CARDIOVASCULAR SYSTEM MODELING: ARTERIAL GROWTH AND REMODELING
By
Hailu Getachew Tilahun

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mechanical Engineering—Doctor of Philosophy
2017

ABSTRACT
CARDIOVASCULAR SYSTEM MODELING: ARTERIAL GROWTH AND REMODELING
By
Hailu Getachew Tilahun
Vascular systems in the human blood circulation regulate their environment when there
are alterations of hemodynamic loading from homeostatic levels, depending on the
magnitude and the duration of the changes. The main focus of this work is to develop a
bio-chemo-mechanical model of arterial growth and remodeling wherein changes in blood
pressure and flow are sustained over weeks or months. Using a two-step kinetic reaction
model of collagen, which is the dominant structural protein in arteries, as a function of
arterial wall stress changes, we track and evaluate the temporal change in mass
deposition and degradation of extracellular matrix. We employ a constrained mixture
model to capture the response of the artery to hemodynamic loadings, leading to strain
energy changes that depend on the stiffness and relative mass ratio of the constituents
of the artery. In so doing, we investigate the temporal changes of the geometry of the
artery over weeks and months. We also explore the possible ranges of the collagen
turnover rates, the coupling between collagen turnover and stresses, and the length of
time it takes for the vascular stresses to return back to the steady homeostatic states
while the artery is still under sustained loadings. The developed mathematical models are
verified by previously published mathematical models and validated by comparing the
mathematical result with animal experiments.
Using reported experimental data, we inversely compute the arterial constituent mass
turnover. After minimizing the total error between the simulated and experimental arterial

thickness values, parameters such as collagen and smooth muscles degradation rates
are estimated. The efficiency of computation is improved by singular value decomposition
and regularization. We also study a lumped whole body model in the cardiovascular
system with baroreflex. In this model, we incorporate the effect of arteriovenous fistula
and can get verifiable results as per reported vascular maturation data.

ACKNOWLEDGMENTS
I would like to express my sincere appreciation to my advisor, Dr. Seungik Baek, for his
kind and continuous support. I am honored to meet such a wonderful and caring person.
I am also grateful for the opportunity I am given to study in the USA, for all the financial
and intellectual support I got, for all safe and secure research and study environment
provided by Michigan State University in general and Mechanical Engineering
Department in particular. All my instructors in the graduate courses I attended are
wonderful. During my research, I got a lot of helpful information from Prof. Jay Humphrey,
Prof. Brain Feeny, and Prof. Abraham Engeda. My teammates in Soft Tissue Continuum
Mechanics MSU-CMUQ Research Consortium are very friendly and very helpful. I
benefited a lot from their readily available advice, guidance, and cooperation. Thank you!
I am grateful to Prof. William F. Jackson, Dr. Jong-Eun Choi and Dr. Sara Roccabianca
for their willingness to serve on my graduate committee and for their constructive
feedback. Finally, I would like to express my gratitude for all the funding agencies that,
directly or indirectly, supported my research, especially for generous support of National
Institute of Health.

iv

TABLE OF CONTENTS
LIST OF TABLES ............................................................................................................vi
LIST OF FIGURES ......................................................................................................... vii
INTRODUCTION ............................................................................................ 1
GROWTH AND REMODELING OF ARTERIES DUE TO SUSTAINED
CHANGES IN HEMODYNAMIC LOADING ..................................................................... 5
Introduction .................................................................................................................. 5
Biochemical Model ..................................................................................................... 10
Biomechanical Model ................................................................................................. 11
Biochemomechanical Model ...................................................................................... 17
Simulations ................................................................................................................ 19
Result ......................................................................................................................... 23
Discussion.................................................................................................................. 31
INVERSE PROBLEM AND PARAMETER ESTIMATION ............................. 37
Introduction ................................................................................................................ 37
An Inverse Problem and Mathematical Formulation .................................................. 37
Simulation ............................................................................................................... 40
Results .................................................................................................................... 42
Parameter Estimation ................................................................................................ 43
Discussion.................................................................................................................. 46
MODELING EFFECT OF AVF ON CARDIOVASCULAR SYSTEM ............. 48
Introduction ................................................................................................................ 48
Incorporating G&R Resistance Changes into CVS .................................................... 49
AVF ............................................................................................................................ 50
CVS Model ................................................................................................................. 52
Results ....................................................................................................................... 54
Further Discussion ..................................................................................................... 60
CONCLUSION AND FUTURE WORK ......................................................... 63
REFERENCES .............................................................................................................. 65

v

LIST OF TABLES
Table 1. The range of biochemical parameters (𝑀𝑊: collagen molecular weight, 𝑎𝑚𝑎𝑥:
collagen maximum age) ................................................................................................ 11
Table 2. Parameter Estimation of Mouse Carotid Artery ............................................... 23
Table 3. Normalized experimental arterial adaptation data from literature for changed
pressure and flow .......................................................................................................... 41
Table 4. Estimated parameters using the combination of inverse problem and
optimization ................................................................................................................... 45
Table 5. Reported vein and artery diameter on different days after AVF ...................... 51
Table 6. Blood flow through AVF at different days (W: Won et al86, T: Toregeani et al78)
...................................................................................................................................... 51

vi

LIST OF FIGURES
Figure 1. A schematic of signaling cascade in arterial wall to regulate ECM turnover
and SMC tone and proliferation due to flow and pressure stimuli ................................... 7
Figure 2. A schematic of the production of collagen ....................................................... 9
Figure 3. Simplified model of collagen production ........................................................ 10
Figure 4. An idealized model for a growth and remodeling of an artery in response to
perturbed hemodynamics. ............................................................................................. 20
Figure 5. Intramural pressure (mmHg) vs outer diameter (Âľm) for a mouse carotid
artery. 30 The passive and active parameters for the arterial model were estimated from
passive (* * *) and active(o o o) experiments. The fit to data with the model is shown by
the solid (passive) and dashed (active) curves. ............................................................ 21
Figure 6. Flowchart indicating the solution for different variables, given pressure (P) or
flow rate (Q). The Newton-Rapson iteration that is used to determine the mean radius, r,
at each time step, ∆t, is the nested internal loop. .......................................................... 22
Figure 7. Variation of a) internal radius, b) arterial thickness, c)wall shear stress and d)
circumferential stress for different mass coefficients for a 30% decrease in volumetric
flow rate ( 𝐾𝜎𝑖 = 0 (solid line); 𝐾𝜎𝑖 = 2(broken line); 𝐾𝜎𝑖 = 6(dotted line); 𝐾𝜎𝑖 =
10(dotted-broken line) and 𝐾𝜇1𝑘 = 𝐾𝜇2𝑘 = 0.01, 𝐾𝜏𝑤𝑐 = 𝐾𝜏𝑤𝑚 = 52,𝑘𝑡𝑜𝑛 = 𝐾𝜏𝑤𝑐3,
µ1 = 0.1, 𝛽2 = 0.2) ......................................................................................................... 25
Figure 8. Variation of a) internal radius, b) arterial thickness, c) wall shear stress and d)
circumferential stress for a step 50% increased pressure (µ1 = 0.0, 𝛽2 = 0.001(dottedbroken line); µ1 = 0.0, 𝛽2 = 0.05(dotted line); µ1 = 0.05, 𝛽2 = 0.05(broken line); µ1 =
0.1, 𝛽2 = 0.2(solid line)) (𝐾𝜎𝑐 = 𝐾𝜎𝑚 = 2, 𝐾𝜇1𝑘 = 𝐾𝜇2𝑘 = 0.01, = 𝐾𝜏𝑤𝑐 = 𝐾𝜏𝑤𝑚 =
3𝑘𝑡𝑜𝑛 = 52) ................................................................................................................... 26
Figure 9. Variation in internal radius for one day as a function of the change in volume
flow rate (from top to bottom 30%, 20%, and 10% increased volume flow rate, 10%,
20% and 30% decreased volume flow rate.) (𝐾𝜇1𝑘 = 𝐾𝜇2𝑘 = 0.01, 𝑘2𝑚 = 13.6, 𝐾𝜎𝑐 =
𝐾𝜎𝑚 = 1, 𝐾𝜏𝑤𝑐 = 𝐾𝜏𝑤𝑚 = 3𝑘2𝑚, µ1 = 0.1, 𝛽2 = 0.1) .................................................. 27
Figure 10. Variation of a) internal radius, b) arterial thickness, c) wall shear stress and
d) circumferential stress for a step 30% increased flow rate (µ1 = 0.0, 𝛽2 =
0.001(dotted-broken line); µ1 = 0.0, 𝛽2 = 0.05(dotted line); µ1 = 0.05, 𝛽2 = 0.05(broken
line); µ1 = 0.1,𝛽2 = 0.2(solid line)) (𝐾𝜎𝑐 = 𝐾𝜎𝑚 = 3, 𝐾𝜇1𝑘 = 𝐾𝜇2𝑘 = 0.01, 𝐾𝜏𝑤𝑐 =
𝐾𝜏𝑤𝑚 = 3𝑘𝑡𝑜𝑛 = 52) .................................................................................................... 29
vii

Figure 11. Active stress (kPa) vs normalized muscle fiber stretch at different days for a)
30% decreased volume flowrate and b) 50% increased pressure (𝐾𝜎𝑐 = 𝐾𝜎𝑚 =
3, 𝐾𝜇1𝑘 = 𝐾𝜇2𝑘 = 0.01, 𝐾𝜏𝑤𝑐 = 𝐾𝜏𝑤𝑚 = 3𝑘𝑡𝑜𝑛 = 52, µ1 = 0.1, 𝛽2 = 0.2) (0
(basal)(solid line), 7(broken line), 14(dotted line) and 300 days(dotted-broken line). The
circles indicate maximas for the respective curves) ...................................................... 31
Figure 12. a) Simulated changes in internal diameter (solid), ideal adaptation
(diamond-dotted line) and experimental data (circles) reported by Sluijter et al. 72, b)
Simulated changes in internal diameter (solid line), and experimental data (circles)
reported by Friedz et al. 26 and c) corresponding thickness (simulated (solid line),
experimental (circles) and ideal adaptation (diamond-dotted line)) (𝐾𝜎𝑐 = 𝐾𝜎𝑚 = 3,
𝐾𝜇1𝑘 = 𝐾𝜇2𝑘 = 0.01, 𝐾𝜏𝑤𝑐 = 𝐾𝜏𝑤𝑚 = 3𝑘𝑡𝑜𝑛 = 52, µ1 = 0.1, 𝛽2 = 0.2). .................... 34
Figure 13. Comparison of inverse solution (solid line) with ideal thickness ratio (i.e.
ℎℎℎ = 𝑃𝑃ℎ)(broken) ...................................................................................................... 43
Figure 14. Flow chart for parameter estimation using inverse solution ......................... 44
Figure 15. Comparison of two inverse solutions (Figure 13 (solid line), simulation using
optimized parameters (broken line)) with ideal thickness ratio (assuming flow is
homeostatic (dotted line)) and experimental thickness ratio (circles) ............................ 46
Figure 16. Cardiovascular System.81 The main cardiovascular elements are 𝐶compliance, 𝑅- resistance and 𝐿- inertance;................................................................. 53
Figure 17. Mean pressure in systemic arteries (a) and skeletal muscle mean flow (b)
under different regulation, pressure, and blood resistance. (0-30 sec: circulation without
baroreflex, 30-300 sec: circulation with baroreflex, 60-70 sec: 50% reduced systemic
arteries pressure, 100-110 sec: 50% increased systemic arteries pressure, 150-200
sec: 30% reduced skeletal muscle bed resistance, 200-250 sec: 70% reduced skeletal
muscle bed resistance) ................................................................................................. 55
Figure 18. Cardiac load: left ventricle (lv) P-V curves for baseline (solid line), 30%
reduced (broken line) and 70% reduced (dotted line) skeletal vascular bed resistance 57
Figure 19. Blood flow through lower arm AVF, simulation (solid line)vs data from
literature (Toregeani et al.78 (circle), Lomonte et al. 51(diamond)) ................................. 58
Figure 20. Flow through upper arm AVF, simulation (solid line) vs data from Toregeani
et al.78(circles) ............................................................................................................... 58
Figure 21. Heart rate (a) and Cardiac output (b) for variable access flow (simulation
(solid line), data from literature9 (circles)) ...................................................................... 59

viii

Figure 22. P-V curve in left ventricle (baseline (solid line); 50% increased mean
pressure in systemic arteries (broken line); 50% increased pressure and 50% reduced
vascular compliance (triangle-dotted line); 50% increased pressure, 50% reduced
vascular compliance and 50% reduced baroreceptor sensitivity (i.e. 𝑐𝑛3 = 0.5 in Eq.
(40)) (circles-dotted line)) .............................................................................................. 62

ix

INTRODUCTION
According to World Health Organization Global Health Estimates 2015, 31 percent of all
deaths worldwide are due to cardiovascular diseases.89 The significant portion of these
diseases like hypertension, ischemia, and stroke are related to variations in
hemodynamics. There is, therefore, a huge interest to study the coupling between the
hemodynamic perturbations and cardiovascular adaptations. For example, many animal
experiments are conducted to study the short term and long term effects of high blood
pressure on arteries. Mathematical models and simulations are also carried out to predict
the cardiac and vascular spatiotemporal changes under different hemodynamic
conditions.
In the cardiovascular system (CVS), perturbations in blood flow from homeostatic levels
may occur due to various reasons. In most cases, these perturbations last for short
durations, from few seconds to few days, and their effect may not show adverse results
in maladaptation. These perturbations are usually corrected by various autoregulation in
the system, such as myogenic effect, coronary autoregulation, cerebral autoregulation
and renal autoregulation.43

