Modeling to link regional myocardial work, metabolism and blood flows

Abstract

Given the mono-functional, highly coordinated processes of cardiac excitation and contraction, the observations that regional myocardial blood flows, rMBF, are broadly heterogeneous has provoked much attention, but a clear explanation has not emerged. In isolated and in vivo heart studies the total coronary flow is found to be proportional to the rate-pressure product (systolic mean blood pressure times heart rate), a measure of external cardiac work. The same relationship might be expected on a local basis: more work requires more flow. The validity of this expectation has never been demonstrated experimentally. In this article we review the concepts linking cellular excitation and contractile work to cellular energetics and ATP demand, substrate utilization, oxygen demand, vasoregulation, and local blood flow. Mathematical models of these processes are now rather well developed. We propose that the construction of an integrated model encompassing the biophysics, biochemistry and physiology of cardiomyocyte contraction, then combined with a detailed three-dimensional structuring of the fiber bundle and sheet arrangements of the heart as a whole will frame an hypothesis that can be quantitatively evaluated to settle the prime issue: Does local work drive local flow in a predictable fashion that explains the heterogeneity? While in one sense one can feel content that work drives flow is irrefutable, there are no cardiac contractile models that demonstrate the required heterogeneity in local strain-stress-work; quite the contrary, cardiac contraction models have tended toward trying to show that work should be uniform. The object of this review is to argue that uniformity of work does not occur, and is impossible in any case, and that further experimentation and analysis are necessary to test the hypothesis.

Introduction

From the times of the early observations using microparticle deposition to estimate regional myocardial blood flow distributions ^{104,14,134,86} (Richmond 1970; Bassingthwaighte 1972; Yipintsoi, 1973b; Marcus 1977), people were curious about the reason for the heterogeneity. The first thought was that the heterogeneity was random variation in the methods due to particle heterogeneity, variability at branch points, and temporal variation. A next thought was that it was due to variations in flows in the branching networks of the coronary arteriolar system. Later, when it had been worked out that the heterogeneity due to the methods amounted to only a few percent of the observed of regional flows, it was thought to be temporal variation, fluctuations in the resistance in smaller arteries and arterioles, as had been seen in skeletal muscle. But King et al⁷⁴ [1985] observed the flow distributions in awake baboons with high resolution and found the same degree of heterogeneity seen earlier with the cruder methods, and further, found that the flows in individual regions were actually rather stable relative to the mean flow for the heart ⁷⁵(King 1989).

Deeper analyses revealed that near neighbor regions tended to have similar flows. The near-neighbor spatial correlation was self-similar, that is, the decrement in correlation diminished proportional to distance in logarithmic fashion. This led to a series of studies documenting the fractal nature of the spatial heterogeneity ¹¹⁹(van Beek 1989), and stimulated analyses of the washout characteristics, which for flow-limited solutes like ¹⁵O-water ²⁸(Beard 2000a) and antipyrine ¹³²(Yipintsoi 1970) turned out also to have fractal, power law, time courses. The washout could be “explained” by models of the branching of the coronary network ^119,29(van Beek89; Beard2000b), and have been observed in the modeling of flows through vascular networks reconstructed from the anatomic data on the vessels ^29,70(Beard 2000b; Kassab93). This raises the question of why the networks had this particular kind of heterogeneity, and whether or not the network was simply adapted to the need to deliver nutrients locally.

It was long known that oxygen consumption and coronary blood flow were increased in proportion to increases in the work of the heart. Such observations don’t distinguish a homogeneous myocardium from a heterogeneous one, but are most easily explained if all regions, homogeneous or heterogeneous, changed flow proportionately.

“Experiments of nature”, most particularly myocardial infarction and left bundle branch block, led to cardiac remodeling, for example, ventricular wall thickening and increasing blood flow in regions called upon to increase their work load in response to the disorder. In LBBB the septal region is activated early compared to the rest of the left ventricle (LV) ¹⁰¹(Prinzen 1990). It also changed metabolic profile, diminishing glucose uptake, even to the extent of raising suspicions that the septum was ischemic ⁵(Altehoefer98), though it turned out later that while glucose uptake was down, flow per gram of tissue was near normal. But in the late-activated LV regions that were doing more work, the glucose uptake increased; the LV free wall thickened, hypertrophied. Starling ⁹⁶[Patterson14] had shown that prestretch of the myocardium by extra filling at the end of diastole evoked a stronger contraction on the next beat. Does this imply that the pre-activation lengthening can lead to remodeling? The left ventricular free wall hypertrophy in LBBB and the hypertrophy of the undamaged myocardium around the edges of an aneurysmal infarcted region are certainly in accord with this possible causation.

The suggestion is therefore that regional cardiac force development and work drives regional responses in metabolism, energy production, and initiates the adaptation to load, the process of remodeling and localized hypertrophy. We review here the cause and effect relationships, and extend the perspective to identifying potential experiments that are needed to test the validity of our current assumptions and concepts.

SECTION I. The broad heterogeneity of regional myocardial blood flows, rMBF

The first observations of the heterogeneity of rMBF were made using gamma camera imaging of deposits of macroaggregated albumin injected into the aorta in isolated blood perfused dog hearts ¹⁰⁴(Richmond70). The interpretation was that the heterogeneity was artifactual, the error being caused by the particles being of widely varied sizes. When polystyrene microspheres were first used the same issue of nonuniformity of size was considered to be the prime source of the variation, though it was clear that a big contributor to the variation was in the statistics of the small numbers of microspheres per unit mass of tissue ¹³⁴(Yipintsoi 1973b). One source of bias was that the larger spheres, over 25μm in diameter, tended to flow in the central streamlines of the penetrating arterioles running perpendicularly through the wall, and so overestimated sub-endocardial flows. When the technology advanced to produce spheres of virtually uniform sizes, comparisons of the depositions of spheres of different sizes led to a standard procedures of using 15 μm spheres: ones of 10 μm were less than 100% trapped, ones of 25 to 50μm appeared biased and those of 12 to 16 μm were trapped and appeared unbiased, though this was not proven until the “molecular microsphere” technique evolved⁸³(Little83). The evidence ¹⁸(Bassingthwaighte 87 spheres) was that microsphere estimates of regional flows were reliable. The data on the heart of dogs, pigs, sheep, baboons, rabbits, guinea pigs, and hamsters all showed that the heterogeneity was about 25 to 30% at a spatial resolution of about 1% of the heart size; this was consistent, independent of cardiac size or species. (Comment: fluorescent microspheres are more reliable than radioactive spheres for studies extending over months because the fluorescent dyes leach out of the spheres more slowly than do the radioactive labels ¹⁰² [Prinzen 2000].)

What was particularly interesting was that the pattern of relative flows in individual hearts was stable over time; repeated microsphere injections gave remarkably similar results each time. A particular study on awake baboons ⁷⁴(King et al 1985) with sequences of 6 repeated microsphere injections during periods of rest, exercise or mild heat stress showed that the regional flows maintained the same distribution, relative to the mean flow, through all of these states. As shown in Figure 1, the density function of regional myocardial blood flows, rMBF, in the left ventricle (dotted line) ranged from about 30% of the mean flow to over twice the mean flow, almost a ten-fold range of flows. In comparison of the flow in any specific region to later or earlier measures within a 24 hour period, a regional flow that was 50% of the mean flow or less was never found later to be as high as the mean flow on 5 subsequent observations. Nor did a region with local flow of 150% of the mean or greater ever decrease to the mean flow on the 5 subsequent observations. This remarkable stability held even through three to four-fold changes in the mean myocardial blood flow ⁷⁵(King and Bassingthwaighte 1989).

Figure 1.

