Overview of mathematical modeling of myocardial blood flow regulation

Abstract

The oxygen consumption by the heart and its extraction from the coronary arterial blood are the highest among all organs. Any increase in oxygen demand due to a change in heart metabolic activity requires an increase in coronary blood flow. This functional requirement of adjustment of coronary blood flow is mediated by coronary flow regulation to meet the oxygen demand without any discomfort, even under strenuous exercise conditions. The goal of this article is to provide an overview of the theoretical and computational models of coronary flow regulation and to reveal insights into the functioning of a complex physiological system that affects the perfusion requirements of the myocardium. Models for three major control mechanisms of myogenic, flow, and metabolic control are presented. These explain how the flow regulation mechanisms operating over multiple spatial scales from the precapillaries to the large coronary arteries yield the myocardial perfusion characteristics of flow reserve, autoregulation, flow dispersion, and self-similarity. The review not only introduces concepts of coronary blood flow regulation but also presents state-of-the-art advances and their potential to impact the assessment of coronary microvascular dysfunction (CMD), cardiac-coronary coupling in metabolic diseases, and therapies for angina and heart failure. Experimentalists and modelers not trained in these models will have exposure through this review such that the nonintuitive and highly nonlinear behavior of coronary physiology can be understood from a different perspective. This survey highlights knowledge gaps, key challenges, future research directions, and novel paradigms in the modeling of coronary flow regulation.

INTRODUCTION

Coronary heart disease (CHD) is a major cause of mortality, with one death every ~90 s in the United States. The coronary circulation supplies nutrients and oxygen to the heart muscle to maintain its physiological function. Failure of the coronary blood supply due to CHD or metabolic diseases results in ischemia, which may trigger chest pain (angina) or lead to myocardial infarction and potential heart failure. The main physiological responses elicited by the coronary tree to altered cardiac metabolism are the dilation or constriction of its vessels to regulate the coronary blood flow (CBF) (30, 41). Flow regulation is most dominant in the microvasculature (17, 22, 45, 46, 69), where there is the greatest resistance to flow in the coronary tree. Coronary blood flow regulation mechanisms comprise metabolic (oxygen dependent) (30), myogenic (pressure dependent) (44, 57), flow (shear stress dependent) (58), myocardial contraction (6), and neural control by α-adrenergic (16) and β-adrenergic (67) factors and vagal stimulation (31). In addition, there are endothelial cell-dependent factors such as nitric oxide, hyperpolarizing factors of hydrogen peroxide, and humoral factors such as serotonin, acetylcholine, bradykinin, and substance P, which independently affect coronary vessel diameter (27, 36). The present review provides an overview of the theoretical and computational models for the first three regulation mechanisms of metabolic, myogenic, and flow control. The premise is that modeling of these control mechanisms is necessary for regions that elude experimental observations such as in the deep myocardial layers.

Although there are a wide variety of models for analyzing the many characteristics of coronary flow, many aspects of the coronary interactions with the myocardium and systemic and pulmonary circulations are not clearly understood. For example, how is coronary flow in the coronary artery regulated by the highly variable metabolic environment in the surrounding myocardium? Theoretical models are conceptual models of the physiological and biological processes involved in flow regulation and have evolved since the pioneering work by August Krogh (56). Computational models are numerical representations of the mathematical forms of the theoretical models, and the generic usage of “models” in this article refers to either of these models. There has been one review article that discussed theoretical models of flow regulation (81), but since then many advances in flow regulation have been reported. The following discussion is focused on existing models of coronary flow and regulation mechanisms that account for the nonlinear interactions. Here, coronary models are reviewed within a limited context of their ability to characterize the spatial variation in myocardial vessel interaction (MVI) and coronary flow characteristics. Key predictions made by existing coronary flow regulation models and their comparison with in vitro and in vivo measurements are presented. Furthermore, the limitations of these models and future modeling directions are presented.

