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6.4 Coronary Flow Regulation 395
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passive (unregulated) state, the model predicts that the maximally achievable perfusion density under a perfusion pressure of 100 mmHg is slightly higher in the
sub-endocardium than the sub-epicardium (1.37 vs. 1.24 mL/min/g) which follows
a similar trend to the data of Fokkema et al. (2005) on the transmural distribution of
the conductance.
6.4.3 Model Sensitivity
The different control mechanisms have a variable effect on the local flow dispersion
(Appendix 2). Flow CV is highest under only myogenic regulation, followed by
myogenic + shear, myogenic + shear + full metabolic activation, and under no
regulation (passive state).
6.4.3.1 Myogenic Sensitivity
The n etwork flow is found to be highly sensitive to the myogenic regulation
(Appendix 2, Fig. 6.1) as evident from the significant reduction in flow levels as
compared with the passive state, and from the large effect of the myogenic amplitude
ρ
(Eq. 6.31) on the flow. The mean flow in terminal order 1 vesselsq
m
from 0.30 10
reference level ρ
3
to 0.08 103mm3/s when ρmis increased by 40% of the
ref
and increased to 1.05 103mm3/s when ρmis decreased by
m
50% from its reference level. Flow dispersion is affected as well. The CV increased
from 0.27 to 0.36 when ρ
increased by 90%.
m
decreased
term
6.4.3.2 Shear Sensitivity
The predicted sensitivity to shear is lower than that for the myogenic regulation
(Fig. 6.1).Themeanterminalflow decreased from 0.63 10
0.44 10
3mm3
to 0.83 when F
CV decreased slightly with increasing F
6.4.3.3 Metabolic Sensitivity
There is a significant sensitivity of the flow to the metabolic regulation as seen by the
large shift of the network flow versus perfusion pressure curves closer to the passive
curves under full metabolic activation (Fig. 6.1). The effect of optimized metabolic
activation increases with increasing target flow to an extent which depends on the
perfusion pressure and on the network location and MVI (Fig. 6.2).
/s when F
increased by 40% of its reference level. The corresponding flow
τmax
is reduced to 50% of the referenceF
τmax
, and vice versa (Namani et al., 2018).
τmax
ref
and increased
τmax
3
to
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6.4.3.4 Order Dependence of the Metabolic Diameter Regulation
The model predicts that as the metabolic demand increased from low level
(q
¼ 1.2 103mm3/s) to a high level (q
target
¼ 2.25 103mm3/s) under
target
perfusion pressure of 120 mmHg, the mean diameter increase is 15% for terminal
order 1, 13% for order 2, 11% for order 3, 10% for order 4, and 9.5% for orders 5 and
6 vessels.
6.4.4 Model Validations
Model validation is assessed by comparison with several observed coronary flow
characteristics (Namani et al., 2018). A comparison of the flow sensitivities to the
three regulation mechanisms (Fig. 6.1) indicated that the myogenic and metabolic
regulations have major effects on the flow, while the flow (shear) mechanism has a
minor effect. These predictions are consistent with experimental observations (Feigl,
1983). Goodwill et al. (2017) showed that nitric oxide (NO)-dependent responses
occur primarily in upstream large arterioles and arteries (100–300 μm). The NO
responses are largely absent in resistance vasculature (<100 μm diameter) (Chilian,
Kuo, DeFily, Jones, & Davis, 1993), a diameter range which includes most of the
resistance vessels in the network which are included in the analysis.
The role of the metabolic regulation is clearly demonstrated in Figs. 6.1 and 6.2,
where the metabolic mechanism is responsible for several essential coronary flow
features. Of special note is the network’s ability to meet the flow levels required to
balance the metabolic demand. In so doing, it also significantly reduces flow
dispersion and transmural flow heterogeneity. Finally, the metabolic mechanism
facilitates autoregulation under increasing perfusion pressure. The analysis has
shown that the metabolic and myogenic regulations alone are adequate to yield
flow autoregulation of a similar pattern to that in Fig. 6.2a (with a larger range of
autoregulation) without the need for shear regulation.