The probability density function of regional myocardial blood flows relative to the whole heart mean flow from the deposition densities of 15 μm microspheres in the hearts of 13 awake baboons. Data were 4 to 6 estimates in each of 2,706 tissue pieces totalling 13,114 estimates of blood flow per gram of tissue. The solid line is the distribution for the whole heart; the dotted lines are for left ventricle (LV), right ventricle (RV) and atria (Atr). The shaded patch outlines one standard deviation of the f_i in each of the 25 classes of relative flows for the whole heart, where each class width is 10% of the mean flow, and the number of distributions was 67. The area under each curve represents the fraction of heart weight: LV 70%, RV 20% and atria 10%. The mean regional flow relative to the heart mean were LV 1.14, RV 0.81, and atria 0.41. From an average of 208 tissue pieces per heart, the relative dispersion, RD, for the flow heterogeneity, defined as the standard deviation divided by the mean, was: whole heart 0.38, for LV 0.30, RV 0.32, and atria 0.17. (From King et al.⁷⁴)

To determine microsphere measurement error, we developed a “molecular microsphere”¹⁸, a molecular marker that was nearly 100% taken up by the heart tissue during single passage: small molecules cannot be biased by fluidic forces at branch points as can hard spheres of nearly arteriolar dimensions, and therefore their deposition regionally should be purely proportional to the flow. The “molecular microsphere” iodo-desmethylimipramine (IDMI), which binds firmly to serotonin receptor sites, and standard 15 μm spheres, were injected simultaneously into the left atrial cavity of dogs so the spheres were well mixed in the blood before entering the coronaries. Using two different IDMI labels and 4 different microsphere labels, the standard deviation of estimated flow for a 200 mg region was about 2% for IDMI and 7% for 15 μm spheres, much less than the variation within each left ventricle, 25 to 30%, at the same piece size resolution. This gave a clear evaluation of the situation; variation due to measurement error is far too low to explain the heterogeneity in rMBF. Shortly thereafter we found that we could replace radioactive tracers by using the microspheres labeled with fluorescent markers. The estimates did not differ, and fluorescent spheres have been in common use since ^55,121,102(Glenny 1993; van Oosterhout 1998; Prinzen 2000). Having learned that the heterogeneity of regional myocardial blood flow was large, and stable, it made sense to try to figure out the nature of that heterogeneity, as a step toward finding its cause.

SECTION II. The fractal nature of regional myocardial blood flows, rMBF

A clue was that the variation was not due to random fluctuations over time, for the flow in individual regions was relatively constant when conditions were not changing, and when there were large changes in cardiac work and coronary blood flow, the tendency was for the flows to change roughly proportionately everywhere.

Heterogeneity is described quantitatively by calculating the variance of the distribution; equally good in reporting is the standard deviation or the relative dispersion, RD. The RD is the standard deviation divided by the mean, and known in statistics as the coefficient of variation. The measurement of variance of something over a spatial domain is problematic in that the choice of unit size is arbitrary. The fundamental problem is that the more refined the spatial resolution used, the greater is the apparent variance, as in Fig. 2. This is true for population densities (by country, province, county, township, square kilometer) and for tissue properties ¹⁷(Bassingthwaighte, Liebovitch, and West 1994). For regional flows in the heart the variations are not random: there is correlation in flows among near-neighbor regions as one might expect from the nature of a dichotomously branching system ^133,23(Yipintsoi 1973a; Bassingthwaighte, Yipintsoi, Harvey 1974). The correlation can be found either by the method of fractal dispersional analysis described by Bassingthwaighte, King and Roger ¹⁶[1989] and by van Beek ¹¹⁹[1989] or by looking at the spatial autocorrelation structure ¹²(Bassingthwaighte and Beyer, 1991). The nearest-neighbor correlation coefficient, r₁, is related to the fractal dimension D, r₁ = 2^3–2D −1. The spatial autocorrelation function between neighbors separated by n − 1 pieces of a given size, r₁, r₂, … r_n, falls off as a power law function of the distance. This power law scaling in the autocorrelation and between the log apparent heterogeneity and the log of the spatial resolution is a fractal, demonstrating the self-similarity. The self-similarity, or correlation fall-off, is independent of actual location. The diminution in apparent heterogeneity with larger tissue samples is the same for any specific percent enlargement, independent of the actual unit size.

Figure 2.

Normalization of heterogeneity estimate. The RD, relative dispersion of regional flows, is the standard deviation of the regional flows in regions of size m in grams divided by the mean flow for all the regions together, in this case the whole left ventricle, LV. The LV mass was 50 g. The dashed vertical line at 0.5 g, 1% of LV mass, intercepts the observed fractal relationship at RD = 0.262. The regression equation is RD(m) = 0.232 (m/m₀)^−0.18 using m₀ = 1 g. The fractal dimension D = 1.18 or H = 0.82. (Data are from 11 sheep. See Bassingthwaighte King and Roger ¹⁶ 1989.)

The fractal dimension, D, is 1.20 to 1.28 for regional myocardial flow distributions, rMBF; the near-neighbor correlation coefficient, r₁, is about 0.6. The Hurst coefficient, ⁶⁷(Hurst 1951) H equals 2 – D. It is named after the British engineer and hydrologist who first analyzed the correlation structure in the Nile River flows recorded over a millennium ¹¹⁴(Sutcliffe, 1979). He calculated it as the range of cumulative deviations from the mean divided by the standard deviation; later this was found to give biased estimates, and improved methods are now used ³⁸(Cannon 1997). High values of H, 0.5 < H < 1, indicate smoothness and a high r₁, that is a high degree of contiguity in successive years, the years of famines and then of floods. Low H, 0 < H < 0.5, indicates roughness and negative r₁, with a high tendency to alternate levels below and above the mean. With H = 0.5 the relationship is purely random, with zero correlation. The description of the diminution in correlation with distance, or the number of intervening units or tissue volume elements, is given by the expression for correlation between the nth units ⁸⁵ [Mandelbrot and van Ness]:

$$r_1=\frac{1}{2}\{{\mid n+1\mid }^{2H}-2{\mid n\mid }^{2H}+{\mid n-1\mid }^{2H}\},$$ (1)

where n is the number of units of any chosen uniform size.

This expression fits cardiac data at two different volume element sizes (Fig. 3). The analysis is a self-similarity test: it tests whether or not the same relationship for the correlation falloff holds for two different piece sizes. It does, thus fulfilling the standard fractal phrasing “self-similarity independent of scale”. The autocorrelation function for n > 2 is a power law function:

$$r_{\text{n}}/r_{\text{n}-1}={[n/(n-1)]}^{2\text{H}-2}.$$ (2)

Figure 3.

Correlation between flows in successive tissue units in a series. Sequence number 1 means adjacent units. The correlation, given by the line described by Eq. 1, falls off similarly for 150-mg units (▲) and 300-mg units (○), even though the latter are actually twice the distance between centers. A value of H = 0.73 or D = 1.27 describes the falloff.

The line shown in Figure 3, when plotted on log-log axis, becomes straight for n > 2. That these regional flows are fractal is not an explanation, but is an important statistical description of the heterogeneity and the correlation, and provides a basis for relating structure to function. A recommendation on how the observed heterogeneity should be reported can be derived by taking a utilitarian view about the spatial resolution: from a point of view of refined measures of regional flows, resolution better than 500 tissue pieces of a heart of 200 g is seldom exceeded; this is 0.2% of total mass. When a heart is cut into 500 pieces the RD can always be calculated for larger pieces by combining nearest-neighbor regions and redoing the calculation. The RD must decrease monotonically with increasing tissue unit size, as in Fig. 2; the aggregation process neglects the variations between subunits since each is regarded as if it were internally uniform. For standardizing the reporting of heterogeneity one could choose a particular unit size, e.g. 1 gram and give the value of D or H. To compare mouse and human hearts, a better standardization is to report RD at a particular fractional mass, e.g., at 1% of the LV mass as in Fig. 2. An alternative is to report both the RD and the fractal dimension as in the expression:

$$\text{RD}(\text{m})=\text{RD}({\text{m}}_0)\ast {(\text{m}/{\text{m}}_0)}^{2-\text{D}}$$ (3)

where m is the particular tissue unit size, grams, m₀ is an arbitrary reference size, e.g., 1 gram, and D is the fractal dimension, giving RD(m) = 0.232 m^−0.178; the value 0.232 is m₀ the RD(m = 1 g). Choosing m₀ as 1% of LV mass would give generality in the reporting since RD(m₀) would give the reference value directly. For the data from the 50g heart in Figure 2 with 1% of mass being 0.5 g, the RD(m = 0.5 g) = 0.262.