CORONARY FLOW REGULATION MODELS

The coronary tree spans over multiple vessel length and diameter scales. In vivo measurements of changes in vessel caliber and flow in the dynamically moving heart are not possible with existing methodologies. As an alternative, considerable effort has been invested in developing theoretical models to describe coronary flow regulation (27, 30, 36, 41). These models of flow regulation can be classified into categories based on their specific attributes of incorporating MVI (1, 70), flow analysis with a symmetric tree structure (12, 18, 61) and an asymmetric tree structure (32, 73), feedback control systems (68, 78), and kinetics of mechanotransduction in smooth muscle and endothelial cells (13, 48, 83, 99). A different class of models that incorporate multiple regulation mechanisms of a single vessel as the basic control unit in the context of coronary network flow analysis are known as integrative physiological models (12, 18, 61, 73). Integrative physiological models can assess the complex nature of flow dynamics in the coronary tree both in a single vessel and at the organ scale simultaneously. The following is a description of the single-vessel models for each flow regulation mechanism (Fig. 1) and the integrated network flow models (Fig. 2).

Fig. 1.

A: a mechanistic view of the upstream conduction of metabolic signal and the shear stress response to blood flow in a single coronary vessel. B: a schematic of a cross-sectional view of a coronary vessel depicting the various constituents that comprise the passive and active properties of the vessel wall that affect coronary flow regulation.

Fig. 2.

A: a schematic of the multiscale reconstructed coronary arterial tree structure encompassing the epicardial conduit vessels and the microvascular resistance vessels in the left ventricle, right ventricle, and septum. B: a view of the upstream propagation of the conducted metabolic signal along a branch of the microvascular tree.

Metabolic Control

There are many metabolites released into the coronary vessels that cause vasodilation during cardiac metabolic changes (36). Theoretical models of the local metabolic response in a single vessel have related the magnitude of this vasodilation to be a sigmoidal function of the local concentration of metabolites such as adenosine (61). Based on data, the sensitivity of the vessel dilation to the local metabolite concentration has been considered by the models to be highest in the small arterioles (61). Another aspect of metabolic regulation is vasodilation based on the oxygen concentration in the blood. Although Fick’s law can be used to estimate the global perfusion increase necessary to meet the arterio-venous gradient in the partial pressure of oxygen (Po₂) due to altered myocardial metabolism, it cannot be used to estimate the effect of oxygen distribution because of coronary vessels’ dilation to metabolites. One of the earliest theoretical models of oxygen delivery was developed by Krogh (56). Since then, there have been several theoretical models of oxygen delivery due to changes in cardiac metabolism (76, 77). A more recent focus of metabolic regulation has been on the erythrocyte acting as an oxygen sensor (Fig. 1A), which releases adenine nucleotides that bind to the purinergic receptors on the endothelial cells (28). Models based on this hypothesis have shown that a conducted metabolic response can produce organ-level increase in the blood flow to meet the oxygen demand (3, 34, 73).

A salient feature of a recent model of the conducted response (73) is that it is dependent on the asymmetric tree structure (Fig. 2). As the conducted signal decays exponentially with network path length (Fig. 2B) (23, 34, 98), the metabolic response was modeled as an exponential function with a characteristic length, L₀, in the range of 0.15–2.5 mm (40). This model (73) considered the average of the conducted signals transmitted along possible flow pathways starting from all terminal arterioles in a subtree of a specific coronary vessel (Fig. 2B). The metabolic signal in the terminal arterioles is directly proportional to the venous Po₂ content, a physiological variable that is representative of the mismatch between oxygen supply and demand. As many metabolites are released during hypoxia with a common goal of eliciting vasodilation, optimization of the terminal arteriole flow to maintain the required metabolic demand under varying perfusion pressure has yielded the autoregulatory behavior in the model.