22

In some cases, though, the perturbations become chronic

and result in significant changes in the CVS and the final consequence could be fatal.40
In the first part of this research, we model adaptation of soft tissues, especially arteries,
to chronic changes in blood pressure and flow. Over the past decade, researchers have
developed various computational models of arterial adaptation in physiological and
pathological conditions (like hypertension82, vasospasm7, and aneurysm6) based on
constrained mixture approach and successfully described and predicted these
1

adaptations over relatively longer periods (months to years) by assuming mass
generation of constituents as functions of stresses. On the other hand, although there are
various experiments over shorter duration (days to weeks) in animal experiments,73

58

there is no mathematical model that is capable of capturing the time course of vascular
adaptation -- the turnover rates of the constituents connecting them with the final vascular
mass and volume changes—during the relatively short period (days to months) compared
to the previous models, which is important for predicting the short period adaptation of
the arteries in medical and clinical problems such as deploying stenting and post-surgery
of artiovenous fistula. One of the major structural constituent in arteries and that plays a
key role in the maintenance of tissue integrity is collagen.60 Hence, we incorporate
collagen turnover rate in the form of biochemical models to previous mechanical models.
The dynamic turnover of collagen in the artery (due to pressure and flow perturbations)
is, however, not well understood, since it involves complex chemo-mechano-biological
pathways. Therefore, a central main objective of this study is to model the turnover of
collagen in three stages: the subcellular synthesis of procollagen, the transition from the
intermediate state of self-assembled microfibrils to matured cross-link, and the collagen
removal that is regulated by wall tension.
The collagen turnover dynamics is combined with constrained mixture model in which
vascular walls are assumed to be made up of different constituents. Mixture theory allows
tracking the production and removal of individual constituents, while still assuming the
constituents of the mixture move together. It also allows the constituents to have distinct
sets of evolving natural configurations. This approach has been already shown to be a
useful tool to simulate arterial adaptations to changes in flow, pressure and axial stretch. 31
2

32

The collagen reaction kinetics within a constrained mixture theory enables us to study

the possible ranges of kinetic parameters (turnover rates) in collagen synthesis and
degradation. For a given change in hemodynamic loading (pressure or flow changes), we
can also determine the resulting variation in arterial geometry and stress distribution by
forward iterative approaches (like Newton-Raphson).
The second objective of this research is developing an alternative parameter estimation
by using experimental reports in the literature. Instead of forward iterative approaches,
we inversely predict the mass turnover from the measured dimension of the artery, the
reported hemodynamic change, and the mixture model. We employ orthogonalization and
regularization to improve the computational efficiency. Comparing the predicted and
measured dimensions, we use an optimization algorithm to compute some of the
(observable) parameters like muscle tone and shear stress coupling, collagen
degradation and smooth muscle cell degradation rates.
Finally, our research aims to expand our computational model toward the whole body
CVS. We have two goals in doing so. First, using lumped model (as opposed to 3-D
models), it is possible to study the coupling between local temporal changes in the arteries
and the corresponding changes in cardiovascular circulation. Geometrical adaptation of
the arteries due to variations in blood pressure and flow modifies other mechanical
properties like arterial compliance, stiffness, and resistance to blood flow. The latter
properties intern influence blood circulation parameters like the energy absorption
character of the arteries, sensing of pressure and cardiac load. Second, patient-specific
diagnostic data can be combined with CVS model and the resulting simulation data may
either complement the data from medical devices or predict hard to measure, yet critical,
3

cardiovascular elements. For instance, vascular and physiologic measurements from
arteriovenous fistula in dialysis patients can be better analyzed after combining with CVS
model. The short term and long term effects of various chemical and mechanical stimuli,
like a concussion and air pollution, on cardiovascular circulation can also be investigated.

4

GROWTH AND REMODELING OF ARTERIES DUE TO
SUSTAINED CHANGES IN HEMODYNAMIC LOADING
Introduction
In the CVS, frequent, short-lived hemodynamic perturbations may occur due to various
causes. Autoregulation, at different levels, in CVS minimizes the effect of the
perturbations so that the cells in the vascular walls function at homeostatic conditions.
For example, if blood pressure is increased, Bayliss myogenic response causes the blood
vessel to constrict.

10

But some hemodynamic loading changes, like hypertension and

hyper-perfusion, last longer. The CVS has to adapt to such sustained changes if the
working environment of vascular cells is to be preserved. The adaptation involves
functional and structural changes of CVS elements. In this research, we focus on
investigating structural change (or growth and remodeling (G&R)) of arteries under
sustained blood pressure or flow changes.
The main structural constituents of arterial wall are an extracellular matrix (ECM) and
smooth muscle cells (SMCs). ECM is composed of elastin, collagen, and many other
proteins. During G&R, vascular growth involves increasing vascular mass (in the form of
deposition of collagen, smooth muscle cells proliferation), whereas the remodeling
involves a change in orientation of collagen families, non-uniform growth, change in the
proportion of constituents or cell migration. The constituent’s turnover is affected by
various stimuli such as diseases, injury or mechanical load; and the individual
constituents have different half-lives. Elastin has a longer half-life (about 50 years) and
slower turnover rate, as slow as 1% per year.84 On the other hand, collagen has a normal
half-life close to 70 days. Its turnover plays a significant role in maintaining the arterial
5

integrity and in G&R due to hemodynamic loads.40 Differences between the rates of
synthesis and degradation of collagen can lead to fibrosis (as in aging or hypertension)
or weakening of the wall (as in dissection or rupture). There is, therefore, strong interest
to evaluate temporal imbalances in collagen turnover.

54 21 15 12 20 73

The common mechanical stimuli in the cardiovascular circulation are blood pressure and
flow. Corresponding to these stimuli, there are various signaling pathways in the arterial
wall. The main pathways are sketched in Figure 1. Perturbation in blood flow rate
changes arterial wall shear stress. The shear stress is sensed by endothelial cells which
then produce vasoactive molecules. The latter molecules modify the tone and the
turnover of SMCs. Increased flow increases shear stress and upregulates endothelialderived nitric oxide synthase, which catalyzes the synthesis and secretion of nitric oxide
(NO). NO activates soluble guanylate cyclase and increases cyclic guanosine
monophosphate (cGMP), which relaxes the contractile apparatus and results in
vasodilation.59

52 43

NO also inhibits proliferation of SMCs.80

17 27

Reduced flow reduces

shear stress and enhances the secretion of endothelin-1 (ET-1), which causes SMCs
contraction and vasoconstriction.47 ET-1 induces SMCs proliferation.50 SMCs production
of collagen appears to be attenuated by NO

46and

augmented by ET-1.67

ECM carries stretch and stress due to pressure. Adhesion junctions and adhesion
molecules, such as integrin, transfer stresses and strains to the cytoskeleton (actin
filaments) in the cells.

19 49 88 87

Transient receptor potential channels (TRPC6) play an

essential role in the regulation of myogenic tone.85 Increased wall stress enhances SMCs
production of ECM such as collagen.68 If the increased pressure persists for days and
weeks, arterial mass increases due to deposition of collagen and other ECM proteins both
6

in the media and adventia.24

12

Excess stretch could also suppress matrix

metalloproteinase (MMP) production and reduce collagen degradation.53
Blood Flow

Q

low shear stress

Flow
induced

high shear stress

MMP

ECs
[ ET-1]

[NO]

relaxation
contraction

synthesis

high tension

Pressure
induced

synthesis

SMCs
ECM
MMP

Figure 1. A schematic of signaling cascade in arterial wall to regulate ECM turnover
and SMC tone and proliferation due to flow and pressure stimuli
The mechanotransduction signals, by which the cells mechano-sense and mechanoregulate the ECM, are complex. The ECM turnover regulation sites are gene transcription,
post-translational modification, secretion, and assembly. The synthesis, deposition, and
degradation of collagen type I are shown in Figure 2. Mechanical stimuli followed by
mechanotransduction upregulates collagen genes (COL1A1, COL1A2). Îą-helix mRNAs
are transcribed in the nucleus and transported to cytoplasmic ribosomes.48

62 11

Thousands of amino acids (glycine (G) and others) are sequenced in (G-X-Y)n motif and
form Îą-chains within about seven minutes. In about eight minutes, three Îą-chains (two
7

Îą1(I) and one Îą2(I) chains) are combined to form procollagen helix in the Golgi apparatus.
10-50% procollagen is degraded inside the cell.57 69 The remaining procollagen is packed
in vesicles, transported to the cell membrane and secreted in about 20 min. Procollagen
changes to tropocollagen in extracellular space after losing propeptides. Tropocollagen
aggregates form within about 20 min.28 Tropocollagen self-assemble and form
microfibrils. 3-10% of microfibrils degrade.74 The remaining microfibrils cross-link 1-8%
per day and form fibrillar collagen. 61 The cross-linking continues as the collagen matures.
It takes from 3 hours to 24 hours for mature collagen to form.55 Though cross-linking
occurs quickly, the degree of cross-linking could change over time as the fibers mature.
Degradation half-life of mature collagen fibers is from 50 days to 100 days.63
degradation could occur via intracellular phagocytotic ingestion

25

The

or extracellular

pathway. MMPs are the major class of enzymes that degrade collagen.18

8

29

a-helix
mRNA

Reactive
a-helices

Intracellular
procollagen
(mp)

10 min - 60 min
(Folding)
Triple-helix
procollagen

3 hr -24 hr
(Mature
collagen
formation)

1-β1
Secretion

Degrade
(10 -50 %)

β1

Tropocollagen
Collagen in
intermediate
state (CI)

Microfibrils

Cross-link
formation

Fibrillar
collagen
(reducible)

β2

Âľ1
Degrade
(3-10% per day)

Cross-linked
collagen
(CF)

Mature
collagen
(irreducible)

Âľ2

Degradation

Extracellular space

½ hr-2 hr
(Exocytosis
of
procollagen)

Synthesis of
procollagen

a-molecules

Intracellular space

~7 min
( Pro-Chain)

Degrade
(Half-life: 50 – 200 days )

Simplified Process

Figure 2. A schematic of the production of collagen
In general, we observe that there are multiple time scales in the G&R of arteries in
response to hydrodynamic loading starting from very fast myogenic effect, to vasodilation
that occurs within minutes of increased flow4, to acute changes for sustained
hydrodynamic loading (days to weeks) and to chronic adaptation (weeks to months).
Although the signaling cascades and the turnover of arterial constituents are well studied,
the detailed kinetics models and specific ranges of reaction rates are still lacking. Rather
than developing a detailed reaction model, with many yet unknown parameters, we
lumped the turnover of collagen into three major categories as shown in the right part of
Figure 2. The extracellular collagen is classified into an intermediate state, which
9

represents self-assembled micro-fibrils, and a final state, which represents fibrillary
collagen with mature cross-links. In the next sections, we will develop the mathematical
model for the simplified synthesis and degradation of collagen, and incorporate into the
constrained mixture model.

Biochemical Model

Stress

A simplified process of collagen synthesis and degradation is sketched in Figure 3.

β1

Collagen in the
intermediate state,
CI

β2

Cross-Linked
Collagen, CF

Âľ2

mp

Âľ1

Cell,

Degradation to other molecules

Figure 3. Simplified model of collagen production
Similar to the work of Niedermuller et al.61, we consider first-order reactions, with an initial
supply 𝑚𝑝 scaled by 𝛽1 (fraction of procollagen that is released to the extracellular space),
a conversion from intermediate to final parameterized by rate 𝛽2, and degradation in each

10

of the two extracellular states parameterized by rates 𝜇1 and µ2 , respectively. The
governing equations are given us

𝜕𝐶𝐼
𝜕𝑡

𝜕𝐶𝐹
𝜕𝑡

(1)

= 𝛽1 𝑚𝑝 − (𝛽2 + 𝜇1 )𝐶𝐼 ,

(2)

= 𝛽2 𝐶𝐼 − 𝜇2 𝐶𝐹 ,

where 𝐶𝐼 and 𝐶𝐹 are the molar concentrations of intermediate and cross-linked matured
collagen. Values of 𝛽1, 𝛽2, 𝜇1 , and 𝜇2 with their corresponding references are summarized
in Table 1. Measurement of collagen turnover without considering reutilization of isotopic
precursors used to label the collagen can result in longer than actual turnover times 75 48,
hence actual rates may be larger than the values indicated in Table 1. The mean age of
1

the mature collagen is thus given by 𝑎𝑚𝑒𝑎𝑛 = 𝜇 = 100 day, with 𝜇2 constant (0.01 day-1).
2

Table 1. The range of biochemical parameters (𝑀𝑊: collagen molecular weight, 𝑎𝑚𝑎𝑥 :
collagen maximum age)
Para.
𝛽1
𝛽2
𝜇1
𝜇2
𝑀𝑊
𝑎𝑚𝑎𝑥