The spatial correlation is self-similar, as is illustrated in Figure 3. The correlation falloff as a function of distance is the same at two different levels of aggregation. The aggregation lumps nearest neighbors together, a process that is repeated to produce the RD plots like Figure 2. The nearest neighbors in the heart are in 3-space so each piece can pair with any of several nearest neighbors to form an aggregate of twice the size. In Figure 3 the correlation falloff is shown for two sizes. The same fractal dimension, D = 1.27, fits both relationships, demonstrating the self-similarity.

The next question is “What gives rise to the fractal characteristics of the flow distributions?”⁷²[Kendal 2001]. A probable cause is the nature of the vascular branching; van Beek^20,119 (Bassingthwaighte and vanBeek 1988, 1989) found that a dichotomous branching system with a small degree of asymmetry in flows in successive branches sufficed to give the observed flow heterogeneities and fractal dimensions. Data on many vascular trees of a variety of organs have been observed to be self-similar, as illustrated in Figure 4 by constancy of the ratios of the lengths of parent to daughter segments of the left anterior descending coronary artery of pigs by Kassab et al ⁷⁰[Kassab1993]. Log-log relationships are also found for diameters. One can expect there to be correlation in flows amongst vessels having common feeder vessels and identical pressure heads. The high degree of correlation r₁ between nearest neighbor regions seems to affirm this because it is most likely that neighboring regions are supplied by common parent vessels.

Figure 4.

Analysis of casts of the coronary left anterior descending arterial tree of a pig heart: length, L, of average element (i.e., segments in series) of a given generation versus the average diameter, d, of the lumen. The logarithmic slope is 1.053. (The data points are average values from Kassab⁷⁰, Table 1, left anterior descending coronary artery.)

But blood vessels grow in response to local needs. Taking this view, it may be better to think of the tissue requirements as the stimulus for vessel growth. A variety of models can be shown to fulfill growth processes ending up with fractal flow distributions. A three-dimensional space filling model ²⁸ [Beard and Bassingthwaighte, 2000a] was based simply on sending out successive branches using exactly the statistics of Kassab’s data⁷⁰ [Kassab 1993] but positioning them for maximal avoidance of previously positioned branches. The result was a network that provided flow distributions with the same fractal dimension as the animal hearts, the same correlation structure, the same density of blood vessels and the same levels of flow. Furthermore the washout characteristics of the network followed the same power law relationship, log tissue residue versus log time, as was found for ¹⁵O-water washout from isolated dog hearts ¹¹[Bassingthwaighte and Beard 1995] Now given that there was no overt growth stimulus from the empty shell into which the vessels grew, one could believe that the vessel growth sufficed as an explanation. The counterargument is of course that the avoidance of established vessels forced the new vessels to be placed in regions devoid of vessels, exactly the same regions that might be hypoxic and sending signals promoting growth. Thus, even if lacking proof, it is logical to expect the requirements for nutrients to find its basis in regional contractile stress, strain and coordinated contraction.

Pries and Secomb ¹⁰⁰[2009] constructed microvascular networks based on the anatomy of thin tissues with low metabolic rates and, from observing that structural adaptation including response to local needs only partially compensated for this heterogeneity, they derived the view that a residual degree of heterogeneity was due to the network structure which could not be further adapted. This may apply to mesentery and cremaster, or even brain ¹⁰⁸(Secomb 2000), but the cardiac adaptations to prolonged dyssynchrony appear relatively complete ¹²²(van Oosterhout 2002) and would argue against extrapolating their ideas to the heart. In contrast, acute changes such as catecholamine infusion (increasing contractility and work), and hypoxia or adenosine infusion (causing coronary vasodilation) result in rapid redistribution of flows within a minute or two ¹³⁵(Yipintsoi 2012)

SECTION III

Oxygen Delivery and Exchange: From Hemoglobin to Cellular Metabolism

Oxidative energy metabolism in muscle cells, including mitochondrial ATP synthesis from carbohydrate and fatty acid substrates, is intimately coupled to the blood flow and O₂ delivery by the microcirculation ³⁴[Beard 08]. ATP is required to maintain the contractile function of the muscle and many other key cellular processes ⁹[Bassingthwaighte 2001]. Since O₂ is highly extracted from the blood and is rapidly consumed in the cells of many tissues, O₂ tension (P_O2) varies spatially in the microvasculature, from about 100 mmHg near the inflow into capillaries to as low as 20–40 mmHg near the outflow in metabolically active organs such as the heart ³⁰(Beard 2001a). Therefore, the transport of O₂ and other key solutes to tissue is an inherently a spatially distributed process requiring partial differential equations. Mathematical modeling and computer simulations of blood-tissue O₂ and solute exchange are used in analyzing experimental data on cell and tissue/organ function and testing hypotheses during changing physiological conditions such as ischemia (low blood flow), hypoxia (low O₂ supply), and exercise (high energy demand)^33,26,44,128 [Beard03;Beard06;Dash06;Wu07]. Such simulations are also helpful in interpreting experimental data from tracer studies involving tracer-labeled ¹⁵O and ¹⁷O-oxygen and other substrates. Modeling the transport of tracer ¹⁵O oxygen and its metabolic byproduct tracer ¹⁵O water is key to interpreting the results from positron emission tomographic (PET) imaging to estimate local perfusion and metabolism in the heart ^46,82[Deussen96; Li 97]. Similarly, understanding the effects of changing blood flow on local blood and tissue O₂ concentration is important in interpreting the physiological significance of blood-oxygen-level-dependent (BOLD) contrast magnetic resonance imaging (CMRI) ^73,94[Kettunen2002; Ogawa93] and ¹⁷O nuclear magnetic resonance imaging (NMRI) ^97.98(Pekar91; Pekar 1995] in the working brain.

Oxygen Transport to Tissue

The process of O₂ transport and metabolism in the microcirculation involves convection, diffusion and hemoglobin-facilitated transport in the blood, permeation across the capillary and cell membranes, diffusion and myoglobin-facilitated transport in the tissue, and finally consumption by many interacting biochemical reactions inside the mitochondria of the muscle cells ^11,99,61[Bassingthwaighte 84; Popel89; Hellums96]. Many mathematical models have been developed for understanding various aspects of this complex problem and for interpreting experimental data and testing physiological hypotheses behind the experiments¹⁵. The classical Krogh tissue cylinder model ⁷⁹ [Krogh, 1919], a single cylindrical tissue unit supplied by a single capillary, has been the basis of most of the theoretical studies on O₂ transport to the tissue over the past decades ^{99,61,53,58,59}[Popel89; Hellums96; Federspiel 86; Groebe90; Groebe and Thews 90] for steady-state analyses and ^46,82,35[Deussen96; Li97; Beyer2002] for tracer transient analyses. Simplified compartmental models have also been developed and used in dynamic PET imaging ^66,88[Huang 86; Mintun84] and BOLD CMRI ^73,94[Kettunen02; Ogawa93] studies of cerebral O₂ utilization. Other models based on non-Krogh geometry have been designed/proposed to reflect the anatomical and morphological structures of the specific vessels and tissues ^{33,24,29,28,30}[Beard03; Beard01b; Beard00b; Beard and B00b; Beard and B01a]. These models are based on the assumption of constant P_CO2, pH, 2,3-diphosphoglycerate (DPG) concentration and temperature in the microcirculation. However, all these variables affect the dynamic transport and metabolism of O₂ in the microcirculation (see below). A descriptive model of the system must consider integration of the anatomical and morphological structures of the specific vessels and tissues along with the physical, physiological, and biochemical processes that govern the transport and metabolism of O₂ in the microcirculation.

