An electrohydrodynamic model of the cardiac myocyte interstitium

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Elements of the cardiac system

The heart is a vital organ in the human body. Its primary role is to supply blood throughout the body. The heart has to adapt to the varying energy needs of the body. Its performance is startling if one considers how the rhythmic beating is maintained over a lifespan without failure or fatigue with never more than a few milliseconds interval between beats. Of equal importance to the inbuilt rhythmicity are those mechanisms which allow recovery of both the electrical and contractile systems between each beat. The normal adult human heart rate at rest is about 70 beats/min but can rise to 200 beats/min during exercise or stress. The volume of blood ejected per beat, called the stroke volume, is around 90 ml at rest and rises to 140 ml during exercise for normal adults. The product of the stroke volume and the heart rate gives the cardiac output, which for normal adults is around 6.5 l/min at rest and 28 l/min during exercise. These values change with ﬁtness and age. When the heart fails to appropriately adapt and meet the expected pumping output, it results in complications that could range from a minor discomfort to death. This chapter describes some of the primary components of the heart and their role in its function. The information provided is a summary of selected topics with the emphasis on components and their behaviour directly related to the pathological conditions whose genesis lies in the electrophysiology.

Chambers

The heart is a synchronized pump that sequentially circulates blood through the body. The circulation can be divided in to two major categories, pulmonary circulation which involves the right side of the heart and systemic circulation which involves the left side of the heart. Some of the anatomical features of the human heart are shown in Figure 2.1. The wall thickness of each heart chamber is related to its function. The atria within which only low pressures are developed is thin-walled. The left ventricle, responsible for systemic circulation, develops high pressure and has a thicker wall compared to the right ventricle which is responsible for pulmonary circulation characterized by a relatively low pressure. Wall thickness changes due to myocardial infarction or from sustained loading. These changes can aﬀect the pressure developed within the chamber and in turn aﬀect the body.

Heart valves

The proper functioning of the heart depends on the eﬃciency of the valves separating the chambers. The atrioventricular valves connect an atrium and a ventricle. When the ventricle relaxes (diastole) the valve opens to allow blood to ﬂow from the atrium to the ventricle and closes when the ventricle contracts (systole). The ventricles are connected to the aorta or pulmonary artery by semilunar values. These values open during ventricular systole and close when the ventricular pressure falls. Failure of the valves to open and close rhythmically causes regurgitation or stenosis which in acute conditions leads to cardiac failure

Coronary vessels

Coronary vessels are responsible for the blood circulation to the heart tissue. Coronary blood ﬂow accounts for 5% of the total cardiac output under normal conditions and raises to nearly 8% during exercise. The delivery of blood to the ventricular tissue is highly specialized as the contraction of the ventricles impedes the blood ﬂow in the coronary vessels. The right and left coronary vessels arise at the base of the aorta, just beyond the semilunar values. The driving force for blood ﬂow along them is the diﬀerence between the aortic pressure and ventricular pressure. The pressure diﬀerence is greatest during diastole and the bulk of the coronary ﬂow occurs during this part of the cardiac cycle. The coronary ﬂow remains low during the systolic period.

