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Appendix 2: Hybrid 1D/Womersley Model (Huo & Kassab, 2007) 405
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Fig. 6.14 Schematic representation of computational domains (main trunk and primary branches)
in right coronary artery (RCA, a) and left anterior descending coronary (LAD) and left circumflex
(LCx) arterial trees (b). (c) Schematic of branching angles. Reproduced from Huo and Kassab
(2007) with permission
406 6 Network Analysis of Coronary Circulation: II. Pulsatile Flow
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Equations (6.35) and (6.36) can be used to determine the pressure–flow relationship
at each junction in the large coronary arteries.
Branching Angles
In order to calculate the loss coefficient, K, the optimum branching angles
(Fig. 6.14c) are calculated using the formulations in Murray (1926) and Zamir
(1978). The optimal branching angles are given as a function of area ratios as:
cos θ
1 þ α
ðÞ
¼
BD
4=3
3
þ 1 α
2=3
21þ α
3
ðÞ
4
and cos θBC¼
1 þ α
ðÞ
4=3
3
þ α4 1
2
1 þ α
ðÞ
2α
ð6:37Þ
2=3
3
where
R
0BC
α ¼
R
R
0BC
and R
are the radii for vessel segments BC and BD (Fig. 6.14c), respec-
0BD
0BD
with R
0BC
< R
0BD
ð6:38Þ
tively. Once the optimum branching angles are calculated, the loss coefficients can
be estimated from Miller (1990).
Material Parameters
The viscosity (μ) and density (ρ) of the solution are selected as 1.1 cp and 1 g/cm3,
respectively, to mimic our experimental cardioplegic solution containing Albumin.
The coronary wall thickness for every order is adopted from Guo and Kassab’s data
((2004); Chap. 3). The static Young’s modulus is calculated as ~7.0 10
2
cm
) as described in Huo and Kassab (2006).
6
(dynes/
Method of Solution
In order to solve the nonlinear hyperbolic 1D equations, the time-centered implicit
(Trapezoidal) finite difference method (stable and second-order in both time and
space) is adopted. The details of the mathematical derivations are outlined in Huo
and Kassab (2007). The criterion for the convergence of the iteration between
different meshes is set to 1 10
(Δx) is selected as 0.05 cm and the time step (Δt) is set to 2.0 10
for the FORTRAN program is approximately one hour using general LU
3
(norm-2 relative error). Finally, the mesh size
3
. The run time
Appendix 3: Myocardial–Vessel Interaction (Algranati et al., 2010) 407
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decomposition method to solve the sparse matrix assembled in the largest computational domain on an AMD Opteron 240 computer. Even for a large sparse matrix
with millions of equations, the matrix can be solved relatively fast by using the LU
decomposition with partial pivoting and triangula r system solvers through forward
and back substitution (SuperLU_dist implemented in ANSI C, and MPI for
communications).
Table 6.1 Various parameters for proximal LAD vessel segments in Fig. 6.14b
Number
(Fig. 6.14)
Diameter
(μm)
Length
(μm)
A
A
mother
Q
Q
mother
Re θ K
pp
0
2
1
ρU
mean
2
1 4418 6132 0.96 0.90 363 5.9 0 2.44 1032.0 10
2 4139 2807 0.88 0.96 370 3.2 0 2.05 1031.6 10
3 3473 1866 0.70 0.96 425 6.5 0 1.10 1031.3 10
4 3451 2142 0.99 1.00 427 0.3 0 1.07 1037.4 10
5 3415 3753 0.98 0.94 404 4.9 0 1.17 1031.6 10
6 3360 2963 0.97 1.00 409 0.7 0 1.11 1032.0 10
A 1439 4351 0.10 0.10 129 74.6 0.25–0.45 2.05 1033.2 10
B 979 3616 0.05 0.04 71 79.4 0.25–0.45 3.10 1034.1 10
C 1196 712 0.08 0.04 47 73.6 0.25–0.45 1.07 1041.2 10
D 1018 2668 0.09 0.06 93 76.1 0.25–0.45 1.96 1037.1 10
.