Factors Affecting Oxygen Transport to Tissue

The delivery of O₂ to tissue is characterized by a complex system of physiochemical processes, as schematized in Figure 5. It depends on simultaneous release of CO₂ from the tissue (a waste-product of oxidative energy metabolism) as well as on bicarbonate (HCO₃⁻) buffering, acid-base balancing, and hemoglobin-mediated nonlinear O₂-CO₂ interactions inside the red blood cells (RBCs) ⁵⁴[Geers 00]. For example, a decrease in pH or an increase in P_CO2 in the systemic capillaries decreases the O₂ saturation of Hb (SHbO₂) and increases the O₂ delivery to the tissue (the Bohr effect: shifting the oxygen saturation curve to higher PO₂s). On the other hand, an increase in P_O2 in the pulmonary capillaries decreases SHbCO₂, the CO₂ saturation of Hb (the Haldane effect), and so in the lung increases the rate of CO₂ removal from the blood to alveoli. Consideration of both the Bohr and Haldane effects is important in establishing the Hb-mediated nonlinear O₂-CO₂ interactions. Raising the temperature, as in working muscle, also decreases the O₂ binding to Hb and so further facilitates O₂ delivery from blood to tissue.

Figure 5.

Schematic diagram of the physiochemical processes that influence the simultaneous blood-tissue exchange of O₂ and CO₂. CA is carbonic anhydrase, the enzyme accelerating the CO₂ hydration reaction inside the RBCs and tissue. The HCO₃⁻/Cl⁻ exchange between plasma and RBCs occurs through the band-3 anion channels on the membrane of RBCs, satisfying the Gibbs-Donnan electrochemical equilibrium principle. [Figure from Geers⁵⁴]

In view of these complex phenomena, the interpretation and understanding of the alveoli-blood or blood-tissue gas exchange, whether being assessed from the observations of arterial and venous O₂ concentrations, or by intra-tissue chemical signals such as BOLD Contrast MRI or NMRI, or by external detection of tracer contents such as ¹⁵O-oxygen and ¹⁵O-water through PET imaging, depends in general on all the factors involved in the dissociation of HbO₂ and HbCO₂. Thus, the modeling of O₂ transport and metabolism must account for the coupled transport and exchange of CO₂, HCO₃⁻ and H⁺ as well as nonlinear O₂-CO₂ interactions inside the RBCs due to the competitive binding of O₂ and CO₂ with Hb.

In order to quantify the Hb-mediated nonlinear O₂-CO₂ interactions and the Bohr and Haldane effects, Dash and Bassingthwaighte ^43,44,45[Dash06;Dash04;Dash10], have developed new mathematical models for the O₂ and CO₂ saturations of Hb (S_HbO2 and S_HbCO2) from the equilibrium binding of O₂ and CO₂ to Hb inside the RBCs. They are in the form of invertible Hill-type equations (as used for Figure 6 ) with the apparent Hill constants K_HbO2 and K_HbCO2 dependent on P_O2, P_CO2, pH, [2,3-DPG], and temperature in the blood. The invertibility of these equations enables analytical calculations of P_O2 and P_CO2 from S_HbO2 and S_HbCO2 and vice-versa. (This is not to be confused with the reversible Hill equation put forward by Hofmeyr and Cornish-Bowden ⁶⁴(1997) for cooperative enzymes.) This is especially important in the computational modeling of simultaneous (dynamic) transport and exchange of O₂ and CO₂ in the alveoli-blood and blood-tissue exchange systems, allowing one to account efficiently for the rapidly changing conditions along the length of the capillary (in lung or metabolizing tissue). The HbO₂ and HbCO₂ equilibrium dissociation curves computed from these models are in good agreement with the previously published experimental and theoretical curves in the literature ^{36,126,71,110}[Buerk 86; Winslow83; Kelman 66; Severinghaus 79].

Figure 6.

Bohr and Haldane effects. Left: Bohr: Increasing P_CO2 decreases the affinity of Hb for O₂, shifting the oxy-Hb dissociation curve down and to the right. Right: Haldane: Increasing P_O2 deceases the affinity of Hb for CO₂ shifting the carboxy-Hb dissociation curve down and to the right. The Bohr effect aids release of O ₂ in metabolizing tissues; the Haldane effect aids release of CO₂ in lung capillaries. (Adapted from Dash⁴⁵)

Simultaneous Oxygen and Carbon Dioxide Transport and Exchange

There have not been many studies in the literature on simultaneous transport and exchange of O₂ and CO₂ in the microcirculation that account for spatial gradients of O₂ and CO₂. Hill et al.^62,63[ Hill73a,73b] and Salathe et al. ¹⁰⁶[1981] have studied the kinetics of O₂ and CO₂ exchange through compartmental modeling by accounting for the physiochemical processes discussed above, including the acid-base regulation, but not network geometry. However, Goldman and Popel ⁵⁶ did in addition account for convection and diffusion in complex networks (including concentration gradients along the capillary length), an important feature of O₂ and CO₂ transport and exchange in metabolically active tissues like the myocardium. Huang and Hellums ⁶⁵[Huang 94] developed a computational model for the convective and diffusive transport and exchange of O₂ and CO₂ and acid-base regulation in the blood flowing in microvessels and in oxygenators by accounting for the Bohr and Haldane effects, the nonlinear O₂-CO₂ interactions quantitated by Dash⁴⁵ (2010) shown in Figure 6. The review by Hellums et al.⁶¹ [Hellums96] captures these phenomena well, even though they do not account for production of CO₂ in muscle and its further facilitation of O₂ release from Hb.

Dash and Bassingthwaighte ⁴⁴[2006] included metabolic O₂ consumption and CO₂ production in modeling the simultaneous transport and exchange O₂ and CO₂ for analyzing experimental data related to muscle cellular energetics. Since the pH variations in blood and tissue influence the transport and exchange of O₂ and CO₂ (via the Bohr and Haldane effects), and since most of the CO₂ is transported as HCO₃⁻ via the CO₂ hydration (buffering) reaction, the transport and exchange of H⁺ and HCO₃⁻ were also simulated along with that of O₂ and CO₂. This model, using invertible Hill equations for the O₂ and CO₂ saturations of Hb (S_HbO2 and S_HbCO2) allows efficient calculation of P_O2 and P_CO2 from S_HbO2 and S_HbCO2 along the length of the capillary during dynamic simulation of the model under dynamically changing conditions (ischemia, hypoxia and exercise).

Analysis of Tracer ¹⁵O-oxygen and ¹⁵O-water in PET imaging

The positron emission tomography (PET) imaging technology using radiolabeled ¹⁵O-oxygen and ¹⁵O-water has largely been developed for discovering information related to local perfusion and metabolism in the myocardium and brain. Modeling the transport of tracer ¹⁵O-oxygen and ¹⁵O-water in blood-tissue exchange systems is therefore important in interpreting the residue and outflow dilution curves from dynamic PET imaging. To model these high temporal resolution tracer transient data, one needs a physiologically realistic and computationally efficient model.

