Angiogenesis and vessel co-option in a mathematical model of diffusive tumor growth: The role of chemotaxis.

This work considers the propagation of a tumor from the stage of a small avascular sphere in a host tissue and the progressive onset of a tumor neovasculature stimulated by a pro-angiogenic factor secreted by hypoxic cells. The way new vessels are formed involves cell sprouting from pre-existing vessels and following a trail via a chemotactic mechanism (CM). Namely, it is first proposed a detailed general family of models of the CM, based on a statistical mechanics approach. The key hypothesis is that the CM is composed by two components: i) the well-known bias induced by the angiogenic factor gradient; ii) the presence of stochastic changes of the velocity direction, thus giving rise to a diffusive component. Then, some further assumptions and simplifications are applied in order to derive a specific model to be used in the simulations. The tumor progression is favored by its acidic aggression towards the healthy cells. The model includes the evolution of many biological and chemical species. Numerical simulations show the onset of a traveling wave eventually replacing the host tissue with a fully vascularized tumor. The results of simulations agree with experimental measures of the vasculature density in tumors, even in the case of particularly hypoxic tumors.

Optimal weekly scheduling in fractionated radiotherapy: effect of an upper bound on the dose fraction size.

This work concerns the optimization of the dose fractionation for cancer radiotherapy schedules of the kind one fraction/day, five fractions/week, assuming a fixed overall treatment time. Constraints are set to limit the radiation damages to surrounding normal tissues, as well as the daily fraction size. The response to radiation of tumour and normal tissues is represented by the classical LQ model, including the exponential repopulation term. We provide a framework to analytically determine the optimal weekly scheme of radiation doses as a function of the tumour type, the fraction upper bound and the normal tissue parameters. For a comparison with the literature, we present some numerical examples of optimal treatment schedules for specific tumour types.

Optimal solution for a cancer radiotherapy problem.

We address the problem of finding the optimal radiotherapy fractionation scheme, representing the response to radiation of tumour and normal tissues by the LQ model including exponential repopulation and sublethal damage due to incomplete repair. We formulate the nonlinear programming problem of maximizing the overall tumour damage, while keeping the damages to the late and early responding normal tissues within a given admissible level. The optimum is searched over a single week of treatment and its possible structures are identified. In the two simpler but important cases of absence of the incomplete repair term or of prevalent late constraint, we prove the uniqueness of the optimal solution and we characterize it in terms of model parameters. The optimal solution is found to be not necessarily uniform over the week. The theoretical results are confirmed by numerical tests and comparisons with literature fractionation schemes are presented.

Response of tumor spheroids to radiation: modeling and parameter estimation.

We propose a spatially distributed continuous model for the spheroid response to radiation, in which the oxygen distribution is represented by means of a diffusion-consumption equation and the radiosensitivity parameters depend on the oxygen concentration. The induction of lethally damaged cells by a pulse of radiation, their death, and the degradation of dead cells are included. The compartments of lethally damaged cells and of dead cells are subdivided into different subcompartments to simulate the delays that occur in cell death and cell degradation, with a gain in model flexibility. It is shown that, for a single irradiation and under the hypothesis of a sufficiently small spheroid radius, the model can be reformulated as a linear stationary ordinary differential equation system. For this system, the parameter identifiability has been investigated, showing that the set of unknown parameters can be univocally identified by exploiting the response of the model to at least two different radiation doses. Experimental data from spheroids originated from different cell lines are used to identify the unknown parameters and to test the predictive capability of the model with satisfactory results.

Optimal number and sizes of the doses in fractionated radiotherapy according to the LQ model.

We address a non-linear programming problem to find the optimal scheme of dose fractionation in cancer radiotherapy. Using the LQ model to represent the response to radiation of tumour and normal tissues, we formulate a constrained non-linear optimization problem in terms of the variables number and sizes of the dose fractions. Quadratic constraints are imposed to guarantee that the damages to the early and late responding normal tissues do not exceed assigned tolerable levels. Linear constraints are set to limit the size of the daily doses. The optimal solutions are found in two steps: i) analytical determination of the optimal sizes of the fractional doses for a fixed, but arbitrary number of fractions n; ii) numerical simulation of a sequence of the previous optima for n increasing, and for specific tumour classes. We prove the existence of a finite upper bound for the optimal number of fractions. So, the optimum with respect to n is found by means of a finite number of comparisons amongst the optimal values of the objective function at the first step. In the numerical simulations, the radiosensitivity and repopulation parameters of the normal tissue are fixed, while we investigate the behaviour of the optimal solution for wide variations of the tumour parameters, relating our optima to real clinical protocols. We recognize that the optimality of hypo or equi-fractionated treatment schemes depends on the value of the tumour radiosensitivity ratio compared to the normal tissue radiosensitivity. Fast growing, radioresistant tumours may require particularly short optimal treatments.

A model with 'growth retardation' for the kinetic heterogeneity of tumour cell populations.

In the present paper we propose a continuous cell population model based on Shackney's idea of growth retardation. Cells are characterized by two state variables: the cell maturity x, 0 < or = x < or = 1, and a state variable T that identifies the rate of maturation along cell cycle. During their life span, cells can change T at random by jump transitions to T values corresponding to slower maturation rates, while at each jump the maturity x is conserved. Both the time evolution of the population and the exponential stationary solution are numerically computed. The distribution of the cell cycle transit time in asynchronous exponential growth is investigated by Monte Carlo simulation. An approximated formula for the distribution of cell cycle time is also provided.