∂u
u
x
x
∂p
∂
1
x
ρ
∂x
2
2
2
3
2
2
2
4
2
3
Diameter and length are obtained from morphometric measurements (Kassab, Berkley, & Fung,
1997). Loss coefficients (K ) depends on many factors, such as the branching angles (θ in degree),
the Reynolds number (Re), the area ratios (A/A
Branching angles (Fig. 6.14c) are estimated from Murray (1926) and Zamir (1978). Since the
branching angles are between 45
and 90, loss coefficients can be estimated based on Miller
(1990). The computed time average ratios of pressure-to-velocity head ( p p
ρ and U
are solution density and mean velocity, respectively) and convective-to-pressure term
mean
), and the flow rate ratios (Q/Q
mother
0
to ρU
2
mean
mother
=2; where
(Eq. 6.31) at the inlet of vessel segments are calculated using the hybrid 1D model
Appendix 3: Myocard ial–Vessel Interaction (Algranati et al.,
2010)
).
Network Reconstruction
Microvascular Network Stochastic network reconstruction is based on statistical
morphometric data (Kassab et al., 1999; Kassab & Fung, 1994; Kassab, Lin, &
Fung, 1994; Kassab, Rider, et al., 1993). The first network (Fig. 6.15) features
174 segments: 20 arterioles, 123 capillaries, and 31 venules. Each network is
supplied by an order 3 (17 μm diameter) arteriole (microvascular inlet) and two
order 3 (30 μm diameter) venules (microvascular outlet). The volume of myocardium tissue supplied by this network (determined from the network dimensions) is
2.4 10
6
mL. The network capillary density is roughly 2800 capillaries/mm2,
consistent with measured data.
408 6 Network Analysis of Coronary Circulation: II. Pulsatile Flow
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Fig. 6.15 A schematic of network reconstruction based on morphometric data (Chap. 2). Upper left
panel: A general layout of transmural distribution of vasculature: Representative microvascular
networks (rectangles) interconnected through arterial and venous trees, at four representative
transmural layers. P
panel: A symmetric arterial transmural tree. Length, diameter, and outlet flow conditions of both
daughter vessels (denoted by asterisks) are mutually equal, except when daughter vessels feed
different wall layers and are subject to different myocardial loading (bifurcations 1–3). Middle
panel: A single microvascular network. A, V denote the microvascular arterial inlet and two venous
outlets, respectively. Thin lines are capillaries while thicker ones are arterioles and venules. Broken
bold lines are the perfused area boundaries. Thin arrows and full and blank circles represent
capillaries that branch out in the network plane, and in the upward and downward directions,
respectively. Bottom panel: Analog circuit for flow analysis in a single vessel segment. P
the segmental inlet and outlet pressures, respectively. P
cular pressures, respectively. ℜ and C are the vessel (nonlinear) resistance and capacitance,
respectively. Reproduced from (Algranati et al., 2010) by permission
are the inlet and outlet boundary pressures, respectively. Upper right
A,PV
and PEVare intravascular and extravas-
IV
, P
are
in
out
Arterial and Venous Trees To significantly reduce the computational load asso-
ciated with a full-scale network (consisting of millions of vessel segments) but still
retain realistic morphometric features, the reconstructed microvascular networks are
placed at four representative transmural locations: at sub-epicardium (myocardial
relative depth, MRD ¼ 0.125), midwall (MRD ¼ 0.325 and MRD ¼ 0.625), and
sub-endocardium (MRD ¼ 0.875). These microvascular networks are interconnected
and linked with the major epicardial vessels via intramyocardial arterial and venous
tree-like networks, taken to be symmetric and dichotomous (Fig. 6.15) up to order
8 vessels. These are reconstructed based on the morphometric data (Kassab et al.,
1994; Kassab, Imoto, et al., 1993), but assigned identical daughter vessels diameters,
lengths, and outlet flow conditions at each bifurcation. The MRD of interconnecting
Appendix 3: Myocardial–Vessel Interaction (Algranati et al., 2010) 409
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vessels is assigned intermediate values, depending on their transmural location. This
reconstructed network had 906 segments, representing the flow in four characteristic
myocardial layers.