The model prediction that under increased metabolic demand, the diameter of
smaller arterial microvessels increases more than those of the higher order ones is in
accordance with the experimental conclusion (Kanatsuka et al., 1989) from studies
with dogs under increased oxygen consumption produced by rapid pacing.
6.4.5 Novel Model Predictions
Two new predictions or hypotheses are inferred from the model. First, failure of the
regulation system to maximally vasodilate the vessels even under profound ischemia
is due to the decaying nature of the conducted metabolic response which thus fails to
maximally vasodilate upstream vessels even under full downstream metabolic
6.4 Coronary Flow Regulation 397
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activation. Second, the extravascular loading by the contracting myocardium (MVI)
enhances the flow reserve under high perfusion pressure and extends the range of
perfusion pressure under which the flow is effectively autoregulated. The model is
hypothesis-generating such that future experiments need to test the two novel
hypotheses. In particular, the second hypothesis on the effect of MVI since it is
counterintuitive.
Coronary flow reserve (CFR) is an important functional characteristic of the
network which has two major components. One derives from the metabolic regulation (the metabolic flow reserve—MFR), the other is due to increased heart rate and
altered waveform (reduced diastolic time fraction) under high metabolic demand.
The model predicts MFR of a factor of 2–3.5 under physiological levels of perfusion
pressure (Fig. 6.1) whereas CFR can be as high as 5-6 in young healthy adults
(Pitkanen et al., 1998). The combined effects of increased heart rate (factor of 2–3.5)
and increased perfusion pressure is expected to agree with the measured CFR.
Clinical observations indicate that even under profound ischemia, flow regulation
fails to maximally dilate the vessels (Hoffman & Spaan, 1990; Sambuceti, Marzilli,
Fedele, Marini, & L’Abbate, 2001). This observation can be accounted for by
recalling that even under full metabolic activation of the terminal vessels, the
predicted flow is lower than the passive one (Fig. 6.1 and Appendix 2 ). Even if
the terminal vessels are maximally vasodilated by the metabolic signal, not all
upstream vessels are maximally dilated due to the decaying nature with path length
of the conducted response towards those upstream vessels.
Under normal physiological inlet perfusion pressure and metabolic demand,
metabolic regulation can yield the required metabolic needs (represented by q
target
and enhanced perfusion homogeneity (Appendix 6, the reference case). Under
extreme levels of demand or perfusion pressure (either very low or very high),
however, a growing difference is found between predicted and target flows, as
well as increased flow dispersion. The mechanistic basis is that under such extreme
conditions, the metabolic regulatio n is either in full activation (all F
P
is extremely low (Fig. 6.2a), or completely suppressed (all F
in
mterm
¼ 1) when
mterm
¼0) whenPinis
extremely high (Appendix 6). A similar metabolic limitation is observed in the
autoregulation response (Fig. 6.2a). Autoregulation has the highest effect in the
physiological range of inlet pressure. As the inlet pressure falls outside of the
physiological range, autoregulation cannot be maintained and the flow deviates
from the metabolic demand (q
).The model-based rationale for the clinical
target
observation that failure of the regulation system to maximally vasodilate the vessels
even under profound ischemia is that it is a direct consequence of the conducted
vasodilatory signaling which implies that even under full metabolic activation of the
terminal vessels, not all the network vessels are maximally dilated to their passive
diameters.
The two novel and apparently counterintuitive model outcomes suggest that the
extravascular loading by the contracting myocardium (MVI) enhances the flow
reserve under high perfusion pressure, and extends the range of perfusion pressure
under which the flow is effectively autoregulated are intriguing. The first (see
Appendix 7) is unexpected in view of the compressive effect of MVI on the vessels.
)
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A closer inspection based on the conceptual framework of the present model reveals
the rationale for this result. In the absence of MVI and under low metabolic demand,
vessels are relatively dilated due to their high trans-vascular pressure which provides
little reserve for further dilation under increased metabolic demand. The resulting
flow reserve (ratio of highest-to-sedentary perfusion levels) is thus low. With MVI,
on the other hand, the trans-vascular pressure is reduced by the extravascular loading
and so are the vessel diameters. Hence, the vessels retain their capacity to significantly dilate under higher metabolic demand and hence increase the flow. The
resulting flow reserve is thus increased.
