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6.3 Myocardial–Vessel Interaction Flow 385
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Fig. 6.10 Schematic of
coronary network and
pressure (mmHg)
distribution in the coronary
vasculature at (a) systole
and (b) diastole. The
extravascular pressure is
given in parentheses (black)
for each of the four
myocardial layers (first top
layer: epicardium; and
bottom fourth layer:
sub-endocardium).
Reproduced from Kassab,
Algranati, and Lanir (2013)
with permission
measured asymmetric branching pattern. This model enabled the assignment of
different extravascular pressures to different microvascular networks, subjected to
varying IMP in different layers of myocardial wall. Both the feeding vessel of the
arterial tree and draining vessel of the venular tree are epicardial and subject to
experimental input and output pressures.
6.3.6.2 Single Vessel Flow Model
The hemodynamics in each vessel segment is modeled as quasi-steady, fully developed laminar flow (Fig. 6.9). A generalized, validated nonlinear capacitor analog
(Fibich, Lanir, & Liron, 1993; Jacobs, Algranati, & Lanir, 2008) is employed to
incorporate the effects of the change of cross-sectional area and axial stretch of each
vessel during the cardiac cycle. This approach is computationally far less demanding
than solving the detailed 3D Navier–Stokes equation but still provides an accurate
prediction of the network bifurcation pressures, flow heterogeneity, and phase shift,
when compared to the full distributive analysis (Fibich et al., 1993). The single
vessel lumped flow model (Jacobs et al., 2008) has been validated with a distributive
model under the relevant flow conditions (Fibich et al., 1993).
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The diameter-dependent apparent blood viscosity in each vessel segment is
calculated at each time step according to the empirical formula of Pries et al.
(1994); Appendix 3, Chap. 5. The vessel compliance, i.e., pressure–diameter relations (PDR), is predicted for each vessel by a validated vessel-in-myocardium stress
analysis (Appendix “Network Reconstruction”). Because of a lack of elasticity data
on the venous vessels, a uniform venous elasticity is assumed and subsequently
varied through a sensitivity analysis.
6.3.6.3 Network Flow Model
Pressure continuity and mass conservation is enforced at each bifurcation junction,
and hence the hemodynamics of the entire coronary network is formulated into a set
of linear differential equations (Appendix “Network Reconstruction”). Both inlet
(arterial) and outlet (venous) pressures are assumed to follow an aortic pressure
pattern, with systolic and diastolic magnitudes as measured in the literature (Klassen,
Armour, & Garner, 1987; Tillmanns, Steinhausen, Leinberger, Thederan, & Kubler,
1981). Each vessel in the network is subjected to a tissue pressure as determined by
the assumed MVI mechanism and myocyte contraction-induced pressure (Brugada
et al., 1991; Nevo & Lanir, 1989). The latter is taken as proportional to the sarcomere
dynamic shortening pattern (Hurst & Logue, 1970; Rabbany et al., 1989; Rodriguez
et al., 1992 ).
6.3.6.4 Model Pre dictions
The model can predict longitudinal pressure distribution, flow, velocity, and diameter changes under normal boundary conditions (i.e., coronary pressure, LV pressure, contractility, heart rate) (Algranati et al., 2010). The following is a brief
summary of some aspects of the model predictions.
Phasic Changes The model pressure predictions for arterioles in the
sub-endocardial and sub-epicardial layers of the myocardium are summarized in
Fig. 6.11. The out-of-phase diameter and velocity measurements (Toyota et al.,
2005) and volume displacements (Kajiya et al., 2008) are consistent with the
model predictions. The phasic changes in diameter are much greater in the
sub-endocardium while the converse is true for the sub-epicardium. The model
predictions are also in good agreement with phasic diam eter measurements of
Hiramatsu et al. (1998) and Yada et al. (1994).