Mechanics of Vessel-in-Myocardium System
Following Vis, Sipkema, and Westerhof (1995), a simplified geometry is considered:
each vessel is surrounded by a myocardial tissue of circular cross-section
(Fig. 6.16e). The myocardial outer diameter is chosen to satisfy the measured 1:7
vessel-to-myocardium area ratio (Aliev et al., 2002; Spaan, 1991). Both tissues are
considered incompressible and hyperelastic, having a common interface. Each
vessel diameter is determined by stress analysis based on the force equilibrium
equations of the two concentric cylinders (vessel wall and myocardial tissue)
under the prescribed loading conditions of dynamic axial stretch λ
ratio between loaded and unloaded lengths, see below) and trans-luminal pressures
ΔP. The model equations are presen ted below, foll owed by analysis of the vessel and
myocardium reference configurations.
Model Equations Kinematics: The axisymmetric mappings between each pair of
configurations i and i+1 (Fig. 6.16 in cylindrical coordinates is (r, θ, z)
prescribed by:
¼ r
r
iþ1
iþ1rj
; θ
¼ OA
ðÞθi; z
iþ1
iþ1
=OA
i
iþ1
¼ Λ
iþ1, i
z
(defined as the
z
! (r, θ, z)
i
i
ð6:39Þ
i+1
where OA denotes the opening angle, L is the cylinder length and the stretch ratio Λ
+1,i
¼ L
. The incompressibility constraint implies that:
i+1/Li
r
iþ1
The Green-Lagrange strain is E ¼ (F
deformation gradient for each mapping between configurations. Assuming no twist,
F is given by:
Explicit expressions for the deformation gradient of each mapping are given in
Algranati et al. (2010).
Equilibrium Equations: The Cauchy stress tensor T is derived from the strain
energy function W of each material (vessel wall and myocardium, see below) via the
hyperelastic relationshipT ¼PI + F (∂ W/∂E) F
the circumferential and axial directions imply that the Cauchy stress components T
and Trzvanish. The radial force equilibrium equation is ∂Trr/∂r +(Trr Tθθ)/r ¼ 0.
r
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
in
¼
r
iþ1
F ¼ diag ∂r
hi
2
þ riðÞ2 r
T
F I)/2, where I is the unit matrix and F is the
=∂riΛ
ðÞð6:41Þ
iþ1
in
ðÞ
i
iþ1, i
2
ðÞ=Λ
OA
i
OA
=OAiΛ
iþ1
T
. The equilibrium equations in
=OA
iþ1
iþ1, i
iþ1, i
i
ð6:40Þ
rθ
410 6 Network Analysis of Coronary Circulation: II. Pulsatile Flow
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Fig. 6.16 Vessel loading configurations (a) Vessel configuration as determined from measure-
ments. The cylinders are exposed to in situ axial stretch and trans-luminal pressure. (b) Externally
unloaded: Vessel and myocardium cylinders are closed and share a common length due to tethering.
