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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3787_Библиотеки_им_академика_М_И_Перельмана
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Fig. 6.7 (a) Flow waves sequentially along the main trunk (TA–TB, different locations along the
trunk) starting from the inlet of the left anterior descending artery (LAD). (b) Flow waves of
primary branches (PA, PB, and PC, different side branches) of the LAD trunk (TA–TB).
Reproduced from Huo and Kassab (2007) with permission
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similar at every primary branch, whi ch implies that the flow is self-similar or scales
as a fractal (Chaps. 2 and 7).
Although the pressure and flow waves have been widely studied from the aorta to
the limb arteries (Nichols & O’Rourke, 1998), the pressure and flow waves along the
main trunk of LAD artery are less well studied. As shown in Fig. 6.7a, the amplitude
of the flow waves decreases gradually with a small phase angle shift of the flow
waves along the main trunk of LAD artery in a relaxed heart. This is because the
primary branches shunt the flow away from the main trunk. Since primary branches
have different CSA, the flow waves at the inlet of primary branches show different
amplitudes. When the flow waves are normalized by the time-averaged flow rate,
however, it is found that the flow waves tend to scale to a single curve except for a
small phrase angle difference as shown in Figs. 6.8a, b. Figure 6.8c shows a
comparison between the normalized flow waves at the inlet of the main trunk and
one primary branch. It is noted that the normalized flow waves have very similar
patterns to each other except for a phase angle shift. This novel observation reflects
an interesting scaling of flow to structure and follows the flow-CSA scaling laws
presented in Chap. 7. This scaling feature is important as it can provide the boundary
condition (i.e., flow) at the desired vessel (diameter obtained from CT or angiogram)
for simulation of flow pattern in patient-specific coronary arteries (Chap. 8).
6.2.2.2 Effect of Energy Loss at Bifurcation
Since vortices or separation of flow often occur at bifurcations, the energy loss
should be considered. The loss of energy at a daughter vessel segment of a bifurcation is affected by many parameters, such as the branching angles (θ), the Reynolds
number (Re), the area ratios (A/A
), and the flow rate ratios (Q/Q
mother
mother
Table 6.1 (Appendix 2) lists these various parameters for the vessel segments in
the first seven bifurcations. It is found that various loss coefficients are estimated in
the range of 0.25 – 0.45 which yields almost identical flow waves as compared to zero
loss coef ficient. Huo and Kassab (2007) compared the flow wave s when loss
coefficients are equal to zero, 0.35 and 1.5. It is found that the computed flow
wave shows a slight apparent dissipation as the loss coefficient increases. Obviously,
the dissipation increases when a larger portion (beyond the seven bifurcations) of the
tree is considered. Furthermore, the loss coefficients are found to be larger in the
daughter vessels with smaller diameters because those vessels have larger bifurcation angles with mother vessels which correlates with the predilection for atherosclerosis (Fung, 1984; Pedley, 1980) as described later in this chapter.
).
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Fig. 6.8 (a) flow waves normalized by time-averaged value in main trunk of left anterior
descending artery (LAD). (b) flow waves normalized by mean value at inlet of primary branches
of LAD. (c) normalized flow waves at inlet of main trunk and one primary branch of LAD.
Reproduced from Huo and Kassab (2007) with permission
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6.3 Myocardial–Vessel Interaction Flow
The mechanism of the effect of the c yclic contraction on coronary blood is complex.
This section will consider the major cardiac mechanical interactions with coronary
blood flow including previous models, current hypotheses, and future directions.
Experimental validation of present and future mechanical interaction models will be
emphasized, as well as the utility of the models to explain the mechanical propensity
of the sub-endocardium to ischemia.
6.3.1 Models of Coronary Vasculature
The coronary vasculature forms the specialized channels for the advection of
oxygenated blood through the myocardium. The function of the coronary circulation
is to continuously supply blood to meet the metabolic demands of cardiac tissue. The
branching network of distensible coronary vessels is an important determinant of
coronary blood flow, and thus a full understanding of the mechanics of this system is
not possible without consideration of the geometry and mechanical properties of the
coronary vasculature. The rationale is that a vascular system, comprised of distensible vessels, strategically distributed and mostly embedded within the myocardium,
should be modeled in as much detail as possible, rather than “lumped.” The detailed
morphometric data (diameters, lengths, and number of vessels) that serve as the
architectural foundation of the coronary circulation models are described in Chap. 2.