The mean arterial intravascular pressure increases as the microcirculation is
subjected to higher extravascular pressures. Intravascular pressure measured by
micro-puncture of the epicardial vessels of rats and cats reveal a close resemblance
between aortic and coronary arteriolar pressure patterns, with an av erage reduction
of 30% in pressure magnitude in arterioles as compared to the aorta (Tillmanns et al.,
1981). Sub-epicardial venous pressures measured in the same study reached the
6.3 Myocardial–Vessel Interaction Flow 387
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Fig. 6.11 Effect of intra-myocardial fluid pressure (IMP) on predicted blood velocity and vessel
diameter. Left: Toyota et al. (2005) measurements of blood velocities and diameters in
sub-endocardial (top) and sub-epicardial (bottom) arterioles. Right: model prediction of extravascular pressure. X-axis: normalized time. Y-axis: velocity (mm/s) at left and diameter (μm) at right.
Reproduced from Kassab et al. (2013) with permission
extreme values of ~25 in late systole and ~5 mmHg in late diastole. Hence, vascular
dynamic pressure patterns in the deeper layers are significantly affected by the
surrounding IMP. Interestingly, IMP increases the intravascular coronary pressure
and alters the pressure wave pattern in both arteries and veins. The model prediction
that blood pressure in sub-endocardial arteries is significantly elevated throughout
the cardiac cycle compared to more superficial layers is previously suggested by
Hoffman, Baer, Hanley, and Messina (1985). The model predicts a short, steep
reduction in this blood pressure as the mitral valve opens and LVP decreases. This
pressure reduction may be less steep, however, depending upon synchronization
between cavity pressure and myocyte contractility which is assumed to be simultaneous in these illustrative calculations.
Test of MVI Mechanisms There has been controversy over the mechanism by
which systolic flow is impeded by intramyocardial fluid pressure (IMP; (Smith &
Kassab, 2001)). Algranati et al. (2010) tested five mechanisms, cavity pressure
model (CPM), varying elasticity model (VEM), intramyocyte pressure model
(IPM), CPM + VEM, and CPM + IPM to determine which model or combination
of models is the most consistent with the phasic velocity and diameter data measured
by Toyota et al. (2005). The predicted velociti es match the measurements fairly well,
as shown in Fig. 6.11, but only with the CPM + IPM mechanism (Algranati et al.,
2010). It should be noted that the LVP (from CPM) alone does not provide sufficient
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extravascular pressure on vessels to reproduce the physiological measurements
without the contribution of contractility generated by surrounding myocytes (IPM)
(Krams, Sipkema, et al., 1989; Krams, Sipkema, Zegers, et al., 1989). The latter
effect is at least 50% of the LVP. Only the model predictions including CEP+SIP
mechanisms agree with various physiological measurements (Table 6.2, Appendix
3).
In summary, the MVI model predictions of vascular pressure, velocity, and
diameter patterns are only in agreement with in vivo measurements when both the
LV pressure and the myocardial contractility are externally imposed on the vessel
wall. This underscores the significance of muscle contractility and suggests that none
of the current hypoth eses (intramyocardial pump, time-varying elastance, etc.) alone
can predict the phasic coronary physiology. Therefore, a combination of the previous hypotheses is required to predict the experimental measurements on coronary
circulation.
6.4 Coronary Flow Regulatio n
Many mechanisms have evolved to regulate the coronary circulation to ensure
adequate perfusion of the heart. The regulation mechanisms can be classified into
physical (or mechanical), metabolic, and neural factors. The physical factors that
affect coronary flow include perfusion pressure, extravascular pressure
(intramyocardial pressure, IMP), diameters and lengths of the various generations
of blood vessels, and blood rheology. Since coronary vessel lengths and blood
viscosity are relatively constant in the normal heart, the main variable is the vessel
diameter. The neural control is mediated by both the sympathetic (vasoconstriction)
and parasympathetic (vasodilatation) components of the autonomic nervous system.
The term autoregulation refers to “the intrinsic tendency of an organ to maintain
constant blood flow despite changes in arterial perfusion pressure” (Johnson, 1964).
Extrinsic effects of nerves or hormones are not included in this definition. The three
hypotheses that have been put forth to explain autoregulation are metabolic, myogenic, and flow. The metabolic hypothesis proposes that the degree of arteriole
smooth muscle vasoconstriction is regulated by the tissue levels of metabolite. It is
well known that coronary blood flow is very sensitive to local metabolic control
when myocardial oxygen consumption is altered. The metabolic controls are very
effective and are even more potent than the neural controls. The myogenic hypothesis states that the arteriolar smooth muscle vasoconstriction is due to the stretch
imposed by the increase in arterial pressure. The vasoconstriction of the arteriole
counter balances the flow increase. Finally, the shear flow hypothesis states that
changes in vessel flow will change endothelial shear stress which will in turn cause
changes in vessel diameter to maintain a constant shear stress on the endothelium for
homeostasis.
