog
carotid artery. Circulation Research, 33, 275–283.
Hershey, A. (1954). The elasticity of an isotropic aggregate of anisotropic cubic crystals. Journal of
Applied Mechanics-Transactions ASME, 21, 236–240.
Hill, R. (1952). The elastic behaviour of a crystalline aggregate. Proceedings of the Physical
Society: Section A, 65, 349–354. https://doi.org/10.1088/0370-1298/65/5/307
Hill, R. (1965). A self-consistent mechanics of composite materials. Journal of the Mechanics and
Physics of Solids, 13, 213–222. https://doi.org/10.1016/0022-5096(65)90010-4
Hollander, Y., Durban, D., Lu, X., Kassab, G. S., & Lanir, Y. (2011a). Experimentally validated
microstructural 3D constitutive model of coronary arterial media. Journal of Biomechanical
Engineering, 133(3), 031007. https://doi.org/10.1115/1.4003324
Hollander, Y., Durban, D., Lu, X., Kassab, G. S., & Lanir, Y. J. B. E. (2011b). Constitutive
modeling of coronary arterial media: Comparison of three model classes. Journal of Biome-
chanical Engineering, 133(6), 061008. https://doi.org/10.1115/1.4004249
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.
https://doi.org/10.1023/A:1010835316564
Holzapfel, G. A., Gasser, T. C., & Ogden, R. W. (2004). Comparison of a multi-layer structural
model for arterial walls with a Fung-type model, and issues of material stability. ASME: Journal
of Biomechanical Engineering, 126, 264–275. https://doi.org/10.1115/1.1695572
Holzapfel, G. A., Sommer, G., Gasser, C. T., & Regitnig, P. (2005). Determination of layer-specific
mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening
and related constitutive modeling. American Journal of Physiology-Heart and Circulatory
Physiology, 289, H2048–H2058. https://doi.org/10.1152/ajpheart.00934.2004
Holzapfel, G. A., & Weizsäcker, H. W. (1998). Biomechanical behavior of the arterial wall and its
numerical characterization. Computers in Biology and Medicine, 28, 377–392. https://doi.org/
10.1016/S0010-4825(98)00022-5
Horowitz, A., Lanir, Y., Yin, F. C., Perl, M., Sheinman, I., & Strumpf, R. K. (1988). Structural
three-dimensional constitutive law for the passive myocardium. Journal of Biomechanical
Engineering, 110, 200–207. https://doi.org/10.1115/1.3108431
304 4 Constitutive Models of Coronary Vasculature
https://t.me/med1917
Humphrey, J. D. (1995). Mechanics of the arterial wall: Review and directions. Critical Reviews in
Biomedical Engineering, 23,1–162. https://doi.org/10.4236/ce.2015.612140
Humphrey, J. D. (1999). An evaluation of pseudoelastic descriptors used in arterial mechanics.
Journal of Biomechanical Engineering, 121, 259–262. https://doi.org/10.1115/1.2835113
Humphrey, J. D., & Na, S. (2002). Elastodynamics and arterial wall stress. Annals of Biomedical
Engineering, 30, 509–523. https://doi.org/10.1114/1.1467676
Humphrey, J. D., & Yin, F. C. (1987). A new constitutive formulation for characterizing the
mechanical behavior of soft tissues. Biophysical Journal, 52, 563–570. https://doi.org/10.
1016/S0006-3495(87)83245-9
Huo, Y., Cheng, Y., Lu, X., Liu, Y., & Kassab, G. S. (2012). Biaxial vasoactivity of coronary artery.
American Journal of Physiology-Heart and Circulatory Physiology, 302, H2058–H2063.
https://doi.org/10.1152/ajpheart.00758.2011
Huo, Y., Zhao, X., Cheng, Y., Lu, X., & Kassab, G. S. (2013). Two-layer analysis of coronary
artery vasoactivity: Theory and experiment. Journal of Applied Physiology, 114(10),
1451–1459. https://doi.org/10.1152/japplphysiol.01237.2012
Hutchinson, J. W. (1976). Bounds and self-consistent estimates for creep of polycrystalline
materials. Proceedings of the Royal Society A, 348, 101–127. https://doi.org/10.1098/rspa.
1976.0027
Ingber, D. E. (2006). Cellular mechanotransduction: Putting all the pieces together again. The
FASEB Journal, 20, 811–827. https://doi.org/10.1096/fj.05-5424rev
Itskov, M., & Aksel, N. (2002). Elastic constants and their admissible values for incompressible and
slightly compressible anisotropic materials. Acta Mechanica, 157,81–96. https://doi.org/10.
1007/BF01182156
Kailasam, M., Ponte Castañeda, P., & Willis, J. R. (1997). The effect of particle size, shape,
distribution and their evolution on the constitutive response of nonlinearly viscous composites.
I. Theory. Philosophical Transactions of the Royal Society A: Mathematical, Physical and
Engineering Sciences, 355(1730), 1835–1852. https://doi.org/10.1098/rsta.1997.0092
Kato, Y. P., Christiansen, D. L., Hahn, R. A., Shieh, S. J., Goldstein, J. D., & Silver, F. H. (1989).
Mechanical properties of collagen fibres: A comparison of reconstituted and rat tail tendon
fibres. Biomaterials, 10(1), 38–42. https://doi.org/10.1016/0142-9612(89)90007-0
Kroon, M., & Holzapfel, G. A. (2008). A new constitutive model for multi-layered collagenous
tissues. Journal of Biomechanics, 41, 2766–2771. https://doi.org/10.1016/j.jbiomech.2008.05.
033
Kwon, H. M., Sangiorgi, G., Ritman, E. L., Lerman, A., McKenna, C., Virmani, R., ... Schwartz,
R. S. (1998). Adventitial vasa vasorum in balloon-injured coronary arteries: Visualization and
quantitation by a microscopic three-dimensional computed tomography technique. Journal of
the American College of Cardiology, 32(7), 2072–2079.
(98)00482-3
Lakes, R. S. (1999). Viscoelastic solids. Boca Raton: CRC Press.
Lanir, Y. (1979). A structural theory for the homogeneous biaxial stress-strain relationships in flat
collagenous tissues. Journal of Biomechanics, 12, 423–436. https://doi.org/10.1016/0021-9290
(79)90027-7
Lanir, Y. (1980). A microstructure model for the rheology of mammalian tendon. Journal of
Biomechanical Engineering, 102, 332–339. https://doi.org/10.1115/1.3138231
Lanir, Y. (1983). Constitutive equations for fibrous connective tissues. Journal of Biomechanics,
16,1–12. https://doi.org/10.1016/0021-9290(83)90041-6
Li, D., & Robertson, A. M. (2009). A structural multi-mechanism constitutive equation for cerebral
arterial tissue. International Journal of Solids and Structures, 46, 2920–2928. https://doi.org/10.
1016/j.ijsolstr.2009.03.017
Liu, Y. (2003). Macroscopic behavior, field fluctuations and texture evolution in viscoplastic
polycrystals. (Ph.D.), University of Pennsylvania, Philadelphia.
Liu, Y., Gilormini, P., & Ponte Castañeda, P. (2003). Variational self-consistent estimates for
texture evolution in viscoplastic polycrystals. Acta Materialia, 51, 5425–5437. https://doi.org/
10.1016/S1359-6454
https://doi.org/10.1016/S0735-1097
Соседние файлы в папке Библиотека им академика М.И. Перельмана