Myogenic Control

The coronary vessel wall tension and its associated diameter are regulated by the vascular smooth muscle cell (21, 44). Some models have used the smooth muscle’s length-tension relationship to describe the myogenic response (14, 19, 60, 73, 93, 101). These models describe how the vascular wall diameter changes in response to steady or unsteady variations of the intravascular pressure in a triphasic manner, where there is an initial dilation at low vessel pressures, followed by a constriction in the physiological pressure range and subsequent dilation beyond the physiological vessel pressures (101). Other models have represented the kinetics of the transmembrane potential at the cellular scale to account for the pressure-dependent changes on the smooth muscle electrophysiology instead of a length-tension response (35). Models of the smooth muscle cell have also considered the activation of stretch-dependent ion channels by vessel wall stress (13, 99, 100), and some have considered the effect of calcium dynamics on the myogenic response (47, 53). These models at the cellular scale are promising avenues and have yet to be implemented in multiscale models of coronary network flow.

Flow Control

Phenomenological models for flow-mediated vessel vasodilation have proposed that the fractional vessel dilation is a hyperbolic function of the luminal blood shear stress (61). An increase in shear stress alters the concentration of nitric oxide released by the endothelial cells, and the accompanying vessel dilation is a sigmoidal function of the nitric oxide concentration (18). Although models of nitric oxide generation and its effect on smooth muscle tone have been proposed at the scale of a single endothelial cell (15), their integration into network flow models to describe changes caused by alterations of the vascular shear stress is very limited (73). Large arterioles and small arteries are more responsive to shear stress and will dilate more than the small arterioles. This effect was accounted for in several flow regulation models (18, 61, 73). It is hypothesized that the capacitance of large coronary vessels (diameters > 100 μm) can also be affected by vasoconstriction due to the α-adrenergic feedforward mechanism. An integrated coronary network flow analysis model that incorporates adrenergic mechanisms in the coronary vessels is not yet available.

Integrative Models of Myogenic, Flow, and Metabolic Control

There are a plethora of models (Table 1) for the analysis of coronary flow that do not explicitly account for flow regulation. These models are of three types: symmetric tree models with MVI (1, 4, 26, 54, 84, 95), distributed models without MVI (7, 43, 49, 94), and distributed models with MVI (59, 70, 73, 74). Some of the symmetric models lump vessels of similar order into compartments, with each segment modeled as a windkessel element representing the vessel’s hydraulic resistance and capacitance. Distributed models of coronary networks (73) whose vessels are modeled as zero-dimensional (windkessel element) offer several advantages over symmetric models in that they can be calibrated against spatial flow dispersion, fractal characteristics, and spatial correlation of near neighboring flows in the microvasculature (7, 74) apart from a comparison with pressure distribution that can be done with these two models.

Table 1.

Summary of past works on mathematical models of coronary flow analysis and a select few among them that have modeled coronary flow regulation

Reference	Description	Strengths	Limitations
(1)	Myocardial vessel interaction (MVI)	Distributive model; realistic MVI	No coronary flow regulation
(7)	Flow analysis in a reconstructed coronary tree	Distributive model; large tree structure	No coronary flow regulation
(70)	Multiscale model of coronary flow	Realistic MVI; wave propagation of blood flow	No coronary flow regulation
(86)	Transient flow in the coronary network	Anatomically realistic tree; dynamic hemodynamic conditions	MVI and flow regulation are absent
(66)	Steady-state flow in the entire arterial tree	Flow analysis in entire arterial tree; flow dispersion prediction	Flow under steady-state conditions; MVI and flow regulation are absent
(43)	Steady-state flow in a realistic tree structure	Flow analysis in entire arterial tree; flow dispersion prediction	Flow under steady-state conditions, MVI and flow regulation are absent
(94)	Steady-state flow in a large tree	Flow in the entire arterial tree	MVI and flow regulation are absent
(96)	Pressure-flow relations in a coronary tree embedded in the myocardium	Flow in the entire tree in a contracting myocardium	Symmetric tree; steady-state flow, no flow regulation
(18)	Integrative model of coronary flow regulation	Modeled the 3 flow regulation mechanisms	Symmetric tree and metabolic control is highly simplified
(61)	Integrative model of coronary flow regulation	Data-based model of coronary flow regulation	Symmetric tree and metabolic control is highly simplified
(3)	Theoretical model of flow regulation	Conducted metabolic response	Symmetric tree; steady-state flow conditions
(73)	Integrative model of coronary flow regulation	Myogenic, flow, and metabolic control in an anatomically realistic coronary tree structure	Small size of the coronary tree

