Deterministic model of steady-state cell proliferation

A new approach to the kinetics of cell proliferation, based on the postulated restriction of the number of cell divisions in an organism gives the possibility to determine the individual lifetimes of cells. In the model, a necessary condition for a steady-state population is that two sister cells have distinct lifetimes. A steady state was obtained as a consequence of constant rate of cell production in each generation, when sister cell divisions alternated. The mean value of the generation time of cells is in the direct proportion to the number of cells in each generation and connected (with a coefficient of 2) with the generation number. In consequence, we attach great importance to the identification of cells belonging to distinct generations. Corresponding mathematical method to determine the cell population parameters is given and conclusions about stem cells' organization have been drawn.     

Topological solution for cell proliferation in intestinal crypt. I. Elastic growth without cell loss

The cells of an intestinal crypt are tightly packed and, consequently, cell renewal must proceed in accordance with topological laws implicit in the hexagonal cell patterns. The division wave is proposed as the simplest way of proliferation, satisfying topological requirements in steady state. Six pentagonal cells, persisting by topological necessity in the crypt bottom, are the sources of division waves for the whole crypt. The positions of the six pentagonal cells specify the order of cell division. The division, reciprocally, changes the positions of the pentagons which, in turn, specify the order of division in the new cells, and so on. The resulting order of cell division accounts for maintenance of the crypt structure, cell movement toward the villus and cessation of division. Since the pattern of elastic growth is dictated entirely by topological considerations, it does not depend on the genetic constitution of the organism. This model is different from conventional models in which the crypt is assumed to be composed of fixed longitudinal cell columns, the cells of the bottom contributing collectively to the proliferative potential of the whole crypt.     

Variation in crypt size and its influence on the analysis of epithelial cell proliferation in the intestinal crypt.

The standard model of epithelial cell renewal in the intestine proposes a gradual transition between the region of the crypt containing actively proliferating cells and that containing solely terminally differentiating cells (Cairnie, Lamerton and Steel, 1965 a, b). The experimental justification for this conclusion was the gradual decrease towards the crypt top of the measured labeling and mitotic indices. Recently, however, we have proposed that intestinal crypts normally undergo a replicative cycle so that at any time in any region of the intestine, crypts will be found to have a wide range of sizes. We show here that if this intrinsic size variation is taken into account, then a sharp transition between the proliferative and nonproliferative compartments of individual intestinal crypts is consistent with the labeling and mitotic index distributions of mouse and rat jejunal crypts. Thus there is no need to invoke the region of gradual transition from proliferating to nonproliferating cells as is done in the standard model. The position of this sharp transition is estimated for both the mouse and rat. Experiments to further test our model are suggested and the significance of the results discussed.     

Epidermal cell proliferation. II. A comprehensive mathematical model of cell proliferation and migration in the basal layer predicts some unusual properties of epidermal stem cells

The clustering of 3HTdR labelled cells in the epidermal basal layer and their changes with time have been modelled mathematically and cannot be adequately fitted by an earlier model of the cell kinetic organisation of the skin. A more refined model analysis was performed based on Monte Carlo computer simulations of cell layers which take cell division, cell aging and lateral as well as vertical cell migration into account. A large variety of hypothetical scenarios was tested to see if each could provide a fit to the clustering data. The analysis provides further support for the concept of a cell kinetic heterogeneity with a stem-transit-postmitotic differentiation scheme. In the best overall model scheme three transit divisions are predicted but unlike in the earlier model it is now postulated that postmitotic cells can be produced at all stages in the lineage rather than only at the end of the amplification scheme. Most important, the model predicts that stem cells and most of the transit cells differ in the way they process 3HTdR label. Grain dilution is an important mechanism to explain the fate of some labelled cells in the tissue, but on its own it can only consistently explain the data if the stem cells have a very low labelling index (LI less than or equal to 1%) which implies a very short biologically unreasonable S-phase. If a higher LI (longer S-phase) is assumed for the stem-cells other mechanisms must be predicted to explain the lack of large clusters and the increase in time of the singles. The selective segregation of chromosomes at mitosis is one such mechanism. However, on its own a large number of cells would have to behave in this way (i.e. both stem and T1 cells). If combined with other assumptions such as some grain dilution this selective segregation may be restricted only to stem cells. In addition the model allows cell production and migration rates to be estimated and the analysis can be related to the EPU-concept. Indeed the model itself would tend to automatically generate an EPU like structure. The model quantitatively reproduces LI, PLM, CL and clustering data.     

