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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.
Our first approach is simply to obtain a description of the open-loop response of a segment of the circulatory system—that is, the response that would occur if indicator introduced at the entrance to a system passed the sampling site at the exit once and only once. This is the classic identification problem. To put it another way, when the input and output functions are known, can one obtain from them a correct description of the response to a very short pulse (or impulse) input? Sometimes this is not easy, and a variety of approaches exist.
Stephenson (2) and later Zierler (3) introduced the stochastic description of intravascular transport of indicator by defining h(t) as the impulse response or the probability density function of transit times or, as I prefer to call it [after Sherman (4)], the transport function (Fig. 1); the unit for h(t) is fraction per second. The response to an ideal impulse injection of indicator at the input is an output concentration-time curve C(t) (in milligrams per liter), which is equal to mi · h(t)/F, where mi is the amount (in milligrams) of indicator injected and F is the flow rate (in liters per second). [See González-Fernández (5) for definitions of appropriate injections and sampling.] Since the area under h(t) is 1.0, F may be calculated:
This is the basis of our standard indicator-dilution method for measuring flow through the heart and lungs (6).
The circulating system is a closed-loop or recirculating system. Hamilton and his associates (7) demonstrated that it was reasonable to assume that the tail of C(t) was monoexponential in the absence of recirculation and, therefore, that monoexponential extrapolation of the downslope of C(t) could be used to obtain the curve for the indicator passing the output point for the first time. Because this extrapolation is a reasonable approximation even when the input is a fairly broad pulse, Eq. 1 can be applied in a wide variety of circumstances. It is required that indicator mass be preserved (that “amount out” be equal to “amount in”) and that the flow be steady. When flow is unsteady, erroneous estimates of flow are obtained, the greatest errors occurring when the frequency of variation in flow is near the reciprocal of the period from the appearance time to late on the downslope of C(t) (8).
Although only the integral (Eq. 1) need be known for the calculation of flow, the actual shape of h(t) is required for the calculation of mean transit time and dispersion. Theoretically, h(t) is obtained by introducing a unit impulse at the entrance to the system. Impulse inputs are actually impossible to produce and difficult to approximate because of dispersion at the injection site. As a result, it has been more practical to sample the bloodstream at both the input and the output end of the segment of the circulation under study in order to obtain h(t) indirectly by mathematical techniques. In a linear system with steady flow, the convolution of the input curve Cin(t) with the transport function provides the output curve Cout(t):
where λ is a variable used for the integration and the asterisk denotes the process of convolution. When Cin(t) is zero prior to the injection at time zero, the integration may be begun at 0 rather than at −∞.
Perhaps the simplest method of arriving at h(t) is by iterative convolution. One assumes some simple form h′(t) as an approximate guess for h(t), convolutes h′(t) with Cin(t), compares the derived output C′out(t) with the recorded experimental curve Cout(t) and then modifies h′(t) and repeats the operation until C′out(t) closely approximates Cout(t). This method has been successfully applied to various segments of the circulation by using unimodal density functions similar in shape to the one shown in the top graph of Fig. 1 (9, 10). The lagged normal density curve (10) shown, and other models which have been used to describe dye-dilution curves—the random walk equation (11), the log-normal curve (12), a gamma-variate (13), and another gamma-variate (14)—are all about equally usable, and description of h(t) by any of them requires a minimum of four parameters. These four parameters are algebraic combinations of the zeroth to the third moment—that is, they are related to the gain [ordinarily 1.0 and equal to ], the mean transit time [which is ], the dispersion, and the skewncss of h(t). Moments higher than the third contain little further information.
Given Cin(t) and Cout(t), one can also use nonparametric models or transforms to describe h(t). The finite Fourier series transform has proved to be more useful in our hands (15) than Laplace or Z transforms. The finite Fourier series transforms of Cin(t) and Cout(t) are obtained numerically; then the latter is divided by the former to obtain the frequency domain representation of h(t), from which we call obtain h(t) by using the inverse Fourier transform (15).
