James B. Bassingthwaighte

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, the 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.

Endothelial cells lining myocardial capillaries not only impede transport of blood solutes to the contractile cells, but also take up and release substrates, competing with myocytes. Solutes permeating this barrier exhibit concentration gradients along the capillary. This paper introduces a generic model, GENTEX, to characterize blood–tissue exchanges. GENTEX is a whole organ model of the vascular network providing intraorgan flow heterogeneity and accounts for substrate transmembrane transport, binding and metabolism in erythrocytes, plasma, endothelial cells, interstitial space and cardiomyocytes. The model is tested here for the analysis of multiple tracer indicator dilution data on purine nucleoside metabolism in the isolated Krebs–Henseleit-perfused non-working hearts. It has been also used for analysing NMR contrast data for regional myocardial flows and for positron emission tomographic studies of cardiac receptor kinetics. The facilitating transporters, binding sites and enzymatic reactions are nonlinear elements and allow competition between substrates and a reaction sequence of up to five substrate–product reactions in a metabolic network. Strategies for application start with experiment designs incorporating inert reference tracers. For the estimation of endothelial and sarcolemmal permeability-surface area products and metabolism of the substrates and products, model solutions were optimized to fit the data from pairs of tracer injections (of either inosine or adenosine, plus the reference tracers) injected under the same circumstances a few minutes later. The results provide a self-consistent description of nucleoside metabolism in a beating well-perfused rabbit heart, and illustrate the power of the model to fit multiple datasets simultaneously.

Modeling is essential to integrating knowledge of human physiology. Comprehensive self-consistent descriptions expressed in quantitative mathematical form define working hypotheses in testable and reproducible form, and though such models are always "wrong" in the sense of being incomplete or partly incorrect, they provide a means of understanding a system and improving that understanding. Physiological systems, and models of them, encompass different levels of complexity. The lowest levels concern gene signaling and the regulation of transcription and translation, then biophysical and biochemical events at the protein level, and extend through the levels of cells, tissues and organs all the way to descriptions of integrated systems behavior. The highest levels of organization represent the dynamically varying interactions of billions of cells. Models of such systems are necessarily simplified to minimize computation and to emphasize the key factors defining system behavior; different model forms are thus often used to represent a system in different ways. Each simplification of lower level complicated function reduces the range of accurate operability at the higher level model, reducing robustness, the ability to respond correctly to dynamic changes in conditions. When conditions change so that the complexity reduction has resulted in the solution departing from the range of validity,detecting the deviation is critical, and requires special methods to enforce adapting the model formulation to alternative reduced-form modules or decomposing the reduced-form aggregates to the more detailed lower level modules to maintain appropriate behavior. The processes of error recognition, and of mapping between different levels of model complexity and shifting the levels of complexity of models in response to changing conditions, are essential for adaptive modeling and computer simulation of large-scale systems in reasonable time.

Objective: The equations for transport of hydrophilic solutes through aqueous pores provide a fundamental basis for examining capillary–tissue exchange and water and solute flux through transmembrane channels, but the theory remains incomplete for ratios, α, of sphere diameters to pore diameters greater than 0.4. Values for permeabilities, P, and reflection coefficients, σ, from Lewellen [ 18 ], working with Lightfoot et al. [ 19 ], at α = 0.5 and 0.95, were combined with earlier values for α < 0.4, and the physically required values at α = 1.0, to provide accurate expressions over the whole range of 0 < α < 1. Methods: The “data” were the long‐accepted theory for α < 0.2 and the computational results from Lewellen and Lightfoot et al. on hard spheres (of 5 different α's) moving by convection and diffusion through a tight cylindrical pore, accounting for molecular exclusion, viscous forces, pressure drop, torque and rotation of spheres off the center line (averaging across all accessible radial positions), and the asymptotic values at α = 1.0. Coefficients for frictional hindrance to diffusion, F(α), and drag, G(α), and functions for σ (α) and P(α), were represented by power law functions and the parameters optimized to give best fits to the combined “data.” Results: The reflection coefficients = {1 − [1 − (1 − φ)2]G′(α)} + 2α²fF′(α), and the relative permeability P/P_max = φF ′(α)[1 + 9α5.5 · (1.0−α5)0.02], where φ is the partition coefficient or volume fraction of the pore available to solute. The new expression for the diffusive hindrance is F′(α) = (1−α²)3/2φ/[1+0.2 · α[p2·(1 − α2)16], and for the drag factor is G′(α) = (1−2α²/3−0.20217α⁵)/(1 − 0.75851α⁵) − 0.0431[1 − (1 − α¹⁰)]. All of these converge monotonically to the correct limits at α = 1. Conclusions: These are the first expressions providing hydrodynamically based estimates of σ (α) and P(α) over 0 < α < 1 They should be accurate to within 1–2%.

