A comprehensive stochastic model of irradiated cell populations in culture

A stochastic model for the normal tissue complication probability (NTCP) and applicationss

The normal tissue complication probability (NTCP) is a measure for the estimated side effects of a given radiation treatment schedule. Here we use a stochastic logistic birth-death process to define an organ-specific and patient-specific NTCP. We emphasize an asymptotic simplification which relates the NTCP to the solution of a logistic differential equation. This framework is based on simple modelling assumptions and it prepares a framework for the use of the NTCP model in clinical practice. As example, we consider side effects of prostate cancer brachytherapy such as increase in urinal frequency, urinal retention and acute rectal dysfunction.

Optimal treatment and stochastic modeling of heterogeneous tumors

In this work we review past articles that have mathematically studied cancer heterogeneity and the impact of this heterogeneity on the structure of optimal therapy. We look at past works on modeling how heterogeneous tumors respond to radiotherapy, and take a particularly close look at how the optimal radiotherapy schedule is modified by the presence of heterogeneity. In addition, we review past works on the study of optimal chemotherapy when dealing with heterogeneous tumors.

REVIEWERS

This article was reviewed by Thomas McDonald, David Axelrod, and Leonid Hanin.

Cancer stem cells: small subpopulation or evolving fraction?

This review discusses quantitative modeling studies of stem and non-stem cancer cell interactions and the fraction of cancer stem cells.

Optimal schedules of fractionated radiation therapy by way of the greedy principle: biologically-based adaptive boosting

We revisit a long-standing problem of optimization of fractionated radiotherapy and solve it in considerable generality under the following three assumptions only: (1) repopulation of clonogenic cancer cells between radiation exposures follows linear birth-and-death Markov process; (2) clonogenic cancer cells do not interact with each other; and (3) the dose response function s(D) is decreasing and logarithmically concave. Optimal schedules of fractionated radiation identified in this work can be described by the following 'greedy' principle: give the maximum possible dose as soon as possible. This means that upper bounds on the total dose and the dose per fraction reflecting limitations on the damage to normal tissue, along with a lower bound on the time between successive fractions of radiation, determine the optimal radiation schedules completely. Results of this work lead to a new paradigm of dose delivery which we term optimal biologically-based adaptive boosting (OBBAB). It amounts to (a) subdividing the target into regions that are homogeneous with respect to the maximum total dose and maximum dose per fraction allowed by the anatomy and biological properties of the normal tissue within (or adjacent to) the region in question and (b) treating each region with an individual optimal schedule determined by these constraints. The fact that different regions may be treated to different total dose and dose per fraction mean that the number of fractions may also vary between regions. Numerical evidence suggests that OBBAB produces significantly larger tumor control probability than the corresponding conventional treatments.

Tumor control probability in radiation treatment

Patients undergoing radiation therapy (and their physicians alike) are concerned with the probability of cure (long-term recurrence-free survival, meaning the absence of a detectable or symptomatic tumor). This is not what current practice categorizes as "tumor control (TC);" instead, TC is taken to mean the extinction of clonogenic tumor cells at the end of treatment, a sufficient but not necessary condition for cure. In this review, we argue that TC thus defined has significant deficiencies. Most importantly, (1) it is an unobservable event and (2) elimination of all malignant clonogenic cells is, in some cases, unnecessary. In effect, within the existing biomedical paradigm, centered on the evolution of clonogenic malignant cells, full information about the long-term treatment outcome is contained in the distribution Pm(T) of the number of malignant cells m that remain clonogenic at the end of treatment and the birth and death rates of surviving tumor cells after treatment. Accordingly, plausible definitions of tumor control are invariably traceable to Pm(T). Many primary cancers, such as breast and prostate cancer, are not lethal per se; they kill through metastases. Therefore, an object of tumor control in such cases should be the prevention of metastatic spread of the disease. Our claim, accordingly, is that improvements in radiation therapy outcomes require a twofold approach: (a) Establish a link between survival time, where the events of interest are local recurrence or distant (metastatic) failure (cancer-free survival) or death (cancer-specific survival), and the distribution Pm(T) and (b) link Pm(T) to treatment planning (modality, total dose, and schedule of radiation) and tumor-specific parameters (initial number of clonogens, birth and spontaneous death rates during the treatment period, and parameters of the dose-response function). The biomedical, mathematical, and practical aspects of implementing this program are discussed.

Saddlepoint approximations to the moments of multitype age-dependent branching processes, with applications

This article proposes saddlepoint approximations to the expectation and variance-covariance function of multitype age-dependent branching processes. The proposed approximations are found accurate, easy to implement, and much faster to compute than by simulating the process. Multiple applications are presented, including the analyses of clonal data on the generation of oligodendrocytes from their immediate progenitor cells, and on the proliferation of Hela cells. New estimators are also constructed to analyze clonal data. The proposed methods are finally used to approximate the distribution of the generation, which has recently found several applications in cell biology.