The case for stochastic models of digesta flow

J. theor. Biol. (1987) 124, 371-376

LETTER TO THE EDITOR

The Case for Stochastic Models of Digesta Flow France et al. (1985) present a mathematical framework for unifying the compartmental models which have been proposed to describe the kinetics of digesta flow. One of the authors' important contributions consists of incorporating a theory of distributed time lags into classical compartmental modeling. They then proceed to show how this elegant generalized framework may be used to generate most of the digesta flow models in current widespread useage. Their article also discusses the Ellis et al. (1979) model and gives an alternative derivation for its regression equation which is fitted to data. However the importance of the Ellis model is not so much the fact that it produces a different regression equation, but rather the fact that it is the first model in common useage which is based on stochastic assumptions of digesta flow. The France et al. derivation is simple, as claimed, however it fails to incorporate, indeed it does not even mention, this very essence of the Ellis model, namely its inherent stochasticity. One purpose of this note is to point out that France et al. have solved a different, deterministic model, rather than the broad, stochastic model which was envisioned in Ellis et al. (1979). Another purpose is to present a general case for stochastic models of digesta flow by discussing how stochasticity may strengthen both the theoretical foundations as well as the practical utility of compartment models. Most recent books on compartmental analysis, including Anderson (1983), Godfrey (1983), and Jacquez (1985), give general reviews of stochastic compartmental models. The general consensus seems to be that "every real system must be considered to be subject to uncertainties of one type or another, all of which are ignored in the formulation of a deterministic model" (Gold, 1977, p. 96). Stochastic compartment models seem particularly appropriate for describing the flow of digesta in ruminants for two reasons. Firstly, it is natural to conceptualize uncertainty, hence stochasticity, in the location of an individual particle over time due to muscle action within the rumen, whereas the premise of the deterministic model which implies the exact predictability of particle location seems untenable. Secondly, whereas most applications of stochastic compartmental analysis in tracer kinetics describe uncertainties on a molecular level (see e.g. Matis & Wehrly, 1985a), the digesta flow modeling is usually at a different level where the conceptual "particles" are orders of magnitude larger in size and smaller in number. As a rule, the smaller the population size, the greater the interest in the inherent (stochastic) variability resulting from the independent actions of individual particles. The strength of the Ellis model is its foundation on assumptions which recognize, explicitly, stochasticity of importance in theoretical biology, and yet which lead to a tractable regression model for subsequent analysis. 371 0022-5193/87/030371 + 06 $03,00/0

© 1987 Academic Press Inc. (London) Ltd

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Some confusion concerning the nature of the Ellis model may have arisen since the Ellis et aL (1979) article was intended only to present some applications of the model, without either deriving it nor discussing its stochastic assumptions. The model was originally derived in Matis (1972), as noted in France et aL, however the model has also been discussed and derived in many subsequent papers. For present purposes, the following approach given in Matis & Wehrly (1985a) for general compartmental models is enlightening and will clarify the issues. Let R~, i = 1. . . . , m, denote the retention time random variable of a random particle in compartment i of an m-compartment irreversible system prior to its transfer to compartment i + 1. The system exterior is denoted as compartment m + 1. Let us assume that each R~ is an exponentially distributed random variable with parameter k~, where all k~ are distinct. The density function for R~ is fR,(t) = k, exp {-k,t}

(1)

for t-> 0 and i = 1 , . . . , m; from whence the instantaneous transfer probability may be obtained as Prob {particle transfers to i + 1 by t + A t given that it was in i at t} = 1 - e x p { - k i A t } = k~At+o(At)

(2)

for a small increment At. The so-called lack o f memory property o f the exponential distribution is apparent in eqn (2). Now let p~(t) denote the probability that the particle is in i at (elapsed) time t given that it started in compartment 1 at time 0, a n d / ~ ( t ) its derivative. The Kolmogorov forward equations then may be obtained (see e.g. Chiang, 1980) as

p~(t) = - k l p l ( t ) p,(t)=k,_lp,_~(t)-k~p~(t),

(3) for i = 2 , . . . , m .