(c) Untethered: both cylinders are closed but allowed to freely stretch/contract. (d) Stress free: The
residual stresses in both cylinders are released by radial cut. (e) Loaded Configuration: as A, but
stretch and pressure depend on the specific MVI mechanism studied. (f) Predicted and measured
pressure–diameter relations of coronary artery: measured diameters in the passive heart, (Hamza
et al., 2003), rectangles), normalized against diameters under zero pressure and stretch. Model
predicted diameters similarly normalized, using passive (solid line) and active (dash-dot) myocardium material laws. Reproduced from Algranati et al. (2010) by permission
By applying the axial and radial equilibrium equations, the external axial force F
and the trans-luminal pressure ΔP can be expressed in terms of the components Tijof
the tissue stress tensor T as follows (Humphrey, 2002):
Z
r
F
¼ π
z
o
2Tzz Trr T
ðÞrdr; ΔP ¼
r
i
θθ
Z
r
o
r
i
Tθθ T
r
rr
dr ð6:42Þ
Vessel and Myocardi um Constitutive Properties Vessel wall mechanics is studied in the left anterior descending (LAD) artery under positive trans-luminal pressures (Wang et al., 2006) and represented by Fung-type exponential material law
(Algranati et al., 2010). The corresponding parameters are estimated for 10 samples.
z
Appendix 3: Myocardial–Vessel Interaction (Algranati et al., 2010) 411
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The passive and active myocardial mechanics are previously studied on 7 samples
and described by invariant based material laws (Lin & Yin, 1998).
To obtain a representative vessel/myocardium pair, each of the 10 vessel parameter sets is combined with each of the 7 parameter sets of the passive myocardial
sample, and properties of each of the vessel/myocardium pairs is used as inputs to
evaluate the passive in vivo pressure–diameter relationship. The pair having the best
fit to the in situ data of swine large coronary arteries (Hamza et al., 2003) (Fig. 6.16f)
is selected for further analysis of MV I mecha nisms.
Reference Configurations Stress analysis requires knowledge of the reference
stress-free configuration of both ve ssel and myocardium. These references have to
be estimated in order to determine the true states of stress and strain. The available
data consists of statistics of the in situ diameter, length, and wall thickness (Guo &
Kassab, 2004; Kassab et al., 1994; Kassab & Fung, 1994; Kassab, Rider, et al.,
1993) taken under vasodilation, no myocardial activation, and fixed stretch and
intravascular pressure (vessel configuration). The data however, do not account for
transmural morphometric heterogeneity. To incorporate the latter, the reconstructed
diameters are first modified by up to 10% from the vessel cast values in a linear
transmural manner to comply with the observed twice higher endocardial than
epicardial flows in diastolic vasodilated hearts (Goto et al., 1991). To obtain the
reference configurations, the vessel/myocardium equilibrium equations (Eq. 6.42)
are solved subject to the relevant loading boundary conditions. The latter are as
follows:
1. Vessel Configuration (Fig. 6.16a): Kassab and co-workers (Guo & Kassab, 2004;
Kassab et al., 1994; Kassab & Fung, 1994; Kassab, Imoto, et al., 1993) measured
vessel diameters under fixed cast pressure. The loading conditions in this con fig-
uration are ΔP
v
+ ΔPm¼ castin g pressure, where v and m superscripts denote
vessel and myocardium, respectively. Each vessel’s specifi c cast pressure is taken
from a steady-state coronary flow analysis.
2. Unloaded Configuration (Fig. 6.16b): The transition to this configuration is
prescribed by the mapping from the vessel cast configuration (Fig. 6.16a). The
v
loading conditions are F
z
m
þ F
¼ 0; ΔPvþ ΔPm¼ 0. The axial stretch λzis
z
taken to remain constant during this mapping.
3. Untethered Configuration (Fig. 6.16c): Unloaded coronary vessels are not stressfree. When myocardial tethering is removed, large epicardial arteries are found to
shorten by 40% in swine (Wang et al., 2006). The untethered configuration is
obtained upon mapping from the tethered unloaded configuration (Fig. 6.16b).
The loading conditions in this untethered state are
v
F
z
¼ 0; F
m
¼ 0; ΔPvþ ΔPm¼ 0, where it is assumed that the two cylinders
z
maintain a common but stress-free interface.