Integration of the microvascular response to pressure, flow, and oxygen is a complex modeling endeavor. Few models perform flow analysis as well as integrate the various flow regulation mechanisms (3, 12, 18, 32, 61, 73). Coronary flow regulation models are mostly comprised of symmetric tree networks with the flow analysis performed under steady-state boundary conditions (3, 18, 61), with only a few models being of a distributed type where the flow analysis has been performed in an asymmetric network (Fig. 2) under dynamic boundary conditions (32, 73). The earliest of these models by Liao and Kuo (61) represented coronary microvessels with diameters from 45 to 200 μm, divided into four compartments of vessel orders 5–7 and connected into segments in series. Liao and Kuo prescribed the vessel’s passive and active properties from in vitro testing of isolated vessels to obtain empirical relationships between vessel diameter and intravascular pressure under steady-state flow conditions with the myogenic, flow, and metabolic control being active. Subsequent work by Cornelissen et al. (18) considered a large tree size with coronary microvessels from 10 to 500 μm divided into nine compartments and incorporated the properties of myogenic, flow, and metabolic regulation mechanisms from Reference (61). In their model, however, vessels selectively responded to either flow + myogenic or metabolic regulation based on their location in the coronary tree.

A subsequent model by Arciero and colleagues considered not only the arteriolar trees but also the capillary and venous trees of the skeletal muscle microvascular bed, albeit idealized as symmetric compartments. This model (3, 12) also considered the three flow regulation mechanisms to be simultaneously active in each vessel while also accounting for the conducted metabolic signal propagating from the venules. A distributed model of coronary flow regulation (73) considered a realistic coronary tree structure reconstructed from morphometric data (51) for the flow analysis (Fig. 2). This distributed model considered the flow regulation mechanisms to be simultaneously active in each vessel (Fig. 1) to allow nonlinear interactions between the myogenic, flow, and conducted responses that are dependent on the asymmetric tree structure (Fig. 2) taking into account the myocardial vessel interaction, which is not available in the previous models. This distributed type model of coronary regulation (73) was able to demonstrate autoregulation, spatial perfusion heterogeneity, and multiscale flow behavior from the individual vessels to a large coronary subtree. A short summary of the coronary flow models with their strengths and limitations is provided in Table 1.

MODELING THE MYOCARDIUM-CORONARY FLOW INTERACTION

Coronary flow regulation occurs under phasic contraction and relaxation of the myocardium. Specifically, intramyocardial pressure (IMP), generated by the contracting myocardium, acts cyclically on the coronary vessels. This extravascular pressure generated by the myocardium, also known as myocardial vessel interaction (MVI), alters the resistance of coronary vessels to flow (97). MVI has a transmural gradient, with a maximum value in the endocardium that decreases to zero in the epicardium. The transmural gradient in MVI has a significant effect on the phasic coronary flow characteristics within a cardiac cycle. Microvascular hemodynamics in a cardiac cycle is different between the subendocardial and subepicardial vessels (92). The effect of MVI on flow regulation that occurs over several cardiac cycles is still poorly understood (42). MVI can be significantly affected by perfusion pressure, heart rate (diastolic time fraction), or changes in left ventricular (LV) contractility. Below is a brief description of the various models for MVI and their effect on coronary flow.