The Kolmogorov equations have the same form as eqns (1) in France et al., and they are easy to solve recursively. For example, for m = 2 the solutions are

pl(t)=exp{-klt}, p2(t) = kl[exp { - k , t } - e x p {-k2t}]/(k, - k2).

(4)

These sums o f exponentials equations are, of course, familiar, however the solutions now represent the underlying probabilities of particle location and hence have a stochastic interpretation. Thus the stochastic model based on the exponential distribution preserves all of the essential biological structure of its analogous deterministic model, and yet also enriches it with the incorporation of some of the uncertainty to which every "real system" is subject. Sample realizations which illustrate the inherent process (stochastic) error may be obtained using Monte Carlo simulation (see e.g. Matis & Hartley, 1971; Kodell & Matis, 1976). The stochastic assumptions of the Ellis model play a much larger role than just broadening its theoretical foundations, they also have fundamental implications in model application and subsequent interpretation. This is illustrated by the completely

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mechanistic interpretation of the France et al. deterministic model. Their derivation assumes an m = 3 compartment system where the first two compartments have identical rates, i.e. where k~ = k2. However the Ellis model does not assume that there are m = 3 compartments, instead it assumes a two compartment model where the retention time random variable in the first compartment has a gamma distribution with density function fR,(t) = X"t"-' exp { - X t } / ( n - 1)!

(5)

for t --- 0, and n = 1, 2 , . . . , . It should be noted that the parameter A is time-invariant. A different model with time-varying rate functions has also been studied extensively (see e.g. Epperson & Matis, 1979, and its references). The Ellis model is solved by using the well known mathematical property that a gamma distributed random variable in eqn (5) may be generated as the sum of n independent exponential variables (see e.g. Johnson & Kotz, 1970). For the present application, one could write (6)

R l = R11 + R 1 2 + . - . + R l n

where the R u ' s are a random sample from an exponential distribution with parameter hi. Through this representation, one may solve the Ellis model by using eqns (3) without the constraint of distinct k;'s. The general solution from Matis (1972) for the assumed gamma lifetimes in eqn (5) is pl(t) = ~ (A~t) '-~ exp { - A ~ t } / ( i - 1)!

(7)

i=1

p2(t) = a " exp { - k 2 t } - e x p {-A~t} ~ (Alt)"-icti/(n -- i)!

(8)

i=l

where c~ = A1/(Ai - k2). Models for many of the various measurements, or endpoints, in passage modeling follow immediately from these solutions. For example, let Co denote the initial pulse dose of marker into compartment 1, whereupon one has the following: (1) the probability, p3(t), that a particle has left the system is p3(t) =

1-pl(t)-p2(t),

(9)

(2) the expected marker in compartment i, denoted X i ( t ) where i = 3 represents the exterior, is X i ( t ) = Cop,(t)

for i = 1, 2, 3,

(10)

(3) the (faecal) recovery rate of marker, Z ( t ) , is Z ( t) = X3(t) = k2X2( t).

(11)

The Ellis et al. and the France et al. papers discuss only the model with n = 2 and an added time lag, r, which from eqns (10) and (8) yield the equation X2(t)=Co[a2exp{-k2(t-r)}-exp{-Al(t-r)}[ct2+aAl(t-r)]],

=0

t<'c.

t>--r

(12)