Mouse Carotid Artery30
0.5 -0.9
0.01-0.2
0.03-0.1
0.01-0.02
4.981617x10-22
350

Units

Reference
11

day-1
day-1
day-1
kg
day

61,48, 75
61
29
2
63,29

Biomechanical Model
Changes in blood flow and/or pressure under physiologic conditions alter arterial wall
stresses modestly (e.g., wall shear stress, circumferential, and axial wall stress), which
yet can change the gene expression profile of the vascular wall cells (e.g., endothelial,
smooth muscle, and fibroblasts) and lead to changes in muscle tone and, if sustained,
11

matrix turnover. Such changes allow the vessel to adapt, which is often homeostatic if the
perturbations are modest.39 For a normal artery in humans, wall shear stress and
circumferential stress tend to be maintained at certain levels, ~1.5 Pa for the mean wall
shear stress4 and ~150 kPa for the mean circumferential stress26. Stress analyses that
include residual stress, basal smooth muscle tone, and nonlinear wall properties suggest
that the distribution of circumferential and axial wall stress tend to be uniform across the
wall in normalcy.40 The mean values of stress are given by:
𝑓

𝜎1 = 𝜋ℎ(2𝑟1 +ℎ),
𝑖

𝜎2 =

𝑃𝑟𝑖
ℎ

𝜏𝑤 =

,

4𝜇𝑄
𝜋𝑟𝑖3

,

(3)

where 𝑓1 , 𝑃, and 𝑄 are the axial force, transmural pressure, and volumetric flow rate; 𝑟𝑖
and ℎ are the inner radius and thickness of the artery; 𝜇 is the viscosity of the blood; and
𝜏𝑤 is the wall shear stress whereas 𝜎1 , 𝜎2 are the axial and circumferential stresses,
respectively.
The continuum theory of mixtures is particularly well suited for describing the behavior of
arteries38, though it is currently not possible to capture the chemical or mechanical
contributions of all of the extracellular matrix components (there are on the order of 100
different proteins, glycoproteins, and glycosaminoglycans within the arterial wall). Hence,
we focus on the three primary structurally significant constituents: elastic fibers, collagen
fibers, and smooth muscle cells.42,38 The actual complexity of arterial mechanobiology, in
view of the large number of constituents that could be tracked, necessitates judicious
choices that will allow us to capture the salient responses of the mixture by following only
the key constituents. Thus, let material properties at each place 𝒙(𝑡) in the mixture

12

configuration 𝜅𝑡 (𝐵) be modeled by assuming that, in a homogenized sense, multiple
constituents co-exist within local neighborhoods and at each time t.
We denote constituents SMCs, collagen fibers (using a 4-fiber model with angle Îą) and
elastin via 𝑖 𝜖 [𝑚, 𝑘, 𝑒], but we assume the same mechanical properties for multiple
families of locally parallel collagen fibers 𝑘 𝜖 [𝑐1 , 𝑐2 , 𝑐3 , 𝑐3′ ]. Following Baek et al.7 and
Valentin et al.82, let the deformation of the ith constituent that was produced at time 𝜏, be
described by the linear transformation 𝑭𝑖𝑛(𝜏) (𝑡):
𝑭𝑖𝑛(𝜏) (𝑡) = 𝑭(𝑡)𝑭(𝜏)−1 𝑮𝑖ℎ (𝜏).

(4)

𝑭(𝑡)is 𝑑𝑖𝑎𝑔[𝜆1 (𝑡), 𝜆2 (𝑡)] where 𝜆1 (𝑡) = 𝑙(𝑡)⁄𝑙ℎ and 𝜆2 (𝑡) = 𝑟(𝑡)⁄𝑟ℎ . 𝑮𝑖ℎ (𝜏) is the deposition
stretch matrix. 𝑭𝑖𝑛(𝜏) transforms vectors that belong to the tangent space at 𝒙𝑛(𝜏) 𝜖 𝜅𝑛(𝜏) (𝐵)
to the tangent space at the point 𝒙 𝜖 𝜅𝑡 (𝐵) for the 𝑖 𝑡ℎ constituent that was produced at
time 𝜏 ≤ 𝑡. Again following prior work, the strain energy is given as
𝑡

𝑤 𝑖 (𝑡) = ∫𝑡−𝑎𝑖

𝑚𝑎𝑥

𝑚𝑅𝑖 (𝑡, 𝜏)𝑞 𝑖 (𝑡, 𝜏)Ψ𝑖 (𝑭𝑖𝑛(𝜏) (𝑡))𝑑𝜏,

(5)

𝑖
where 𝑚𝑅𝑖 (𝑡, 𝜏)>0 is newly produced constituent 𝑖 at time 𝜏, 𝑎𝑚𝑎𝑥
is the maximum age of

constituent 𝑖. Notice that any quantity that is denoted by (∙)𝑅 is expressed in the fixed
reference configuration. 𝑞 𝑖 (𝑡, 𝜏) ∈ [0,1] is the fraction of constituent 𝑖 that is produced at
time 𝜏 and survives to time 𝑡. It is inspired by population dynamics76 and could be
expressed as
𝑡

(6)

𝑞 𝑖 (𝑡, 𝜏) = 𝑒𝑥𝑝 (− ∫𝜏 𝜇2𝑖 (𝑠)𝑑𝑠),

13

𝑚𝑅𝑖 (𝑡, 𝜏) and 𝜇2𝑖 (𝑠) will be derived in the next section. Finally, Ψ𝑖 (𝑭𝑖𝑛(𝜏) (𝑡))is the strain
energy of a constituent 𝑖 per unit mass and it is computed via
Ψ𝑚 =

𝑐1𝑚
2

2

2

𝑚
((𝜆𝑚
𝑛,1 ) + (𝜆𝑛,2 ) +

𝑐𝑐

2

𝑐𝑚

1
2

𝑚
(𝜆𝑚
𝑛,1 ) (𝜆𝑛,2 )

2

2

2

− 3) + 4𝑐2𝑚 (𝑒𝑥𝑝 (𝑐3𝑚 ((𝜆𝑚
𝑛,2 ) − 1) ) − 1),

(7)

3

2

(8)

Ψ𝑘 = 4𝑐2𝑐 (𝑒𝑥𝑝 (𝑐3𝑐 ((𝜆𝑘𝑛,2 ) − 1) ) − 1),
3

Ψ𝑒 =

𝑐1𝑒
2

2

2

((𝜆𝑒𝑛,1 ) + (𝜆𝑒𝑛,2 ) +

1
2
2
(𝜆𝑒𝑛,1 ) (𝜆𝑒𝑛,2 )

− 3).

(9)

𝜆𝑖𝑛 is the stretch of constituent 𝑖 from its natural to the current configuration and derived
from Eq.(4). Multiple families of collagen fibers are defined by their present different fiber
alignments. Let 𝒚𝒌 (𝜏) be a unit vector in the direction of the kth collagen fiber family that
was produced at time v. Let the angle between𝒚𝒌 (𝜏) and the first principal direction at
time τ be denoted by 𝜶𝒌 (𝜏). The unit vector 𝒀𝒌 (𝜏) in the reference configuration that
corresponds to 𝒚𝒌 (𝜏) is:

𝑭(𝜏)−1 𝒚𝑘 (𝜏)

𝒀𝑘 (𝜏) = |𝑭(𝜏)−1 𝒚𝑘 (𝜏)|.

(10)

The stretch in the fiber direction from the reference to the current configuration is thus:

1/2

𝜆𝑘 (𝑡) = (𝑭(𝑡)𝒀𝑘 (𝜏). 𝑭(𝑡)𝒀𝑘 (𝜏))

(11)

,

and the stretch of the collagen fiber from its natural to the current configuration is given
by

14

𝜆𝑘 (𝑡)

𝜆𝑘𝑛(𝜏) (𝑡) = 𝐺ℎ𝑐 𝜆𝑘 (𝜏),

(12)

where 𝐺ℎ𝑐 is a homeostatic value of the deposition stretch (i.e. assuming collagen fibers
are always deposited at the same stretch), and 𝜆𝑘 (𝜏) is stretch of the unit vector in the
fiber direction in the reference configuration relative to the configuration at time 𝜏.
A similar approach can be adopted for smooth muscle cells, which are mainly in the
circumferential direction. The homeostatic deposition of smooth muscle is assumed to be
𝑮𝒄𝒉 = 𝐺ℎ𝑐 𝑰. For elastin, which is thought to be distributed isotropically, the homeostatic
deposition is assumed to be 𝐺ℎ𝑒 = 𝐺ℎ𝑒 𝐼. Because the cross-linked elastin is synthesized
primarily at early stages of development, it is difficult to trace its production time. We
̃ 𝒆 = 𝑭−𝟏 (𝑣)𝑮𝒆𝒉 = diag(𝐺̃1𝑒 , 𝐺̃2𝑒 ) which represents a mapping from the
define a new tensor 𝑮
natural configuration of elastin to the reference configuration. Note that 𝐺̃ 𝑒 is not a
deposition stretch tensor for elastin, whereas 𝐺ℎ𝑐 and 𝐺ℎ𝑚 are deposition stretches for
𝑒
collagen and smooth muscle, resp., and 𝐹𝑛(𝑣)
= 𝐹(𝑡)𝐺̃ 𝑒 .

Because newly produced material could have a different material symmetry than that
which is in place, the material properties of Ψ𝑖 can change with time. The newly produced
material depends on the stress and the history of the deformation. The constitutive
relation for the Cauchy membrane stress (force per deformed length) is:
1

𝜕𝑤

(13)

𝑻 = 𝐽 𝑭 𝜕𝑭𝑇 ,

15

1 𝜕𝑤

which gives: 𝑇11 = 𝜆

2

𝜕𝜆1

1 𝜕𝑤

, 𝑇22 = 𝜆

1

𝜕𝜆2

and 𝐽 = 𝜆1 𝜆2. Denoting the membrane stress due to

vascular smooth muscle tone by 𝑻𝑎𝑐𝑡 , the total Cauchy membrane stress of a vasoactive
vessel is assumed to be:
𝑻 = 𝑻𝑝𝑎𝑠𝑠 + 𝑻𝑎𝑐𝑡 ,

(14)

where 𝑻𝑝𝑎𝑠𝑠 is the passive Cauchy membrane stress and 𝑻𝑎𝑐𝑡 is computed from: 66,34
𝑎𝑐𝑡
𝑻𝑎𝑐𝑡 = 𝑆𝜆𝑎𝑐𝑡
2 𝑓(𝜆2 )𝒆2 ⨂𝒆2 ,

(15)

𝑀𝑚 (𝑡)

𝜏 (𝑡)

𝑆(𝑡) = 2𝑆𝑏𝑎𝑠𝑎𝑙 𝑀𝑅𝑚 (0) (1 + tanh (−𝑘𝑡𝑜𝑛 ( 𝜏𝑤

𝑤ℎ

𝑅

𝑓(𝜆𝑎𝑐𝑡
2 )= 1−(

− 1) −

Ln(3)
2

)),

𝜆𝑀 −𝜆𝑎𝑐𝑡
2
)2 ,
𝜆𝑀 −𝜆0

(16)

(17)

where 𝑆𝑏𝑎𝑠𝑎𝑙 is the basal vasoactive tone,𝑀𝑅𝑚 is the total mass of SMCs, 𝑘𝑡𝑜𝑛 is scaling
constant and𝜏𝑤ℎ homeostatic wall shear stress. Moreover 𝜆𝑀 and 𝜆0 are the stretches at
which the contraction is maximum and zero, respectively, and 𝜆𝑎𝑐𝑡
is an active stretch
2
that is computed from:7
𝑑𝑟 𝑎𝑐𝑡
𝑑𝑡

𝑟

(18)

= 𝐾𝑎𝑐𝑡 (𝑟 − 𝑟 𝑎𝑐𝑡 ), 𝜆𝑎𝑐𝑡
2 = 𝑟 𝑎𝑐𝑡

where 𝑟 is the mean radius, with 𝐾𝑎𝑐𝑡 is constant.
The normal Cauchy stress equations in Eq.(3) can be re-written as:
𝜎1 =

𝑇11
ℎ

,

𝜎2 =

𝑎𝑐𝑡
𝑇22 +𝑆𝜆𝑎𝑐𝑡
2 𝑓(𝜆2 )

ℎ

16

.

