Modeling cellular systems and aging processes: I. Mathematics of cell system models-a review

Chemotherapy in conjoint aging-tumor systems: some simple models for addressing coupled aging-cancer dynamics

Background

In this paper we consider two approaches to examining the complex dynamics of conjoint aging-cancer cellular systems undergoing chemotherapeutic intervention. In particular, we focus on the effect of cells growing conjointly in a culture plate as a precursor to considering the larger multi-dimensional models of such systems. Tumor cell growth is considered from both the logistic and the Gompertzian case, while normal cell growth of fibroblasts (WI-38 human diploid fibroblasts) is considered as logistic only.

Results

We demonstrate, in a simple approach, how the interdependency of different cell types in a tumor, together with specifications of for treatment, can lead to different evolutionary patterns for normal and tumor cells during a course of therapy.

Conclusions

These results have significance for understanding appropriate pharmacotherapy for elderly patients who are also undergoing chemotherapy.

How square is the survival curve of a given species?

An investigator-independent parameter, the prolate rectangularity index kappa for describing the so-called rectangularity of biological population survival curves, is introduced, developed, and applied to realworld survival datasets. This new rectangularity parameter is constructed using an intrinsic time scaling that places the intrinsic inflection point time at a value of unity so that species populations may be compared independently of their extrinsic life span distributions. The analytical expressions for the prolate rectangularity index of the theoretical Gompertz and Weibull continuous models are obtained, as are numerical values of this index for discrete experimental population survival data sets from two dissimilar species with orders of magnitude difference in extrinsic life span range. The values of the parameter are also compared for populations of a single species having differing dietary regimens, and for human demographic populations at decade intervals in extrinsic chronological time during the current century. It is found that scaling time, using the survival inflection point, appreciably collapses extrinsic survival profile dispersion among similar populations and allows a more meaningful comparison of profiles among dissimilar populations. Using this method of scaling, demographic populations within the United States are seen to have rectangularity parameter values that have been slowly drifting during this century toward values indicating a higher degree of rectangularity. In recent decades, however, the trend appears to be stabilizing with kappa values indicating no approach towards the theoretical maximum rectangularity. This apparent submaximal stabilization of kappa supports a hypothesis of no genetically pre-determined maximum life span in human populations. Or, if such a maximum exists, we are not currently near it.

A model of proliferating cell populations with inherited cycle length

A mathematical model of cell population growth introduced by J.L. Lebowitz and S.I. Rubinow is analyzed. Individual cells are distinguished by age and cell cycle length. The cell cycle length is viewed as an inherited property determined at birth. The density of the population satisfies a first order linear partial differential equation with initial and boundary conditions. The boundary condition models the process of cell division of mother cells and the inheritance of cycle length by daughter cells. The mathematical analysis of the model employs the theory of operator semigroups and the spectral theory of linear operators. It is proved that the solutions exhibit the property of asynchronous exponential growth.

A return to time, cells, systems, and aging: III. Gompertzian models of biological aging and some possible roles for critical elements

In this paper, I continue my investigation into the modeling of senescence in biological hierarchies. Making use of my previous discussion on non-reestablishable biological components, I derive a mathematical model which has Gompertzian-like dynamics. I show how this model may be approximated, in certain instances, by a Gompertzian equation. I then demonstrate how our approach yields a biological interpretation for the parameters in the Gompertzian equation. I then demonstrate how changes in the parameter values may be interpreted in light of the biology. Subsequently, I review the literature on the allometry of aging, and I demonstrate how my reliability model may be used to obtain--in a qualitative manner--some of the lifespan curves found in the literature. I close my discussion by constructing a more complex reliability model which incorporates the deterministic failure of biological components with stochastic aspects of senescence.

Probability of monozygotic twinning as a reflection of the genetic control of cell development

The stability of the incidence of monozygotic twinning (MZT) suggests that its origin is genetically, rather than environmentally, controlled. Available data, though scant, supports our hypothesis that the MZT probability is (1/2)K, where K is a species-specific integer parameter. For humans MZT occurs in about four of 1000 births, which is close to one occurrence in 2(8) births, i.e. K = 8. The environmental factors are not the cause of MZT, but may influence its expression. When this influence is in effect under some extreme experimental conditions the above form of MZT probability is observed. Binary structure of the MZT probability provides insight into genetic control mechanism of cell division.

Biological system interactions

Mathematical modeling of cellular population growth, interconnected subsystems of the body, blood flow, and numerous other complex biological systems problems involves nonlinearities and generally randomness as well. Such problems have been dealt with by mathematical methods often changing the actual model to make it tractable. The method presented in this paper (and referenced works) allows much more physically realistic solutions.

Late skin damage in rabbits and monkeys after exposure to particulate radiations

Skin biopsies were taken from the central regions of the ears of New Zealand white rabbits following localized exposure of one ear of each rabbit to 530 MeV/amu Ar or 365 MeV/amu Ne ions. The unirradiated ears served as controls. Biopsies were taken also from the chests and inner thighs of rhesus monkeys after whole-body exposure to 32 MeV protons and from unirradiated control animals. The linear energy transfers (LET infinity's) for the radiations were 90 +/- 5, 35 +/- 3, and approximately 1.2 keV/micrometer, respectively. In the rabbit studies, explants were removed with a 2 mm diameter dermal punch at post-irradiation times up to five years after exposure. Similar volumes of monkey tissue were taken from skin samples excised surgically 16-18 years following proton irradiation. Fibroblast cultures were initiated from the explants and were propagated in vitro until terminal senescence (cessation of cell division) occurred. Cultures from irradiated tissue exhibited decreases in doubling potential that were dependent on radiation dose and LET infinity and seemed to reflect damage to stem cell populations. The implications of these results for astronauts exposed to heavy ions and/or protons in space include possible manifestations of residual effects in the skin many years after exposure (e.g. unsatisfactory responses to trauma or surgery).