Another approach is to make use of the moment-generating function (16). In this method, one simply takes the moments of the input and output functions and calculates from them the moments of the transport function:
where is the mean transit time, µn is 1/A , and A is the area of C(t). To do this one must extrapolate the downslope of C(t) to the base line, excluding recirculated indicator. It is fortunate that the first few moments suffice, because higher moments are notoriously inaccurate, but the method is workable for simple systems such as indicator transport through the arterial system of the human leg (10, 17). A useful parameter calculated from the moments is the relative dispersion, σ/ or , which is the standard deviation divided by the mean transit time (18).
These types of analysis are based on the applicability of superposition, which means that the system must be linear (the responses sum) and stationary (the flow and the distribution of flow are both constant). The usual test for linearity is to obtain the response C1out(t) to an input C1in(t), to obtain a second input-output pair C2in(t) and C2out(t), and then to test to find whether a summed input C1in(t) + C2in(t) produces the output C1out(t) + C2out(t). Because simple conservation of mass is not a very critical test and because flow in the vascular system is almost never steady, not even in the veins, the following test of superposition was devised.
After an injection of the test substance at some position upstream, indicator dilution curves were recorded simultaneously at three sites in the vascular system. The transport functions were described for the segment between the first and second sites, between the second and third sites, and between the first and third sites. The test was then to perform the convolution of the transport functions of the two short segments and to compare the result with the transport function estimated for the whole system between the first and third sites. In this way, all the observational data were acquired simultaneously, it was reasonable to ignore any slow changes in flow that would invalidate the traditional test, and, when the final comparison was close the system could be said to be linear and, for the period of data recording, stationary. Systems so tested were the aorta of the dog (19) and the circulatory system through the lungs and heart of the dog (20). The results indicated that superposition was applicable.
Observations on transport through the human leg artery at widely different flow rates in the same subject and in different subjects showed (10) that, over a range of Reynolds numbers (Re) from about 500 to over 6000 (21), the transport functions retained their shape with great consistency. This indicated that there was no transition from laminar to turbulent flow at around Re = 2300, contrary to statements commonly made in the literature. The consistency of shape is obvious when the transport functions are normalized and plotted together (Fig. 2). Multiplying h(t) by and plotting a0ainst t/ preserves unity area in the transformation. A11 the data shown in Fig. 2 were obtained from one young man with flow rates in the external iliac artery (which supplies essentially the whole leg) that varied from 300 to more than 3000 milliliters per minute.
The concept that the transport function is linearly flow-dependent may be expressed as follows.
If, at constant flow F1,
then, when the flow is changed to F2
The implication is simply that the spatial dispersion in a constant-volume system is constant; it is related to distance traveled and is otherwise independent of flow or time.
Constant relative dispersion at varied flows has also been observed for the circulatory transport functions of organs with multiple parallel pathways, which suggests that there was little redistribution of flow with changes in total organ flow (10, 19, 20). The relative dispersion of the transport function in the lung and in the kidney is about 2½ times that for a branching single artery; a broad distribution in path lengths and stream velocities among the many pathways causes the great dispersion occurring in capillary networks.
This brings up the question of relative regional flows and the influences of regional variation on the quantity of nutrient material delivered to the tissue per unit mass or volume of tissue. Useful information can be obtained by applying the stochastic approach to capillary networks because these cal usually be considered parallel pathway systems. The transport function for the whole organ, h(t), is a summation of the fractional flow times the transport function through each of the individual classes of pathways:
Here I is the total number of classes or groupings of of width Δi which are used to describe the system; i is the index of each class; fi is the fraction of indicator passing through pathways having between i − Δi/2 and i + Δi/2; and hi(t) is the transport function of the ith class of pathway.
Greenleaf and his associates (22) have developed a method for ascertaining the fi of Eq. 6, the relative frequency of utilization of each class of pathway, by assuming that each pathway through the organ has a transport function with relative dispersion and skewness similar to those for an artery. This is reasonable because most of each pathway is composed of vessels larger than 50 microns in diameter. Then the transport function of the ith pathway is related to that of the jth pathway by
This is more general than Eq. 5 because both volume and flow might contribute to a change in , and it results in a great simplification of Eq. 6,in which only one of the hj(t) need be known, for example, h1(t):
In using digital computer techniques for finding the fi which can be considered the continuous density function f(), an I of less than 40 sufficed. An example of a solution obtained with I = 50 is shown in Fig. 3. The fi, are given by the triangles in the bottom graph, each triangle representing the relative areas under the hi, each of which has the form of a lagged normal density curve (10 and Fig. 1) with a relative dispersion of 0.2. The sum of the hi is h(t) (bottom graph), and the convolution of h(t) with the recorded input curve r(t) resulted in a curve C′(t) (triangles, top graph) which closely fits the recorded output curve C(t).