(19)

Biochemomechanical Model
To integrate the biochemical model into the biochemical-mechanical model, we have to
use a common computational configuration. A fixed reference configuration 𝜅𝑅 (𝐵) (at
reference position X) is utilized and mapped to configurations (e.g., 𝜅𝑡 (𝐵), in vivo
configuration) at different times during the G&R. The use of fixed-reference configuration
allows us to use the material description in modeling kinetics of mass and strain energy
without considering the volume and area changes.
We let the synthesis rate of intracellular procollagen and proliferation of SMCs 𝑚𝑝𝑅 (t) (Eq
(1) ) be given by a stress-dependent scalar function depending on time t. We also
consider different rates of collagen production by the two primary cell types, fibroblasts in
the adventitia and smooth muscle cells in the media:
𝑘 (𝑡)
𝑚𝑝𝑅
=

𝑐 (𝑡)
𝑀𝑅
(𝐾𝜎𝑐
𝑐
𝑀𝑅 (0)

𝜎𝑘 (𝑡)

(

𝜎ℎ

𝜏𝑤 (𝑡)

− 1) − 𝐾𝜏𝑐𝑤 (

𝜏𝑤ℎ

− 1) + 𝑚𝑝𝑘 (0)),

(20)

𝑘
where 𝜎ℎ is homeostatic normal stress, and 𝐾𝜎 and 𝐾𝜏𝑤 are gain-type parameters; 𝑚𝑝𝑅
(𝑡)

is procollagen mass turnover and 𝑀𝑅𝑐 (𝑡) is collagen mass.
𝑘
𝑘
After the synthesis of 𝑚𝑝𝑅
(𝑡), the change of intermediate collagen, 𝐶𝐼𝑅
, is given by a

differential equation (Euler method at each time step ),
𝑘
𝑑𝐶𝐼𝑅
(𝑡)

𝑑𝑡

(21)

𝑘 (𝑡)
𝑘 (𝑡).
= 𝛽1 𝑚𝑝𝑅
− (𝛽2 + 𝜇1 )𝐶𝐼𝑅

In order to utilize the reaction equations from intermediate to mature collagen for arterial
mechanics, we consider the production of matured collagen, meaning by converting from

17

the intermediate collagen, which is soluble, to the cross-linked collagen fiber, k, where
the mass production rate 𝑚𝑅𝑘 (𝜏) at the production time τ, given by
(22)

𝑚𝑅𝑘 (𝜏) = (𝑀𝑊)𝑐 𝛽2 𝐶𝐼𝑅 (𝜏),

where 𝑀𝑊 𝑐 is the molecular weight of collagen. Written in this way, information from the
‘reaction kinetics’ can be incorporated directly within the mixture formulation, thus
providing additional guidance on reasonable constitutive relations for constituent
production and removal. The mass of matured cross-linked collagen fiber family 𝑘 (and
the total mass of collagen, 𝑐) per unit reference area is given by
𝑡

𝑀𝑅𝑘 (𝑡) = ∫𝑡−𝑎

𝑚𝑎𝑥

𝑚𝑅𝑘 (𝜏)𝑞 𝑘 (𝑡, 𝜏)𝑑𝜏,

𝑀𝑅𝑐 (𝑡) = ∑4𝑘=1 𝑀𝑅𝑘 (𝑡).

(23)

The scalar measure of wall stress and thickness in the collagenous fiber families are given
by:
𝜎 𝑘 (𝑡) =

𝑀𝑐 (𝑡)

|𝑻𝑐 𝒚𝑘 (𝑡)|
ℎ𝑐 (𝑡)

ℎ𝑐 (𝑡) = (1−∅ 𝑅)𝜌𝜆

,

𝑓

1 𝜆2

,

(24)

where 𝑻𝑐 = ∑𝑘 𝑻𝑘 , 𝒚𝑘 is unit vector in the direction of collagen family 𝑘, ∅𝑓 is the mass
fraction of fluid in the vessel (70%). Although the relative percentages and mechanical
properties of elastin and collagen change with age70, we consider adaptations that occur
over much shorter time scales. Hence, we take the baseline state as homeostatic.
The rate of degradation 𝜇2𝑖 (𝑡) can be given as a constitutive function of stress. It is also
related to the relative tension of collagen fiber k, as, for example:
(25)

𝑘
𝑘 (𝜁 𝑘
𝜇2𝑘 (𝑡) = 𝐾𝜇1
+ 𝐾𝜇2
(𝑡) − 𝜁𝑐𝑘 )2 ,

18

𝜁 𝑘 (𝑡) =

𝜕𝑤𝑘
(𝜆𝑘
𝑛(𝜏) (𝑡))
𝜕𝜆𝑘
𝑛(𝜏)
𝜕𝑤𝑘
(𝐺ℎ𝑘 )
𝜕𝜆𝑘
𝑛(𝜏)

(26)

.

In this work, we assume that a soft tissue has been in a homeostatic state for a long time
until 𝑡 = 0. Thus, the condition for a steady state for 𝑘 𝑡ℎ collagen family in a tissue is (from
𝑘
Eq. (1), (2) and (22)), and observing that 𝑀𝑅𝑘 (𝑡) = (𝑀𝑊)𝑐 𝐶𝐹𝑅
(𝑡),

𝑚𝑅𝑘 (0) = 𝑚ℎ𝑘 = 𝜇2𝑘 𝑀𝑅𝑘 (0),
1

𝑘 (0)
𝐶𝐼𝑅
= (𝛽 𝑐 )

𝑘 (0)
𝜇2𝑘 𝑀𝑅

(𝑀𝑊)𝑐

2

(28)

,

𝑘 (0)
𝛽 𝑐 +𝜇 𝑐 𝜇2𝑘 𝑀𝑅

𝑚𝑝𝑘 (0) = ( 𝛽2𝑐𝛽𝑐1 )
1 2

(27)

(𝑀𝑊)𝑐

(29)

.

Similarly, SMCs mass will be computed from,
𝑀𝑚 (𝑡)

𝑚𝑅𝑚 (𝑡, 𝜏) = 𝑀𝑅𝑚 (0) (𝐾𝜎𝑚 (

𝜎𝑚 (𝜏)

𝑅

𝜎ℎ

𝜏 (𝜏)

− 1) − 𝐾𝜏𝑚𝑤 ( 𝜏𝑤

𝑤ℎ

(30)

− 1) + 𝑚𝑅𝑚 (0)),

where 𝑚𝑅𝑚 (𝑡, 𝜏) is the turnover rate of SMCs and 𝑀𝑅𝑚 (𝑡) is to total mass of SMCs at time
t. The rate of loss of smooth muscle cells (apoptosis) is assumed to be constant, 𝜇2𝑚 =
𝑚
𝐾𝜇1
= 0.01 𝑑𝑎𝑦 −1. Similar to Eq.(27), 𝑚𝑅𝑚 (0) = 𝜇2𝑚 𝑀𝑅𝑚 (0). The scalar measure of the

stress is set to:
𝜎 𝑚 (𝑡) =

𝑎𝑐𝑡
𝑇22 +𝑆𝜆𝑎𝑐𝑡
2 𝑓(𝜆2 )

ℎ𝑚

𝑀𝑚 (𝑡)

ℎ𝑚 (𝑡) = (1−∅𝑓𝑅)𝜌𝜆

,

1 𝜆2

.

(31)

Simulations
It has been shown previously that 2D (i.e., membrane) models can capture many salient
features of arterial G&R33,34, hence we use the same approach here. Homeostatic
dimensions

(wall

thickness,ℎℎ ,

diameter, 𝐷ℎ )
19

and

corresponding

stresses

(circumferential, 𝜎ℎ , and wall shear, 𝜏𝑤ℎ ) change when the homeostatic pressure, 𝑃ℎ , or
volumetric flow rate, 𝑄ℎ , are perturbed from homeostatic levels. Depending on the
duration(∆𝑡) and magnitude of the perturbations (𝛿𝑃 or 𝛿𝑄), the vascular dimensions
(𝐷(𝑡), ℎ(𝑡)) and stresses (𝜎(𝑡), 𝜏𝑤 (𝑡)) evolve in time (Figure 4).

D (t) is measured

D(t0)

P(t0) + δP(t0+∆t)
Q(t0) + δQ(t0+∆t)

P(t0) + δP(t0)
Q(t0) + δQ (t0)
h(t0)
h(t) is measured

∆t

Figure 4. An idealized model for a growth and remodeling of an artery in response
to perturbed hemodynamics.
Parameters used in the simulation are listed in Table 2. The homeostatic mass fractions
𝑘
𝑓
(∅𝑒0 , ∅𝑚
0 , ∅ , ∅0 ), dimensions (roh, hh, 𝛼ℎ ), stresses (𝜎ℎ , 𝜏ℎ ), pressure (𝑃ℎ ), parameters for

the strain energy ( 𝐺ℎ𝑐 , 𝐺ℎ𝑚 , 𝐺1𝑒 , 𝐺2𝑒 , 𝑐1𝑒 , 𝑐2𝑐 , 𝑐3𝑐 , 𝑐1𝑚 , 𝑐2𝑚 , 𝑐3𝑚 ) and smooth muscle tone (𝜆𝑀 , 𝜆0 ,
𝑆𝑏𝑎𝑠𝑎𝑙 ) are determined by optimization and curve fitting to a mouse carotid artery data. 30
Letting all material and biochemical parameters be the same for the four collagen fiber
families, Figure 5. shows optimization results for intramural pressure vs. outer diameter.
Then, assuming that blood pressure and volumetric flow are independent input variables,
the artery is assumed to be in the homeostatic state at and before 𝑡 = 0; this state is then
perturbed by either changing the pressure or the flow from arterial homeostatic values.
20

We then compute changes in the other arterial parameters as the time progresses. Using
Eq. (13) and (14), the mean blood pressure and circumferential membrane stress for a
vasoactive vessel are calculated from the equilibrium equation in the circumferential
direction:
(32)

𝑎𝑐𝑡
𝑃𝑟 = 𝑇22 + 𝑆𝜆𝑎𝑐𝑡
2 𝑓(𝜆2 ),

where pressure (𝑃) is assumed to be known. After each time step (𝑡 + Δ𝑡), we determine
the radius using Newton-Rephson iteration, where the solution procedure is shown in
Figure 6.

Figure 5. Intramural pressure (mmHg) vs outer diameter (Âľm) for a mouse carotid
artery. 30 The passive and active parameters for the arterial model were estimated
from passive (* * *) and active(o o o) experiments. The fit to data with the model is
shown by the solid (passive) and dashed (active) curves.

21

start
Parameters: energy, etc
Homeostatic values: CI, mass turnover, mass fraction, stresses, hh, rh
t=t+∆t
P, Q
Iteration1=Iteration1+1; update stresses
Iteration2=Iteration2+1; rold=r; Îť2=r/rh
Compute: әw/әλ2 , Tact
fun(r) = әw/әλ2 + Tact - rP
r=rold - fun(r )/dfun(r )/dr
error2=|1-r/rold|
Yes
error2 > Tol Iteration2<Maxit
Compute: mass turnover, stresses, h
error1=difference between new stresses and old stresses
error1 > Tol
Iteration1<Maxit
Iteration2<Maxit
Yes

Yes

t<tfinal
stop

Figure 6. Flowchart indicating the solution for different variables, given
pressure (P) or flow rate (Q). The Newton-Rapson iteration that is used to
determine the mean radius, r, at each time step, ∆t, is the nested internal loop.

22

Table 2. Parameter Estimation of Mouse Carotid Artery
Para.
roh
hh
𝐺ℎ𝑐
𝐺ℎ𝑚
𝐺1𝑒
𝐺2𝑒
𝜎ℎ
𝜏ℎ
∅𝑒0
∅𝑚
0
∅𝑓
∅𝑘0
𝑃ℎ
𝜆𝑀
𝜆0
𝐾𝑎𝑐𝑡
𝑆𝑏𝑎𝑠𝑎𝑙
𝛼ℎ
𝑐1𝑒
𝑐2𝑐
𝑐3𝑐
𝑐1𝑚
𝑐2𝑚
𝑐3𝑚

Mouse Carotid Artery30

Units
mm
Îźm

0.328
21.2
1.07
1.25
2.19
1.64
120
1.50
0.06
0.09
0.70
0.15*{0.111,0.0349,0.427,0.427}
102
1.65
0.65
0.10
0.862
47.6
657.9
3090
18.9
103
11.5
0.794

kPa
Pa

Reference
Para Est
Para Est
Para. Est
Para. Est
Para. Est
Para. Est
Para. Est
44,79

mmHg

day-1
N/m
degree
J/kg
J/kg
J/kg
J/kg

Para. Est
Para. Est
Para. Est.
Para. Est.
Para. Est.
Para. Est
Para. Est
7

Para Est
Para Est
Para Est
Para Est
Para Est
Para Est
Para Est
Para Est

Result
When the blood flow rate reduces over a longer period, the blood vessel geometry
permanently changes. We can investigate the corresponding variation of the stresses by
varying stress sensitivity parameters, for example: 𝐾𝑔𝑐 , 𝐾𝑔𝑚 (see Eq. (20) and (30)). Many
studies have shown that wall shear stress tends to return to original values in response
to changes in flow42,40. If we also assume that the normal stress values return back to
hemostatic levels, given information about flow perturbation (for example: 30% reduced
volume flow rate and constant pressure, as shown in Figure 7), then, using Eq. (3), we
𝜏

𝜎

can predict the final dimension of the artery, i.e. lim 𝜏 𝑤 = lim 𝜎 2 = 1,𝑃 = 𝛾𝑃ℎ , and 𝑄 =
𝑡→∞ 𝑤ℎ

23

𝑡→∞ 2ℎ

𝜀𝑄ℎ yield

𝑟𝑖
𝑟𝑖ℎ

ℎ

= 𝛾ℎ = 𝜀 1/3 and resulting in
ℎ

𝑟𝑖
𝑟𝑖ℎ

ℎ

= ℎ = 0.89 for 𝛾 = 1 and 𝜀 = 0.7. Our
ℎ

simulation complies with this ideal prediction to a various degree depending on the a
value of stress sensitivity parameters we picked. When the mass production was
assumed to be independent of normal stress (i.e., 𝐾𝜎𝑐 = 𝐾𝜎𝑚 = 0), the internal radius
adapts slowly while reducing beyond the expected ideal lumen radius, Figure 7 (a),
indicative of a weak luminal radius regulation. The corresponding shear stress, Figure 7
(c), passed beyond its homeostatic level and continued to increase. Without the stressregulation, the thickness increased excessively, Figure 7 (b). The circumferential stress
similarly did not achieve its homeostatic value, as seen in Figure 7 (d), but instead
continued to reduce as the simulation progressed. In contrast, when the strength of the
mechano-regulation increased (i.e. increasing values of 𝐾𝜎𝑐 and 𝐾𝜎𝑚 ), the internal radius
and shear stress returned to their corresponding homeostatic levels quicker. Yet, too
much increment of 𝐾𝜎𝑐 and 𝐾𝜎𝑚 result in bigger overshoot in thickness change and similarly
larger undershoot in hoop stress change.
Similarly, when the pressure was increased 50% above its homeostatic value (and
constant volume flow rate), the simulation indicates that the internal radius, wall shear
stress, and hoop stress returned back close to the homeostatic level (Figure 8(a, c and
d)).