Effect of MVI

The earliest models of MVI assumed a linear distribution of IMP across the transmural wall thickness, with the endocardial IMP equal to the LV pressure and the epicardial IMP equal to zero (26, 88, 97). Systolic flow impediment was attributed to a “vascular waterfall mechanism” due to hypothesized vessel collapse (26), whereas an alternative model attributed it to the existence of an “intramyocardial pump” (88). Lumped models of the coronary tree (55, 102) with the latter MVI model have not been able to predict the transmurally dependent microvascular hemodynamics.

A more recent MVI model in a cylindrical LV geometry (1) has demonstrated the importance of specifying IMP as an additive contribution of 1) cavity-induced extravascular pressure and 2) myocyte shortening-induced pressure. This hypothesized model reproduced experimental velocity profiles in both the epicardial and endocardial arterioles (1). A distributed model of coronary flow regulation (73) has incorporated this IMP axiom and applied the relevant boundary conditions of ventricular cavity pressure and myocyte shortening-induced pressure from data. Other models of MVI have formulated the extravascular pressure on the coronary vessels from the mechanical stresses in the myocardial wall estimated from finite element models (64, 84, 85). Finite element models of the myocardium can incorporate patient-specific geometries of the ventricles, have a realistic myofiber orientation (82), and predict IMP in regions with large curvatures such as in the LV apex, the septum, and the thin right ventricle (RV). Finite element models (75) can be coupled to the systemic circulation to evaluate the effects of preload and afterload characteristics as well as the myocardial contractility and passive properties on the spatial distribution of flow regulation (73). The various outcomes of MVI on flow regulation are discussed in the next section.

Effect of Heart Rate

Physical exercise rapidly increases the heart rate, arterial pressure, and cardiac output. The increase in metabolic demand (oxygen consumption) due to exercise is mostly met by an increase in coronary flow (27). Heart rate increase not only affects coronary flow but also LV contractility, work done, systolic time fraction, and the MVI, all of which alter the magnitude and phasic characteristics of coronary flow and the transmural distribution. Models based on control theory have considered coronary flow changes due to an increase in heart rate (20, 68). An MVI model of a symmetric coronary tree (1) found that an increase in heart rate alters the transmural distribution of coronary flow, causing hypoperfusion in the subendocardium (50). As coronary blood flow is tightly coupled to oxygen consumption, regional hypoperfusion due to an elevated heart rate makes the subendocardium more vulnerable to ischemia. However, distributed models have yet to account for the heart rate effects on flow regulation in the microvasculature and to identify the role of changes in systolic time fraction due to heart rate increase versus changes in LV contractility.

Effect of Contractility

Myocardial contractility is known to alter the pulsatility of coronary flow. Models of coronary flow have incorporated altered contractility and quantified their effect on the pulsatility of coronary flow (8) and the pressure-area relationship of arterioles (95). Contractility affects MVI by altering both the myocyte shortening-induced pressure and cavity-induced extravascular pressure, and models have explained the effect of contractility on the transmural redistribution of coronary flow (2). There is nearly a linear relation between regional contractility and regional coronary flow, and since the spatial distribution of myocardial contractility is heterogeneous, high-resolution three-dimensional models (75) of the myocardium with an accurate description of contractility would be invaluable in generating maps of regional distribution of coronary flow and flow dispersion. More importantly, this will aid in a more accurate description of the spatial distribution of flow reserve and autoregulation behavior. Such information would be invaluable in understanding ischemic heart conditions when regional contractility, and hence the regional flow, is impaired and flow dispersion is drastically altered.

Effect of Ventricular and Perfusion Pressures

Since LV pressure is a major component of the cavity-induced extravascular pressure on the coronary vessel, mathematical models have found that coronary flow dynamics is sensitive to its changes (41, 50). The systemic arterial pressure defines the inlet boundary condition of the coronary network flow models and has a strong effect on the longitudinal pressure distribution. An increase in the perfusion pressure improves perfusion but also creates conditions for a stronger myogenic response particularly in the epicardial vessels (12, 19). These effects are not yet tested in the coronary network flow models. It is known that the LV pressure is nearly four times the right ventricular (RV) pressure, and hence the MVI component in the coronary tree perfusing the RV will be drastically different from that of the LV. Mathematical models have, however, yet to simulate flow regulation in the RV.