This type of parallel pathway analysis is potentially very useful because f() can be unimodal or multimodal and its variation with changes in total flow or with the administration of drugs may be informative. Of course, such an analysis will not always be sufficient. Further experiments would be needed to ascertain whether the individual pathways are appropriately described, or which anatomic regions are associated with which fi.
The technique of Greenleaf and his associates can also be used to determine h(t) when Cin(t) and Cout(t) have been recorded. When hi(t) with relative dispersion much less than that of h(t) is used, the form of h(t) becomes virtually independent of the form of hi(t). The extreme case occurs with hi(t) equal to a unit impulse δ(t), and h(t) is simply f(). Numerical solutions require the use of functions that are dispersed relative to δ(t); however, the same h(t) is obtained independent of the shapes of the hi(t) over a wide range of shapes. Of course this is not true for f(), whose form becomes independent of the shape of the hi only when their relative dispersion is small.
The transport of diffusible and of nondiffusible substances can be considered identical within large blood vessels; however, during its passage through the capillary, the diffusible indicator can leave the blood and enter interstitial fluid and cells. Thus, the system volume is larger for sodium than for albumin injected at the same time into the artery supplying an organ. The exchanges across the capillary wall are governed mainly by diffusion, and the use of double- or triple-indicator techniques can yield information on capillary wall permeability and some indication as to whether the route of exchange is by way of small aqueous “pores” between the cells of the capillary endothelium or across the cells themselves.
Appropriate terms have been defined and used in studies of chemical reactors (23, 24). From h(t), several useful functions can be derived (Fig. 1). The expression H(t), the integral of h(t), is the probability of a single particle’s remaining in the system for a time t or less and is termed the “cumulative residence time distribution function” or, more simply, the “cumulative residence time distribution.” It is the fractional amount of indicator which would be collected by time t in a bucket retaining all of the outflowing fluid—that is, it is the fraction of the injected indicator which has passed the outflow point by time t. The expression H*(t) is its complement and is the fraction of the injected indicator which resides within the system at time t after an impulse injection. I call this the “residue function” because it is the residual indicator which has not yet emerged or escaped from the system. The bottom graph of Fig. 1 illustrates the emergence function, or fractional escape rate, η(t), which is the rate of emergence from the system of indicator particles still retained in the system at time t:
In life insurance work, η(t) is the risk function, the death rate among those still living at a given age. The function was a natural outgrowth of the conceptual approach provided by Danckwerts (25), who came close to defining it. Shinnar and Naor (23) called it the “intensity function,” and they and others—for example, Aris (24)—have used it fruitfu11y in exploring the behavior of chemical reactors of various types. For biologists the term intensity function is burdened with a host of conflicting connotations; I prefer emergence function or escape rate, which seem to convey the meaning. A reference standard for η(t) is provided by the single mixing chamber: with complete, instantaneous mixing, the emergence of a residual particle of indicator is as likely for a particle that has just entered as it is for one that has been there for a long time. In this case the rate of escape is the flow divided by the volume (the reciprocal of the time constant of washout). Ordinarily it is most convenient to display the emergence function after multiplying by , the mean transit time of h(t); for the mixing chamber the product • η(t) is always 1.0.
Zierler (26) has shown that H*(t), the residue function, is essentially the curve that one obtains by recording from an isotope detector collimated over an organ after introducing a radioisotope into the inflow to the organ. For very highly diffusible indicators, the blood-tissue exchanges and the intertissue diffusional exchanges are so rapid that the curve H*(t) is the same whether a small volume of indicator is injected into the tissue itself with a fine, non-traumatic needle or whether the indicator is injected into the arterial inflow to the organ. Because is exactly the mean transit time (27) the flow per unit volume of tissue can be calculated correctly from the curve recorded from the isotope detector:
Here F is the flow (in milliliters per second), V is the volume of distribution of the substance within the tissue, and C*(t) represents the actual recorded curve (in counts per second); H*(t) is equal to C*(t)/C*(0). This approach is useful because no sampling of the bloodstream is necessary in order to obtain valuable information even though the organ may be inaccessible, and no assumption of any specific model is necessary for the analysis. What is required, however, is that the concentration of indicator in the inflowing arterial blood be zero (that is, that there be no recirculating indicator). Of course by monitoring arterial concentration independently, appropriate corrections can be made for recirculation. The method is most readily applied to flow estimation through the use of indicators such as krypton and xenon, which are eliminated in the air expired from the lungs and whose arterial concentration is very close to zero.