Evidence for cell generation controlled proliferation in the small intestinal crypt

In this paper we discuss the hypothesis that cell proliferation is controlled by the number of generations after leaving an 'eternal' stem cell. The theory is based on a simulation of the kinetic behaviour of cells in the intestinal crypts. There is evidence of three, four and five generations of cells which are allowed to enter mitosis in the lower and upper part of the normal intestinal tract, and in some disease states, respectively. We suggest an internal proliferation control: some kind of knowledge that cells carry from generation to generation. It is an open question what sets and changes the generation counter: internal genetic information or external influences such as growth factors or chalones. The geometric shape of the epithelial tissue in the intestinal tract can be understood as the steady state of a highly dynamic process. Age and death are determined from the beginning; cell-cell interaction or communication is not necessary and can be neglected. Our theory will be illustrated using the intestinal crypts as they are easily accessible, of a simple structure and completely described in the literature.     

Intestinal crypt proliferation. II. Computer modelling of mitotic index data provides further evidence for lateral and vertical cell migration in the absence of mitotic activity

The position-dependent mitotic index before, and 1, 2 and 3 h after vincristine was scored. The accumulation of cells in mitosis leads to an increase in the mitotic index from 0.06 to 0.34 at crypt positions 8-12. Surprisingly, the leading edge of the position-related mitotic index distribution moves to higher crypt positions although cell division was stopped. In addition, the vertical clustering of mitotic figures in sections was recorded. The data were examined using a previously described computer crypt model. We conclude: the average mitotic phase duration is about 0.7 h (40 min) and varies little with cell position; the geometrical correction factor for overscoring mitoses in crypt sections is about 0.6-0.7 and adjacent cell columns can merge. Lateral cell displacement after mitosis, as predicted in a previous model analysis, would be a mechanism to counteract other forces that tend to reduce the crypt circumference. In the normal steady state merging and expansion processes would just balance each other. This would not follow if one mechanism was blocked. Thus we propose a new concept in which the crypt geometry would be dynamically determined by cell proliferative activity in connection with lateral positioning of new cells on one hand and contracting forces on the other hand.     

Matrix simulation of duodenal crypt cell kinetics. I. The steady state.

Steady state crypt cell kinetics have been simulated using matrix algebra. The model crypt cell population is distributed through two proliferation compartments (P1 and P2) and a quiescent state (Q). Under steady state conditions half the daughter cells produced on completion of P1 enter G1 of P2 and half enter G1 of P1. Both P2 daughter cells enter Q. Cells in Q are non-dividing but retain the potential to divide. On completion of Q, cells lose the potential to divide and move up onto the villi. The model has been developed by simultaneously simulating the following biological data: (1) the per cent labeled mitosis (PML) curve, (2) the number of labeled cells per crypt as a function of time following an injection of 3H-thymidine, and (3) the total number of cells per crypt.     

A nonlinear mathematical model of cell turnover, differentiation and tumorigenesis in the intestinal crypt

We present a development of a model [Tomlinson, I.P.M., Bodmer, W.F., 1995. Failure of programmed cell death and differentiation as causes of tumors: Some simple mathematical models. Proc. Natl. Acad. Sci. USA 92, 11130-11134.] of the relationship between cells in three compartments of the intestinal crypt: stem cells, semi-differentiated cells and fully differentiated cells. Stem and semi-differentiated cells may divide to self-renew, undergo programmed death or progress to semi-differentiated and fully differentiated cells, respectively. The probabilities of each of these events provide the most important parameters of the model. Fully differentiated cells do not divide, but a proportion undergoes programmed death in each generation. Our previous models showed that failure of programmed death--for example, in tumorigenesis--could lead either to exponential growth in cell numbers or to growth to some plateau. Our new models incorporate plausible fluctuation in the parameters of the model and introduce nonlinearity by assuming that the parameters depend on the numbers of cells in each state of differentiation. We present detailed analysis of the equilibrium conditions for various forms of these models and, where appropriate, simulate the changes in cell numbers. We find that the model is characterized by bifurcation between increase in cell numbers to stable equilibrium or explosive exponential growth; in a restricted number of cases, there may be multiple stable equilibria. Fluctuation in cell numbers undergoing programmed death, for example caused by tissue damage, generally makes exponential growth more likely, as long as the size of the fluctuation exceeds a certain critical value for a sufficiently long period of time. In most cases, once exponential growth has started, this process is irreversible. In some circumstances, exponential growth is preceded by a long plateau phase, of variable duration, mimicking equilibrium: thus apparently self-limiting lesions may not be so in practice and the duration of growth of a tumor may be impossible to predict on the basis of its size.     