MODEL PREDICTIONS AND THEIR PHYSIOLOGICAL RELEVANCE

Integrative coronary network flow models have quantified the hemodynamics locally in the microvessels and globally at the organ scale due to the active responses of the myogenic, flow, and metabolic mechanisms. Specifically, the effect of the three flow regulation mechanisms on the flow dynamics could be uncoupled by the models. An early model hypothesized that the heterogeneous properties of flow regulation enhance the coronary tree response to metabolites (61). A subsequent model (18) demonstrated the following: 1) the combined interaction between flow regulation properties of the upstream vessels and metabolic properties of the downstream vessels in the coronary tree is essential; 2) the myogenic response has a significant effect on coronary flow; and 3) flow regulation due to shear stress reduces autoregulation of the coronary tree. Because of the computational challenges involved in the simulation of flow regulation, most coronary flow models are of symmetric networks. A distributed model of flow regulation integrated the different flow regulation models to quantitatively determine the confounding effects of MVI and vascular tone on coronary flow reserve and autoregulation (73). The model predicted that MVI increased the perfusion pressure range at which autoregulation occurs (Fig. 3B) and decreased the flow dispersion (73). Furthermore, metabolic heterogeneity in the precapillaries under variable metabolic demands has been determined by these models. The model explains why under low perfusion pressures and extreme metabolic demand the metabolic activation in the precapillaries is saturated. On the other hand, under high perfusion pressures the metabolic activation in the precapillaries is suppressed (Fig. 3A).

Fig. 3.

Key results from a distributed model of flow regulation (73) are presented as a schematic. A: frequency distribution of terminal arterioles as a function of the level of their metabolic activation and the perfusion pressure. The perfusion pressure increases from a low value of ~70 mmHg to a high value of ~140 mmHg (top to bottom). At a low perfusion pressure and a high metabolic demand, the metabolic activation in the terminal arterioles is exhausted, whereas at a high perfusion pressure, there is negligible metabolic activation. B: autoregulation curves at 3 different levels of metabolic demand: high (dashed), basal (dash dotted), and low (dotted). The flows under passive vessel conditions (*), at maximum metabolic activation (○), and without metabolic activation (+) are shown as a function of perfusion pressure.

There are three main characteristics of coronary flow regulation observed at the organ level, i.e., 1) the presence of a flow reserve in the coronary tree, 2) the ability to control (i.e., autoregulate) blood flow over a wide range of perfusion pressures, and 3) the spatial heterogeneity in myocardial perfusion. These characteristics and their physiological implications are as follows.

Coronary Flow Reserve

The coronary tree can scale up myocardial perfusion based on metabolic demand. Coronary flow reserve (CFR) is quantified as the ratio of maximum flow (under complete vasodilation) to the flow under basal conditions supplied by the coronary tree (39) and is a clinically useful measure for diagnosis of coronary microcirculation (38). CFR is strongly related to the pressure-flow relationship of the coronary tree, is nearly uniform in the pressure range of autoregulation, and decreases with an increase in heart rate or the cardiac preload (27). Models of flow regulation in a symmetric tree (18, 61) have predicted CFR of ~2.0, which is lower than the typical values found in vivo between 3.0 and 6.0. A distributed model of flow regulation estimated metabolic flow reserve that is independent of heart rate to be 2.0–3.5 and also found it to be transmurally heterogeneous (73). CFR is spatially heterogeneous, with a far greater reduction in the subendocardium than in the subepicardium due to CHD (41) or metabolic syndromes (11, 72, 80).