4. Stress-Free (Reference) Configuration (Fig. 6.16d): The tethered-free vessel and
myocardium are still not stress-free, but rather loaded by internal residual stress.
Their magnitudes are quantified by the measured opening angles (OA) of the
corresponding cylinders when cut open radially (Chap. 3). For coronary vessels,
412 6 Network Analysis of Coronary Circulation: II. Pulsatile Flow
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OA are specimen dependent. For the myocardium OA ¼ 2.75 rad. (Lanir et al.,
1996). The stress-free reference configuration is obtained by mapping between
configurations A and D (Fig. 6.16). Stress and strain analysis is possible if the
radii and stretch ratios at each configuration are known. Since they are not, they
are solved by applying the vessel-in-myocardium model equations under the six
loading boundary conditions listed above (one in section i, two in ii and three in
iii), subject to the assumptions of incompressibility of and common interface
between cylinders (Algranati et al., 2010a). The solution of this highly nonlinear
system is obtained by MATLAB
MATLAB
®
fsolve.
®
ga code for genetic algorithm search and
Loaded Vessel Diameter (Fig. 6.17e) With the stress-free configuration deter-
mined, the vasodilated vessel diameters are evaluated (using the above MATLAB
codes) to optimally satisfy force equilibrium (Eq. 6.42), subject to the vessel internal
and external (MVI dependent) loading conditions. This is done for arteries, capillaries, and veins, under prescr ibed conditions of trans-luminal pressure, dynamic
axial stretch, and myocardial activation. To obtain the corresponding in vivo diameters required for the network flow analysis, modifications are needed to account for
the vessel dynamic axial stretch during the cardiac cycle, and for the autoregulatory
myogenic response and myocardial state of activation, as discussed below.
Dynamic Vessel Stretch (λ
) Vessels are dynamically stretched during the cardiac
z
cycle, together with the surrounding myocardium. This stretch, added to the constant
in situ tethering stretch, affects the vessel diameter. This effect can be evaluated by
solving the vessel-in-myocardium model at each time point and for each vessel. To
reduce computational load, the stretched diameter is evaluated by linear interpolation
between the vessel highest and lowest levels of stretch. The maximum and mean
errors associated with this interpolation are found to be 4% and <1% respectively, in
all vessels at all pressures, in either passive or fully active myocardium. The dynamic
vessel stretch depends on the vessel orientation. Vessels oriented along myocytes
stretch in proportion to the myocytes stretch, whereas vessels perpendicular to
myocytes stretch in proportion to myocytes thickening, and therefore shorten during
myocyte elongation. Hence, penetrating vessels (orders 5–8) are assumed to stretch
in proportion to the myocardial wall thickening, whereas capillaries (except for
cross-connections) are assumed to stretch in proportion to the sarcomere stretch
ratio. Other vessels are assumed to be randomly oriented, and therefore unaffected
by myocardium contraction (i.e., no dynamic stretch).
®
Myocardial Activation This is accounted for by calculating the loaded diameters
under passive and fully active myocardium (D
under intermediate activation is determined as a linear interpolation between these
two states.
Autoregulation The entire complexity of tone regulation is not explicitly
accounted for. Yet, in order to compare predictions with in vivo (autoregulated)
data, arterioles diameters are reduced by 30% from their calculated dilated values, in
line with data for coronary arteries of <200 μm diameter (Kuo et al., 1995).