Control of coronary vessels

Under conditions of normal health, coronary blood ﬂow is well adapted to the metabolic needs of the heart. With severe exercise, myocardial oxygen consumption increases. As the extraction of oxygen from blood is practically maximal in the coronary system, blood ﬂow has to increase more or less proportionally to this increased oxygen demand. When oxygen consumption is kept constant, coronary blood ﬂow is constant as well and independent of coronary arterial pressure variations. When pressure is kept constant, ﬂow varies linearly with variations in oxygen consumption [3]. Measurements of oxygen saturation in venules with cryogenic spectroscopy show a wide distribution, with oxygen saturation varying from below 5% up to 60% [4]. The vascular system seems not to distribute the ﬂow homogeneously, at least not in all conditions. The microvascular bed gives rise to heterogeneous distribution of blood ﬂow and oxygen and therefore some areas are more prone to ischemia than others. The coronary vascular volume forms a dynamic entity and is composed of larger and smaller arteries, capillaries and smaller and larger veins.
Coronary vasculature system
Nutrient supply and waste removal is a signiﬁcant process that inﬂuences the function of the heart. The tissue anatomy and the perfusion network therefore play a signiﬁcant role in cardiac function. All intramural vessels are distensible and deformable, since stiﬀ vessels would make cardiac contractionand hence its functionimpossible. Studies havedemonstrated that thedegree of ﬁlling of the vascular bed impacts cardiac function through the contractile and metabolic functions of the myocytes [5]. Especially in the dilated coronary vascular bed, an increase in perfusion pressure will increase the oxygen consumption of the heart at a constant level of mechanical performance. This is called the Gregg eﬀect or the water hose eﬀect. It is therefore important to maintain the distending blood pressure in the myocardium as low as possible. This principle provides a rationale for the high oxygen extraction in the coronary circulation. A high oxygen extraction corresponds with high arteriolar resistance, thereby reducing ﬂow to a level that is just suﬃcient for heart function. A lower arteriolar resistance would make the heart mechanicallylesseﬃcient. Thisinteractionbetweentheperfusionandtheheartfunctionishighly sophisticated as it not only manages feedback from metabolism to the tone of smooth muscle in the walls of resistance vessels, but also the eﬀect of perfusion of capillaries on metabolism [6]. The heart controls its blood ﬂow at the organ level and there are two primary methods it uses as described below.
Metabolic control
The heart uses approximately 11% of the body’s oxygen supply and this increases further during exercise. The oxygen demand is met by increasing the amount of blood ﬂow as the oxygen extraction process is around 70% of blood oxygen volume during normal activity and 90% of blood oxygen volume during exercise. The blood ﬂow is increased by the dilation of the coronary vessels (vasodilation). Chief agents causing vasodilation are metabolites. The adenosine released during the hydrolysis of ATP (Adenosine Triphosphate) acts on receptors of coronary arterioles causing the dilation of these vessels. Nitric Oxide also plays a role in coronary vasodilation.
Nervous control
Sympathetic nerve ﬁbres innervate the coronary vessels as they do the rest of the heart. The vasoconstrictor eﬀects due to the action of noradrenaline on α-adrenoreceptors are overridden in exercise by metabolic vasodilation of coronary vessels. Circulating adrenaline will also counteract vasoconstriction by action on β2-receptors which relax vascular smooth muscle. Coronary vasoconstriction can predominate in certain emotional states such as anger.

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Coronary Heart Disease

Coronary disease(CHD)can be deﬁned as ather osclerotic disease of the coronary circulation. The hall mark of CHDis a theromatous plaque. Small plaque stend to lead to myocardialinfarcts while larger plaques lead to angina. Plaques can also narrow coronary vessels leading to myocardial ischemia.
Myocardial ischemia
CHD leads to myocardial ischemia or the lack of adequate blood ﬂow to the heart muscle. This deprives the myocytes (cardiac cells) of nutrients and allows waste to accumulate which usually causes cell damage. Myocardial ischemia is characterized by contractile failure.

Coronary blood ﬂow and cardiac contraction

The coronary vascular system is subjected to the forces of cardiac contraction and its blood volume is lower in systole than in diastole. The average per beat blood volume depends not only on perfusion conditions, such as the degree of vasodilation and perfusion pressure, but also on parameters related to cardiac contraction. Since coronary inﬂow occurs predominantly during diastole, time-averaged intramural blood volume decreases with increasing heart rate, which shortens diastole. Coronary blood ﬂow is pulsatile and displays a biphasic pattern. In the left coronary arteries, systolic ﬂow is lower than diastolic ﬂow despite the higher inlet pressure during systole. Systolic ﬂow can even be retrograde when intramural pressure exceeds coronary root arterial pressure [7]. These variations in coronary arterial ﬂow are directly coupled to the contraction-related intramural blood volume variations. During cardiac muscle contraction, a decrease in volume of an artery will be accompanied by an increase in volume of the corresponding vein [8]. Obviously, when microvascular resistance (MR) changes are imposed by the ﬂow control of the myocardium, this will aﬀect not only the time-averaged ﬂow but also the ﬂow pulsatility. At rest, the dilation of the resistance vessels could compensate for the increased resistance induced by cardiac contraction. However, in the presence of vascular disease, e.g. a proximal stenosis, the resistance vessels have to dilate to compensate for the pressure loss in the diseased vessel. At rest, this prevents ischemia, but not during exercise when demand for ﬂow increases and the increase in ﬂow will further increase the pressure drop over the stenosis. Hence, the myocardial region supplied by the diseased artery can become compromised. Diastole is needed to reﬁll the intramural blood volume expelled during systole. Hence, the more time spent in diastole, the higher the intravascular volume and the higher the vascular conductance. The relative duration of diastole within a heartbeat is expressed as diastolic time fraction (DTF). DTF is a major determinant of subendocardial perfusion.

