The aim of the cardiac Physiome Project is to develop theoretical models to simulate the functional behaviour of the heart under physiological and pathophysiological conditions. Heart disease continues to be the leading cause of morbidity and mortality in industrialised countries and throughout much of the world, and better methods for cardiovascular disease management are sorely needed (Ricotta et al., 2008
). The overall clinical goal of in silico
modelling is the development of patient-specific predictive models to improve diagnosis, therapy planning and treatment of cardiovascular diseases (Siebes and Ventikos, 2010
). However, the achievement of this objective will also necessarily be underpinned by characterisation of the underlying physiological mechanisms derived from fundamental scientific investigation.
Using a multi-scale, multi-dimensional, and multi-disciplinary approach, theoretical modelling has the potential to predict clinical outcomes in order to achieve more effective healthcare. The aim is to develop individualised computer simulations that exploit patient-specific clinical visualisation modalities and experimentally obtained material properties in combination with solid mechanics and fluid dynamic models. Detailed knowledge about physiological (control) mechanisms and pathophysiological processes is necessary to arrive at clinically relevant decision-making tools. Ultimately, these models must account for processes operating at different time scales, ranging from transient behaviour of pressure and flow during a cardiac cycle, to effects of altered physiological demands or therapeutic interventions, through to much longer time-scale processes involving growth and remodelling due to disease progression and ageing (Lieber et al., 2005
). New diagnostic methods evolving from this approach should allow better patient selection, targetted interventions, therapy assessment and predictions of therapeutic outcomes (Ricotta et al., 2008
Heart function is critically dependent on the availability of an adequate blood supply to the myocardium. The network of coronary vessels must bring oxygenated blood within a small distance of every point in the tissue to meet the varying metabolic demands of the individual myocytes. In building mathematical models capable of simulating functional heart behaviour, it is therefore necessary to develop models for the coronary vasculature, in addition to models for sub-cellular function, cellular excitation–contraction coupling and cardiac tissue mechanics (see Clayton et al., 2011
; Nordsletten et al., 2011
). This article focuses on theoretical models for the coronary vascular system.
The ability of the coronary vasculature to meet the metabolic needs of heart tissue is facilitated by its dynamic structure, which is regulated on a short time scale via the active contraction and dilation of the small arteries and arterioles, and capable on a longer time scale of generating new vessels and remodelling existing vessels according to changing physiological and pathophysiological conditions. In addition to this wide range of temporal dynamics there are also many components of the coronary vasculature that interact over a range of spatial scales from sub-cellular to whole organ scale (see ). To build mathematical models capable of simulating functional heart behaviour, we thus need to understand in detail the individual components of the coronary vascular system (including vascular structure and mechanics, fluid flow and mass transport, regulation and remodelling, and cellular biomechanics) and how they work in an integrated way to respond to the ever-changing demands placed upon the coronary vascular system.
Examples of typical length and time scales encountered in constructing models of the coronary vascular system.
Embedding of current physiological understanding within mathematical frameworks is an approach that has already led to a number of important contributions for understanding coronary circulation over many years. We start by presenting a brief synopsis of the research that has provided the foundation for the current models. Within this historical summary the interested reader is referred to a number of recent and more comprehensive reviews. However, it is important to note that the large body of work in this area means that, rather than providing a thorough review, our goal in the sections which follow is to examine the current status of key theoretical models for each constituent component and their integration, and assess the modelling challenges that are currently defining the cardiac Physiome in the context of the coronary circulation.
Unique to the coronary circulation is the continuous and rhythmical compression of the blood vessels as the heart contracts, combined with the necessity to provide continuous perfusion to match a wide range of metabolic rates. This squeezing effect, or systolic flow impediment, was first proposed by Scaramucci in 1695 and has subsequently been investigated by Porter (1898)
, Anrep et al. (1927)
, Downey and Kirk (1975)
, Spaan et al. (1981)
, Bruinsma et al. (1988)
, Krams et al. (1989a
, among others. In the models of Downey and Kirk (1975)
and of Spaan et al. (1981)
the compression effects of intramyocardial pressure were assumed to be equal to ventricular pressure at the endocardium and to decrease linearly to zero at the epicardium. Arts (1978)
developed an integrated model of cardiac wall mechanics and the coronary circulation, in which the coronary microvessels were loaded by an intramyocardial pressure that was related to myofibre contraction through left ventricular pressure (Arts et al., 1979
; Arts and Reneman, 1985
). Later models also considered a direct interaction between the myocardium and the microvasculature, through stiffening of the myocardium during systole alone (Krams et al., 1989a
), or in combination with radial passive tissue stress (Beyar et al., 1993
; Huyghe et al., 1992
; Zinemanas et al., 1994
; Vis et al., 1997
; Bovendeerd et al., 2006
). In particular, Krams et al. (1989a
suggested a more limited role of ventricular pressure, and applied the “time varying elastance concept” of Suga et al. (1973)
to explain systolic flow impediment. This concept emphasises the effect that time-varying ventricular wall stiffness, which is assumed to be independent of ventricular pressure, has on coronary blood volume. The theory of Krams et al. (1989a
is based on the observation that flow impediment is similar for isovolumic (high systolic ventricular pressures) and low after load isobaric (low systolic ventricular pressures) contractions. However, the elastance concept does not explain why epicardial flows are not inhibited to the same degree as endocardial flows (Goto et al., 1991
; Spaan, 1995
) and subsequent studies again suggested that ventricular pressures have a significant effect on time-varying coronary flow.