6.4 Coronary Flow Regulation 389
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6.4.1 Coronary Autoregulation
Autoregulation of coronary blood flow implies relatively constant flow regardless of
changes in coronary perfusion pressure, i.e., flow is autoregulated to match myocardial metabolic demand under a broad range of physical activity (Duncker & Bache,
2008). Flow regulation is achieved by diameter control of resistance vessels
(<400 μm in diameter) through vasoreactivity of vascular smooth muscle cells,
VSMC (Goodwill, Dick, Kiel, & Tune, 2017). Three major mechanisms of VSMC
regulation have been proposed as follows: (1) Myogenic (Johnson, 1980; Kuo,
Chilian, & Davis, 1990; McHale, Dube, & Greenfield Jr., 1987; Miller Jr.,
Dellsperger, & Gutterman, 1997) where vessel diameter changes in direct response
to trans-luminal pressure, (2) Shear (Holtz, Forstermann, Pohl, Giesler, & Bassenge,
1984; Jones, Kuo, Davis, & Chilian, 1995; Kuo, et al., 1990; Kuo, Davis, & Chi lian,
1995) where diameter is regulated by nitric oxide (NO) in response to wall shear
stress, and (3) Metabolic regulation that affects the diameter of small arterioles
(Feigl, 1983; Jones, Kuo, Davis, Defily, & Chilian, 1995 ; Kanatsuka, Lamping,
Eastham, Dellsperger, & Marcus, 1989) through metabolic vasoactive agents. Previous studies have shown that vascular responsiveness to these three regulation
mechanisms is heterogeneous whereby (a) the Adenosine (a putative vasodilator)
effect is significant in downstream microvessels (diameters < 150 μm) and decreases
with incre ase in diameter (Jones, et al., 1995; Kanatsuka et al., 1989; Kuo, et al.,
1995); (b) Shear-induced dilation is more prominent in upstream larger microvessels
(Kuo et al., 1995), and (c) Myogenic control is negligible in small microvessels due
to their relatively low intravascular pressures (Chilian, Layne, Klausner, Eastham, &
Marcus, 1989; Tiefenbacher & Chilian, 1998).
Coronary blood flow regulation raises important questions regarding the impact
of each control mechanism and respective interactions as well as the role of extravascular loading (IMP) by the surrounding myocardium on flow regulation (Muller,
Davis, & Chilian, 1996). Clinically, the circumstances that lead to failure of flow
regulation remain unclear, e.g., why does the system fail to maximally vasodilate
even under profound ischemia (Hoffman & Spaan, 1990)?
Such questions cannot be experimentally studied in vivo due to the difficulties in
separating the vascular response between shear and pressure, and in measuring
in vivo flow and pressure in the deeper layers of the beating heart. Although
model simulation is a powerful hypothesis-generating approach to study the coronary circulation, there are several major challenges. The understanding of molecular
and transport mechanisms in flow regulation is still incomplete (Chen & Popel,
2006; Dash & Bassingthwaighte, 2006; Feigl, 1983
single-vessel models based on isolated vessel studies (Liao & Kuo, 1997). The
heterogeneities in coronary vasculature (Kassab, Rider, Tang, & Fung, 1993;
VanBavel & Spaan, 1992) and intramyocardial pressure along with vascular responsiveness to regulatory mechanisms, however, require simulations that consider
regulation mechanisms in each arterial segment in the framework of an integrated
vascular network dynamic flow analysis.
). Past studies adopted empirical
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6.4.2 Models of Autoregulation
Previous models of coronary flow regulation (Carlson, Arciero, & Secomb, 2008;
Cornelissen, Dankelman, VanBavel, & Spaan, 2002; Liao & Kuo, 1997) did not
incorporate various characteristics of the coronary system, such as distributive
anatomy and mechanical properties. The network structure considered is assumed
to be symmetric and orders are assigned to vessel groups such as “small arterioles,”
“large arterioles,” and “small arteries.” In addition, these past models considered
isolated networks having no interaction with the surrounding myocardium. Finally,
flow analysis is steady-state without consideration of the dynamic effect of
coronary flow.