Summary

This concludes an overall description of the cardiac system. The following chapters will focus on the primary components of the cardiac system related to excitation-contraction of the cardiac muscle tissue. These components will be discussed in detail, laying the foundations for the mathematical models presented in Part II.

Contents
Contents
1 Introduction
1.1 Overview
1.2 Contributions
1.3 Public dissemination of this work
I Review of Cardiac biology
2 Elements of the cardiac system
2.1 Chambers
2.2 Heart valves
2.3 Coronary vessels
2.4 Summary
3 Synchronization in the heart
3.1 Genesis of the myocyte excitation
3.2 Voltage sensitive mechanisms
3.3 Gap junctions
3.4 Stretch sensitive mechanisms
3.5 Summary
4 Molecular biology of myocyte membrane
4.1 Membrane composition
4.2 Membrane-Protein interactions
4.3 Interacting membrane proteins and cooperativity
4.4 Energy landscape of Membrane proteins
4.5 Mutations and the energy landscape of Membrane proteins
4.6 Mutations and cardiac pathologies
4.7 Drug interactions and cardiac pathologies
4.8 Summary
5 Multicellular organisation
5.1 Tissue structure
5.2 Ventricular cell types
5.3 Emergent electrophysiological dynamics
5.4 Summary
II Mathematical Modelling
6 Review of cardiac models
6.1 Mathematical models of the myocyte
6.2 Simpliﬁed models of cardiac myocytes
6.3 Biophysically detailed models
6.4 Modelling cardiac geometry
6.5 Cardiac Mechanics
6.6 Summary
7 Reduction of computational cost
7.1 Problem scope and Characteristics
7.2 Model order reduction
7.3 Outline
7.4 Results
7.5 Discussion
7.6 Conclusions
8 Critique on synthetic modelling
8.1 Synthetic modelling
8.2 Analytic modelling
9 A mean-ﬁeld model of ventricular muscle tissue
9.1 Introduction
9.2 Overview
9.3 Background
9.4 Theoretical basis
9.5 Parameter estimation and Model ﬁtting
9.6 Physical meaning of geometric parameters
9.7 Material parameter variation
9.8 Modelling force-calcium relationship of cardiac muscle tissue
9.9 The role of ECM cross-linking topology in myocardial elasticity
9.10 Discussion
10 An electrohydrodynamic model of the cardiac myocyte interstitium
10.1 Hypothesis
10.2 Theoretical model
10.3 Simulations
10.4 Role of material parameters in ﬂow stability
10.5 Discussion
11 Myocardium stability and protein expression
11.1 Overview
11.2 Outline
11.3 Results
11.4 Stability constraints on a myocyte in a tissue matrix
11.5 Physical implications of stability constraints
11.6 Discussion
III Analysis
12 Non-linear models for deformation
12.1 Motivation
12.2 Overview
12.3 The structure of the model
12.4 Observations and more general models
12.5 Persistence of the interface
12.6 Linear deformations
12.7 Discussion
13 Abridgement
13.1 The piecewise phase space approximation (PPSA) method
13.2 Analysis on mathematical representations
13.3 Analytic Models
13.4 Nonlinear models of deformation
13.5 Conclusion
A The ten Tusscher epicardial cell model
B Derivation of free energy expression
C Minimal elastic model of the myocardium
D Models, scales and the scale-spectrum
Conclusion
Bibliography