Autoregulation

Coronary flow regulation is of significant clinical importance as it maintains a steady rate of blood flow to the myocardial tissue under variations in perfusion pressure. One of the first models predicted only weak autoregulation from their network flow analysis (18, 61). A strong autoregulation response, where the flow is independent of perfusion pressure, i.e., a nearly flat region of the steady-state pressure-flow curve (Fig. 3B), has been predicted by a few models (12, 32, 73). The pressure-flow autoregulation curves from distributed models (32, 73) have been found to depend on transmural depth, with both metabolic and myogenic regulation mechanisms exerting a strong effect on autoregulation. Our group recently used an integrative coronary model to demonstrate that turning off the myogenic response significantly diminishes autoregulation (25). Current models cannot predict autoregulation behavior at different spatial locations of the LV, the septum, and the RV.

Spatial Heterogeneity of Coronary Flow

Unlike lumped models, distributed models can produce flow dispersion in the terminal arterioles under flow regulation and MVI conditions (73). The model outcome that MVI reduces flow dispersion can be corroborated with data from microsphere experiments that found that coronary flow dispersion in a beating heart is significantly lower than in an arrested heart (6). The flow dispersion of 10–15% predicted by the model under autoregulation (73) is similar to the 14–19% measured by microsphere measurements (6). Myocardial perfusion is spatially heterogeneous (5, 6) at both the macroscale (52) and the microscale (63). Under autoregulation conditions, a distributed model determined that flow dispersion is similar in both the subepicardium and subendocardium at baseline perfusion pressure conditions. Coronary flow dispersion is well documented in the LV transmural direction, and flow dispersion is highly sensitive to changes in the myocardial contractility (27), the diastolic time (9, 65), and the LV’s afterload and preload (41). These characteristics of flow dispersion under flow regulation have yet to be simulated by the mathematical models.

KEY CHALLENGES

Theoretical and Computational Challenges

One of the challenges in the modeling of flow regulation is the pulsatile nature of coronary flow, with highly nonlinear interactions between different flow regulation mechanisms. Since analytical solutions for such a highly nonlinear system with multiple feedback loops remain intractable, computational methods have been the ideal tools to overcome these limitations. Computational models can account for the longitudinal heterogeneity in the flow regulation mechanisms across the coronary tree. A solution of the flow regulation problem for a large coronary network is challenging, however, as the computational time increases exponentially with the number of terminal vessels (network size). Novel algorithms for the computational model of flow regulation are required to achieve solutions that can be hypothesis-generating. For instance, some simplifications in the computational model have been made by formulating the flow regulation problem as an incremental solution of steady states where the flow dynamics of the entire network can be solved under small perturbations around these steady states (73). Beyond the theoretical advancement in the development of flow regulation algorithms, novel computational strategies using high-performance parallel computing, machine learning, and data analytics are required for patient-specific coronary flow analysis. Further complexity in the models arises when combining models of the heart with the coronary network flow models to evaluate their integrated behavior under varied demands (e.g., exercise conditions). Another major limitation in the construction of these models is the lack of data for estimating the parameters of the flow regulation model in the various microdomains of the coronary tree. As such, many parametric studies and sensitivity analyses must be performed because of the limited data available to calibrate the models, which can greatly increase the computational load.

Gaps in Mechanistic Understanding of Autoregulation

Models are very important to integrate many fundamental questions and serve as hypothesis-generating in coronary physiology regarding the mechanistic aspects of coronary flow regulation. There are unanswered questions that can benefit from computational modeling: 1) What is the respective contribution of myogenic tone and the metabolic regulation to coronary flow reserve and autoregulation (24)? 2) What is the role of MVI in coronary flow regulation and how does it relate to the regional variation in CFR (37)? 3) What is the quantitative relationship between regional work done by the myocardium and regional coronary autoregulation and how does it vary under hypoxia due to either exercise or coronary disease? 4) How does the heterogeneous distribution of each of the myogenic, flow, and metabolic regulation mechanisms affect the specific location of the autoregulation curve and coronary flow reserve? Some other major knowledge gaps that could be addressed by models are to quantify the effects of MVI on coronary flow regulation and flow dispersion in the RV, which has been much less studied than the LV.