Multiscale modeling is essential to integrating knowledge of human physiology starting from genomics, molecular biology, and the environment through the levels of cells, tissues, and organs all the way to integrated systems behavior. The lowest levels concern biophysical and biochemical events. The higher levels of organization in tissues, organs, and organism are complex, representing the dynamically varying behavior of billions of cells interacting together. Models integrating cellular events into tissue and organ behavior are forced to resort to simplifications to minimize computational complexity, thus reducing the model's ability to respond correctly to dynamic changes in external conditions. Adjustments at protein and gene regulatory levels shortchange the simplified higher‐level representations. Our cell primitive is composed of a set of subcellular modules, each defining an intracellular function (action potential, tricarboxylic acid cycle, oxidative phosphorylation, glycolysis, calcium cycling, contraction, etc.), composing what we call the “eternal cell,” which assumes that there is neither proteolysis nor protein synthesis. Within the modules are elements describing each particular component (i.e., enzymatic reactions of assorted types, transporters, ionic channels, binding sites, etc.). Cell subregions are stirred tanks, linked by diffusional or transporter‐mediated exchange. The modeling uses ordinary differential equations rather than stochastic or partial differential equations. This basic model is regarded as a primitive upon which to build models encompassing gene regulation, signaling, and long‐term adaptations in structure and function. During simulation, simpler forms of the model are used, when possible, to reduce computation. However, when this results in error, the more complex and detailed modules and elements need to be employed to improve model realism. The processes of error recognition and of mapping between different levels of model form complexity are challenging but are essential for successful modeling of large‐scale systems in reasonable time. Currently there is to this end no established methodology from computational sciences.

New mathematical model equations for O₂ and CO₂ saturations of hemoglobin (S _{HbO 2} and S _{HbCO 2}) are developed here from the equilibrium binding of O₂ and CO₂ with hemoglobin inside RBCs. They are in the form of an invertible Hill-type equation with the apparent Hill coefficients _{K HbO 2} and K _{HbCO 2} in the expressions for S _{HbO 2} and S _{HbCO 2} dependent on the levels of O₂ and CO₂ partial pressures (P _{O 2} and P _{CO 2}), pH, 2,3-DPG concentration, and temperature in blood. The invertibility of these new equations allows P _{O 2} and P _{CO 2} to be computed efficiently from S _{HbO 2} and S _{HbCO 2} and vice-versa. The oxyhemoglobin (HbO₂) and carbamino-hemoglobin (HbCO₂) dissociation curves computed from these equations are in good agreement with the published experimental and theoretical curves in the literature. The model solutions describe that, at standard physiological conditions, the hemoglobin is about 97.2% saturated by O₂ and the amino group of hemoglobin is about 13.1% saturated by CO₂. The O₂ and CO₂ content in whole blood are also calculated here from the gas solubilities, hematocrits, and the new formulas for S _{HbO 2} and S _{HbCO 2}. Because of the mathematical simplicity and invertibility, these new formulas can be conveniently used in the modeling of simultaneous transport and exchange of O₂ and CO₂ in the alveoli-blood and blood-tissue exchange systems.

Systems for describing myocardial cellular metabolism with appropriate thermodynamic constraints on reactions have to be on the basis of estimates of intracellular and mitochondrial concentrations of metabolites as driving forces for reactions. This requires that tissue composition itself must be modeled, but there is marked inconsistency in the literature and no full data set on hearts of any species. To formulate a self-consistent set of information on the densities, contents, or concentrations of chemical components and volumes of tissue spaces, we drew on information mostly on rats. From the data on densities, volumes, volume fractions, and mass fractions observed mainly on left ventricular myocardium, cytoplasm, and mitochondria and from morphometric data on cellular components and the vasculature, we constructed a matrix based on conservation laws for density, volume, and constituent composition. The four constituents were water, protein, fat, and solutes (or ash). To take into account the variances in the observed data sets, we used a constrained nonlinear least squares optimization to minimize the differences between the final results and the data sets. The results provide a detailed estimate of cardiac tissue composition, previously unavailable, for the translation of whole tissue concentrations or concentrations per gram protein into estimated local concentrations that are relevant to reaction processes. An example is that the concentrations of phosphocreatine and ATP in cytosolic water space are twice as high as their mean tissue concentrations. This conservation optimization method is applicable to any tissue or organ.