Limitations of a TCP model incorporating population heterogeneity.

The variation between individuals in their dose-response characteristics complicates attempts to extract estimates of radiobiological parameters (e.g. alpha, beta, etc) from fits to clinical dose-response data. The use of 'population' dose-response models that explicitly account for this variability is necessary to avoid obtaining skewed parameter estimates. In this work, we evaluated an example of a 'population' tumour control probability (TCP) model in terms of its ability to provide reliable parameter estimates. This was accomplished by performing fits of this population model to 'pseudo' data sets, which were generated with Monte Carlo techniques and based on preset values for the various radiobiological parameters. The fitting exercises illustrated considerable correlations between the model parameters. Especially significant was the large correlation observed between the parameter mu=alpha/sigmaalpha used to characterize the level of population heterogeneity in radiosensitivity and the alpha/beta parameter typically used to describe the response to fractionation. The results imply that fits to clinical data may not be able to distinguish between tumours exhibiting a high degree of heterogeneity and a strong beta-mechanism and those containing little heterogeneity and having a weak beta-mechanism. One implication is that basing the design of optimal fractionation regimes on such fitting results may be error-prone. If in vitro assays are to be used to independently determine biologically reasonable ranges for parameter values, an accurate knowledge of the relationship between in vitro and in vivo dose-response characteristics is required.

Investigating the effect of clonogen resensitization on the tumor response to fractionated external radiotherapy.

In this work we further develop the modeling of tumor dynamics by proposing a mechanism of tumor resensitization that is based on the process of reoxygenation. Reoxygenation is modeled using the concept of nonstationary diffusion of oxygen. This leads to the derivation of an explicit expression for the radiosensitivity parameter that predicts a radiosensitivity that increases with time. To account for the resensitization mechanism, the time-dependent expression for the radiosensitivity is then incorporated within a tumor control probability (TCP) model that already includes tumor cell repopulation and repair. We fit a set of experimental animal TCP curves corresponding to several different fractionation regimes using both the modified (with resensitization) and unmodified (without resensitization) versions of the TCP model. In comparison to the unmodified model, the modified model produces statistically superior fits, and is able to describe an "inverse" dose-fractionation behavior present in the data.

Investigating the effect of cell repopulation on the tumor response to fractionated external radiotherapy.

In this work we study the descriptive power of the main tumor control probability (TCP) models based on the linear quadratic (LQ) mechanism of cell damage with cell recovery. The Poisson, binomial, and a dynamic TCP model, developed recently by Zaider and Minerbo are considered. The Zaider-Minerbo model takes cell repopulation into account. It is shown that the Poisson approximation incorporating cell repopulation is conceptually incorrect. Based on the Zaider-Minerbo model, an expression for the TCP for fractionated treatments with varying intervals between two consecutive fractions and with cell survival probability that changes from fraction to fraction is derived. The models are fitted to an experimental data set consisting of dose response curves that correspond to different fractionation regimes. The binomial TCP model based on the LQ mechanism of cell damage solely was unable to fit the fractionated response data. It was found that the Zaider-Minerbo model, which takes tumor cell repopulation into account, best fits the data.

A study of objective functions for organs with parallel and serial architecture.

An objective function analysis when target volumes are deliberately enlarged to account for tumour mobility and consecutive uncertainty in the tumour position in external beam radiotherapy has been carried out. The dose distribution inside the tumour is assumed to have logarithmic dependence on the tumour cell density which assures an iso-local tumour control probability. The normal tissue immediately surrounding the tumour is irradiated homogeneously at a dose level equal to the dose D(R) delivered at the edge of the tumour. The normal tissue in the high dose field is modelled as being organized in identical functional subunits (FSUs) composed of a relatively large number of cells. Two types of organs--having serial and parallel architecture are considered. Implicit averaging over intrapatient normal tissue radiosensitivity variations is done. A function describing the normal tissue survival probability S0 is constructed. The objective function is given as a product of the total tumour control probability (TCP) and the normal tissue survival probability S0. The values of the dose D(R) which result in a maximum of the objective function are obtained for different combinations of tumour and normal tissue parameters, such as tumour and normal tissue radiosensitivities, number of cells constituting a normal tissue functional unit, total number of normal cells under high dose (D(R)) exposure and functional reserve for organs having parallel architecture. The corresponding TCP and S0 values are computed and discussed.

A new method for optimum dose distribution determination taking tumour mobility into account.

A method for determining the optimum dose distribution in the planning target volume is proposed when target volumes are deliberately enlarged to account for tumour mobility in external beam radiotherapy. The optimum dose distribution is a dose distribution that will result in an acceptable level of tumour control probability (TCP) in most of the arising cases of tumour dislocation. An assumption is made that the possible shifts of the tumour are subject to a Gaussian distribution with mean zero and known variance. The idea of a reduced (mean in ensemble) tumour cell density is introduced. On this basis, the target volume and dose distribution in it are determined. The tumour control probability as a function of the shift of the tumour has been calculated. The Monte Carlo method has been used to simulate TCP distributions corresponding to tumour mobility characterized by different variances. The obtained TCP distributions are independent of the variance of the mobility because the dose distribution in the planning target volume is prescribed so that the mobility variance is taken into account. For simplicity a one-dimensional model is used but three-dimensional generalization can be done.

A variational approach to the problem of optimizing the radiation dose distribution in tumours.

This paper presents a precise mathematical formulation of a biological criterion by which the radiation dose distribution in tumours homogeneous or heterogeneous in cell density and radiosensitivity can be optimized. The criterion is formulated as search for a dose distribution that would minimize the mean dose delivered to the tumour under the constraint that the tumour control probability reaches a given desired value. Using a method from the calculus of variations it has been proven that a homogeneous dose distribution is the solution in case of tumours homogeneous in radiosensitivity independent of their cell spatial density status. Thus the usual requirement for homogeneous dose distribution in case of homogeneous tumours is proven if the leading clinical criterion is the described one. The formula for the dose distribution in case of tumours heterogeneous in cell radiosensitivity is given too.

A mathematical approach to optimizing the radiation dose distribution in heterogeneous tumours.

This paper offers a general mathematical approach to dose distribution optimization which allows tumours with different degrees of complexity to be considered. Two different biological criteria - A) keeping the control probability of the different parts of the tumour (local tumour control probability) uniform throughout the tumour and B) minimizing the mean dose delivered to the tumour are studied. For both criteria we impose the requirement that the whole tumour control probability be kept on a certain desired level. It is proved that the adoption of the first criterion requires a dose distribution logarithmic with the cell density and proportional to the inverse of the cell radiosensitivity while the adoption of the second criterion necessitates a homogeneous dose distribution when the cell radiosensitivity is constant. The corresponding formula for the dose distribution in case of heterogeneous cell radiosensitivity is also given. The two criteria are compared in terms of local tumour control probability and mean dose delivered to the tumour. It is concluded that maintaining constant local tumour control probability (criterion A) may be of greater clinical importance then minimizing the mean dose (criterion B).