Closely linked to this issue is the role of vascular resistance and compliance in determining coronary flow. The “vascular waterfall mechanism” proposed by Downey and Kirk (1975)
sought to explain reduced coronary inflow by the increase in resistance resulting from the collapse of vessels embedded in the myocardium. However, since this throttling effect would impede both arterial inflow and venous outflow, this theory alone could not explain the increased venous outflow during systole. The introduction of the “intramyocardial pump model” (Spaan et al., 1981
) accounted for the role of vascular compliance. In this model compliant vessels are filled from the high-pressure arterial side in diastole and then discharged through the low-pressure venous side in systole. This concept has since been extended in a number of lumped parameter mathematical models of coronary circulation, e.g. Bruinsma et al. (1988)
, to account for the variation in vascular resistance and compliance throughout the coronary network with the temporal change associated with variation in intramyocardial pressure.
Key to understanding and unravelling the role of myocardial contraction on coronary blood flow has been the development and application of experimental measurement techniques to determine the temporal dynamics of myocardial blood flow across a range of vessel sizes. These experimental observations now include flows in the microcirculation (see reviews of Kajiya et al. (2008)
and van den Akker et al. (2010)
) and flows subject to varying mechanical conditions. These latter studies in particular have been instrumental in quantifying phasic variations throughout the cardiac cycle in both animal models (Kimura et al., 1992
; Kajiya et al., 1989
) and more recently humans in clinical contexts (see reviews of Spaan et al. (2006
, Knaapen et al. (2009)
In addition to capturing the dynamic interactions of flow and myocardial contraction, a further understanding of the control of coronary flow in both normal and pathological conditions is required; specifically the regulation of vessel resistance to match perfusion with the metabolic demands of the heart, in spite of fluctuating perfusion pressure. This tendency is termed autoregulation, and is the result of a large number of different physiological mechanisms (see reviews of Rubio and Berne (1975)
, Feigl (1983)
, Jones et al. (1995)
, Deussen et al. (2006)
, Duncker and Bache (2008)
, Zhang et al. (2008)
A quick outline of some established concepts of autoregulation can be summarised as follows. In the heart, microcirculation plays a key role in regulation of flow since the majority (~70%) of the resistance, and the greatest capacity to adjust it, resides in the microvessels (Chilian et al., 1989
). Previous studies have discovered many different mechanisms of regulation, including the major effectors of myogenic response, flow-induced dilation, metabolic control and conducted responses, as discussed in turn below.
The myogenic response is caused by contraction of vascular smooth muscle cells which respond directly to distending pressure in the lumen (Bayliss, 1902
), with a typical time scale in the order of tens of seconds to minutes. Under normal flow conditions, the myogenic response provides the basal tone, producing the vasodilatory reserves which can be exploited by other regulatory mechanisms. Reduction in perfusion pressure has been observed to produce dilation predominantly in microvessels (Kanatsuka et al., 1989
; Chilian and Layne, 1990
), and graded responses were observed in vessels of different diameters, with the most sensitive myogenicity found in intermediate arterioles of diameter around 60 mm in pig coronary vessels (Liao and Kuo, 1997
The metabolic control hypothesis proposes that coronary flow remains constant when subject to a fixed level of metabolic demand, as autoregulation is governed by a myocyte-produced substance which diffuses to the vascular smooth muscle cells via interstitium. Originally it was suggested that adenosine was the main substrate for metabolic control (Berne, 1963
), but subsequent experimental results have failed to confirm this. There have been many other mediators proposed for the role, including bradykinin, CO2
, potassium and endothelin. However, due to the redundant design of the metabolic control system in which blocking of any one of these substances fails to abolish the control mechanism, it is now widely held that the metabolic control is achieved via a combination of many different mediators (Zhang et al., 2008
The conducted responses in flow control and the oxygen sensing mechanism of the red blood cells have been the focus of some of the more recent modelling studies. In addition, it should be noted that coronary autoregulation is achieved via integrated interactions of the aforementioned mechanisms. The quantitative investigation of such a system is an ongoing challenge in the cardiac Physiome project, and is described in greater detail in Section 6.