The second prediction on the effect of MVI on the effective autoregulation
pressure range (Fig. 6.13) can be similarly rationalized. Under low perfusion pressure, both with and without MVI, the metabolic regulation is fully activated to
maximize the vessel diameters and the associated flow. Under increasing perfusion
pressure without MVI, the vessel diameters rapidly increase under their high transvascular pressure. Thus, the vascular metabolic activation is not needed and is
therefore shut down, resulting in increased flow and loss of ability to auto-regulate.
With MVI, even under high perfusion pressure, the vessel trans-vascular pressure is
low, and the metabolic acti vation is distributed within the limits of 0–1 to retain the
ability to auto-regulate.
In summary, the integrated myogenic and flow regulation mechanisms, combined
with the conducted metabolic response, reproduce the experimentally observed flow
features including significant flow reserve, autoregulation of flow under a range of
perfusion pressures, and lower dispersion of regulated flow except under extreme
levels of metabolic demand. The predictions are also in agreement with observations
that the myogenic and metabolic flow regulations are dominant, with lesser contribution from shear regulation. These findings are consistent with observed features of
the coronary system (e.g., (Austin et al., 1994; Kanatsuka et al., 1989; Matsumoto &
Kajiya, 2001)). Therefore, this model can be very useful for understanding the
various regulatory mechanism that become very important under conditions of
compromise blood flow (e.g., coronary artery disease and ischemia) or increase
metabolic demand (e.g., exercise, hypertrophy, heart failure).
Appendix 1: Womersley Model (Huo & Kassab, 2006)
Governing Equations
The blood flow is assumed to be impermeable, incompressible, Newtonian, and
laminar. The blood flow in a vessel segment is taken as the flow in an axisymmetric
cylinder. Therefore, the velocity of the fluid inside the vessel is denoted by u
and u
(r, x, t), where r is the radial coordinate, x as the position along the length of
x
vessel, t is time, u
sectional area of the vessel A(n) is calculated from the diameter of the vessel D(n)
(r, x, t)
r
is the radial velocity, and uxis the axial velocity. The cross-
r
Appendix 1: Womersley Model (Huo & Kassab, 2006) 399
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(n is the vessel number) consistent with experimental measurements. The density ρ
and viscosity μ are assumed constant.
Wave propagation in a tube is governed by wave equations for the pressure p(x,t)
and volume rate of flow q(x,t). In order to derive the wave equations, the continuity
(mass conversation) and momentum equations for tube flow may be written as
follows:
∂qx; tðÞ
∂uxr; x; tðÞ
∂t
þ
þ
∂t
1ρ∂px; tðÞ
∂x
¼
¼ 0 ð6:1Þ
∂x
μρr∂
∂uxr; x; tðÞ
r
∂r
∂r
ð6:2Þ
It is noted that there are two equations and three independent variables (A(x, t),
qx; tðÞ¼2π
R
RnðÞ
uxr; x; tðÞrdr, and p(x, t)). Therefore, an additional equation in the
0
form of constitutive relation is needed to solve for the three unknown variables.
The constitutive equation is based on a pressure–CSA relation for every vessel.
From Laplace’s law, the following holds:
px; tðÞp
where τ
t) p
ðÞRnðÞ
τ
¼
θ
is the mean circumferential stress that relates the transmural pressure p(x,
θ
(intravascular pressure minus external pressure which is assumed zero in
0
hnðÞ
0
ð6:3Þ
diastole), the vessel radius R(n) ¼ D(n)/2, and the wall thickness h(n) for each order
of vessel n. If the stress–strain relation is assumed linear, we may write:
τ
¼ EnðÞ
θ
where E(n) is Young’s modulus for each order vessel and
R2 R2nðÞ
2
nðÞ
R
R2R2nðÞ
2
nðÞ
R
ð6:4Þ
is the circumfer-
ential strain. If Eqs. (6.3) and (6.4) are combined, we o btain the following constitutive equation:
Ax; tðÞ
RnðÞ
AnðÞ
1
ð6:5Þ
px; tðÞp
EnðÞhnðÞ
¼
0
where the radius is expressed in terms of cross-sectional area. At this point, a
pressure–CSA relation is needed as provided in Chap. 3 (Kassab & Molloi, 2001).