Cell cycle analysis by the relative movement approach: effect of variability across S-phase of DNA synthesis rate.

Cell populations pulse-labelled with BrdUrd, and sampled at increasing times after the pulse, yield DNA-BrdUrd distributions from which the relative movement (RM) and the depletion function (DF) of labelled, undivided cells can be calculated. In this paper we present an extension of the equation for the time course of RM, given by White and Meistrich (Cytometry 1986, 7, 486-490), to the case in which the rate of DNA synthesis changes across S-phase. Some modalities of cell loss were also considered. Computer simulations showed that different patterns of DNA synthesis rate across S-phase can result in appreciably different RM curves. An analytical expression of the RM curve, in which the variability across S-phase of the rate of DNA synthesis is accounted for by only one parameter, was proposed. This expression was used for the simultaneous fitting of time sequences of RM and DF data of U937 cells, in order to estimate the phase transit times TS and TG2+M, and the potential doubling time Tpot. The use of the extended model gave better results than those obtained under the assumption of constant rate of DNA synthesis across S-phase.

Steel's potential doubling time and its estimation in cell populations affected by nonuniform cell loss.

The in vivo infusion of the thymidine analogue bromodeoxyuridine (BrdUrd). followed by delayed biopsy and bivariate DNA-BrdUrd flow cytometry, makes it possible to estimate Steel's potential doubling time (Tpot) of human tumors. In the present paper, the expression of Steel's Tpot for a rather general cell population model, in which the distribution of cell loss is assumed to be nonuniform, is derived in terms of the model parameters. We show that Steel's Tpot of a population can be markedly different for the doubling time that would be exhibited by the population in the absence of cell loss. These doubling times, on the contrary, are equal when loss is uniform. Moreover, we studied the influence of modes of cell loss different from the uniform random loss on the estimation of Tpot by using the labeling index or the nu-function, quantities that can be determined from the bivariate DNA-BrdUrd distribution.

Reoxygenation and split-dose response to radiation in a tumour model with Krogh-type vascular geometry.

After a single dose of radiation, transient changes caused by cell death are likely to occur in the oxygenation of surviving cells. Since cell radiosensitivity increases with oxygen concentration, reoxygenation is expected to increase the sensitivity of the cell population to a successive irradiation. In previous papers we proposed a model of the response to treatment of tumour cords (cylindrical arrangements of tumour cells growing around a blood vessel of the tumour). The model included the motion of cells and oxygen diffusion and consumption. By assuming parallel and regularly spaced tumour vessels, as in the Krogh model of microcirculation, we extend our previous model to account for the action of irradiation and the damage repair process, and we study the time course of the oxygenation and the cellular response. By means of simulations of the response to a dose split in two equal fractions, we investigate the dependence of tumour response on the time interval between the fractions and on the main parameters of the system. The influence of reoxygenation on a therapeutic index that compares the effect of a split dose on the tumour and on the normal tissue is also investigated.

Cell resensitization after delivery of a cycle-specific anticancer drug and effect of dose splitting: learning from tumour cords.

After a single dose of an anticancer agent, changes due to cell death are expected to occur in the distribution of cells between proliferating and quiescent compartment as well as in the oxygenation and nutritional state of surviving cells. These changes are transient because tumour regrowth tends to restore the pretreatment status. The reoxygenation due to the decrease of oxygen consumption is expected to induce cell recruitment from quiescence into proliferation, and consequently to increase the sensitivity of the cell population to a successive treatment by a cycle-specific drug. In previous papers we proposed a model of the response of tumour cords (cylindrical arrangements of tumour cells growing around a blood vessel of the tumour) to single-dose treatments. The model included the motion of cells and oxygen diffusion and consumption. On the basis of that model suitably extended to better account for the action of anticancer drugs, we study the time course of the oxygenation and of the redistribution of cells between the proliferating and quiescent compartments. By means of simulations of the response to a dose delivered as two spaced equal fractions, we investigate the dependence of tumour response on the spacing between the fractions and on the main parameters of the system. A time window may be found in which the delivery of two fractions is more effective than the delivery of the undivided dose.

Cell loss and the concept of potential doubling time.

The in vivo infusion of Bromodeoxyuridine (BrdUrd), followed by delayed biopsy and bivariate DNA-BrdUrd flow cytometry, allows the potential doubling time (Tpot) of human tumors to be estimated. According to Steel, the mathematical definition of Tpot is Tpot = ln 2/Kp, where Kp is the rate constant of cell production. All the operative formulas which allow the estimation of Tpot from flow cytometric data derive from this definition. Most authors, however, identify the potential doubling time as the doubling time that the same cell population would exhibit if cell loss were removed. We denote here as T(d)noloss this quantity. Although these two definitions are equivalent in the case of uniform random cell loss, we show, in the framework of Steel's theory of growing cell populations, that Tpot and T(d)noloss become distinct kinetic quantities when cell loss is not uniform, i.e., when loss differently affects the quiescent and the proliferative compartment. We discuss the validity of the two formulas currently used for the calculation of Tpot, one based on LI and the other on the v-function, in conditions of non-uniform cell loss. Moreover, we propose two formulas for the estimation of the cycle time T(C), which require, in addition to T(S) and LI, that a measure of the growth fraction be available.