Namani, Kassab, and Lanir (2018) provided an integrated flow regul ation model
using a structure-based multiscale framework for the analysis of dynamic flow
regulation in a realistic coronary network subject to heterogeneous transmural
variation of the extravascular loading by the contracting myocardium and incorporated the longitudinal heterogeneity of passive and active vascular mechanical
properties. This model is used to elucidate the impact of each regulation mechanism,
the nature of the interaction between them, and the interaction of flow regulation
with the extravascular myocardial contraction. The model descriptions are outlined
in Appendix 4.Briefly, the simulation is rooted in experimental measurements
namely: (1) Network structure is reconstructed from measured morphometric data
on coronary vasculature (Kassab & Fung, 1994; Kassab, Rider, et al., 1993);
(2) Effects of both the transmural extravascular myocardial loading and the vessel
tethering to the myocardium are incorporated, as opposed to isolated, externally
unloaded networks; (3) Models for the three regulation mechanisms are adopted
from a detailed experimental study on coronary microvessels; (4) Conducted nature
of the metabolic regulation is considered; (5) Simulations incorporated the dynamics
of coronary flow using realistic boundary condition; and (6) Network dynamic flow
analysis including the mechanisms for the vessel–myocardium interaction (MVI) are
also considered.
6.4.2.1 Perfusion Dispersion
The coefficient of variation (CV) in pre-capillary flow rates is a measure of flow
dispersion which is important to understand the level of perfusion for different
regions of the myocardium. Under reference inlet and outlet pressures, and passive
vessel conditions, the CV is ~25% while under full metabolic activation, CV is
~29% (Appendix 5). Under only myogenic regulation, the CV increased to ~35% in
the sub-endocardial and 46% sub-epicardium networks. When the metabolic regulation is optimized to yield the target terminal flow, the flow dispersion reduced to
between ~10% and ~30%, depending on perfusion pressure and target flow (Appendix 6). In general, the flow dispersion is high under extreme conditions where the
metabolic demand q
is low and the perfusion pressure is high, and when the
target
6.4 Coronary Flow Regulation 391
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metabolic demand is high and perfusion pressure is low (Appendix 6). In these cases,
the metabolic control is either shut down (all F
capacity (all F
¼ 1) and the flow CV is very high. This can occur under extreme
mterm
¼ 0) or exhausted to its full
mterm
conditions of ischemia where flow reserve is exhausted.
6.4.2.2 Transmural Perfusion Heterogeneity
The model revealed transmural perfusion differences, as exhibited by the terminal
flow rate and dispersion, by the metabolic flow rate (MFR) and the effects of
myocardial–vessel interaction (MVI), and by the effects of metabolic demand
(indexed by the target terminal flow) and perfusion pressure. The predicted flow
rate is higher in the sub-epicardium under all regulation conditions (Appendix 5)
compared to the endocardium. These transmural flow differences reduce under
optimized metabolic regulation aimed to achieve a set target flow (Appendix 6)to
levels which are flow rate dependent. Under optimized metabolic activation (Appendix 6), the CV is ~10% in both the sub-endocardial and sub-epicardium networks for
the reference target terminal flow of 1.50 10
3mm3
/s. This transmural trend
changes significantly, however, with the target flow and perfusion pressure. Under
all target flow rates studied, the dispersions are lower in the sub-epicardium than in
the sub-endocardium under low perfusion pressures but are higher than in the
sub-endocardium under high perfusion pressures.
6.4.2.3 Metabolic Flow Reserve (MFR)
MFR is variably affected by the input perfusion pressureP
depending on the
in
network transmural location and MVI (Appendix 7). In the in vivo case (with MVI)
under low perfusion pressure, it is higher in the sub-epicardium while under medium
and highP
, it is higher in the inner sub-endocardial layer. In the sub-epicardium,
in
MFR had a similar trend as in the sub-endocardium under no MVI (Appendix 7). In
the sub-epicardium, MFR decreases monotonous ly with increasing perfusion pressure from 3.25 atP
sub-endocardium, MFR is highest (3.19) underP
¼ 60 mmHg to 1.45 atPin¼ 180 mmHg. In the
in
¼ 120 mmHg and drops under
in
both lower and higher perfusion pressures (Appendix 7).