The anatomical models along with the physical laws governing blood flow and the
appropriate boundary conditions in a dynamic model of the beating heart, based on
cardiac mechanics, form the basis for a rational approach to understanding the
myocardial–vessel interaction (MVI).
6.3.2 Intramyocardial Pressure (IMP)
The intramyocardial vessels are elastic and hence compliant, i.e., the lumen diameter
can change with pressure. Hence, their volume is a function of the transmural
pressure (difference between coronary blood pressure and intramyocardial pressure,
IMP). IMP increases as the heart contracts, which causes an increase in coronary
systolic blood pressure. This, in turn, decreases the pressure gradient o n the arterial
side of the coronary network and increases the longitudinal pressure gradient on the
venous side. The change in longitudinal pressure gradient decreases the systolic flow
of blood through the arterial side and increases blood flow through the venous side.
The blood volume reduction, due to net outflow in areas where IMP is high, results in
reduced vessel diameter, which may produce a dramatically higher resistance and
thereby further impede blood flow. This squeezing effect, or systolic flow
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impediment, is first proposed by Scaramucci in 1695 and has subsequently been
investigated by Porter (1898), Anrep, Cruickshank, Downing, and Sabba Rau (1927)
and later by Downey and Kirk (1975), Spaan, Breuls, and Laird (1981) and Krams,
Sipkema, and Wsterhof (1989); Krams, Sipkema, Zegers, and Westerhof (1989),
among others.
In addition to the squeezing effect, the intramural coronary blood vessels are
embedded in the myocardium, where the interaction of blood pressure, vessel
elasticity, smooth muscle tone, and tissue stress leads to complex blood flow
patterns. This interaction between contraction and coronary flow has been a subject
of great interest to coronary physiologists. The systolic extravascular resistance
model (Gregg & Green, 1940; Sabiston & Gregg, 1957), the waterfall model
(Downey & Kirk, 1975; Permutt & Riley, 1963), the nonlinear intramyocardial
pump model (Arts, Kruger, Van Gerven, Lambregts, & Reneman, 1979; Bruinsma,
Arts, Dankelman, & Spaan, 1988), and the variable elastance model (Krams,
Sipkema, & Wsterhof, 1989) have all been proposed to explain the interaction
between contraction and coronary blood flow.
There are two major contributions to the IMP that couples the myocardial and
blood flow interaction. The first is the transmission of the ventricular pressure
through the ventricular wall. At the endocardium, this contribution is equal to
ventricular pressure while at the epicardium, it is equal to pericardial pressure. The
second contribution is the effect of time-varying, muscle fiber contractions that
increase the effective stiffness of the myocardium. While these contractions are
primarily developed to act against ventricular pressure, the results of Allaart and
Westerhof (1996) indicate that significant internal myocardial stress is developed
solely as a consequence of the deformation and contraction of muscle fibers (crossfiber and normal to the myocardial sheet directions).
There is considerable debate over the relative impact of each contribution. The
models of the cardiac ventricle can be used to calculate IMP along each vessel
segment. These calculated results show an approximately linear variation in IMP
from ventricular pressure at the endocardium to atmospheric pressure at the epicardium (Gregg & Green, 1940; Nielsen, Le Grice, Smaill, & Hunter, 1991; Vetter &
McCulloch, 1998), which are experimentally supported by Heineman and Grayson
(1985) and Mihailescu and Abel (1994). Mihailescu and Abel (1994) measured an
IMP which is higher than LV pressure in the endocardium, which is reasonable given
the contribution of muscle contraction.
The transmission of IMP to the coronary circulation occurs as follows: IMP
affects the vessel transmural pressure, which in turn, affects the vessel diameter,
which controls the vascular resistance. The resistance determines the pressure
distribution, which in turn affect the transmural pressure. It should be noted that
the compressive effects on intramural vessels are highly dependent on vessel
orientation relative to the local myocardial fiber direction. This raises an interesting
question: What are the functional and mechanical implications of the existing
relationship between muscle and vessel orientation?