Compartmental, stirred tank models assume uniform concentrations along the length of a capillary-tissue exchange region and cannot fit high temporal resolution data. Therefore axially-distributed models ^8,13,15,22,[Bassingthwaighte et al 74; 89; 92] as in Figure 7 were developed to handle solute exchanges in a general way. Li et al.’s ⁸²[Li97] nonlinear model is computationally efficient and has been used in the analysis of ¹⁵O-oxygen and ¹⁵O-water tracer transient data [time activity curves (TAC) obtained through the region of interest (ROI) analysis] in dynamic PET imaging to estimate the regional myocardial blood flow (MBF) and regional metabolic rate of oxygen (MRO₂). While it has long been generally accepted since the work of Starling ⁹⁶(Patterson 1914) and Evans and Matsuoka ⁵⁰(1915) myocardial oxygen consumption is proportional to overall cardiac work, it is not yet demonstrated that this holds at the regional level. Figure 8 shows that MBF and MRO₂ correlate linearly under varying physiological conditions. The model of Li et al. ⁸²(1997) was based on the assumptions of (1) equilibrium binding of oxygen with hemoglobin and myoglobin, and (2) constant levels of P_CO2, pH, 2,3-DPG and temperature. Dash ⁴⁴[Dash06] augmented the linear models to take into account for the dynamic system behavior under varying physiological conditions, including the detailed kinetics of oxygen binding to hemoglobin and myoglobin, and the effects of P_CO2, pH, 2,3-DPG and temperature on the binding ⁴⁵[Dash 2010]. An additional factor comes into play when high specific activity ¹⁵O-oxygen coming straight from the cyclotron is injected or inhaled: the ¹⁵O-oxygen concentration is not at tracer levels but is so high that it competes with the ¹⁶O-oxygen for the binding sites on Hb. The solution to this is to use a dual model, accounting for both ¹⁶O-oxygen and ¹⁵O-oxygen and their competition for the Hb binding sites. A breath of ¹⁵O-oxygen at high pO₂ displaces the ¹⁶O-oxygen from hemoglobin.

Figure 7.

Schematic diagram of a four-region, axially distributed, blood–tissue exchange (BTEX) model for advection, diffusion, permeation, reaction, and binding of oxygen (O₂), carbon dioxide (CO₂), bicarbonate (HCO₃⁻), and hydrogen ions (H⁺) in the heart. Endothelial cells region in the capillary membrane were ignored. Detailed kinetics of the binding of O₂ and CO₂ to hemoglobin (Hb) in blood (nonlinear O₂-CO₂ interactions) and the binding of O₂ to myoglobin (Mb) in the parenchymal cells were considered. For the four species, there are four four-region models in parallel. Consumption of O₂ and production of CO₂ occurred only in the parenchymal cells, ignoring O₂ utilization elsewhere. Detailed cellular metabolism leading to consumption of O₂ and production of CO₂ was ignored. The rates of CO₂ production and O₂ consumption were phenomenologically related to respiratory quotient (RQ) which was set 1.0 for glucose metabolism, 0.8 for fat metabolism, and 0.7 for protein metabolism, averaging about 0.8 in normal physiological conditions. F denotes blood flow, V denotes anatomical volumes, V′ denotes the effective water spaces (including binding spaces), PS denotes permeability surface area product, D denotes the effective axial diffusion (dispersion), and G denotes the rate of O₂ consumption and CO₂ production; rbc: red blood cells, pl: plasma, isf: interstitial fluid, pc: parenchymal cells. (The figure is taken from Dash and Bassingthwaighte⁴⁴ with permission.)

Figure 8.

Correlation between blood flow (rMBF) and oxygen consumption (rMRO2) in regions of interest (size 0.9 ± 0.3 g) of the LV myocardium of a closed-chest dog. (From Li et al.⁸²)

Blood flow through the microvasculature is intimately controlled to match the dynamics of O₂ release from Hb, diffusion into the surrounding tissue, and consumption in the tissue cells. The specific details on how this regulation is influenced by vascular and cellular level dynamics are still the subject of much conjecture. The vascular regulatory response to local stimuli such as pressure and flow has been well documented, but how cellular metabolism affects the control of vascular blood flow is less well understood. In myocardial tissue, studies of coupled O₂ transport and energy metabolism have been made, however a detailed quantitative analysis of the complex dynamics including interactions between the release of O₂ from RBCs, utilization of O₂ in the cell and the response of the vascular system to O₂ demand has not been explored. A major reason is the difficulty in obtaining the detailed information needed to integrate realisitically all the model components.

SECTION IV. Modeling and Analysis of Cardiac Metabolism

The next phase is to develop computational models to describe regulation of blood flow and the transport of O₂ and other key substrates (e.g., glucose, lactate, fatty acids) from the microvasculature into the surrounding tissue of the myocardium as demanded by the underlying cellular and subcellular metabolic mechanisms. Such models will account for the axially-distributed (radially-lumped) advection and diffusion of species in the blood along with diffusion and reaction of the species in the tissue. They must incorporate the mechanisms of cooperative binding of O₂ with hemoglobin (Hb), non-cooperative binding of CO₂ with Hb, and nonlinear O₂-CO₂ counterinfluences as well as bicarbonate buffering, and pH. In cytosol and mitochondria they must account for the linked metabolic pathways for ATP hydrolysis, creatine kinase, adenylate kinase, glycolysis, fatty acid metabolism, tricarboxylic acid (TCA) cycle, and oxidative phosphorylation. Regulation at the vascular level will include local responses to pressure and shear stress in the vessel in addition to ATP and NO release from the RBCs which can trigger an upstream conducted dilatory response. These models will further advance our understanding of the influence of mitochondrial metabolic mechanisms on microvascular blood flow and O₂ transport and will have future applications for the description of flow heterogeneities in cardiac tissue exhibiting physiological and pathophysiological mitochondrial function.

The rates of contraction, ATP utilization, and oxygen consumption in the myocardium all vary about three- to five-fold from resting conditions to strenuous exertion in the healthy mammalian heart. ATP to drive contraction, ion handling and other processes, is generated to match demand over this range of work, primarily by mitochondrial oxidative ATP synthesis ^21,10(Bass and Vinnak2004; Bass 2008). This variation persists in spite of the fact that all the cardiomyocytes are activated each beat as excitation spreads over the whole heart. Cardiac action potential generation is probably the best characterized integrated cellular phenomena ^92,93(Noble1962, 2012); while ATP is required for regaining ionic balance after each beat, this component of ATP usage is smaller and probably more uniform than is the consumption for force development.

A long-term imbalance over months to years between the force of contraction demanded for function and the ability of the cardiomyocytes to generate the contractile force needed can lead to failure. In the short-term cardiac metabolism changes between rest and exercise, using relatively more glucose at higher levels of work. For example in cardiac dyssynchrony, as occurs in left bundle branch block, LBBB, or pacing from the right ventricular outflow tract ¹²²(van Oosterhout 2002), the metabolism changes toward a lower work state in the septum and toward a high work state in the LV free wall. The reason for this is that the early septal activation results in a rapid contraction since there is almost no load to oppose septal fiber shortening; this contraction stretches the not-yet-activated fibers in the lateral free wall opposite the septum before it contracts. About a hundred milliseconds later the now-activated free wall begins a longer, strongly preloaded contraction. Continued dyssynchrony of this sort results in remodeling of the heart. (A relatively new treatment for LBBB is to use dual electrode stimulation, attempting to provide cardiac resynchronization by stimulating the late-activated part of the ventricle earlier than occurs with the slow spread of activation through the myocardium.)