It is found that the differences in the P-CSA relation of vessels proximal to 1.5 mm in
diameter (order 11, 10, 9) are relatively small (see Table 3.1 in Appendix 1, Chap. 3).
Therefore, the mean of the P-CSA data for the 1.3 and 2.8 mm diameter vessels are
adopted here. As determined by linear least squares fits of data, the experimental data
may be expressed as follows:
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Ax; tðÞ
¼ 2:5 10
AnðÞ
6
Px; tðÞP
ðÞþ1:06 ð6:6Þ
0
which can be approximated as:
Ax; tðÞ
Px; tðÞP
¼ 4:0 105
0
If Eqs. (6.5) and (6.7) are combined, we obtain
1
AnðÞ
EnðÞhnðÞ
¼ 4:0 105. The thickness-
RnðÞ
ð6:7Þ
to-radius ratio for the 1.3 and 2.8 mm vessels is very similar and is used to calculate
the Young’s modulus E. When R(n) and h(n) are obtained from Guo and Kassab
(Guo & Kassab, 2004; Appendix 3, Chap. 3), the static Young’s modulus E(n) is
determined as ~8.0 10
6
(dynes/cm2). In these simulations, it is assumed that this
value is constant through the coronary arterial tree.
The dynamic Young’s modulus must be considered because the current model
calculates the impedance, pressure, and flow under various frequencies after a
Fourier transform. Previous studies (Douglas & Greenfield, 1970; Gow, Schonfeld,
& Patel, 1974) have investigated the dynamic elastic properties of canine coronary
artery. Here, the ratio of E
dyn/Estat
(Gow et al., 1974) is used to adjust the dependence
of Young’s modulus on frequency.
Assuming the blood flow is harmonic and quasi-steady state, the variables can be
written as:
iωt
iωt
iωt
iωt
ð6:8Þ
ð6:9Þ
ð6:10Þ
ð6:11Þ
u
r; x; tðÞ¼Uxr; x; ωðÞe
x
px; tðÞ¼Px; ωðÞe
qx; tðÞ¼Q
x; ωðÞe
x
Ax; tðÞ¼Ax; ωðÞe
where ω is the angular frequency, also P(x, ω), U
be simplified as P, U
and (6.2), we obtain:
and
The velocity profile is first obtained by Womersley (1955) for pulsatile flow in a
rigid tube as a solution to Eq. (6.13) in the form:
(r, x, ω), Qx(x, ω), and A(x, ω) may
x
, Q, and A. If Eqs. (6.5)– (6.11) are substituted into Eqs. (6.1)
x
iωU
x
iωP
þ
AnðÞR
1ρ∂P
∂x
Eh
∂Q
þ
¼ 0 ð6:12Þ
∂x
μρr∂
¼
∂r
∂U
x
r
∂r
ð6:13Þ
Appendix 1: Womersley Model (Huo & Kassab, 2006) 401
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Ux¼
1
∂P
iωρ
∂x
()
J0i3α2r=RnðÞ
1
i3α
J
0
2
ð6:14Þ
ffiffiffiffiffiffiffiffiffiffiffi
where α ¼ RnðÞ
ωρ=μpis the Womersley number, and J
is zero-order Bessel
0
function. The volume flow rate is the same as that obtained by Duan and Zamir
(1992). Integrating the flow velocity over the cross-sectional area, the following
wave equation is obtained:
AnðÞρ∂P
where F
10
αðÞ¼
3=2
i
2J1i
αJ0i
iωQ þ
3=2
α
ðÞ
, and J
3=2
α
ðÞ
1
∂x
1 F
αðÞ½¼0 ð6:15Þ
10
is the first-order Bessel function. Finally, using
Eqs. (6.12) and (6.15), the wave equations for the pressure P and volume rate of flow
Q may be written as follows:
c
∂Q
0
iωP þ
where c
iωQ þ 1 F
q
ffiffiffiffiffiffiffi
Eh
p
ffiffiffiffi
¼
0
is the wave velocity without the viscous effect, and Y
ρR
10
characteristic admittance. Also we define Z
¼ 0 ð6:16Þ
∂x
Y
0
∂P
αðÞ½c0Y
¼ 1/Y0as the characteristic impedance.
