Tumor control probability in hypofractionated radiotherapy as a function of total and hypoxic tumor volumes.

Clinical studies in the hypofractionated stereotactic body radiotherapy (SBRT) have shown a reduction in the probability of local tumor control with increasing initial tumor volume. In our earlier work, we obtained and tested an analytical dependence of the tumor control probability (TCP) on the total and hypoxic tumor volumes using conventional radiotherapy model with the linear-quadratic (LQ) cell survival. In this work, this approach is further refined and tested against clinical observations for hypofractionated radiotherapy treatment schedules. Compared to radiotherapy with conventional fractionation schedules, simulations of hypofractionated radiotherapy may require different models for cell survival and the oxygen enhancement ratio (OER). Our TCP simulations in hypofractionated radiotherapy are based on the LQ model and the universal survival curve (USC) developed for the high doses used in SBRT. The predicted trends in local control as a function of the initial tumor volume were evaluated in SBRT for non-small cell lung cancer (NSCLC). Our results show that both LQ and USC based models cannot describe the TCP reduction for larger tumor volumes observed in the clinical studies if the tumor is considered completely oxygenated. The TCP calculations are in agreement with the clinical data if the subpopulation of radio-resistant hypoxic cells is considered with the volume that increases as initial tumor volume increases. There are two conclusions which follow from our simulations. First, the extent of hypoxia is likely a primary reason of the TCP reduction with increasing the initial tumor volume in SBRT for NSCLC. Second, the LQ model can be an acceptable approximation for the TCP calculations in hypofractionated radiotherapy if the tumor response is defined primarily by the hypoxic fraction. The larger value of OER in the hypofractionated radiotherapy compared to the conventional radiotherapy effectively extends the applicability of the LQ model to larger doses.

Dynamics of pathologic clot formation: a mathematical model.

Recent studies have provided evidence of a significant role of the Hageman factor in pathologic clot formation. Since auto-activation of the Hageman factor triggers the intrinsic coagulation pathway, we study the dynamics of pathologic clot formation considering the intrinsic pathway as the predominant mechanism of this process. Our methodological approach to studying the dynamics of clot formation is based on mathematical modelling. Activation of the blood coagulation cascade, particularly its intrinsic pathway, is known to involve platelets. Therefore, equations accounting for the effects of activated platelets on the intrinsic pathway activation are included in our model. This brings about a considerable increase in the values of kinetic constants involved in the model of the principal biochemical processes resulting in clot formation. The purpose of this study is to elucidate the mechanism of pathologic clot formation. Since the time window of thrombolysis is 3-6h, we hypothesize that in many cases the rate of pathologic clot formation is much lower than that of haemostatic clot. This assumption is used to simplify the mathematical model and to estimate kinetic constants of biochemical reactions that initiate pathologic clot formation. The insights we gained from our mathematical model may lead to new approaches to the prophylaxis of pathologic clot formation. We believe that one of the most efficient ways to prevent pathologic clot formation is simultaneous inhibition of activated factors ХII and ХI.

Non-linear dynamics of the complement system activation.

The complement system (CS) plays a prominent role in the immune defense. The goal of this work is to study the dynamics of activation of the classic and alternative CS pathways based on the method of mathematical modeling. The principal difficulty that hinders modeling effort is the absence of the measured values of kinetic constants of many biochemical reactions forming the CS. To surmount this difficulty, an optimization procedure consisting of constrained minimization of the total protein consumption by the CS was designed. The constraints made use of published data on the in vitro kinetics of elimination of the Borrelia burgdorferi bacteria by the CS. Special features of the problem at hand called for a significant modification of the general constrained optimization procedure to include a mathematical model of the bactericidal effect of the CS in the iterative setting. Determination of the unknown kinetic constants of biochemical reactions forming the CS led to a fully specified mathematical model of the dynamics of cell killing induced by the CS. On the basis of the model, effects of the initial concentrations of complements and their inhibitors on the bactericidal action of the CS were studied. Proteins playing a critical role in the regulation of the bactericidal action of the CS were identified. Results obtained in this work serve as an important stepping stone for the study of functioning of the CS as a whole as well as for developing methods for control of pathogenic processes.

Dynamics of granzyme B-induced apoptosis: mathematical modeling.

Apoptosis is mediated by an intracellular biochemical system that mainly includes proteins (procaspases, caspases, inhibitors, Bcl-2 protein family as well as substances released from mitochondrial intermembrane space). The dynamics of caspase activation and target cleavage in apoptosis induced by granzyme B in a single K562 cell was studied using a mathematical model of the dynamics of granzyme B-induced apoptosis developed in this work. Also the first application of optimization approach to determination of unknown kinetic constants of biochemical apoptotic reactions was presented. The optimization approach involves solving of two problems: direct and inverse. Solving the direct optimization problem, we obtain the initial (baseline) concentrations of procaspases for known kinetic constants through conditional minimization of a cost function based on the principle of minimum protein consumption by the apoptosis system. The inverse optimization problem is aimed at determination of unknown kinetic constants of apoptotic biochemical reactions proceeding from the condition that the optimal concentrations of procaspases resulting from the solution of the direct optimization problem coincide with the observed ones, that is, those determined by biochemical methods. The Multidimensional Index Method was used to perform numerical solution of the inverse optimization problem.

The University of Rochester model of breast cancer detection and survival.

This paper presents a biologically motivated model of breast cancer development and detection allowing for arbitrary screening schedules and the effects of clinical covariates recorded at the time of diagnosis on posttreatment survival. Biologically meaningful parameters of the model are estimated by the method of maximum likelihood from the data on age and tumor size at detection that resulted from two randomized trials known as the Canadian National Breast Screening Studies. When properly calibrated, the model provides a good description of the U.S. national trends in breast cancer incidence and mortality. The model was validated by predicting some quantitative characteristics obtained from the Surveillance, Epidemiology, and End Results data. In particular, the model provides an excellent prediction of the size-specific age-adjusted incidence of invasive breast cancer as a function of calendar time for 1975-1999. Predictive properties of the model are also illustrated with an application to the dynamics of age-specific incidence and stage-specific age-adjusted incidence over 1975-1999.

A stochastic model of tumor response to fractionated radiation: limit theorems and rate of convergence.

The iterated birth and death Markov process is defined as an n-fold iteration of a birth and death Markov process describing kinetics of certain population combined with random killing of individuals in the population at moments tau 1,...,tau n with given survival probabilities s1,...,sn. A long-standing problem of computing the distribution of the number of clonogenic tumor cells surviving an arbitrary fractionated radiation schedule is solved within the framework of iterated birth and death Markov process. It is shown that, for any initial population size iota, the distribution of the size N of the population at moment t > or = tau n is generalized negative binomial, and an explicit computationally feasible formula for the latter is found. It is shown that if i --> infinity and sn --> 0 so that the product iota s1...sn tends to a finite positive limit, the distribution of random variable N converges to a probability distribution, which for t = tau n turns out to be Poisson. In the latter case, an estimate of the rate of convergence in the total variation metric similar to the classical Law of Rare Events is obtained.