Altehoefer ⁵(1998) reviewed early evidence that in LBBB there was reduction in flow and glucose uptake in the septum. We know also from the work of Eisner ⁴⁸(Eisner 2005), for example, that glucose uptake is increased in the LV free wall and reduced in the early-activated septal wall. Thus glucose metabolism and local cardiac work diminish in the early-activated unloaded region, and both increase in the late-activated region. This situation is an ideal one to study for there is an inbuilt control: two regions with all the same conditions except for the timing of excitation and the load-time sequence. The regions have the same blood composition and the same cycle time. In normal dogs the uptake of fatty acid by the heart is in each region in virtually direct proportionately to flow ³⁷(Caldwell94), and the proportionality factor is so high that the observation cannot be explained by simple membrane permeation but requires that uptake kinetics are relatively more facilitated in high flow regions. The mechanism is not clear, but the suggestion is strong that it is receptor or transporter mediated ⁸⁹[Musters 2006]. Fatty acid uptake is not really slowed by hypoxia or ischemia, but its metabolism is slowed. The uptake of tracer-labeled fatty acid was used as an early method of infarct hot-spot imaging because the tracer fatty was not metabolized but retained preferentially in the hypoxic regions ^51,52[Evans 1963, 1965]

What is not known is the time course of the shifts in metabolism after initiation of LBBB. Kinetically, biochemical systems are designed for fast responses and require no changes in gene expression. Even inserting stored GLUT4 glucose transporters into cell membranes takes only minutes ¹⁰⁹(Sedaghat 2002). One would like to know how long it takes for the glucose uptake shifts to occur, but we know from Prinzen ¹⁰¹(1990) and subsequent studies [e.g.^121,122 van Oosterhout, 1998, 2002] that the patterns of timing and strain are stable in a very few minutes after changing the activation pattern. Such changes should be evident by MR tagging ^42,84(Clarysse,2000; Maier et al 1992) to show changes in strain both acutely and chronically or by ultrasound ⁴⁷(D’Hooge 2000). Dogs paced asynchronously for 6 months at normal rates do not go into heart failure ¹²²(van Oosterhout 2002), but there is massive remodeling, as measured by wall thickness, and it begins early, being measurable at the cellular level in hours and days after inducing either dyssynchrony or a regional infarction. See reviews by Swynghedauw¹¹⁵ (1999) and by Spinale ¹¹²(2007).

Unlike in skeletal muscle, ATP utilization in the heart does not normally exceed the capacity for oxidative synthesis: ATP production rate is tightly synchronized with the ATP utilization rate in vivo. Early investigations into the control of oxidative phosphorylation pointed to ADP as the critical signal stimulating ATP production in a feedback mechanism where increasing rate of ATP utilization leads to build-up the hydrolysis product ADP, which in turn, stimulates mitochondrial uptake and phosphorylation of ADP ^39,6[Chance 1956; Atkinson 1968]. Indeed, the theory of ADP-mediated feedback control in skeletal muscle has stood the test of time with the advent of ³¹phosphate magnetic resonance spectroscopy (³¹P-MRS) to assay phosphate metabolites in muscle in vivo. Using ³¹P-MRS in experiments on human subjects thirty years after his pioneering in vitro studies, Chance et al. ⁴⁰[1985] could confidently conclude that in skeletal muscle “the primary control” is “exerted by ADP.” More recent studies built on and refined these concepts ^69,123[Jeneson 96; Vicini 2000]. Yet, while the advent of ³¹P-MRS technology facilitated clear progress in the field of skeletal muscle energetics by helping to validate and refine existing hypotheses and ruling out competing alternatives, the early in vivo ³¹P-MRS measurements on heart could not be explained based on the same mechanism apparently at work in skeletal muscle. Specifically, experiments in animal models failed to show a measureable change in estimated ADP concentration in the myocardium over a range of work rates ⁷[Balaban 86]. These observations led to the establishment of the “metabolic” hypothesis that, in the heart, ATP hydrolysis products are maintained at constant levels and thus cannot facilitate feedback-mediated control of ATP synthesis in vivo. Since feedback is excluded as a primary controller, the metabolic stability hypothesis implicitly invoked open-loop control (through an unknown mechanism) as the primary mechanism.

More recent studies have excluded the metabolic stability hypothesis and pointed to inorganic phosphate (Pi) as a key feedback signal regulating oxidative phosphorylation in the heart in vivo ^{127,128,129,130,131,25,32,26,34} [Wu 2007a,2007b, 2008, 2009a, 2009b; Beard 2005a,b, 2006, 2008]. Indeed, going beyond demonstrating that Pi does vary in vivo in the myocardium over the range associated with stimulation of oxidative phosphorylation, Beard’s group has attempted to quantify how much of the regulation could be accounted for by feedback, versus other mechanisms such as open-loop stimulation by calcium ion or other signals. We conclude that Pi-mediated feedback not merely contributes, but probably dominates the control in vivo. The conclusion from the computational modeling showed that when calcium-mediated control is left out of simulations of oxidative phosphorylation, the in vivo data are reproduced, and thus any potential additional calcium-mediated effect must be small compared to the feedback mechanism. This finding is demonstrated by what is perhaps the simplest extant model of oxidative phosphorylation that can predict the experimental data ²⁷[ Beard 2011], illustrated in Figure 9. Additional independent tests of the alternative hypothesis that change calcium levels can stimulate oxidative phosphorylation reveal that over the expected in vivo range of calcium concentration demonstrate that calcium modulation of mitochondrial dehydrogenase activity cannot provide adequate stimulation of mitochondrial oxidative phosphorylation to serve as an open-loop control mechanism of oxidative phosphorylation in the heart ¹²⁴[Vinnakota et al 2011].

Figure 9.

Feedback control of oxidative phosphorylation in the heart. Diagram in panel A illustrates feedback-mediated control of oxidative phosphorylation in the heart. Panels B and C illustrate the simple two parameter model P_i- and ADP-driven control developed in Beard²⁷. The two adjustable parameters in the model are the apparent Michaelis-Menten constants for the ADP and the Pi stimulation of ATP synthesis; they were measured in vitro and not adjusted but, as shown, fit the data well within the experimental variance. Data in B and C are adapted from Atkinson⁶, Chance⁴⁰;Jeneson⁶⁹; Vicini¹²³; Balaban⁷ ; Wu¹²⁸. Figure reprinted from Beard²⁷ with permission.

While a solid theoretic understanding (and associated computational models) of mitochondrial energy metabolism, and particularly of oxidative phosphorylation, in the heart has emerged in recent years, the overall picture of substrate utilization in the heart is not completely understood. The heart is capable of utilizing both carbohydrates and fatty acids to maintain phosphorylation potential necessary to sustain function under various physiological conditions. Randle ¹⁰³[1998] proposed a conceptual model for the interactions between glucose and fatty acid utilization wherein each substrate inhibits the utilization of the other. One of the component mechanisms is the inhibition of phosphofructokinase and the consequent diminished fraction of glycolytic flux that enters the citrate acid cycle. The more modern refinement is yet more complex, but still not completely clear, as seen for example in the nice isotopomer studies by Metallo et al ⁸⁷(2012) relating the shifts in use of glucose, fatty acid and glutamine that occur with hypoxia. Glutamine itself makes a contribution to maintaining energy balance even while supplying AcylCoA that is used for fatty acid formation and incorporation into membrane lipid.

Garfinkel and coworkers ^{1,2,3,76,77,78}[Achs 68, 77, 82; Kohn 79, 83a,83b] pioneered the development of computational models of glucose and fatty acid utilization in the heart, based on catalytic mechanisms of the enzymes in those pathways. The primary aim of their studies was to explain the observed experimental behavior of metabolite concentrations on the basis of enzyme kinetic mechanisms that were used to describe the reaction fluxes, which constituted their working hypotheses. For example, their simulation of palmitate metabolism in perfused hearts led to the conclusion that citrate did not inhibit glycolysis, simply because citrate is mainly in the mitochondria not the cytosol where glycolysis occurs ^77,78[Kohn 83a,b]. At the same time, their model was interpreted to be consistent with the creatine kinase shuttle concept for intracellular fluxes of high energy phosphate from sources to sinks inside the cell, whereas an alternative hypothesis treating the creatine kinase reaction at equilibrium could explain data on phosphoenergetics in the intact working heart ^127,131[Wu 2008, 2009a].