A key step to characterising these autoregulator responses is a mechanistic understanding of endothelial function at the lower spatial scale of the cell. This in turn defines a central challenge of developing theoretical multi-scale models for the coronary circulation, that is, to understand the role of endothelial cells lining all blood vessels in vascular physiology and pathophysiology, and how they sense and modulate their function when exposed to changes in their local biochemical and biomechanical environment. The endothelial cell is particularly sensitive to fluid dynamical forces such as shear stress and pressure, in response to which they produce biochemical signals during the process of mechanotransduction (Dewey et al., 1981
; Davies et al., 1984
; Levesque and Nerem, 1985
; Florian et al., 2003
; Weinbaum et al., 2003
; Mochizuki et al., 2003
; Tarbell and Pahakis, 2006
). The surface of endothelial cells has two important specialisations that factor into mechanotransduction and solute transport: the glycocalyx composed of membrane bound highly charged macromolecules regularly distributed over the luminal surface (comprehensively reviewed by Reitsma et al. (2007)
and Weinbaum et al. (2007)
) and primary cilia – one per cell – that can project beyond the luminal surface as membrane bound continuations of the cytoskeleton (Kojimahara, 1990
; van der Heiden et al., 2008
). Both may play a role in endothelial mechanotransduction and the glycocalyx also acts as a transport barrier and as a porous hydrodynamic interface in the motion of red and white cells in microvessels (Weinbaum et al., 2003
). These cellular elements are more extensively outlined in section 8.
This inherently integrative nature of coronary investigation combines experimental measurement and modelling. Such work is focused on understanding, arguably, one of the most complex vascular systems in terms of regulation, mechanical interaction and clinical pathologies. Below we aim to outline many of the research challenges faced in developing integrated mathematical models to describe the coronary vascular system which, if overcome, will also be invaluable in developing models to understand other organ systems.
As already highlighted, in order to build computational tools capable of simulating functional heart behaviour, we are faced with the challenge of integrating models for physical processes at disparate spatial scales, e.g. incorporating micro-scale flow and mass transport processes in a macro-scale model for myocardial tissue. The challenge is to introduce small-scale information into larger-scale models without the resulting models becoming computationally intractable. One approach is to use a lumped representation where fine-scale structures, e.g. blood vessels, smaller than a certain size are represented by a single compartment with uniform properties. A limitation of this simplified approach is that significant spatial variations may exist within this compartment, which are not represented by the model. An intermediate approach between detailed representation of fine-scale structure and a lumped approach is provided by homogenisation theory. In this theory, a local spatial averaging of fine-scale structure is achieved by exploiting asymptotic techniques to estimate macro-scale properties, based on explicit solutions in smaller-scale subunits (Huyghe et al., 1989a
; Vankan et al., 1997
; Chapman et al., 2008
; Shipley and Chapman, 2010
; Shipley et al., submitted for publication). Although homogenisation techniques have been used for many years in modelling the mechanical properties of the myocardium, the technique is now starting to be used more widely in cardiovascular fluid dynamics modelling, and we highlight this methodological approach in Section 9.
Because of the inherently multi-scale nature of the system we have chosen to present the research ideas by application (flow, mass transport, etc) rather than by modelling methodology, and stress that many of the same theoretical techniques are used in the development and solution of the component models. To model coronary flows, including the microcirculation and the large arteries and veins, we must understand the geometry of the flow domain, and the mechanical environment within which the vessels find themselves (determined both by the properties of the vascular wall and the surrounding myocardium). Such aspects are considered in Sections 2 and 3. We then consider flow and mass transport in Sections 4 and 5. Finally, we consider how coronary vasculature networks evolve on both short (regulation) and long (adaptation) time scales in Sections 6 and 7. The mechanisms by which endothelial cells sense fluid mechanical forces and produce biochemical signals in the process of mechanotransduction are considered in Section 8. Each section highlights the current state of the art of modelling in the field, before going on to explore open research challenges. In section 9 we discuss how the component models may be integrated.