24

a

b

c

d

Figure 7. Variation of a) internal radius, b) arterial thickness, c)wall shear stress
and d) circumferential stress for different mass coefficients for a 30% decrease in
volumetric flow rate ( 𝐾𝜎𝑖 = 0 (solid line); 𝐾𝜎𝑖 = 2(broken line); 𝐾𝜎𝑖 = 6(dotted
𝑘
𝑘
line); 𝐾𝜎𝑖 = 10(dotted-broken line) and 𝐾𝜇1
= 𝐾𝜇2
= 0.01, 𝐾𝜏𝑐𝑤 = 𝐾𝜏𝑚𝑤 = 52,𝑘𝑡𝑜𝑛 =
𝐾𝜏𝑐𝑤
3

, µ1 = 0.1, 𝛽2 = 0.2)
25

a

b

c

d

Figure 8. Variation of a) internal radius, b) arterial thickness, c) wall shear stress
and d) circumferential stress for a step 50% increased pressure (µ1 = 0.0, 𝛽2 =
0.001(dotted-broken line); µ1 = 0.0, 𝛽2 = 0.05(dotted line); µ1 = 0.05, 𝛽2 =
𝑘
𝑘
0.05(broken line); µ1 = 0.1, 𝛽2 = 0.2(solid line)) (𝐾𝜎𝑐 = 𝐾𝜎𝑚 = 2, 𝐾𝜇1
= 𝐾𝜇2
=
𝑐
𝑚
0.01, = 𝐾𝜏𝑤 = 𝐾𝜏𝑤 = 3𝑘𝑡𝑜𝑛 = 52)
26

On the other hand, the thickness is increased 50% above its homeostatic level (Figure 8
(b)), but it is sensitive to mechano-regulation and collagen turnover parameters. As the
combination of 𝛽2 and 𝜇1 was increased from 0.001 to 0.3, the thickness approached the
final value faster. Lumen radius and shear stress are relatively insensitive to collagen
turnover when pressure increases and flow rate is constant.
We also studied the sensitivity of luminal changes to a wide range of blood volumetric
flow rates with mechano-regulation included. When the volume flow rate increased by
10% or 30% (over a relatively short period), the internal radius increased by about 1.8%
and 2.1%, respectively, whereas if it decreased by the same magnitude, the internal
radius decreased by about 4.5% and 8.2%, respectively (Figure 9).

Figure 9. Variation in internal radius for one day as a function of the change in volume
flow rate (from top to bottom 30%, 20%, and 10% increased volume flow rate, 10%,
𝑘
𝑘
20% and 30% decreased volume flow rate.) (𝐾𝜇1
= 𝐾𝜇2
= 0.01, 𝑘2𝑚 = 13.6, 𝐾𝜎𝑐 = 𝐾𝜎𝑚 =
1, 𝐾𝜏𝑐𝑤 = 𝐾𝜏𝑚𝑤 = 3𝑘2𝑚 , µ1 = 0.1, 𝛽2 = 0.1)

27

From this early change (e.g., within one day in Figure 9) we see that initial luminal control
depends in large part on the vasodilatory versus vasocontractility capacity. This finding is
qualitatively consistent with previous reports, including Valentin et al. 82 who modeled
cerebral arteries. Figure 10 shows a case for a 30% increase in flow for 300 days. An
ideal adaptation, where wall shear stresses returns back to the hemostatic level, Eq. (3)
𝑟

reveals that the final value of the internal radius should be 𝑟 𝑖 =1.09. Figure 10 (a) shows
𝑖ℎ

a result consistent with an ideal adaptation but with a decreasing settling time for an
increase in collagen turnover constants (starting from about three months for 𝜇1 + 𝛽2 =
0.001 to about a month for 𝜇1 + 𝛽2 = 0.3). The peak values for internal radius shift to the
left and decrease in magnitude as turnover constants increase. Although the final
simulated thickness is almost equal to the homeostatic value, Figure 10(b), the transient
undershoots are very sensitive to the collagen turnover constants. As collagen turnover
increases, the magnitude of the undershoot reduces and shifts towards the left. Figure 10
(c and d) shows corresponding variations in the stresses for different collagen turnover
constants. Overall, the different panels in Figure 10 show that, as collagen turnover
increases, the adaptation of the artery to a flow perturbation is faster.

28

a

b

c

d

Figure 10. Variation of a) internal radius, b) arterial thickness, c) wall shear stress and
d) circumferential stress for a step 30% increased flow rate (µ1 = 0.0, 𝛽2 = 0.001(dottedbroken line); µ1 = 0.0, 𝛽2 = 0.05(dotted line); µ1 = 0.05, 𝛽2 = 0.05(broken line); µ1 =
𝑘
𝑘
0.1,𝛽2 = 0.2(solid line)) (𝐾𝜎𝑐 = 𝐾𝜎𝑚 = 3, 𝐾𝜇1
= 𝐾𝜇2
= 0.01, 𝐾𝜏𝑐𝑤 = 𝐾𝜏𝑚𝑤 = 3𝑘𝑡𝑜𝑛 = 52)

29

Intrinsic changes in muscle tone can play important roles over different periods. Inspired
by research by Valentin et al.82, we simulated variations in active stress,𝜎𝑎𝑐𝑡 =

𝑻𝑎𝑐𝑡
ℎ𝑚

, for

different circumferential stretches on different days, as G&R progresses (Figure 11). For
a 30% decrease in volumetric flow rate, the changes in muscle tone are relatively large
over a period of days (Figure 11 (a)). As the circumferential stretch (Îť2) increases from
0.65, the active stress increases and the peak active value is at stretches of 2.1, 2.0, 1.9,
𝑀𝑚 (𝑡)

and 1.8 for muscle mass fractions (𝑀𝑅𝑚 (0)) taken at basal, 7, 14 and 300 days, respectively.
𝑅

Clearly, there is a left-ward shift of the peak active stress as the artery adapts to the
perturbation in flow. On the other hand, for a 50% increase in transmural pressure, the
simulation suggested, Figure 11 (b), that the peak active stress occurs at constant
circumferential stretch of 2.1, while the contribution of active stress declines very rapidly,
from 820 kPa at 7th day to 437 kPa in 300 days.
The active stretch range in Figure 11 is similar to the experimental report by Butler et al.
16

And the magnitude of active stress is not too large compared to research report of Price

et al. 65

30

a

b

Figure 11. Active stress (kPa) vs normalized muscle fiber stretch at different days
for a) 30% decreased volume flowrate and b) 50% increased pressure (𝐾𝜎𝑐 = 𝐾𝜎𝑚 =
𝑘
𝑘
3, 𝐾𝜇1
= 𝐾𝜇2
= 0.01, 𝐾𝜏𝑐𝑤 = 𝐾𝜏𝑚𝑤 = 3𝑘𝑡𝑜𝑛 = 52, µ1 = 0.1, 𝛽2 = 0.2) (0 (basal)(solid line),
7(broken line), 14(dotted line) and 300 days(dotted-broken line). The circles indicate
maximas for the respective curves)

Discussion
Although prior models were able to predict changes in response to altered blood pressure
and flow rate for a longer time (more than 6 months or years)83 7, those models do not
incorporate intrinsic (shorter) timescales. Hence, we replace a simple macroscopic mass
turnover rate of the previous model by a biochemical collagen turnover process: from the
synthesis rates of pro-collagen to cross-linked collagen fibers or rates of removal of
intermediate collagen within the extracellular space. This biochemical kinetic rate model,
that integrates vascular mechanical model, allows constituent turnover to be a function of
mechanical stress, with the associated continuum equilibrium problem enabling us to
31

track temporal changes in stress and geometry of the artery during adaptations to
changes in hemodynamics. By employing mass fractions rather than volume fractions,
the changes in mass were easily coupled with the stress variations over time.
Overall, this new biochemomechanical model has successfully predicted salient aspects
of arterial G&R, including adaptations over periods from weeks to a few months.
Outcomes from the simulations tend to be consistent with observations and to be intuitive;
yet, the complex couplings among shear and normal stress-mediated G&R, as well as
differential synthesis and degradation rates, renders it difficult to intuit the underlying
reasons for the overall (mal)adaptation. A computational model can be additionally useful
in this regard. For example, one can assess differences in increased mass fractions of
collagen and smooth muscle by studying parametrically the relative contributions of 𝐾𝜎𝑐 or
𝐾𝜎𝑚 to G&R (Eq.(20),(30)). Similarly, one can study potential effects of differential
degradation rates for intermediate and mature collagen or conversions of intermediate
into mature collagen (this can be modeled by varying 𝛽2 or µ1 in Eq. (1)), noting that early
and long-term mechanisms may differ (Figure 8).
The simulation results of Figure 7 and Figure 10 show the temporal prediction in vascular
adaptation for two cases: a step-change of decreased blood flow versus a step-change
of increased blood flow, respectively. For step decreases in flow the luminal area
decreases, wall thickness increases for the first few months but then returns close to the
homeostatic level within three months. In contrast, for step increases in blood flow, the
simulation predicts an increased luminal area, increased shear and normal stresses, but
a reduction in wall thickness. The latter two reversely approach the homeostatic level
within four months. These differences stem primarily from the differential effects of NO
32

and ET-1 on collagen synthesis. The transient reduction in wall thickness during
increased flow is indicative of hyperperfusion syndrome which could result in
hemorrhage.1 When pressure increased, Figure 8, the simulation showed an increase in
wall thickness with a preserved luminal area, consistent with most previously reported
results.26 82 An increased thickness can help restore wall stress toward normal but it can
increase the structural stiffness that influences the hemodynamics, which may not be
favorable.3 41
Simulation parametric studies should be checked against experimental data whenever
possible; in cases where data are not available, such models can also guide the
experimental plan. Figure 12 (a) compares simulated results for diameter and wall shear
stress against data from carotid ligation studies. The measured blood volumetric flow rate
over 21 days was used as input in the simulation, which yielded results comparable to
that which was reported experimentally.72 Since there was a lack of pressure
measurements over the reported period, we assumed that pressure remained constant.
The ideal lumen change (the dotted line) is smaller the experimental report, and our
simulation predicts values that are in between the two sets of values after 21 days.

33

a

b

c

Figure 12. a) Simulated changes in internal diameter (solid), ideal adaptation
(diamond-dotted line) and experimental data (circles) reported by Sluijter et al. 72, b)
Simulated changes in internal diameter (solid line), and experimental data (circles)
reported by Friedz et al. 26 and c) corresponding thickness (simulated (solid line),
𝑘
experimental (circles) and ideal adaptation (diamond-dotted line)) (𝐾𝜎𝑐 = 𝐾𝜎𝑚 = 3, 𝐾𝜇1
=
𝑘
𝑐
𝑚
𝐾𝜇2 = 0.01, 𝐾𝜏𝑤 = 𝐾𝜏𝑤 = 3𝑘𝑡𝑜𝑛 = 52, µ1 = 0.1, 𝛽2 = 0.2).
34

The simulated change in shear stress is smaller than the reported change in shear stress.
Deviations between the reported experimental values and our simulations could be due
to additional effects, including inflammation, that are not included in the mathematical
model and unknown differences across species (e.g., rabbits versus mice).
Comprehensive data are clearly needed to form the model. We also used variable
pressure, Figure 12 (band c), as input over 56 days. The simulation result is similar to
reported findings from animal models. Even though the blood pressure is increased in
large magnitude (more than 65%), the internal diameter stays almost constant. The
internal diameter stays slightly below the homeostatic level as per the reported 26
experiment. Compared to the finding of Fridez, et al26, our mathematical model
overestimates the chronic thickness of the vascular wall, which in turn could be a result
of excess generation of collagen or smooth muscle mass. Assuming the blood volume
flow rate is constant, our simulation matches the ideal adaptation after about a month.
More experiment is needed to verify the true source of the deviation. When the pressure
is increased by 50%, the circumferential stress reaches its homeostatic level in 14-16
days, and its chronic level (which is slightly lower than the homeostatic stress) within 45
days. This result is consistent with the finding of Hu, et al,36 wherewith they indicated that
the circumferential stress returns back to the homeostatic level as early as 2 weeks after
aortic coarctation to induce hypertension.
Some of the deviations of our results from reported experiments could be ascribed to the
limitation of our model. For example, we did not include any proteolytic degradation of or
mechanical damage to the elastin, which could happen in both marked increases in flow
and pressure. We also did not account for any possible viscoelastic effects, noting that

35

viscoelastic stress relaxation and G&R stress recovery could appear similar
experimentally. Given the timescales involved, however, we feel that viscoelasticity
should be negligible. We also used a 2D (membrane) rather than a 3D model of the wall,
which has been shown previously to be able to capture salient changes in geometry and
structural stiffness but not details such as residual stress.45 Given that residual stress
related opening angle data were not available in the experimental reports we used, this
was not a key issue but 3D certainly should offer greater insight. Finally, the biochemical
model was simplified to only two steps and we neglected possible stress-mediated
dynamics of the proteolytic enzymes. Again, therefore, it is clear that more data are
needed to inform greater model sophistication. We submit, however, that the present
model both recovers prior simulation results based on phenomenological descriptors and
enables greater information on collagen turnover to be incorporated. The next major
challenge will be to link such changes in turnover to specific stress-mediated changes in
gene expression.