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The important point which we have always stressed when using this method of solution (see also, e.g., Matis & Wehrly, 1985b) is that the representation of a gamma variable as a sum of n exponentials is merely a convenient mathematical artifice used to solve the desired problem. Our use of eqn (6) does n o t imply either the mechanistic description involving n subcompartments (the conceptualization in France et al. Fig. 5) nor the alternative completely deterministic model based on distributed lags (the conceptualization in France et al. Fig. 6). The general random retention time formulation, by preserving the essential biological structure of two compartments without the inflexibility of any particular set of additional mechanistic assumptions, thus has much broader application than its deterministic counterparts. The far-reaching practical impact of the differences between these two model approaches is even more apparent in the sequel to the Ellis model with n = 2. Consider the problem o f estimating the rate parameter k2 of the terminal compartment in a passage model from data which are not adequately fitted by either the sum of exponentials models in eqn (4) or the Ellis model in eqn (12). The approach suggested by France et al., and further elucidated in Dhanoa et al. (1985), is to create a deterministic n-compartment model whose n transfer rates are given by the identity k,+,= k , + 8 (13) where 8 is a small positive constant. The cited papers show that this model, which may be parameterized either as (n, kt, and k2) or as (n, 3, and k2), yields a mathematically elegant regression equation which is sufficiently rich to describe many data sets where the previously mentioned equations have failed. However the regression equation is predicted on a very complex mechanistic model. In practice, estimates of n have ranged between 10 and 60 (Dhanoa et al., 1985), hence the user is asked, in effect, to accept a conceptual model with a very large number of irreversible compartments whose rates satisfy eqn (13). The set of "real world" applications which may be envisioned to result from such an elaborate mechanistic model is likely to be very small, if not null. Hence, although the equation may fit data successfully, the parameters n and 3 may neither be of direct interest nor of any practical interpretation. Thus the underlying model conceptualization creates a significant stumbling block in the acceptability of its regression equation. On the other hand, the present stochastic approach would instead fit a regression equation given in eqns (9)-(11), in which ks is a model parameter. The other parameters, n and At, are used merely to define the lifetime distribution in eqn (5) which is simple to describe and interpret for the user and which may be estimated from the data. This approach does not attempt to define a detailed causal mechanism which may be irrelevant to the problem of interest, namely the estimation of ks. Of course, not every data set can be successfully fitted by the Ellis model with n = 2, as one might expect and as also apparent in the example given in Matis (1972). However experience has shown that in the cases where the model with n = 2 is not adequate, a model with slightly larger n has always been sufficient (see e.g. Pond, 1982). Stochastic models may also enhance the statistical analysis of kinetic data. Often passage data are gathered as part of a designed experiment to test whether particular

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treatments have had noticeable effects on the passage of digesta. Stochastic models such as the Ellis model may be very useful in such applications by suggesting other parameterizations, such as mean residence time (MRT), which may be utilized as response variables in the analysis. This approach is widely used in pharmacokinetics (see e.g. Gibaidi & Perrier, 1982) and in physiological modeling (see e.g. Covell et aL 1985). Recent experience has shown that tests for treatment effects may be much more powerful statistically when based on estimated mean residence times than on estimated rate coefficients (see e.g. Matis et al., 1985). For example, in Matis et al. (1983) the analysis of some data on cholesterol kinetics using estimated rate coefficients is clearly inferior to the analysis based on MRT's. However, it is difficult to give a biologically meaningful interpretation of such statistical concepts as the mean and variance of a residence time without the notion of residence time random variables, which in turn introduces a stochastic basis for a model. Moreover, the literature, unfortunately, has many examples where modelers have proposed incorrect residence time moments, even MRT's, directly from deterministic models (see, e.g., Chanter, 1985). Of course, most such expressions suggested directly from deterministic models are correct, yet their derivations are generally heuristic thus being difficult to follow and introducing a greater likelihood of major errors. Clearly, the stochastic models provide not only a natural interpretation for the increasingly popular and useful MRT's, but also the appropriate theoretical basis for their derivation. In summary, the France et al. theory of distributed lags is a very elegant and useful generalized framework. It provides a simple, alternative derivation of the regression equation associated with the Ellis model, however the derivation is not based on the inherent stochastic assumptions of the model. Without the stochastic assumptions, the Ellis model would lose much of its theoretical appeal, would be confined to a very narrow mechanistic interpretation, and would be limited in its practical utility in data analysis. The regression equation from the original Ellis model with its stochastic assumptions also has a simple derivation which is outlined in this note and which is easy to generalize. The principal contribution of the Ellis model, which should not be overlooked, is its introduction of much of the recent progress in the theory and application of stochastic compartmental analysis into many practical applications associated with digesta flow modeling. Based on other recent developments cited in this article, it is expected that other stochastic models will follow which will continue to enrich the areas associated with passage kinetics. J. H. MATIS

Department of Statistics, Texas A & M University, College Station, Texas 77843, U.S.A. (Received 10 December 1985, and in revised form 2 September 1986)

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