0
¼ 0 ð6:17Þ
0
∂x
0
AnðÞ
¼
is the
ρc
0
If Eqs. (6.16)and(6.17) are combined, we obtain:
2
ω
Q þ 1 F10αðÞ½c
When the viscous effect is incorporated into the wave velocity, the wave velocity can
be defined as: c ¼
Defining Y
1
¼ Y
and (6.19) yields:
∂2Q
2
¼ 0orω2P þ 1 F10αðÞ½c
0
2
∂x
∂2P
2
¼ 0 ð6:18Þ
0
2
∂x
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 F
2
ω
Q þ c
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 F
0
αðÞpc0. Equation (6.18) can then be written as follows:
10
∂2Q
2
¼ 0orω2P þ c
2
∂x
∂2P
2
¼ 0 ð6:19Þ
2
∂x
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
αðÞpand Z1¼ Z0=
10
1 F10αðÞp, and solving Eqs. (6.16)
Qx; ωðÞ¼a cos ωx=cðÞþb sin ωx=cðÞ ð6:20Þ
Px; ωðÞ¼iZ
a sin ωx=cðÞþb cos ωx=cðÞ½ð6:21Þ
1
402 6 Network Analysis of Coronary Circulation: II. Pulsatile Flow
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where a and b are arbitrary constants of integration, c ¼
c0¼
Z
0
Z
1
ffiffiffiffiffiffiffi
q
Eh
ffiffiffiffi
p
is the wave velocity, Y
ρR
¼ 1/Y0the characteristic impedance, Y1¼ Y
¼ Z0=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 F10αðÞp. Following the transmission line method (TLM)
AnðÞ
¼
0
the characteristic admittance,
ρc
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 F
0
1 F10αðÞp c
αðÞp, and
10
(Christopoulos, 1995), the impedance and admittance can be defined as:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Zx; ωðÞ¼
Px; ωðÞ
Qx; ωðÞ
iZ1a sin ωx=cðÞþb cos ωx=cðÞ½
¼
a cos ωx=cðÞþb sin ωx=cðÞ
Yx; ωðÞ¼
1
Zx; ωðÞ
ð6:22Þ
ð6:23Þ
In a given vessel segment, at x ¼ 0 and x ¼ L, the following inlet and outlet
impedance apply:
Z 0; ωðÞ¼
iZ1b
a
ð6:24Þ
and
ZL; ωðÞ¼
iZ1a sin ωL=cðÞþb cos ωL=cðÞ½
a cos ωL=cðÞþb sin ωL=cðÞ
ð6:25Þ
If Eqs. (6.24)and(6.25) are combined, we obtain:
0
Equation (6.26) is used to calculate the impedance/admittance in the entire coronary
tree from inlet to the capillary vessels.
Method of Solution
The characteristic impedance, characteristic admittance, and velocity (including the
viscous effect) are first calculated for every vessel segment in the entire coronary
arterial tree. There are two or more vessels that emanate from the jth junction point
anywhere in the current tree structure. Mass is conserved at each junction and
pressure is continuous at the junct ion, which may be written as:
Z 0; ωðÞ¼
iZ1sin ωL=cðÞþZL; ωðÞcos ωL=cðÞ
cos ωL=cðÞþiY
Q mother; ωðÞ¼
ZL; ωðÞsin ωL=cðÞ
1
X
Q daughters; ωðÞ ð6:27Þ
ð6:26Þ
Appendix 2: Hybrid 1D/Womersley Model (Huo & Kassab, 2007) 403
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P mother; ωðÞ¼P daught ers; ωðÞ ð6:28Þ
From Eqs. (6.27) and (6.28), we may write:
X
YLmotherðÞ; ωðÞ¼
Y 0 daughtersðÞ; ω½ð6:29Þ
Once the terminal impedance/admittance of the first capillary is compu ted, we
proceed backwards to iteratively calculate the impedance/admittance in the entire
coronary tree by using Eq. (6.26) (obtained from Eqs (6.22) and (6.29)). The
sawtooth pulsatile pressure produced experimentally by the piston pump is
discretized by a Fourier transformation to determine the constants a and b in
Eqs. (6.20) and (6.21). The flow and pressure are then calculated by using
Eqs. (6.20) and ( 6.21).