Kroll and colleagues⁸⁰ have measured and modeled myocardial phosphoenergetics in perfused rabbit hearts with normal perfusion and during underperfusion showing how well the phosphorylation potential is maintained with the system buffering by PCr. Later work from this lab showed that repeated underperfusion, like preconditioning, led to downregulation of AMP hydrolysis. Recent modeling of whole heart metabolism by Zhou et al. ^136,137,138[2005, 2006, 2007] have utilized phenomenological approaches for describing substrate utilization and phosphoenergetics to analyze limited data from perfused hearts. Zhou et al’s model analyses required the hypothetical parallel activation wherein multiple enzymatic fluxes including those of mitochondrial electron transport chain are modulated simultaneously in response to demand in an open loop manner ¹³⁷[Zhou 2006]. However, the existence of such control in mitochondrial electron transport and phosphorylation is contradicted by the arguments of Beard and coworkers ²⁷[2011]. This illustrates the difficulty in mechanistic interpretation when data of several types such as intermediate metabolite concentrations from chemical measurements, blood flow and oxygen consumption and phosphate metabolite data from NMR spectroscopy from comparable experiments are either not available or are not analyzed with appropriate theoretical models.

A reasonable strategy for tackling the problem of substrate utilization might necessitate the use of constraint-based methods, where one could identify sites of enzymatic regulation between different modes of operation ³²[Beard and Qian, 2005]. Constraint-based analysis uses physicochemical principles combined with the specific experimental data to define limits to the particular arrangement of fluxes to provide an objective function (e.g., maximization of ATP generation flux). Detailed kinetic models with appropriately designed experiments are useful in investigating the kinetic and molecular bases of the predicted regulatory sites. These will necessarily be complex, as they must account for multiple cell types, e.g. the different influences of endothelial and muscle cells in purine transport¹⁰⁷.

SECTION V. Regional cardiac work and its measurement

The work of the heart is usually measured globally as “external work”, for example as cardiac output times systolic pressure, or the heart rate times the integral of the flow-pressure product over the time when the aortic valve is open, or less formally as heart rate times the maximum rate of rise of left ventricular pressure. Because myocardial cells and fibroblasts form a syncytium via intercellular gap-junctional connections ¹⁰⁵[Sachse 08], the cells are activated almost in synchrony as the excitation spreads throughout the heart; this has given rise to the false notion that all cells work equally hard. While it is reasonable to believe that the rates of ATP use for cell maintenance and ion pumping (to maintain the cytosolic ionic milieu) might be similar throughout the heart, the data on regional myocardial blood flows prove that cardiac oxygen use and ATP turnover cannot be uniform. The evidence is clear: the low flow regions of the heart do not receive enough oxygen to sustain an average oxygen consumption for the heart, therefore the ATP turnover in these regions has to be less than average for the heart ⁷⁴[King89]. The longer action potential duration in endocardial regions has long been recognized ⁹⁵[Pandit2001]; put together with the observation that the subendocardial regions are especially vulnerable to infarction, one might have expected the subendocardium to have lower blood flows, but this is not borne out by the data. Endocardial flows in the normal heart are the same as the average for the heart. The low flow regions are scattered throughout the heart, and in a given heart remain consistently lower than average despite changes in the mean flows ^74,75[King85, 89].

Regional strain, stress and work have been estimated mathematically through the use of detailed finite element models of cardiac function ^90,118,117[Nickerson05; Usyk03; Trayanova11]. However the validity of local model-based estimates of work has not been tested against local measurements of tension and stress ⁹¹[Niederer09]. More detailed measurements can be obtained on local strains from tagged MRI and ultrasound images ^84,42,47[Maier 92 Clarysse 00; D’Hooge00]; the question is how to infer from these the local stress and work. The difficulty is that calculating the local stress from local strain requires assumptions about dynamic tissue properties in the local and the neighboring regions in the complex 3D geometry, a process fraught with error, but probably achievable. Parameterizing models having regional contractile performance, so that local strain and stress are outputs of the computation, to match the detailed data on strain patterns is an alternative approach. This is computationally expensive for it requires the detailed anatomy of fiber and sheet arrangements as well as accurate MRI- or X-ray-CT strain data. Since there will be many outputs from the model, local strain measurements, the overall cardiac shapes and the ventricular volumes, pressures, and flows, the estimates of stress will be relatively constrained and should be meaningful. The end result should allow local work to be estimated for comparison with blood flow measurements (microspheres or MRI contrast agents) and perhaps Positron Emission Tomographic estimates of oxygen consumption. These kinds of observations would provide large amounts of data on statistical associations among local flows, strain and work, and oxygen uptake.

To take the question a level deeper, one needs to incorporate into the 3-D models the particular hypothesized relationships between local work, ATP consumption and oxygen consumption, preferably at first in steady state situations. To start with, simple mechanistically based computational models of stress and strain development in sarcomeres in series have been used to represent asynchronous activation ¹⁰[Bassingthwaighte 08]. See Figure 10. This is analogous to left bundle branch block, LBBB, and the temporal asynchrony in cardiac excitation. The two elements represent the early- and late-activated portions of the LV, arranged in tandem so that each pulls upon the other. The early septal contraction stretches the free wall segments prior to their activation, so in accord with the Frank-Starling law the free wall generates extra force, and maintains cardiac output. The greater initial length of the prestretched segment leads to unequal regional force generation. The model mimics the relationships among shortening strain, strain velocity, stress, and ATP consumption. The link between the contractile events and regional flow would presumably be mediated through local metabolic demands, consumption of substrates and oxygen used to form ATP.

Figure 10.

Energy use in two muscle segments in series. Black: Early activated contraction. Dashed blue: Late activated and prestretched segment, with the result that contraction is stronger and more energy used. (From Carlson in Bassingthwaighte¹⁰)

The hydrolysis of ATP at the cross-bridge drives the utilization of substrates and oxygen that supply the ATP. The model result in Figure 10 showed that the ATP use is over 10% higher in the late activated region. We have chosen, for simplicity, to use the stoichiometric kinetic model of van Beek ¹²⁰[2007] that is based on the idea that the ADP levels in the cytosol and in the mitochondrial intermembrane space are the primary drivers for mitochondrial ATP generation. Beard’s ²⁵[2005a] is the best detailed model for mitochondrial oxidative phosphorylation. Though our premise lacks Beard’s mechanistic biochemical detail, the two-muscles-in-series model qualitatively defines the rate of conversion of ATP to ADP and the resultant ADP levels to drive O₂ demand. At high workloads there is release of adenosine into the interstitial space, activating A₂ receptors and inducing smooth muscle vasodilatation. The inference is that in regions where there is high demand there is, at least transiently, sufficient ATP breakdown in excess of the mitochondrial capacity for the phosphorylation of ADP to provide a vasodilatory signal. The vasodilatory response to interstitial adenosine is ultrasensitive ¹¹³[Stepp96], that is, the increase in coronary blood flow per unit adenosine concentration is switch-like, having a Hill coefficient of nearly 7.

This type of model can be extended to represent temporal asynchronous activation of sarcomeres or fiber bundles in series and in parallel to represent a 2D slice of the ventricular wall. Models of higher complexity must take the 3D muscle fiber orientation into account. Models at this intermediate level of complexity could be parameterized to data in a computationally tractable manner to provide a step toward validating models of cardiac stress, strain and work to experimental measurements.