The disadvantage of having to record for an infinitely long time, or until all of the indicator has escaped, can be overcome by a minor mathematical manipulation based on the fact that the emergence function for highly diffusible indicators is very nearly flat at the tails of the curves (27). The modified formula is
Here T is the time at which recording was terminated. For indicators like antipyrine and xenon in the heart, T is a time when H*(T) is between 0.3 and 0.05, concentrations which are reached in 2 to 6 minutes.
When the volume of distribution of the indicator per unit mass of tissue is known, then Eq. 10 may be rewritten
Here λ is a partition coefficient (the ratio of the solubility in the tissue to the solubility in the blood), and ρ and W are the specific gravity and the mass of the organ, respectively. For practical clinical purposes, the absolute value of the flow is not essential because F/W gives a good indication of the adequacy of the flow for delivering the required nutrients to the tissues.
When a brief injection has been made at the inflow, from Cout(t) the concentration in the outflowing venous blood, one can obtain the absolute flow, F, from Eq. l, and the mean transit time. Their product is the effective volume of distribution with which the indicator each changes:
Equation 1 holds for both permeating and nonpermeating indicators, for which the subscripts D and N, respectively, are used. The intravascular volume VN, exclusive of any stagnant pools, is F • N. With a diffusible indicator such as sodium, F • D or VD is the total volume of distribution of sodium and includes blood and all of the tissues into which sodium exchanges (even tissues with which the exchange is very slow). The extravascular volume of distribution of the permeating diffusible indicator is VD — VN and is therefore a measure of the size of the tissue reactor. When diffusion is slow, the tail of hD(t) is low and prolonged, rendering the calculation of D inaccurate and the interpretation of VD risky.
It is common practice to analyze curves of h, H, or H* in terms of multicompartmental systems, stirred mixing chambers, or stirred tank reactors in serial and parallel arrangements. When the time for mixing or diffusion throughout a region is short as compared to the time that a particle resides in a region or to the turnover time in a reaction, then first-order approximations are reasonable (28), and such analyses provide estimates of exchange or reaction rates and of compartment volumes or reactant or product mass.
The extraction of material from the blood passing continuously through an organ is of interest for several reasons. In the steady state, when the material is being consumed by the tissue at rate R, it must be extracted from the blood at the same rate:
The classic Fick method of estimating F is to calculate it from experimental observations of R, Cin, and Cout; the procedure gives a correct result. when all the variables of Eq. 14 are constant but a misleading one when they are not (29).
The steady-state approach is of no value when a nonmetabolized indicator is used, but the transient response to a tracer injection when the system is in a steady state can provide useful information on the rate of exchange between blood and tissue. An example is the calculation of the dynamic extraction E(t), as illustrated in Fig. 4, which is defined as
in which N denotes the nonpermeating reference tracer and D, the permeating diffusible tracer. When intratissue reaction and diffusion are relatively rapid, the extraction is related to capillary flow and permeability and to the surface area of the capillary membrane. The experimental approach is to inject a bolus containing a nonpermeating reference substance and a permeating diffusible indicator into the arterial inflow to the organ. The recorded curves for venous concentration relative to time are normalized to h(t) by multiplying by F/mi, defined as in Eq. 1, and E(t) is then calculated from Eq. 15. Crone (30) and Martín and Yudilevich (31) found E(t) to be fairly constant in the early part of the curves and then to diminish when the back-diffusion from tissue into blood became significant (Fig. 4).