Appendix 2: Hybrid 1D/Womersley Model (Huo & Kassab,
2007)
Governing Equations
The details of the mathematical derivations are outlined in Huo and Kassab (2007).
Briefly, the governing equations for flow and pressure may be expressed through
conservation of mass and momentum as:
∂A
∂q
þ
¼ 0 ð6:30Þ
∂x
stath0
R0A
∂A
¼8πv
∂x
0
q
∂2q
þ v
∂x
2
A
ð6:31Þ
∂q
∂t
þ
∂∂x43q
2
þ
A
∂t
AρE
where A is the cross-sectional area of the vessel, q is the volumetric flow rate, ρ is the
density, E
is the static Young’s modulus, h is the wall thickness, R0, h0, and A0are
stat
the original radius of the vessel, original wall thickness, and cross-sectional area,
respectively, and v ¼ μ/p is the kinematic viscosity. Equa tion (6.31) can be rewritten
as follows (Huo & Kassab, 2007):
∂q
∂t
83qA∂q
þ
∂x
43q
2
∂A
AρE
∂x
þ
A
stath0
R0A
∂A
¼8πν
∂x
0
q
∂2q
þ v
∂x
2
A
ð6:32Þ
The constitutive equation for a coronary vessel can be expressed as Huo and Kassab
(2006):
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p p0¼
E
stath0
A
1
A
R
0
0
ð6:33Þ
where p and p
are the internal pressure and external pressures, respectively.
0
Equations (6.30), (6.32), and (6.33) are used to calculate the pulsatile blood flow
in each segment of the large r arteries (e.g., the main trunk and primary branches as
shown below in Figs. 6.14a, b). The viscosity is conside red constant at a value of
1.1 cp to mimic the cardioplegic solution used in the experiments. Below, the
relevant boundary conditions will be established in order to extend the model to
the entire coronary arterial tree.
Boundary Conditions
The inlet pressure boundary condition was obtained from experimental measurements (Huo & Kassab, 2006). The impedance boundary conditions by (Olufsen,
1999, 2000) is adopted as the outlet boundary conditions at the terminals of 1D
model, as shown in Figs. 6.14a, b. The Womersley’s theory as outlined in Appendix
1 is applied to the morphometric trees to represent the distal vascular beds for each of
the outlets of the numerical domain. The impedance/admittance (Z(x, ω)/Y(x, ω)) is
calculated at each outlet of the numerical domain. By inverse Fourier transformation,
z(x, t)/y(x, t) may be obtained from Z(x, ω)/Y(x, ω). Using the convolution theorem,
the new outflow boundary conditions may be obtained as follows:
Z
t
px; tðÞ¼
or
qx; tðÞ¼
qx; τðÞzx; t τðÞdτ
tT
Z
t
px; τðÞyx; t τðÞdτ
tT
ð6:34Þ
Equation (6.34) was used to prescribe the pressure–flow boundary conditions at each
outlet. For the junction boundary condition, mass is conser ved at each junction:
Vortices created at the bifurcations can result in loss of energy. When the effect of
gravity is neglected, a loss coefficient K (Miller, 1990; Olufsen, 2000; Sherwin et al.,
2003) can be incorporated into Bernoulli’s equation as follows:
P
daughter
x; tðÞ¼P
Q
mother
x; tðÞ¼XQ
mother
x; tðÞþ
Kρ
2
x; tðÞ ð6:35Þ
daughter
hi
ρ
u
motherðÞ
x
2
u
motherðÞ
x
2
uxdaughterðÞ
2
2
ð6:36Þ
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