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6.3.3 Lumped Models
Given the difficulties in the simultaneous measurement of IMP and myocardial
blood flow (especially in the deeper layers of myocardial wall), lumped parameter
mathematical models have provided a valuable framework to understand the dynamics of coronary circulation. These models treat the coronary vasculature or subgroups
of vessels as single entities whose whole behavior has been characterized by number
of lumped parameters. There are inherent shortcomings in using these types of
models, however, to describe all aspects of time-varying coronary blood flow (Lee
et al., 1984; Mates, 1993). For example, lumped models are not related to actual
structural and physical entities but instead constitute empirical model selection and
parameter optimization. Moreover, lumped models cannot be extended to incorporate distributed local phenomena such as myocardial–vessel interaction and mechanisms of autoregulation. A model which is made up of individual single vessels
allows for quantifying resistance and capacitance based on measurable diameters
and lengths through physical laws. In a lump ed model, resistance dependency of
volume is based on empirical curve fitting which is correct only under the conditions
of the measurements. It is not generally possible to maintain consistent experimental
conditions, however, when determining every parameter of a lumped model (Mates,
1993). Consequently, it is desirable to reduce the model inputs to the fundamental
physical properties of the system that are more likely to be independent of other
parameters and less sensitive to experimental conditions. By definition, this is not
possible in the "lumped parameter" approach. Although Mates (1993) has discussed
the practicalities of comparing experiment and theory for lumped models, the major
issue is that each parameter represents a number of mechanisms which typically
affect more than one parameter. Hence, there is considerable overlap and interaction
between the parameters which may cause a degree of ill-conditioning in the parameter estimation.
The three-element Windkessel linear lumped model (two resistors and a capacitor, RCR) has been used extensively to model the coronary circulation (Hoffman &
Spaan, 1990; Spaan, 1991). Although nonlinear lumped models have also been used
that overcome some of the limitations of the linear models (Spaan, Cornelissen,
Chan, Dankelman, & Yin, 2000), the fundamental issue is that the models are
empirical and not based on structure (e.g., vessel diameter which changes with
pressure). An additional inherent limitation for both linear and nonlinear lumped
models is that the empirical value of the parameters is valid only for the measured
boundary conditions. Physical laws, howe ver, must be valid regardless of boundary
conditions.
For the microcirculation, mixture or porous media models for blood perfusion
have been proposed (Cookson et al., 2012; Huygh, Oomens, & Van Campen, 1989;
Huygh, Oomens, Van Camp en, & Heethaar, 1989; Vankan et al., 1997). Huygh et al.
(1989) developed an analysis as an extension of Darcy-type theory which involves
averaging of hemodynamic quantities over number of vessels in a continuum. In this
approach, a distinction between arterioles, capillaries, and venules is preserved by
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means of an arterio-venous parameter. Hence, the domain of the mixture model can
be defined by the arterio-venous parameter and consequently the total number of
vessels can be reduced appropriately. In this way, arterial and venous trees feed and
drain, respectively, a porous continuum of microcirculation and substantially
reduced the computational cost.
6.3.4 Distributive Models
In a distributed model approach to coronary circulation, the blood and coronary
vessels are treated as individual tubes, and the mechanics of blood flow in each tube
are described using differential equations. For this type of model, the governing
equations are formulated in terms of physical parameters. The development of a
complete distributed model of coronary blood flow coupled to whole organ myocardial contraction, however, has had four main difficulties. First, the computational
time required to solve for the flow through each individual segment of the coronary
network embedded in a continuum model of the heart has been very large. With the
rapid increase in the size and speed of high-performanc e computers, this obsta cle has
been reduced and is likely to continue to improve in the foreseeable future. Second,
the mathematical models of IMP generated by contraction have not been available.
The ventricular mechanics model of Nevo and Lanir (1989), Nash (1998), and Vetter
and McCulloch (1998), using the detailed mathematical models of the cardiac
ventricles, now provides data on the spatial and temporal variation of IMP. Third,
up until recently the anatomical data to construct realistic coronary networks have
not been available; however, now the necessary data have been made available
(Chap. 2). Finally, the experimental data to validate the whole organ model at the
spatial scale of individual vessels are also needed. Techniques deve loped by Kajiya
et al. (1993) and Yada et al. (1993, 1994) using video-microscopes have provided
in vivo observations of blood flow in the coronary microcirculation. From these data,
temporal changes in blood flow and vessel diameter in intact, beating hearts can be
compared to model results for vessels as small as 100 μm in diameter. Furthermore,
the longitudinal pressure model predictions can also be compared with the epicardial
and endocardial pressure measurements reported in the literature as described in
earlier sections of this chapter.