When the capillary bloodstream velocity is high, the time of exposure of the membrane to the flowing bolus of indicator is reduced and extraction diminishes, as one might expect. Figure 5 shows a set of extractions obtained for sodium in the heart of a dog and another set obtained by using a mathematical model of a concurrent-flow capillary-tissue system (32). Data from the model fit the experimental data, and thus the model provides estimates of capillary permeability. The models of Perl and Chinard (33) and of Johnson and Wilson (34) should also be used to explore the behavior of E(t). In the concurrent-flow model the extraction near zero flow is closer to unity than it is to 1.0 minus the ratio of capillary blood volume to total organ volume of distribution; this reflects the cascading of fractional extractions as the bolus progresses through a capillary. Such analysis of E(t) is restricted to situations in which there is relatively low capillary permeability, but the analysis is as applicable to sodium, whose extravascular volume of distribution is about 30 percent of tissue volume, as it is to potassium, for which the extravascular pool is very much larger.
The use of the emergence function η(t) for the interpretation of biologic data is now beginning to be explored. Shinnar and Naor (23) have illustrated its behavior in serial-stirred-tank systems with internal reflux. This is somewhat analogous to the flowing blood in a capillary: the stream is segmented by erythrocytes with internal circulation in each segment (35), fluxes downstream are convective plus diffusive, and fluxes upstream are diffusive only. The situation is much more complex if the indicator diffuses through the capillary membrane.
With a highly diffusible indicator one might intuitively expect behavior similar to that in a first-order mixing chamber, but this does not occur. Figure 6 shows experimental curves for two highly diffusible substances, xenon and antipyrine, and for sodium, whose exchange is slowed by limited permeability of the capillary membrane. Antipyrine is distributed throughout the water of the tissues of the heart, and its washout is flow-limited, not diffusion-limited, even at the highest blood flows (27). The shape of η(t) shows that the distribution of antipyrine is not homogeneous and, therefore, that regional perfusion rates differ.
It may be possible to relate the fractional flows (Eq. 6) to specific types of tissue within an organ. Antipyrine has much the same solubility in various tissues within the organ, but xenon is 10 to 14 times more soluble in fat than in muscle. Comparison of the η(t) or H*(t) for these two indicators offers the possibility of estimating the relative regional flows through fatty tissue and muscle in the heart. Analysis in terms of two mixing chambers has been found to give erroneous results, the estimates of blood flow in fat being too large by at least two orders of magnitude (27). However, analysis in terms of flows through multiple parallel pathways is essential. Introducing parallel pathways tends to reduce the uniqueness of the set of parameters which provide a good description, but this disadvantage can be offset by using a priori information on the numbers and types of pathways.
Sodium washout is affected not only by a low capillary permeability but also by diffusion of sodium into cells, its active transport out of cells, and its stagnancy in the sarcoplasmic reticulum and transverse tubular system. The low permeability results in early emergence of sodium which has not left the bloodstream, as shown by the early peak of η(t) for sodium. It is anticipated that analysis of the tails of such curves at various flows will provide some information on the kinetics of exchange with the stagnant regions.
When convective and diffusive fluxes can be described, one has basic information on which to superimpose descriptions of biochemical reactions in tissue, such as the kinetics of oxygen uptake and utilization. Schmidt (36) has written a detailed mathematical exposition of the problem, but it is so complicated that no real advances were made over the simplified, but not simple, classical approaches of Krogh (37), Hill (38), and Roughton (39) until digital computer techniques were utilized. Capillary tissue models, similar to Krogh’s model, which consists of a right cylinder of tissue with an axial capillary, have been used, concurrent and equal velocity flow in neighboring capillaries being assumed (40). These models illustrate the phenomenon of a minimum in tissue concentration of oxygen at the downstream outer end of the tissue cylinder, the so-called lethal corner. The model is clearly most applicable to tissues with parallel capillaries. From the teleologic point of view, such a bad design would be unlikely; from the observational point of view capillary flows change direction frequently, and anatomic arrangements of capillaries indicate that some inter-capillary countercurrent exchanges must occur.
Some approaches, mainly stochastic, to the analysis of indicator exchanges in blood and the tissues of organs have been reviewed. The first goal of such analyses is to provide a basis for describing the behavior of the reactor—the organ—and its components. A more distant goal is the incorporation of behavioral descriptions, or of deterministic models when obtainable, into description of the closed-loop, well-controlled systems which operate in the body to maintain a relatively stable state. It is clear that the systems are complex and that our successes are limited. Eventually, mathematical approaches combining stochastic and continuous systems analysis or other types of analysis not generally available to us now will be used in the solution of such problems.