6.4.2.4 Effect of Regulation on the Coronary Flow
The total flow increases with the average inlet pressure (P
) under all flow regulation
in
mechanisms, both with and without myocardial vessel interaction MVI (Appendix 6,
Fig. 6.12). Comparison of flow distributions in the pre-capillary (order 1) terminal
vessels under four flow regulation conditions shows that under solely myogenic
activation (when the vessel diameters are at their lowest values), the distribution
shifts to the extreme left of low flow levels. Under myogenic and shear regulations
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Fig. 6.12 Effects of myocardial–vessel interaction (MVI), regulation mechanisms, and transmural
network location on the network flow. (a–d) Sub-endocardial network flow, Q as a function of mean
inlet pressure,P
(c) and without MVI (d). Apart fromP
are their reference values (Appendix 6.6:1, Table A1). In the sub-endocardial network, MVI has a
significant effect on the flow under both myogenic and myogenic + shear activations. These effects
of MVI are smaller in the sub-epicardial network. The four curves in each subplot are under the
following regulation conditions: (*) only myogenic; (diamond) myogenic and shear; (open circle)
myogenic, shear, and full metabolic (all F
Reproduced from Namani et al. (2018) by permission
, with MVI (a) and without MVI (b); and sub-epicardial network flow with MVI
in
, the vessels’ properties and pressure boundary conditions
in
¼ 1.0); and (open triangle) passive conditions.
mterm
(no metabolic regulation), the distribution shifts to slightly higher flow levels. Full
metabolic activation shifts the flow distribution to the right (towards high flow
levels), closer to their passive levels.
Under optimized metabolic activation with the reference level of perfusion
pressure, the average terminal flow becomes closer to the set target flow q
target
but
deviates significantly from it under both lower and higher perfusion pressures
(Appendix 6). As expected, comparison with the results in Appendix 5 indicates
that a significantly lower flow CV is obtained under optimized metabolic activation
(set level of q
) as compa red to cases of either full or no metabolic activation.
target
6.4.2.5 Model Pre dictions
Simulations show that autoregulation occurs under increasingP
, as evident from
in
the flow plateau region in the flow–pressure curves (Fig. 6.13). The pressure range
where autoregulation is effective is higher under higher target flows. In the absence
of MVI (Fig. 6.13b), the flows under all conditions increase as compared to the flow
with MVI. Also, MVI increases the effective autoregulation pressure range for all
target flows. In the sub-epicardium network with MVI, the flow–pressure curves
(Fig. 6.13c) under all conditions (including autoregulation) are similar to those of the
sub-endocardial network without MVI (Fig. 6.13c).
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Fig. 6.13 Flow autoregulation. (a–c) Total network flow under three levels of metabolic demand
) in the sub-endocardial network with myocardial–vessel interaction (MVI) (a) and without
(q
target
MVI (b) and sub-epicardial network with MVI (c). The vessel properties and boundary conditions
are at their reference values (Appendix 4, Table 6.3). The total flow in the presence of MVI remains
nearly constant over the physiological autoregulation pressure range. MVI expands the perfusion
pressure range for which autoregulation is effective. The autoregulatory behavior in the
sub-epicardial tree is similar to the sub-endocardial tree without MVI. Reproduced from Namani
et al. (2018) by permission
The distribution of the metabolic activation (F
reference target terminal flow q
of 1.5 103mm3/s depends on the input
target
perfusion pressure. The distribution shifts to lower values under highP
) under flow optimization and
mterm
(Namani
in
et al., 2018). On the other hand, the metabolic activation in the terminal vessels is at
the maximum level (all F
The distribution moves towards the lowest possible level (all F
’ 1) when inlet pressure is low (Pin¼ 75 mmHg).
mterm
mterm
’ 0) asP
increases and becomes zero in all vessels whenPinis higher than 135 mmHg
(Namani et al., 2018).