6.3.5 Vessel Elasticity
The elasticity of the coronary vessels has been widely studied as described in
Chaps. 3 and 4. The in vivo mechanical properties of coronary arterioles have
been studied by Hiramatsu et al. (1998) in the sub-endocardium during prolonged
diastole. In situ, Kassab and Molloi (2001) determined the compliance of the first
several generations (orders 9–11) of pig coronary arteries in ex vivo hearts using a
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video-densitometric technique. The pressure–diameter relation of epicardial coronary capillaries is also determined in ex vivo hearts (Kassab, Le, & Fung, 1999). In
isolated vessels (without the surrounding myocardium), a two-layer, three-dimensional (3D) stress–strain relation is validated by triaxial tests (inflation, axial extension, and torsion) of intact porcine coronary arteries of order 11 and corresponding
intima-media and adventitia layers ex vivo (Lu, Pandit, & Kassab, 2004 ; Pandit, Lu,
Wang, & Kassab, 2005; Wang, Garcia, Lu, Lanir, & Kassab, 2006). The Fung-type
exponential strain energy function is used to describe the 3D strain–stress relation for
each layer and the intact wall. The validated 3D constitutive model served as a
foundation for analysis of myocardial–vessel interaction.
The mechanical properties of small arteries have also been extensively studied in
isolated vessels (Giezeman, VanBavel, Grimbergen, & Spaan, 1994; Kuo, Davis, &
Chilian, 1990). Although the isolated vessel studies provide a wealth of data on the
intrinsic properties of blood vessel wall, the mechanical response of the coronary
arteries also depends on the properties of neighboring tissue (Hamza et al., 2003).
Since the myoca rdium (both passive and active) contributes substantially to the
pressure–diameter relation (PDR), a mechanical interaction model is needed to
integrate both the intrinsic properties of the vessel as well as those of the myocardium, as outlined below.
6.3.6 MVI Model
There have been two primary controversies associated with coronary flow dynamics.
First, the mechanism for the generation of IMP and its physical temporal and spatial
basis remains unclear. In the models of Downey and Kirk (1975) and Spaan et al.
(1981), IMP is assumed to be equal to ventricular pressure at the endocardium and to
decrease linearly to zero at the epicardium. Krams et al. (Krams, Sipkema, et al.,
1989; Krams, Sipkema, Zegers, et al., 1989) have suggested a more limited role of
ventricular pressure by applying the “time-varying elastance concept” of Suga,
Sagawa, and Shoukas (1973) to explain the systolic flow impediment. This concept
emphasizes the effect of time-varying ventricular wall stiffness on coronary blood
volume, which is assumed to be independent of ventricular pressure. The theory of
Krams et al. (Krams, Sipkema, et al., 1989; Krams, Sipkema, Zegers, et al., 1989)is
based on the observation that flow impediment is similar for isovolumic (high
systolic ventricular pressures) and isobaric (low systolic ventricul ar pressures)
contractions. The elastance concept does not, however, explain why epicardial
flows are not inhibited to the same degree as endocardial flow (Spaan, 1995).
Other studies suggest that ventricular pressure has a significant effect on timevarying coronary flow as well as myocardial contraction (Doucette, Goto, Flynn,
Husseini Jr., & Hoffman, 1993; Kouwenhoven, Vergroesen, Han, & Spaan, 1992;
Mulligan, Escobedo, & Freeman, 1993). Furthermore, studies of epicardial lymph
pressure (Mihailescu & Abel, 1994) and of coronary arterial pressure and flow
(Kouwenhoven et al., 1992) suggest that rather than inducing flow impediment,
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systolic stiffening shields intramural vessels from the effects of ventricular pressure
during part of systole (Spaan, 1995). Another hypothesis attributes IMP to a
combination of ventricular pressure derived interstitial fluid pressure, and
intramyocyte pressure induced by active contraction (Rabbany, Funai, &
Noordergraaf, 1994; Rabbany, Kresh, & Noordergraaf, 1989). Hence, this issue
has not yet been resolved nor is the physical origin of IMP clear (Beyar, Manor,
Zinemans, & Sideman, 1993).