A dynamic model of proliferation and differentiation in the intestinal crypt based on a hypothetical intraepithelial growth factor

A widely accepted model of the temporal and spatial organization of proliferation and differentiation in intestinal epithelia is based on a cellular pedigree with all cells descending from a few active stem cells and undergoing a sequence of transitory divisions until the non-proliferating maturing cell stages develop. Model simulations have shown that such a pedigree concept can explain a large variety of data. However, so far there is neither a direct experimental proof for the existence of an intrinsic age structure in the transitory proliferative cell stages nor for the distinction between stem and transitory cells. It is our objective to suggest an alternative model which is based on evidence for intercellular communications such as might be mediated through gap junctions. We consider the diffusion of a hypothetical intraepithelial growth factor in a chain of cells which are connected via gap junctions. Individual cells can divide if a critical growth factor concentration is exceeded. Simulation studies show that the model is consistent with many observed features of the small intestinal crypt in steady state and after perturbation.     

The influence of adrenoceptor activity on cell proliferation in colonic crypt ipithelium and in colonic adenocarcinomata.

The effects of chemical sympathectomy and of the injection of amines or amine-receptor blocking drugs on cell proliferation in colonic crypts and in dimethylhydrazine-induced colonic carcinomata is examined in rats using a stathmokinetic technique. In animals which had been chemically sympathectomized by injection of 6-hydroxydopamine cell proliferation essentially ceased in the colonic crypts but continued at a normal rate in the tumours. Stimulation of alpha-adrenoceptors by metaraminol, a drug with properties similar to noradrenaline, caused acceleration of cell proliferation in colonic crypts but not in tumours. Conversely, blockade of alpha-adrenoceptors by phentolamine inhibited cell proliferation in crypts but not in tumours. Injection of adrenaline, predominantly a beta-adrenergic agonist, inhibited cell proliferation in the tumours but not in colonic crypts whereas blockade of beta-adrenoceptors by propranolol accelerated cell proliferation in tumours but not in colonic crypts. It is postulated that cell proliferation in the crypts of Lieberkühn in rat colon resembles that in rat jejunum in being controlled by the autonomic nervous system. However, tumour cell proliferation does not appear to be subject to such regulation.     

Modelling the spatio-temporal cell dynamics reveals novel insights on cell differentiation and proliferation in the small intestinal crypt

We developed a slow structural relaxation model to describe cellular dynamics in the crypt of the mouse small intestine. Cells are arranged in a three dimensional spiral the size of which dynamically changes according to cell production demands of adjacent villi. Cell differentiation and proliferation is regulated through Wnt and Notch signals, the strength of which depends on the local cell composition. The highest level of Wnt activity is associated with maintaining equipotent stem cells (SC), Paneth cells and common goblet-Paneth cell progenitors (CGPCPs) intermingling at the crypt bottom. Low levels of Wnt signalling area are associated with stem cells giving rise to secretory cells (CGPCPs, enteroendocrine or Tuft cells) and proliferative absorptive progenitors. Deciding between these two fates, secretory and stem/absorptive cells, depends on Notch signalling. Our model predicts that Notch signalling inhibits secretory fate if more than 50% of cells they are in contact with belong to the secretory lineage. CGPCPs under high Wnt signalling will differentiate into Paneth cells while those migrating out from the crypt bottom differentiate into goblet cells. We have assumed that mature Paneth cells migrating upwards undergo anoikis. Structural relaxation explains the localisation of Paneth cells to the crypt bottom in the absence of active forces. The predicted crypt generation time from one SC is 4-5 days with 10-12 days needed to reach a structural steady state. Our predictions are consistent with experimental observations made under altered Wnt and Notch signalling. Mutations affecting stem cells located at the crypt floor have a 50% chance of being propagated throughout the crypt while mutations in cells above are rarely propagated. The predicted recovery time of an injured crypt losing half of its cells is approximately 2 days.