E. Trucco

Mathematical models for cellular systems. The von foerster equation. Part II

This is the continuation of Part I, which was published in the September, 1965, issue of theBulletin. The birth rate, α(t), is now assumed to be a linear functional of the age density,n. This gives a simple model of self-replenishing stem cell compartments, and leads to a necessary condition for the existence of a steady state. Some examples are presented to illustrate the formalism. They include: (a) An equivivant population with life spanD and no losses from death or migration. The total number of cells is multiplied by 2 in each time intervalD. As a special case, frequently realized in practice, the population may be increasing exponentially with time (“log-phase” of growth). (b) A compartment with “random” emigration of cells and gamma distribution of life spans. (c) An oversimplified version of L. G. Lajtha’s model describing stem cell kinetics. In section IV a simple case in which the loss function depends explicitly onn is discussed very briefly.

RADIATION EFFECTS ON CELL POPULATIONS IN THE INTESTINAL EPITHELIUM OF MICE AND ITS THEORY

Mice were exposed to 1000 R of X‐rays to their trunks and sacrificed every day up to the tenth day after exposure. Cell counts were made on histological sections of the duodenum. The cell counts in the crypts were reduced to about 50% of the control value on the first day after exposure. The cell counts began to recover on the third day and an overshoot of 170% was observed on the fourth day; thereafter the crypt cell counts tended to return to the control level. The cell counts on the villi reached their minimum value on the third day after exposure. Following an overshoot on the sixth day, the villus cell counts returned to the control level by the tenth day. The above experimental results were analysed using a two‐compartment model with a feedback term. A logistic proliferation was assumed for the proliferative crypt cells, while for the postmitotic villus cells the compartment was assumed to be a first in‐first out type. The calculated results with this model are in general consistent with the experimental ones. The model seems to possess some essential features of the dynamics of cell renewal in the intestinal mucosa.

The determination of cell-cycle parameters from measurements of the fraction of labeled mitoses

A method is described for estimating cell-cycle parameters from experimental fraction-of-labeled-mitoses measurements. The method is closely related to that of J. C. Barrett (1970) but is based on the analysis of Brockwell and Trucco (1970) which takes into account population growth in the calculation of theoreticalFLM-functions. Several sets of experimental data are analyzed, among them the data for the Marshall tumor considered by Barrett. It is found that population growth has a small but nevertheless detectable effect on the estimates of the cell parameters.

A note on the dispersionless growth law for single cells

A population of initially synchronized cells is considered wherein each cell grows according to a dispersionless growth law and the probability of cell division is determined by cell age. The first and second moments of the distribution of birth volumes are considered as functions of time and it is shown that it is impossible for both moments to approach finite, nonzero limits ast→∞. This implies that the volume distribution of the population will not approach a limiting distribution on any finite, nonzero volume interval and that the population will not attain balanced exponential growth. An illustrative example is worked out in detail. The distribution of birth volumes is also analyzed as a function of generation number and it is found that the logarithm of the birth volume in thejth generation is normally distributed asj→∞, with an unbounded variance. Generalizations and implications of these results are briefly discussed.

On the average cellular volume in synchronized cell populations

As was done by Sinclair and Ross (1969(, we consider a cellular population that consists initially (at time zero) ofN0 newborn cells, all with the same volumevo. It is assumed that the occurrence of cell division is determined only by a cell’s age, and not by its volume. The frequency function of interdivision times, τ, is denoted byf(τ). If cell death is negligible, the expected number of cells,N(t), will increase according to the laws of a simple age-dependent branching process. The expression forN(t) is obtained as a sum over all generations; thevth term of this sum, in turn, is a multiple convolution integral, reflecting the life history ofvth generation cells (i.e., the lengths of thev successive interdivision periods plus the age of the cell at timet). Assuming that cell volume is a given function of cell age, e.g., linear or exponential, and that cellular volume is exactly halved at each division, it is possible to calculate the volume of a cell with a given life history, and thus the average cellular volume of the whole population as a function of time. If at time zero the volumes differ from cell to cell, the final equation must be modified by averaging over initial volumes. In the case of linear volume increase with age, a very simple asymptotic expression is found for the average cellular volume ast→∞. The case of exponential volume increase with age also leads to a simple asymptotic formula, but the resulting volume distribution is unstable.

ON THE FOKKER-PLANCK EQUATION IN THE STOCHASTIC THEORY OF MORTALITY.II.

This is the continuation of part I, which was published in the September, 1963, issue ofThe Bulletin. Section 5 treats the special case in which the left absorbing barrier recedes to −∞, leaving essentially only one barrier at a finite distance Λ (>0) from the origin. The eigenfunctions are now parabolic cylinder functions. The limiting cases Λ→+∞ and Λ→0 are also considered. Though meaningless for practical applications to our problem, they are of interest, mathematically, because the Green’s function for the solution of the Fokker-Planck equation assumes a particularly simple form. In section 6 we study, by means of an example, how the “force of mortality” may vary with time before attaining its final asymptotic value. Section7, still dealing with only one absorbing barrier, shows that our results for “strong homeostasis” are identical with those derived by Chandrasekhar for the escape of particles through a potential barrier in the limiting case of quasi-static flow. Precise conditions are given for the validity of both the quasi-static and the Smoluchowski approximations to the Fokker-Planck equation. Finally, in section 8, a brief mention is made of Gevrey’s method for the solution of parabolic partial differential equations.

Monte carlo simulation of PLM-curves and collection functions

A description is given of the two computer programs developed at Argonne National Laboratory for the simulation of the curves describing the fraction of mitoses labeled as a function of time after a single injection of tritiated thymidine (“PLM-curves”). This work was mentioned in two previous papers (Trucco and Brockwell,J. Theoret. Biol.,20, 321–337, 1968; Brockwell and Trucco,J. Theoret. Biol.,26, 149–169, 1970) which also contain the theory of the PLM-process. Computer-generated collection functions may be obtained in a similar manner. Examples of simulated PLM-curves and collection functions are provided, and additional work in this field is briefly outlined.

SELF-ABSORPTION IN SPHERES AND CYLINDERS OF RADIOACTIVE MATERIAL

According to thereciprocity law, the total dose absorbed in a specimen irradiated by a point source can be inferred from the reciprocal situation, where the absorber is replaced by a source of the same shape and the radiation is measured at the position of the original point source.

On the use of the Von Foerster equation for the solution and generalization of a problem studied by S. A. Tyler and R. Baserga

In a population of cells labeled with a single injection of tritiated thymidine at timet=0, it is assumed that a constant fraction, 1−z, of the cells which are potentially able to divide fail to do so, and that the cells which do divide all have identical generation time,D. Death and emigration of cells are neglected. In mitosis, the partitioning of label among the two daughter cells is supposed to follow the binomial probability law. Using the formalism developed by H. Von Foerster the fraction of labeled cells in the total population is computed as a function oft, the time after injection of label. Ift is an integral multiple ofD the results coincide with those of S. A. Tyler and R. Baserga.