With cardiac systole the increased tissue pressure, effectively the extravascular pressure, acts to compress coronary vessels, slowing or even reversing arterial inflow and markedly enhancing venous outflow. Since flow usually reverses in the early phase of systole in the LAD, for example, this implies a major increase in resistance and a decrease in the vessel sizes within the transmural segments, presumably due to compression by flattening them. While Vis and Westerhof ¹²⁵[Vis97] showed that compression of the venae comitantes, the pair of venules accompanying each arteriole, can provide some protection against the arteriolar compression, the flow reversal is nevertheless present. The increased pressure is greatest in the LV endocardium and is a reason for the higher incidence of subendocardial infarctions. Algranati et al. ⁴[Algranati 2010] have described this myocardium-vessel interaction effect as shown in Figure 11. Gorman and Feigl ⁵⁷[Gorman2012] feel that arteriolar α-adreno receptors upstream are activated during increased workload in an effort to minimize the reduction in arterial flow during systole and to reestablish flow immediately after each systole. These vascular regulatory effects coupled with the well established response of the vasculature to intraluminal pressure, flow-induced shear on the vessel wall and metabolic signals conducted upstream from regions of low oxyhemoglobin saturation in the venules ^57,49(Gorman2012; Ellsworth2009] will provide insights into how regional blood flow heterogeneity can be related to regional stress, strain and work.

Figure 11.

Myocardium-vessel interaction analysis of diameter, D vs. transluminal pressure, ΔP. A: Myocardium and vessel are stress-free with opening angle OA. B: The closed, unloaded (not stress-free) configuration. C: The system is loaded by external (myocardium induced, P_myocard) and internal (blood pressure induced, P_lumen) pressures. D: The predicted D(ΔP) relationship (solid line) compared to data (rectangles). (Adapted from Algranati⁴ with permission).

When LBBB is initiated the strain, stress and work, are changed locally, not globally. The remodeling of the vascular system and the cardiac muscle are probably activated simultaneously. Mechanical stress is decreased in the septum, increased in late activated regions, and remains normal in regions in between. Estimating stress levels at millimeter resolution of fiber bundle and sheet structure in heart will be needed to interpret the similarly high-resolution data on regional flows and work. Dissection methods may be required for fiber direction if Diffusion Tensor Imaging (DTI) in vivo cannot provide sufficient spatial resolution. Likewise, the spatial resolution in current techniques for measuring regional flows or strain measurements in 3D heart using MRI is at the edge of practicability.

The mechanics of cardiac contraction begins with actin/myosin interactions at the cross-bridge, when contractile force is developed in the presence of Ca²⁺-binding to troponin. The kinetics of the events leading to myofilament shortening were demonstrated for skeletal muscle and provided with a prescient hypothetical explanation by Huxley ⁶⁸(1957). The recent explanations take into account the several different myofilament proteins and their interactions ¹¹⁶(Tanner et al 2008) and extend the details to account for the Starling effect, that lengthening the fibers prior to contraction produces a stronger contraction ¹¹¹[Smith et al 2009], whose model accounts for the amount of ATP hydrolysis, as was attempted in the earlier models of Landesberg and Sideman ⁸¹[1994].

SECTION VI. Tracking the relationships from organ to genome

What more is needed to demonstrate that regional stress and work actually provide the drive for regional metabolism and remodeling of the musculature and of the vascular system serving the muscle cells? What are the mechanisms by which this might occur? How should we approach determining the cause-and-effect relationships? These nagging questions remain. From the point of view of capturing a systematic quantitative understanding of the mechanisms of cell maintenance and of the remodeling that follows modification of behavior or the disorder of disease, one tends to be “cell-centric”. This is the “middle out” approach to understanding the biology that we recommend as a central strategy for the Physiome Projects ¹⁹[Bassing, NobleHunter09]. “Middle out” implies that we should start at the cell level where we have the most quantitative data and many integrative model descriptions of those data; from the cell we can work “upward” to tissue, organ and organism, and “downward” to subcellular systems, the proteome, transcriptome and genome. From the point of view of normal physiological remodeling it is the modulation of the transcriptome that is central: this means undertaking a cell-to-genome approach.

A top down, cell-to-genome, approach is quite the opposite of the strategy of many molecular biologists. Their emphasis has been on a bottom up approach, finding associations in mRNA and protein profiles, the transcriptome and the proteome, in order to define the pathways for synthesis, proteolysis, and regulation of fluxes. But proteolysis, or autophagy, cleaning out old proteins and recycling their amino acids, is just as important as manufacturing proteins ⁴¹[Ciechanover, 2012]. Generalized autophagy is stimulated by exercise ⁶⁰[He 2012] and is probably one of the reasons that exercise regimens prolong life. Genes for proteases are therefore as beneficial as genes for structural and enzymic proteins. In general, the robustness of the body’s responses to perturbations is centered at the cell level, and the responses to stresses are brought about via regulation at the cell level.

Changes in transcription rates are initiated indirectly by high-level directives, from the brain or from the environment. The mind says, “start exercising”. Exercise training builds more contractile machinery into cells. The intracellular response to the increased workload is signaled down to the gene: at the nuclear level the rates of transcription of particular proteins are increased. A particular chromosome is uncoiled and a particular region becomes the site of ribosomal activity. The response is not general; it only occurs in muscle cells doing extra work. The signaling and the response are at the cell level. Thus to determine how the cell is signaling to a particular part of the genome is a real puzzle.

It is easier to track the response. In the particular case of myofilament production, one would track the gene products or transcriptome for actin and myosin (and other components of the contractile apparatus). The general principle is to provoke the cell in a highly specific manner so as to get the most narrowly defined response, and then to characterize that response. Proteomic techniques, such as following the time course of new myosin formation along with other proteins changing in coordinated fashion, should help to define the pathway from transcription to installation as part of a sarcomere. Presumably as the muscle building reaches steady state the levels of the signals diminish. On this basis one would look for components of the metabolome, substrates or analytes or proteins that rise when the exercise training starts and diminish when steady state is reached, and do the opposite when training is suddenly stopped. Careful attention to the timing of responses should augment the power of the Boolean inference engines used in proteomic searches for correlations.

CONCLUSION

Cardiac electrophysiological modeling, from cell to organ, has been rather well defined through the detailed biophysical measurements and analyses made over the last 5 decades since Noble’s ^92,93(1962, 2012) pioneering initiation of the field. This particular field is probably the most precisely determined set of physiological processes; it sets a standard for quality and depth in the analysis in terms of model representation. Modeling is developing well for cardiac mechanics, from the point of view of myofilament interactions, where the emphasis now is on the combination of the biophysics and biochemistry providing force generation. The modeling appears accomplishable for energetics and biochemical processes as the data accumulates and the hypotheses are expressed in ever more complete forms. All of these are related to, or governing, the heterogeneities in flows in the heart. Biochemical events will be to some extent identifiable in intact hearts using optical probes. It is in the conceptual integrating and practical modeling that the advances occur in specific experimental or analytical studies ³¹(Beard and Kushmerick 2009) or in the all embracing strategies guiding the Physiome Projects ¹⁹(Bass, Noble Hunter 2009), where major successes are occurring to link cell, organ and systems behavior. On the other hand cell-to-gene regulation is poorly defined, and signaling pathways are still difficult to identify, let along quantitate. The gene-to-cell to phenotype is being explored and beginning to be uncovered through statistical inference methods, but the complexity of the system and the low statistical resolution in defining network structure and topology makes progress slow. Identifying the signaling pathways for regulating transcription and autophagy appear to be the big hurdles. Nevertheless the cardiac Physiome is clearly accomplishable, but it will be a long adventure.