Future Modeling Directions

Coronary heart disease (CHD) affects the large and small blood vessels of the coronary tree, disrupting blood flow to various regions of the heart. Atherosclerosis of epicardial conduit vessels obstructs the supply of oxygen and nutrients to the heart muscle and alters the transmural distribution of coronary flow, making the subendocardium more susceptible to ischemia (41). Diagnostic imaging (29, 62, 80) revealed that coronary microvascular dysfunction (CMD) can also cause myocardial ischemia in patients without any obstruction of the epicardial arteries (10). As there are multiple factors that can cause CMD, animal models (87) have attempted to delineate structural and functional changes such that therapies for CMD (90) can be precisely targeted. Computational models of coronary flow regulation that can incorporate the impaired metabolic, flow, myogenic, and MVI mechanisms can not only identify the specific contribution of each mechanism to the diminished coronary flow reserve in patients with CMD (79) but also motivate the design of therapies that are currently not available for targeting CMD.

Computational models of flow regulation in the future can provide answers to fundamental questions such as the mechanisms underlying subendocardial ischemia (2) such as why the entire coronary flow reserve cannot be fully utilized under elevated metabolic demand when CMD is present, why there is elevated coronary flow in patients with CMD under resting conditions, and how CMD affects autoregulation. The effect of cardiac-coronary coupling is also of fundamental importance to CMD, and computational models that account for the bidirectional coupling between cardiovascular and coronary systems can evaluate the effect of the altered MVI state and cardiac oxygen demand under resting and exercise conditions on flow reserve and autoregulation. Such models are invaluable for designing resynchronization therapies (89) to improve cardiac-coronary-coupling and myocardial perfusion in heart failure. Computational models of flow regulation can not only be applied to understand CMD but also be applied to study coronary flow changes in the right ventricle and the septum (33) due to pulmonary hypertension (71). There is a vast amount of data available from published data on pharmaceutical interventions to treat coronary heart disease with a plethora of drugs such as dipyridamole, adenosine, beta-blockers, calcium blockers, statins, and nitroglycerin. Model-based analysis of the effects of these drugs on global perfusion, flow dispersion, and local hemodynamic changes in the coronary vessels is not yet available.

A new class of computational models that incorporate coronary flow regulation mechanisms can provide a paradigm shift in understanding how altered flow regulation responses in the coronary tree can be related to the observed poor clinical outcomes in patients with CMD (91). Computational models of coronary flow regulation in the future may identify regulatory mechanisms of the coronary tree that are potentially more important to focus on in animal models, assess diagnostic protocols for preclinical screening of CMD, and design clinical therapies for hypertension, left ventricle hypertrophy, diabetes, and angina pectoris.

CONCLUSIONS

Multiscale models of flow control that are physiological and mechanistic offer new perspectives and avenues to propose and explore new hypotheses in coronary physiology. There is much-needed work to extend the existing multiscale models of coronary flow regulation to encompass physiological processes at different spatial and temporal scales from the cell to the organ. Such models can then shed light on the causative mechanisms underlying disease progression leading to ischemia and possibly evaluate pharmacological, surgical, or device-based therapies for personalized medicine.

GRANTS

The funding for this work is provided by National Institutes of Health Grants U01-HL-119578, R01-HL-134841, and HL-133359 and American Heart Association Grant 474 SDG (17SDG33370110).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

R.N. conceived and designed research; R.N. prepared figures; R.N. drafted manuscript; R.N., Y.L., L.C.L., and G.S.K. edited and revised manuscript; R.N., Y.L., L.C.L., and G.S.K. approved final version of manuscript.