36

INVERSE PROBLEM AND PARAMETER ESTIMATION
Introduction
In Chapter 2, and as indicated in Figure 6, vascular adaptation to change in
hemodynamics has been modeled using forward prediction that involves iterations using
Newton-Raphson and predictor-corrector methods. These iterations enable to determine
the dynamic changes of the arteries over different durations, given blood pressure and
flow rate as inputs. Yet, these combined iterations’ efficiency is low. For example, to
predict the G&R of an artery for 56 days using the membrane model, it takes 28 min
simulation time. Additionally, the forward predictive model relies on assumed parameters
like 𝛽2, 𝜇1 , 𝜇2 , 𝐾𝜎𝑐 and𝐾𝜏𝑐𝑤 . To improve computation efficiency, it is necessary to explore
alternative mathematical tools.
In this research, a linear inverse problem solution is employed. Using known animal
model carotid artery adaptation (for changed pressure or flow), specifically using
measured arterial internal radius and wall thickness, observation matrix and modified
vascular tensions are computed. From the latter matrices, the corresponding mass
turnover of the main arterial constituents (collagen and SMCs) during the same period is
inversely estimated after orthogonalization of observation matrix, selecting dominant
vectors, and applying constraints on mass turnover elements (regularization).

An Inverse Problem and Mathematical Formulation
Inverse problem solution has a general character of evaluating causal factors from
observed or measured data. Many experiments, such as Fridze et al.26 and Sluijter et al.
73,

report the adaptation of arteries in terms of geometric changes over time for measured
37

hemodynamic loading. The geometrical and hemodynamic loading information can be
employed to inversely determine some of the unknown parameters in the
biochemomechanical model. For example, pressure, vessel wall thickness, and lumen
diameter data can be used to compute the rate of constituent mass turnover.
Revisiting the biochemomechanical model in Chapter 2, whereas axial stretch and rate
of degradation were assumed to be constant (i.e. 𝜆1 = 1, 𝜇2𝑖 =constant), the combined
equations (5) to (19), and Eq. (32) could be re-written as,
𝑔(𝑡) = 𝑓(𝑡, 𝜏),

(33)

where
2 −𝑐16 𝑡
𝑐11
𝑒
𝑀𝑚ℎ (𝑐1 (1 − 𝑐 6

1

2

11 𝜆2

4

𝜆4 𝑐7 (1 + 𝑡𝑎𝑛ℎ (𝑐15 (1 − 𝜏
𝑔(𝑡) = 𝜆2 𝑟ℎ 𝑃 −

2
) + 𝑐2 (𝑐11
𝜆2 2 − 1) (𝑒 𝑐3 (𝑐11 𝜆2

𝜏𝑤
)
𝑤 (0)

2
2
𝑐12
𝑐4 𝑒 −𝑐17𝑡 𝑀2ℎ (𝑐12
𝜆2 2

ln(3)

−

2

2

−1)

2

)) +

2

𝑐 −𝜆

)) (1 − (𝑐8 −𝑐4 ) ) 𝑒 −𝑐16 𝑡 +
8

9

2

− 1) (𝑒

2 𝜆 2 −1))
𝑐3𝑐 ((𝑐12
2
2

2

)+

,

2

2
2 2
2𝑐12
𝑐4 𝑠𝑖𝑛2 (𝛼3 )𝑒 −𝑐17𝑡 𝑀3ℎ (𝑐12
𝜆3 − 1) (𝑒 𝑐5 (𝑐12 𝜆3 −1) ) +
2
𝑀𝑒 𝑐6 𝑐14
(1 − 𝑐 2

{

1

4
4
13 𝑐14 𝜆2

)

}

𝑓(𝑡, 𝜏) = 𝐴1 + 𝐴2 + 𝐴3 ,

𝑡 𝑚𝑚 (𝜏)

2
𝐴1 = 𝑐11
∍0

𝑐7 𝜆4
𝑀𝑚ℎ

𝜆2 (𝜏)2

𝜆

2

𝜆2
2
2 (𝜏)

− 1) (𝑒

2

𝑡

2
𝑒 𝑐16 (𝜏−𝑡) (𝑐1 (1 − 𝑐 62 𝜆 4 ) + 𝑐2 (𝑐11
𝜆

(1 + 𝑡𝑎𝑛ℎ ( 𝑐15 (1 − 𝜏

11 2

𝜏𝑤
𝑤

)−
(0)

ln(3)
2

2

𝑡 𝑚2 (𝜏) 𝑐 (𝜏−𝑡)
2 𝜆2
17
(𝑐12
2𝑒
(𝜏)
𝜆2 (𝜏)2
2

2
𝐴2 = 𝑐12
𝑐4 ∫0 𝜆

(𝜏)4

𝑐 −𝜆

2
𝑐3 (𝑐11

2
𝜆2 2
−1)
𝜆2 (𝜏)2

)) (1 − ( 𝑐8 −𝑐4 ) ) ∫0 𝑚𝑚 (𝜏)𝑒 𝑐16(𝜏−𝑡) 𝑑𝜏,
8

− 1) (𝑒

9

2
𝑐5 (𝑐12

38

2
𝜆2 2
−1)
𝜆2 (𝜏)2

) 𝑑𝜏,

)) 𝑑𝜏 +

𝐴3 =

𝑡 𝑚 (𝜏)
𝜆23
2
2
2𝑐12
𝑐4 𝑠𝑖𝑛2 (𝛼3 ) ∫0 𝜆 3(𝜏)2 𝑒 𝑐17 (𝜏−𝑡) (𝑐12
𝜆3 (𝜏)2
3

− 1) (𝑒

2
𝑐3𝑐 (𝑐12

2
𝜆2
3 −1)
𝜆3 (𝜏)2

) 𝑑𝜏,

𝜆3 = √𝑐𝑜𝑠 2 (𝛼3 ) + 𝜆2 2 𝑠𝑖𝑛2 (𝛼3 ) , 𝛼3 = 𝑡𝑎𝑛−1 (𝜆2 𝑡𝑎𝑛(𝛼ℎ )),
𝜆4 = 𝜆𝑎𝑐𝑡
2 , and all 𝑐𝑖 are constants and their values are given in Table 2.
𝑐1 = 𝑐1𝑚

𝑐2 = 𝑐2𝑚

𝑐3 = 𝑐3𝑚

𝑐4 = 𝑐2𝑐

𝑐5 = 𝑐3𝑐

𝑐6 = 𝑐1𝑒

𝑐7 = 𝑆𝑏𝑎𝑠𝑎𝑙

𝑐8 = 𝜆𝑀

𝑐9 = 𝜆0

𝑐10 = 𝐾𝑎𝑐𝑡

𝑐11 = 𝐺ℎ𝑚

𝑐12 = 𝐺ℎ𝑐

𝑐13 = 𝐺1𝑒

𝑐14 = 𝐺2𝑒

𝑐15 = 𝑘𝑡𝑜𝑛

𝑐16 = 𝜇2𝑚

𝑐17 = 𝜇2𝑐

𝑀𝑖 and 𝑚𝑖 (𝜏) mass of the arterial constituents (𝑀𝑚 : SMCs mass:, 𝑀2 : circumferential
collagen mass, 𝑀3 : helical collagen mass, and 𝑀𝑒 :elastin mass) and corresponding rate
of mass synthesis at time 𝜏, respectively. Circumferential stretch, 𝜆2 , could be a function
of current time t or a function of constituent production time τ. In Eq. (33), we showed
𝜆2 (𝑡) simply as 𝜆2 for the sake of clarity.
Both 𝑔(𝑡) and 𝑓(𝑡, 𝜏) are functions of time. For 𝑁 data points, then we can discretize Eq.
(33) over time as:
𝒈 = 𝑨𝑴,

(34)

where 𝒈 is 𝑁 × 1 vector, 𝑨 is 𝑁 × 3𝑁 matrix, 𝑴 is 3𝑁 × 1 vector. M is rate of mass
synthesis vector, 𝑚𝑚 (𝜏), 𝑚2 (𝜏) and 𝑚3 (𝜏) for SMCs in circumferential direction, collagen
in the circumferential direction and for collagen in helical direction, respectively, each with
Nx1 vector size.
Observing the dimensions of the 𝒈 and 𝑴vectors and the matrix 𝑨, the solution of 𝑴 can’t
be unique. We can still find approximate solution after singular value decomposition of𝑨

39

and computing orthogonal unitary vectors and singular values. If 𝑨 is decomposed to𝑼, 𝚺,
and 𝑽 where 𝑼 is 𝑁 × 𝑁 unitary matrix, 𝚺 is 𝑁 × 3𝑁 diagonal matrix with singular entries
𝑠𝑘 , 𝑽 is 3𝑁 × 3𝑁 unitary matrix, and 𝑨 = 𝑼 𝚺𝑽𝑻 , then we can compute 𝑴 from
𝑠 𝐺

𝑘 𝑘
𝑴 = ∑𝑁
𝑘=1 𝑠2 +𝛾 𝑽𝒌 ,

(35)

𝑘

where 𝐺𝑘 and 𝑽𝒌 are the 𝑘 𝑡ℎ element of vector 𝑼𝑻 𝒈, and the 𝑘 𝑡ℎ column of matrix 𝑽,
respectively. 𝛾 is a regularization parameter, and Eq. (37) is Tikhonov regularization
equation.77
Simulation
It is difficult to get specific vascular dimension data for hypertension patients. Instead,
studies are conducted using animal models. In this research, therefore, hypertensive
animal data for carotid artery study from published literature26 is employed (See Table 3).
Usually, in addition to the change in blood pressure, many other vascular parameters
change during the study. For example, the response of the arteries to changes in pressure
is dependent on the age of the animal model. As reported by Fridez et al.

26,

the internal

diameters of both the control and the hypertensive rat carotid artery were increasing
during the period of study. A similar change is also observed in the data of arterial wall
thickness. Therefore, changes of diameter and thickness in the hypertensive animal
groups are subtracted from those of the control group so that the aging effect during the
experiment is minimized and those adjusted data are used in the simulations. Similar
adjustments are made in the flow change data from Sluijter et al. 73

40

Table 3. Normalized experimental arterial adaptation data from literature for changed
pressure and flow
day
0
1
2
7
8
21
56

Data from Fridez et al. 26
𝑟𝑖
𝑃⁄
ℎ⁄
⁄𝑟𝑖ℎ
𝑃ℎ
ℎℎ

Data from Sluijter et al. 73
𝑟𝑖
𝑄
ℎ⁄
⁄𝑟𝑖ℎ
⁄𝑄
ℎℎ

1
0.98
0.98

1
1.63
1.67

1
1.06
1.05

1
0.85
0.83
0.65

1
0.28
0.50
0.62

1

0.99

1.65

1.15
0.60

0.42

1.03

0.97

1.50

ℎ

1.28

The regularization parameter (𝛾) in Eq. (35) is chosen from the combination of the
information captured from generalized cross validation and the fact that production rate
of each arterial constituent(𝑚𝑖 (𝜏)), at each time 𝜏, should be positive or zero, i.e.

𝛾 ∗ = argmin

𝑠𝑘𝐺𝑘
∑𝑁
)
𝑘=1( 2
𝑠𝑘 +𝛾

𝛾 𝜖 ℝ+ (∑𝑁 𝑠𝑘𝐺𝑘 )
𝑘=1 2

2

2

(36)
∩ arg [𝑴(𝛾): ∀𝑚𝑘 ≥ 0],
𝛾 𝜖 ℝ+

𝑠𝑘 +𝛾

From rate of mass production, vascular thickness is computed
ℎ(𝑡)
ℎℎ

𝑀

(𝑡) 𝑟

(37)

ℎ
= 𝑀 𝑡𝑜𝑡𝑎𝑙(0) 𝑟(𝑡)
,
𝑡𝑜𝑡𝑎𝑙

1

where 𝑀𝑡𝑜𝑡𝑎𝑙 (𝑡) = 1−∅ (𝑀𝑅𝑐 (𝑡)(𝑡) + 𝑀𝑅𝑚 (𝑡) + 𝑀𝑅𝑒 ), 𝑀𝑅𝑐 (𝑡) and 𝑀𝑅𝑚 (𝑡) are computed from
𝑓

mass turnover vector 𝑴 in Eq. (35) and using Eq.(23). ∅𝑓 is a mass fraction of fluid in the
vascular wall. The computed thickness ratio is compared with the experimental thickness
ratio and the solution is further improved by minimizing the difference.

41

Results
In a situation in which we have no information about the flow rate, for example as in Fridez
et al.

26

the first approximation is to assume the volume flow rate is the same as the

homeostatic flow. From Table 3 (column 2), we can see that the internal radius is almost
the same as the control. In other words, the increased blood pressure didn’t change the
lumen area very much, i.e.