6.4.2.6 Effect of MVI
MVI enhances the effect of myogenic and metabolic regulation in the sense that it
induces a higher reduction in the myogenic flow and increases metabolic flow
recovery. But it has a minor effect on the shear regulation (Fig. 6.12). MVI has
significant andP
-dependent effect on the metabolic flow reserve (MFR) in the
in
sub-endocardial network (Appendix 7 ). Under low perfusion pressure
(60–90 mmHg) MVI reduces the level of MFR. Under medium to high pressure
(100–180 mmHg in Appendix 7), MVI significantly increases the MFR level. MVI
has a much lower effect on the sub-epicardium network flow. This is evident
from the closeness of the MFR levels in the sub-epicardium with MVI to that of
the sub-endocardium without MVI except under very low perfusion pressures
(Appendix 7).
in
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MVI affects the network pressure–flow relationship (Fig. 6.12) primarily under
myogenic and myogenic + shear regulations but seems to have a small effect on the
passive and full metabolic flows. MVI also enhances the MFR (Appendix 7) for all
cases except under low inlet perfusion pressures. Another important effect is on the
flow autoregulation (Fig. 6.13), i.e., the presence of MVI significantly increases the
pressure range of effective autoregulation. Under in vivo conditions (with MVI), the
autoregulation pressure range is expected to be even wider than in Fig. 6.13 since the
coronary perfusion pressure increases with increasing LV pressure. The latter is a
key determinant of the magnitude of MVI extravascular loading.
The large spatial flow dispersions of ~24% observed in the passive networks in
both sub-endocardium and sub-epicardium compares well with measured flow
dispersion of ~30% in a beating heart without tone (Austin, Smedira Jr., Squiers,
& Hoffman, 1994). The predicted flow dispersion under physiological levels of
perfusion pressures (~10–15%) under autoregulation (with vascular tone) is slightly
higher than 5–10% reported in one study (Matsumoto & Kajiya, 2001) but within the
range of 14–19% reported in another (Austin et al., 1994). The differences between
these previous experimental studies are likely methodological (double-tracer digital
autoradiography vs. microsphere), species (rabbit vs. canine), and sample size
(1 mg. vs. 150 mg). The estimated tissue volume perfused by one terminal arteriole
is on the order of 1 mg. Extrapolation of flow dispersion from a large sample size of
1 g (Matsumoto & Kajiya, 2001) to a small sample of 1 mg based on fractal scaling
(Bassingthwaighte, King, & Roger, 1989) yields a flow dispersion in the small
sample that is significantly higher than that measured by Matsumoto and Kajiya
(2001). In addition, microsphere measurements are known to overestimate flow
dispersion, and microspheres of 15 μm can interfere with the vessel wall and tone
in the small arterioles. Higher flow dispersion can lead to local oxygen deficit and
local hypoxic conditions in the myocardium. To summarize this point, the predicted
flow dispersion of ~10–15% under autoregulation seems reasonable, and in close
agreement with data of Matsumoto (Matsumoto & Kajiya, 2001).
Transmural flow heterogeneity is found with higher pre-capillary network flow
rates in the sub-epicardium as compared to sub-endocardium (Appendix 2). This is
attributed to the reduced sub-epicardium MVI allowing vessels to dilate to a greater
degree than sub-endocardial vessels. During autoregulation, however, the network
flow rate in the presence of MVI in the normal physiological state is found to be
similar across transmural regions (Appendix 6, Figs. 6.2a and 6.4c). An additional
important transmural difference is in the perfusion density (perfusion per tissue
volume). This measure is affected by both the terminal flow and by the vessel
density in the myocardial tissue. Capillary density is found to be ~30% higher in
the sub-endocardium as compared to the sub-epicard ium (Breisch, White, Nimmo,
McKirnan, & Bloor, 1986; Gerdes & Kasten, 1980; Lee et al., 2009). Under in vivo
autoregulation conditions, measured perfusion density is found to be ~20% higher in
the sub-endocardium than in the sub-epicardium (Feigl, 1983). The model predictions are in close agreement with these data showing that for three levels of
metabolic demand (q
¼ 1.25, 1.5, 1.75 103mm3/s), the endocardium/
target
epicardium perfusion density ratios are 1.1, 1.26, and 1.26, respective ly. In the
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