Blood flow and diffusion through mammalian organs.

Abstract

In recent years a quiet revolution has been occurring in the medical sciences. While intuitive and empirical therapeutics will retain its important role in the art that the individual physician applies to the practice of medicine, the capabilities and the incentives to apply scientific approaches increase each day. The appropriate tools are being provided by the physical scientists—primarily engineers—but also by theoreticians. Because of the complexity of biologic systems, it is difficult to describe them in terms of deterministic models in which one attempts to define the behavior of the physical and chemical components by sets of equations. Complexity and accuracy must be sacrificed to simplicity and approximation in order to produce working hypotheses. One approach that physiologists and many others have found satisfactory is to describe components of a system in terms of their responses to known inputs. The response to an impulse or a brief pulse input may be an output function which has a magnitude varying with time after the input. Thus, the response is expressible as a probability density function of magnitudes at a sequence of times; the empirically determined impulse response is a “stochastic” description of the behavior of a system. There are other types of stochastic responses—for example, those which occur after a constant time interval but with variable magnitudes. Until we can fathom the nature of the components of such systems, we must work with various empiric stochastic descriptions of their behavior. As knowledge of a system grows, parts of it become describable in a deterministic fashion, usually in terms of simplified mathematical models, while other parts remain stochastic. One reason that biologic systems are so resistant to description by deterministic models is that the responses to a given input change from moment to moment: the systems are nonstationary. The “nonstationarities,” and, indeed, apparent nonlinearities, are usually due to uncontrolled and unrecognized inputs or to cyclic fluctuations in sub-systems. Such unrecognized inputs may be of a wide variety of physical or chemical forms and may be continuous or quantized variables or even pulse trains of varying frequency. The system with which we are ultimately concerned is the intact human, but this presentation concerns transport by way of the circulatory system. The function of the circulation is to convey material to and from the tissues, so it is necessary to describe intravascular transport. In order to participate in reactions within cells, reactants must cross capillary membranes, diffuse through tissue spaces, and traverse cell walls; therefore, intraorgan transport must be described. Only when these transport mechanisms are defined and the reactions are known can the behavior of the system be understood. The state variables (1) which are pertinent to this understanding are the concentrations of substances in small-volume units of the blood, membranes, and tissues. The driving forces in the passive parts of the system are concentration gradients and convective forces; in the active parts, the energy is supplied by metabolic processes. If we were able to define a deterministic model, then the rates of solute and solvent exchange would be describable by sets of equations whose parameters describe such properties as capillary wall permeability, capillary size and separation, diffusion coefficients in the extracellular space of organs and within cells, and rates of chemical reaction inside and outside of cells. Failing this, we can apply stochastic methods to gain insight into the behavior of a system and to obtain hints of the specific nature of its components.