The second controversy associated with coronary flow dynamics relates to the
role of compliance and vascular resistance on coronary circulation. Downey and
Kirk (1975) proposed the “vascular waterfall mechanism” to explain the reduced
coronary in-flow by the increase in resistance from the collapse of intramural vessels.
This throttling effect, however, would impede both arterial inflow and venous
outflow. Thus, this theory alone cannot explain the increased venous outflow during
systole, nor has vessel collapse been directly observed in the coronary vasculature.
To explain the phase lag between coronary arterial and venous flows, Spaan et al.
(1981) introduced the “intramyocardial pump model” which accounts for the role of
vascular compliance. Compliant vessels are filled from the high-pressure arterial side
in diastole and then discharge through the low-pressure venous side in systole. This
concept has been extended in a number of lumped parameter mathematical models
of coronary circulation to account for the variation in vascular resistance, compliance, and temporal/spatial variation of IMP (Bruinsma et al., 1988; Chadwick,
Tedgul, Michel, Ohayon, & Levy, 1990; Judd et al., 1991; Kresh, Fox, Brockman,
& Noordergraaf, 1990).
To address these controversies, a structure-based analysis can elucidate the
myocardial–vessel interaction (MVI) in a realistic anatomical model of the coronary
vasculature nested in a dynamic model of the heart (Algranati, Kassab, & Lanir,
2010). The model is used to test the hypothesis that only a specific combination of
MVI mechanisms can account for all observed coronary flow features (Algranati
et al., 2010). Three basic interaction mechanisms (cavity-induced extravascular
pressure, varying elasticity, and shortening-induced intramyocyte pressure), and
their combinations are analyzed based on physical principles in realistic data-based
vascular and myocardial models. An overview of the approach is presented below.
6.3.6.1 Anatomical Model
For illustration purposes, the dynamic flow is considered in the microvascular
networks in several transmural locations of the LV wall (Algranati et al., 2010). A
sample reconstruction of the coronary microvasculature (order 3 arteriole to order -3
venule) consisting of 174 segments and 115 internal nodes is considered, as shown
in Fig. 6.9. One arteriole, of order 3 (~20 μm), served as the source of the network,
and two venules, of order -3 (~30 μm), drained it. The network is positioned at
4 different LV transmural wall locations: superficial or epicardial, sub-epicardial,
midwall, and sub-endocardial. All four microvascular networks are connected to
arterial and venous trees on the respective ends (Figs. 6.9 and
6.10). The arterial and
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Fig. 6.9 Schematic of the myocardial–vessel interaction (MVI) simulation platform. (a) Network
anatomy is reconstructed based on statistical morphometric data of porcine coronary vasculature.
(b) Each vessel flow analysis is carried out based on a nonlinear analog circuit where P
are intravascular and extravascular pressures, respectively. ℜ and C are the vessel (time-varying)
resistance and capacitance, respectively. (c) In situ vessel mechanics is determined from a nonlinear
finite deformations stress analysis of the vessel inside a cylinder of myocardium. The analysis
accounts for the measured stress-free configurations of the two cylinders (unloaded and free of
residual stress, left panel). Right panel—predicted sigmoid pressure–diameter relations (PDR)ina
representative vessel (solid line), and comparison with experimental data (error bars). Dash—the
linearized PDR of that vessel. X-axis is trans-vascular pressure (in mmHg); Y-axis—the diameter
normalized relative to zero-pressure diameter. (d) Each network vessel is subject to an extravascular
pressure stemming from both left ventricle cavity pressure (LVP, left) and the contraction-induced
intramyocyte pressure. (e) Heart rate, contractility, hematocrit and dynamic left ventricle, perfusion
and outlet pressures (LVP, P
evaluated in terms of transmural distribution of flow and dynamic transvascular pressures ΔP.
Reproduced from Algranati et al. (2010) by permission
venous trees are considered symmetric in branching to create a simplified full tree
where the large arteries and veins are idealized (symmetric branching where daughter vessels had the same diameters and lengths based on mean measurements) but the
microvascular networks (order 3 to +3 incl uding order 0 vessels) are based on
and P
IV
, and PV, respectively) are inputs. (f) The model predictions are
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