𝑟(𝑡)
𝑟(0)

~1. As we have already indicated in Chapter 2, for ideal

arterial adaptation and assuming the stresses return back to homeostatic condition after
𝑷𝑟

ℎ𝜎

ℎ

hemodynamics perturbation, lim 𝑷(𝟎)𝑟𝑖 (0) = lim ℎ(0)𝜎2 (0) = ℎ(0) . From this we can see that,
𝑡→∞

𝑖

𝑡→∞

2

given enough time and assuming ideal growth and remodeling of arteries, the thickness
ratio will asymptotically approach blood pressure ratio. Consistent with this assumption,
the inverse solution result (thickness ratio), approaches the pressure ratio as shown in
Figure 13. This is also consistent with the assumed homeostatic flow, which will be given
𝑄

𝑃(0) ℎ

by 𝑄(0) = 1 = (

𝑃

𝟑

) .
ℎ(0)

42

ℎ

Figure 13. Comparison of inverse solution (solid line) with ideal thickness ratio (i.e. ℎ =
𝑃

𝑃ℎ

ℎ

)(broken)

Parameter Estimation
As indicated in the simulation section, it is possible to improve the solution by minimizing
the error between the simulated thickness and the experimental thickness. The previous
inverse solution is based on assumed vascular flow rate (𝑄/𝑄(0)), active muscle tone
constant (𝑘𝑚 ), collagen and SMC mass degradation rates (𝜇𝑓𝑐 ,𝜇𝑓𝑚 ). These parameters
can be optimized so that the difference between the computed arterial wall thickness and
the experimental result is as small as possible as shown in Figure 14.

43

start
Growth and remodeling data from literature
(thickness, lumen area, pressure or flow data)

Hemeostative values: Parameters that are computed from pressure vs
stretch curves like stresses, mass fraction, and strain energy coefficients

Initial value, lower bound and upper bound of parameters to be
estimated like degredation rate and muscle tone coefficient

Iteration=Iteration+1; update estimated parameters
Compute: g , A
Find Îł*
Compute: Minverse (Eq. (35))
Compute: Mforward (Eq.s (1), (20) and (22))

error1 =difference between computed thickness and experimental thickness
error2 =Minverse -Mforward

||[error1 error2]||> Tol
and Iteration<Max iter?

Yes

No
stop

Figure 14. Flow chart for parameter estimation using inverse solution

The combined result from Tikhonov regularization, Eq.(35), and the error minimization
using Matlab optimization algorithm (fminsearchbnd function) is shown in Figure 15
(simulation 2). The Tikhonov regularization parameter (𝛾 ∗ ), as indicated in Figure 14, is
automatically calculated using Eq.(36) during each optimization loop. The computed
parameters are summarized in Table 4.

44

Table 4. Estimated parameters using the combination of inverse problem and
optimization
𝑄/𝑄_ℎ
𝑃/𝑃_ℎ
𝑘𝑡𝑜𝑛
𝜇2𝑐
𝜇2𝑚
𝜇1𝑚 + 𝛽2𝑚
𝜇1𝑐 + 𝛽2𝑐
𝐾𝑔𝑚
𝐾𝑔𝑐
𝑚
𝐾𝑠ℎ
𝑐
𝐾𝑠ℎ
Error
Iteration

Increased pressure
0.51
From Fridez et al.26
6.60
0.036
0.013
0.260
0.320
19.0
2.20
33.0
25.0
0.78
1425

Decreased flow
From Sluijter et al.72
0.50
495
0.050
0.010
0.585
0.000
0.002
482
0.113
46.8
3.0
1425

Comparing estimated parameters for increased pressure (Table 4, column 1) with
experimentally observed ranges (Table 1) of mass turnover rate constants, the upper
range of 𝜇1𝑐 + 𝛽2𝑐 is 0.30, which is very close to the estimated value (0.32). Assuming
SMCs turnover involves two step production and degradation as in Eq. (1)and (2), 𝜇1𝑚 +
𝛽2𝑚 is 0.26 which is in the observed range of collagen turnover rate constants (0.04-0.3).
The estimated half-life of collagen, which is 𝐿𝑛(2)/𝜇2𝑐, is about three weeks, while that of
SMCs, which is 𝐿𝑛(2)/𝜇2𝑚 ,

is about two months. This result is consistent with the

argument of Hu, et al 36 which claim that collagen half-life is shorter during hypertension,
and it could be as short as a week in basilar artery. From the errors, we can say that the
decreased flow parameter estimation is not as good as increased pressure parameter
estimation. In the former case, the arterial thickness information from the experiment is
45

very limited (only the initial and final thicknesses are reported). The parameters, therefore,
may be given with much more reliable confidence intervals if more experimental data are
available and after computing the Jacobean relative to each parameters.
The interesting result is as time progresses, all the four curves in Figure 15 approach
each other. This may be indicative of slow arterial adaptation that is not yet complete after
56 days.

Figure 15. Comparison of two inverse solutions (Figure 13 (solid line), simulation using
optimized parameters (broken line)) with ideal thickness ratio (assuming flow is
homeostatic (dotted line)) and experimental thickness ratio (circles)

Discussion
The application of Tikhonov regularization to compute mass production gives good results
as shown in the previous section. The required computation time is also very small,

46

compared to the usual forward solution that usually requires long iterations. It has been
shown that by using relatively simple procedure it is possible to approximately determine
the intrinsic adaptation of the arteries during pressure or flow changes. The computation
time, which was about 28 min in forward iterative prediction, now is drastically reduced to
only about a minute (57.8 sec).
Knowledge of the mass production rates of the main constituents of the artery helps to
further investigate the deposition rate of extracellular matrix in the artery, the proliferation
rate or apoptosis rates of SMCs, the effect of different enzymes on arterial constituents,
etc. Since each constituent has its own mechanical and chemical property, the
information on the turnover of these constituents enables us to determine the evolution
of the dominant properties in the arterial wall. For example, if too much collagen is
deposited (more specifically, if the elastin to collagen ratio changes), the artery becomes
stiffer.3 A stiffer artery absorbs less energy, which in turn affects the pressure gradient in
the cardiovascular system.
This work can be made more reliable by observing the uncertainty involved in
measurement, the instantaneous changes in blood pressure and flow changes, and the
coupling between pressure and flow. Statistical methods may be appropriately employed
to further improve the solution.
The main limitation in this computation is that we have to have information about the
dimension of the artery, apriori. The solution is heavily dependent on the 𝑨 matrix in
Eq.(34). The strain energy equations that are used to compute 𝑨, though they are widely
employed by various researchers, still require further verification. Of course, more
experimental data are required for better modeling and understanding.
47

MODELING EFFECT OF AVF ON CARDIOVASCULAR
SYSTEM
Introduction
Once we successfully develop the G&R of arteries for perturbed pressure and flowchart
under physiologic conditions, we next aim to incorporate results of such model to general
CVS and investigate their influence as a whole. From the previous chapters, in a condition
where there is a sustained change of pressure or flow, the wall stresses tend to return
back to the homeostatic level within weeks or few months, while there would be
permanent geometrical changes, as shown in Figure 7 and Figure 8. The latter have
secondary consequences since vascular compliance, stiffness, and resistance are
functions of the changed geometry. For example, an increased pressure would increase
the vascular wall thickness for the same lumen area. The thicker vessel will be stiffer and
less flexible. The stiffer vessel absorbs less dynamic load, especially the damping of the
pulse pressure would be lessened, which causes the pulsatile pressure easily pass to the
downstream of the blood vessels like arterioles and capillaries. 64 Stiffer vessel such as
carotid artery can also affect the baroreflex sensitivity and the related autoregulation.14
The autoregulation can then affect the different parameters of CVS such as cardiac
output.
Due to the important wide application of medical equipment to support CVS patients, we
also investigate the long-term (weeks to months) effects of other external stimuli on CVS
by observing local changes and their consequences on whole CVS. There are a wide
variety of reported problems where we can apply our expertise and may find an
acceptable solution. One of such practical and important problems, which has some
48

characters similar to results from our mathematical model, is arteriovenous fistula (AVF).
Another problem of sport-related concussion as reported by Bishop et al.

13

, where the

autonomic function is dysregulated following mild traumatic injury and the corresponding
variations in heart rate, needs further investigation to find out the long-term effects of this
variation on the whole cardiovascular system. The widely prevalent air pollution, for
example, as reported by Shannnahan et al.71, there can be a direct coupling between
inhaled nanoparticles, heart rate variability and induced hypertension. If we model effects
of air pollution as cardiopulmonary interaction, inflammation, chemoreflex and baroreflex
sensitivity, we may predict the corresponding CVS responses. On the other hand, cardiac
output supporting devices like left ventricle assisting device (LVAD) come with significant
problems. LVAD affects systolic blood pressure, pulse pressure, heart rate and
sympathetic nerve activity. It can result in pulmonary hypertension, regurgitation and
aortic insufficiency35. Herein, we, however, focus on AVF and present results of our
research.

Incorporating G&R Resistance Changes into CVS
Blood vessel resistance to flow is the ratio of pressure gradient and blood flow rate. From
the Poiseuille’s equation and assuming laminar flow, resistance can be given in terms of
blood viscosity (𝜂), blood vessel length in axial direction (∆𝑧) and lumen radius (𝑟𝑖 )
8𝜂 ∆𝑧

(38)

𝑅 = ( 𝜋 ) 𝑟4 .
𝑖

If 𝑅ℎ is homeostatic resistance, assuming ∆𝑧 is the same for perturbed condition and
homeostatic resistance, and 𝜂 is constant then

49

𝑅(𝑡)
𝑅ℎ

𝑟

4

(39)

𝑖ℎ
= (𝑟 (𝑡)
) .
𝑖

AVF
AVF is a direct passageway between an artery and a vain. In this research, we focus on
AVF that is surgically created for hemodialysis purposes such as wrist or elbow AVFs.
When AVF is formed, the downstream resistance at the corresponding arterioles and
capillaries will be absent. This leads to decrease in peripheral resistance and increase in
peripheral blood flow. For example, a baseline flow that is 21 ml/min changes to 179
ml/min in 10 minutes after radio-cephalic fistula operation 86. The increased flow increases
wall shear stress and it initiates vascular dilation signal cascade. The dilation increases
the lumen diameter. The flow keeps on increasing for about a week and slightly
stabilizes78 (Table 5 and Table 6).The baseline shear stress 5-10 dyn/cm2 increases to
25 dyn/cm2 in the first week of adaptation and decreases to 18 dyn/cm2 at the end a
month and 10 dyn/cm2 after three months23.

50

Table 5. Reported vein and artery diameter on different days after AVF
Day

Wrist
AVF, Elbow
AVF, Forearm
vein Internal
Diameter,
brachial artery(mm)51
cephalic vein cephalic vein diameter 9
dia (mm) 78
dia (mm) 78

0
1
7
14
21
28
84

3.24

3.71

4.73
4.86
4.84
5.01

5.84
6.02
6.10
6.15

4.3
5.4
5.7

4.4

5.3
6.9

6.1

Table 6. Blood flow through AVF at different days (W: Won et al86, T: Toregeani et al78)
Day

Flow, Wrist AVF Flow,
Elbow Flow, forearm Flow, Wrist AVF,
(ml/min)
AVF (ml/min) 78 vein 23
brachial
artery(ml/min) 51

0
0.0069
1
7
14
21
28
84

20.9 (W)
174(W)
493.63(T)
521.3(T)
458.8(T)
556.8(T)

976.3
1137.8
1151.6
1031.6

56.2

539

640
750

365
438.4
720.4

One purpose of cross-checking the trend of AVF flow (access flow) and diameter is to
verify ‘mature fistula’ before hemodialysis. For example, when mean cephalic vein
diameter increases from 2.3mm to 6.3mm, it is considered mature and ready for
cannulation.5 If we can mathematically model temporal vascular change over time after
the AVF operation, we can complement the functions of devices such as Doppler
ultrasound. It is also possible to predict hard to access CVS variables such as cardiac
output. To validate our mathematical model, we use the temporally changing diameters

51

of the AVF, predict the access flow and compare it with corresponding reported flow
measurements.

CVS Model
Ursino Model81, including the reported baseline parameters, is used to model CVS. As
shown in Figure 16, the Ursino model uses systemic circulation, pulmonary circulation,
and cardiac circulation. The systemic circulation is further subdivided into splanchnic,
extra-splanchnic, skeletal muscle and brain vascular bed. The blood flow, pressure, and
vascular volume variations are related based on mass conservation and Newton’s
Second Law. The baroreflex senses carotid pressure (peripheral artery pressure) and
regulates peripheral resistance (splanchnic, extra-splanchnic), venous unstressed
volume, heart rate and stretches.
The frequency of firing of afferent neurons (𝑓𝑎𝑓𝑓 ) in response to systemic artery pressure
(𝑃𝑠𝑎 ) is modeled as sigmoidal function

𝑓𝑎𝑓𝑓 =

𝑓𝑎𝑓𝑓,𝑚𝑖𝑛 +𝑓𝑎𝑓𝑓,𝑚𝑎𝑥 𝑒 𝑓(𝑃𝑠𝑎 )
1+𝑒 𝑓(𝑃𝑠𝑎)

(40)

,

where 𝑓𝑎𝑓𝑓,𝑚𝑖𝑛 and 𝑓𝑎𝑓𝑓,𝑚𝑎𝑥 are the lower and upper saturation of the frequency (𝑓𝑎𝑓𝑓 ),
𝑓(𝑃𝑠𝑎 ) =

𝑃𝑚𝑒𝑎𝑛 −𝑐𝑛1
𝑘𝑏

,

𝑐𝑛2

𝑑𝑃𝑚𝑒𝑎𝑛
𝑑𝑡

= 𝑐𝑛3 (𝑃𝑠𝑎 + 𝑐𝑛4

𝑑𝑃𝑠𝑎
𝑑𝑡

) − 𝑃𝑚𝑒𝑎𝑛 , 𝑘𝑏 =

𝑓𝑎𝑓𝑓,𝑚𝑎𝑥 −𝑓𝑎𝑓𝑓,𝑚𝑖𝑛
4𝐺

, 𝐺 is

maximum baroreceptor gain and 𝑐𝑖 are constants. Similar sigmodal function is used to
model efferent neural pathways. The coupling between efferent neuron firing and CVS
parameter regulation like resistances and unstressed volumes is modeled by logarithmic
and low-pass first-order derivative functions.