and Da, respectively). The diameter
p
Appendix 3: Myocardial–Vessel Interaction (Algranati et al., 2010) 413
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Fig. 6.17 MVI mechanisms. (a) Varying elasticity (VE): flow is affected by the activation mediated
changes in myocardial stiffness (represented by a spring); (b) Shortening-induced intracellular
pressure (SIP): flow is regulated by the difference between intravascular and contraction-induced
myocyte intracellular pressures (arrows); (c) Cavity-induced extracellular pressure (CEP): extravascular pressure (P
1985) from cavity pressure (LVP) at the endocardium to atmospheric pressure (P
epicardium. Reproduced from Algranati et al. (2010) by permission
) is the interstitial pressure which varies linearly (Heineman & Grayson,
EV
) at the
atm
The Computational Scheme Equations (6.39)–(6.42) are used to evaluate the
dependence of vessel diameter D on the cast value D
pressure ΔP. These values are presented as surfaces of D ¼ D(D
12 cases: one for each combination of vessel type (artery, capillary, vein), lowest
and highest axial stretch, and passive and active myocardium. Based on the experimental results of Hamza et al. (2003), the above response surfaces are fitted for each
vessel by a sigmoid function expressed by:
D
ΔPðÞ¼
loaded
where D
and ΔP
and Ddenote maximum inflation and deflation diameters, respectively,
+
is the trans-luminal pressure corresponding to the average of D+and D.
1/2
These parameters where estimated to best fit the diameter vs. pressure curve of each
vessel. The maximum and mean errors associated with this fit relative to the exact
solution is found to be 6% and <1%, respectively, in all vessels under all simulated
loading conditions.
hi
DþD
1 þ
D0D
Dþ D
ðÞ
0
exp ln
cast
DþD
D0D
hi
0
, and on the trans-luminal
, ΔP) for
cast
ΔP
ΔP
þ D
1=2
ð6:43Þ
414 6 Network Analysis of Coronary Circulation: II. Pulsatile Flow
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MVI Network Flow Analysis
Conservation of mass requires that the difference between in and out discharges of
each vessel n (Fig. 6.15, bottom panel), Q
time derivative of the vessel’s volume (V
n
P
tðÞP
in
n
ℜ
n
dV
¼
¼
dt
where P
n
in
Q
and P
n
in
n
tðÞQ
n
out
tðÞ¼
out
denote vessel inlet and outlet pressures, respectively. The
n
in
n
), i.e.:
tðÞ=2
ddtπD2tðÞLtðÞ
n
and Q
IV
, respectively, should equal the
out
n
tðÞ
n
P
out
þ
tðÞP
n
ℜ
n
tðÞ=2
n
IV
tðÞ
4
ð6:44Þ
hydraulic capacity, C(t),isdefined as:
CtðÞ
P
tðÞPEVtðÞ
IV
V
π=4 DtðÞ2LtðÞ
¼
ΔP
ð6:45Þ
Equation (6.45) extends the common definition of capacity in nonlinear
intramyocardial pump models (e.g., C(t) d(Volume)/d(P
(t) PEV(t)) to the
IV
case in which changes in vessel lumen volume are caused not only by trans-luminal
pressure, but also by changes in axial stretch and myocardial activation.
Flow in each vessel is analyzed using a three-element Windkessel model
consisting of two identical nonlinear resistors and one nonlinear capacitor
(Fig. 6.15 lower panel). This lumped segment flow model is previously validated
(Jacobs et al., 2008) against a distributive vascular model (Fibich et al., 1993). At
each network bifurcation, mass conservation implies that the sum of discharges Q
should vanish, i.e.:
jk
3
X
Qjk¼
j¼1
Here P
j
denotes the intravascular pressure in each of the 3 vessels composing the
IV
kth bifurcation, and P
is calculated from Poiseuille's law, i.e.:
where L, D, and μ are the vessel length, diameter, and blood apparent viscosity,
respectively. The latter is taken to vary with diameter, following Pries et al. (1994).
j
3
X
P
j¼1
k
is the bifurcation pressure. The vessel hydraulic resistance ℜ
bif
ℜ tðÞ
P
IV
ℜj=2
k
bif
¼ 0 k ¼ 1,2, ..., Bifurcation No: ð6:46Þ
P
in
QtðÞ
tðÞP
out
128μ tðÞLtðÞ
¼
tðÞ
πDtðÞ
4
ð6:47Þ
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