52

Rpp

Cpv

Cpp

Rpa

Pulmonary Circulation
Lpa

Rpv
Cardiac Circulation

Rra

Cra

Cla

Crv

Rla

Clv

Cpa

Rrv

Rlv
Csa
T, Emax,rv, Emax,lv

Lsa

Systemic circulation
Rhv

Rbv

Rev

Rsv

Rsa

Rmv

Csv

Csp

Rsp

Baroreflex Control
Rep

Cep

Cev

Cmp
Cmv

Cbv

Chv

Rmp

Psa

Cbp

Rbp

Chp

Rhp

Figure 16. Cardiovascular System.81 The main cardiovascular elements are 𝐶compliance, 𝑅- resistance and 𝐿- inertance;
53

Figure 16 (cont’d)
The different blood circulation elements are identified by subscripts: 𝑠𝑎- systemic
artery, 𝑠𝑣-splanchnic venous, 𝑠𝑝-splanchnic peripheral, 𝑒𝑣-extrasplanchnic venous, 𝑒𝑝extrasplanchnic peripheral, 𝑚𝑝-peripheral skeletal muscle vascular bed, 𝑚𝑣-venous
skeletal muscle vascular bed, 𝑏𝑝- peripheral brain vascular bed, 𝑏𝑣- venous brain
vascular bed, ℎ𝑝-peripheral coronary vascular bed,ℎ𝑣-venous coronary vascular
bed, 𝑝𝑝-pulmonary peripheral, 𝑝𝑣-pulmonary venous, 𝑟𝑎-right atrium,𝑟𝑣-right ventricle,
𝑙𝑎-left atrium and 𝑙𝑣-left ventricle. The baroreflex regulates cardiac period ( 𝑇 ), slope of
right ventricle end-systolic P-V relationship (𝐸𝑚𝑎𝑥,𝑟𝑣 ), slope of left ventricle end-systolic
P-V relationship (𝐸𝑚𝑎𝑥,𝑙𝑣 ), extrasplanchnic peripheral resistance (𝑅𝑒𝑝 ) and splanchnic
peripheral resistance (𝑅𝑒𝑝 ). The highlighted skeletal muscle vascular bed
resistances,𝑅𝑚𝑣 and 𝑅𝑚𝑝 , are functions of AVF.

Results
We model changing lumen diameter in AVF as a function of time in such a way that an
exponential curve,𝐷𝑓 − 𝐷(𝑡) = 𝑎𝑒 −𝑏𝑡 , is fitted to the data in Table 5, where 𝐷𝑓 is the final
lumen diameter (steady value), 𝐷 is transient lumen diameter (in mm) , 𝑡 is time in days,
𝑎 and 𝑏 are unknown constants. After curve fitting and using data reported by Toregeani
et al.

78,

we get

ln(a)=0.055, b=0.11, 𝐷(𝑡) = 5.01 − 1.0565𝑒 −0.11𝑡 for wrist AVF, and

ln(a)=0.55, b=0.18, 𝐷(𝑡) = 6.15 − 1.7333𝑒 −0.18𝑡 for elbow AVF.
Using equation (39) and the curve fitting equations, the change in resistance can be given
as
3.9535

4

(41)

Wrist AVF: 𝑅(𝑡) = 𝑅ℎ (5.01−1.0565𝑒 −0.11𝑡) ,
4.4167

4

Elbow AVF: 𝑅(𝑡) = 𝑅ℎ (6.15−1.7333𝑒 −0.18𝑡) .

54

a

b

Figure 17. Mean pressure in systemic arteries (a) and skeletal muscle mean flow (b)
under different regulation, pressure, and blood resistance. (0-30 sec: circulation without
baroreflex, 30-300 sec: circulation with baroreflex, 60-70 sec: 50% reduced systemic
arteries pressure, 100-110 sec: 50% increased systemic arteries pressure, 150-200
sec: 30% reduced skeletal muscle bed resistance, 200-250 sec: 70% reduced skeletal
muscle bed resistance)

Using baseline parameters and different initial conditions, the simulation is checked if it
is stable. The result is as shown in Figure 17. Without the baroregulation, the heart rate
is higher than the baseline (~70 beats/min) and the blood pressure is also higher. We
perturbed the mean pressure by 50% for 10 seconds and the CVS returns back to the
homeostatic level (baseline) immediately when the perturbation is removed. The
corresponding flow changes in systemic circulation is minimal, which indicates powerful
regulation. We also varied resistance in skeletal muscle, and the pressure in systemic
arteries and flow in systemic arteries are little affected, whereas we see significant
55

changes in flow through skeletal muscles. The immediate drop in resistance after AVF
operation would directly affect the skeletal muscle resistance. The wrist AVF is modeled
as 30% drop in skeletal muscle resistance (
Figure 16, 𝑅𝑚𝑣 𝑅𝑚𝑝 ), whereas elbow AVF is modeled as 70% drop in skeletal muscle
resistance. The temporal variation of theses resistances is as shown in Eq. (41). The
cardiac load due to these changes in resistance can be observed from ventricular P-V
curve (Figure 18). When the resistance drops in 30%, the PV shifts slightly to the left
from the baseline, i.e. end diastolic P-V is almost the same, while ventricular diastolic
filling starts early, end systolic pressure is lower. This trend is exacerbated when the
resistance drops by 70%. This indicates that for larger drop in skeletal muscle, though the
cardiac stroke is higher, it can’t achieve the baseline pressure. The autoregulation tries
to compensate this shortage by increasing heart rate.

56

Figure 18. Cardiac load: left ventricle (lv) P-V curves for baseline (solid line), 30%
reduced (broken line) and 70% reduced (dotted line) skeletal vascular bed resistance

The simulated access flow rate is compared with data in Table 6 to validate the result as
shown in Figure 19and Figure 20.The result is well within the reported measurements.

57

Figure 19. Blood flow through lower arm AVF, simulation (solid line)vs data from
literature (Toregeani et al.78 (circle), Lomonte et al. 51(diamond))

Figure 20. Flow through upper arm AVF, simulation (solid line) vs data from Toregeani
et al.78(circles)
58

The cardiac load for different access flow is predicted and compared with reported
literature (Figure 21). Access flow increases from 0.3 l/min to 2 l/min, the heart rate
increases from 74 beats/min to 76 beats/min. The cardiac output is within the reported
measurement.9 When the access flow is above 2 l/min, the simulation fails to predict the
cardiac output. In reality, as indicated by Basile et al. 9, the heart failure may occur for
access flow that is larger than 2 l/min.

a

b

Figure 21. Heart rate (a) and Cardiac output (b) for variable access flow (simulation
(solid line), data from literature9 (circles))
59

Further Discussion
As vascular G&R takes place, not only does the vascular resistance changes, but also
the vascular compliance and inertance may vary. Commonly blood vessel compliance is
given as the ratio of strain and pulse pressure8. The strain may be computed from
distensibility. One option to compute distensibility is using the ratio of systolic crosssectional area and diastolic cross-sectional area minus one. In our membrane model,
where we employed mean pressure to predict G&R, we may not implement such
formulation directly. Instead, analogs to the common definition of compliance and since
we can compute stretch at different pressure for different levels of the adaptation of the
vessel, it is possible to get compliance from the slope of stretch versus pressure curve,
i.e. the compliance will be the ratio of change in the stretch to corresponding change in
pressure. On the other hand, using the approach developed by Hughes et al.

37,

and

defining compliance, 𝐶, as a ratio of change in volume and change in pressure, then 𝐶 =
2𝜋∆𝑧𝑟𝑖 3
𝐸ℎ

, where 𝐸 is elastic modulus. The change in compliance over longer time relative

to the basal compliance would be
𝐶(𝑡)
𝐶ℎ

𝑟 (𝑡) 3 ℎℎ ∆𝑧(𝑡) 𝐸ℎ

= ( 𝑟𝑖 )
𝑖ℎ

(42)

.

ℎ(𝑡) ∆𝑧ℎ 𝐸(𝑡)

Based on our analysis in Chapter 2 and Chapter 3, hypertensive arteries will be thicker
and there will be more collagen deposition (hence, increased overall elastic modulus of
the artery), while the lumen area is almost the same as the arteries that carry homeostatic
blood pressure. Increased thickness and elastic modulus reduce compliance as shown in
Eq. (42). Reduction of compliance is especially important for arteries that host
baroreceptors, like carotid artery, since the sensed pressure is a function of the
60

compliance of the artery. Reduced compliance down-regulates the sensed pressure; for
example, the studies of Mattace-Raso et al.56 showed that when carotid artery
distensibility reduces, cardiovagal baroreflex sensitivity also reduces. This distorts the
whole chain of the baroreflex signaling cascade starting from reducing the firing (spikes)
rate in afferent fibers to changes in vascular resistance and cardiac load, finally increasing
pressure (positive feedback loop) and creating additional hemodynamic load.81 The
variation of cardiac load, when vascular compliance and baroreceptor sensitivity change,
is shown in Figure 22. For 50% step increased pressure, the ideal chronic thickness of
the artery would also increase by 50%. From Eq. (42) and assuming the lumen area, axial
stretch and elastic modulus would stay at homeostatic level after adaptation, then, the
compliance would reduce by 50%. In reality, the compliance would reduce more than 50%
due to deposition of additional collagen which is stiffer than elastin. We have to also
remember that in blood vessels, the variation of compliance also occurs due to dilation or
constriction when there is variation in wall shear stress.
In general, although the normalized simulation result follows the reported clinical data
(Figure 19, Figure 20), using simplified lumped CVS model, we can’t claim the actual
simulated compliance and resistance changes are the same us the clinical data. To be
more precise, for example, the CVS model should have more branches specific to the
location of the AVF, and the clinical data should be taken in a controlled manner.

61

Figure 22. P-V curve in left ventricle (baseline (solid line); 50% increased mean
pressure in systemic arteries (broken line); 50% increased pressure and 50% reduced
vascular compliance (triangle-dotted line); 50% increased pressure, 50% reduced
vascular compliance and 50% reduced baroreceptor sensitivity (i.e. 𝑐𝑛3 = 0.5 in Eq.
(40)) (circles-dotted line))

62

CONCLUSION AND FUTURE WORK
In this research, it is shown that collagen turnover reaction kinetics could be incorporated
to constrained mixture theory and predict the growth and remodeling of arteries. This
opens a subtle domain where chemical reaction models could be coupled with
mechanical models. Expanding these models further to the realms of different drug
reactions in blood vessels, corresponding responses of the constituents of CVS like the
variability of the half-life of the vascular constituents and the potency of the different
enzymes in vascular wall, and the parallel changes in vascular mechanical properties
would be a medically fertile application area to explore.
In Chapter 2, it shown that during sustained low or medium perturbations in
hemodynamics like 50% increased blood pressure or 30% reduced flow, it takes the
arteries about three months to recover the homeostatic stresses by adapting their
geometry. During this G&R process, for increased pressure, the lumen area stays almost
constant while the steady arterial wall thickness is proportional (and almost equal) to the
pressure increment. On the other hand, for increased flow rate, the lumen area increases
and the arterial wall thickness doesn’t change too much. When the flow decreases, the
lumen area decreases. It is also shown that these adaptations are sensitive to the strength
of the coupling between stresses and arterial constituent mass turnover. The arterial
muscle tone is also sensitive to changes in hemodynamics and it varies as the vessel
adapts to the changes. In Chapter 3, using limited data from literature and mixture theory,
it is demonstrated that it is still possible to inversely estimate the arterial constituents. The
inverse solution is also combined with optimization to estimate the G&R parameters.
Unlike the forward iterative prediction, the efficiency of inverse solution process could be
63

improved by orthogonalization. Finally, in Chapter 4, the possible application of the
models in previous chapters are shown using CVS and AVF. Based on patient data on
arteriovenous maturation after AVF, the variation of these vascular resistances are
computed and incorporated into full body model of CVS. The resulting access flow is
simulated and verified with patient data.
This research, of course, has many limitations. The major part of this research relies on
membrane model for the deposition of the arterial constituents. In the G&R simulation,
the axial stretch changes are kept constant. Incorporating the collagen kinetic model in
this research to spatial 3D arterial models could give more realistic results. Elastin mass
is constant in the previous simulations and this is not always true. More detailed model
(as opposed to the simplified two-step model) of collagen deposition and the appropriate
turnover reaction for elastin, SMCs and other proteins in the vascular wall could enrich
our research finding.
The inverse mathematical models that are presented in this research can be made more
reliable by including more experimental data, by including the uncertainty in the
measurements and by regressive statistical analysis such as Monte Carlo methods.
Combining predictive models and inverse models can give more applicable and versatile
results. Rigorous analysis is required to improve the computational efficiency.
The integration of G&R models in full body CVS models has huge medical application.
The lumped CVS models that are used in this research couldn’t really capture the spatial
variations of different physical properties. Detailed analysis of the spatiotemporal changes
of the parameters in the CVS model during G&R is necessary.

64

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