Volume effects in the TCP for hypoxic and oxygenated tumors.

PURPOSE: Clinical studies in radiation therapy with conventional fractionation show a reduction in the tumor control probability (TCP) with an increase in the total and hypoxic tumor volumes. The main objective of this article is to derive an analytical relationship between the TCP and the hypoxic and total tumor volumes. This relationship is applied to clinical data on the TCP reduction with increasing total tumor volume and, also, dose escalation to target tumor hypoxia. METHODS: The TCP equation derived from the Poisson probability distribution predicts that both (a) an increase in the number of tumor clonogens and (b) an increase in the average cell surviving fraction are the factors contributing to the loss of local control. Using asymptotic mathematical properties of the TCP formula and the linear quadratic (LQ) cell survival model with two levels of hypoxic and oxygenated cells, we separated the TCP dependence on the total and hypoxic tumor volumes. The predicted trends in the local control as a function of total and hypoxic tumor volumes were evaluated in radiotherapy model problems with conventional dose fractionation for head and neck and non-small cell lung cancers. Tumor-specific parameters in the LQ model and the density of clonogens in the TCP model were taken from published data on predictive assays and the plating efficiency measurements, respectively. RESULTS: Our simulations show that, at the dose levels used in conventional radiation therapy for head and neck and non-small cell lung cancers, the TCP dependence on the total tumor volume is negligible for completely oxygenated tumors. However, the presented results demonstrate that tumor hypoxia introduces a significant volume effect into estimates of the TCP. The extent of tumor hypoxia is a plausible mechanism to explain the TCP reduction with increasing total tumor volume observed in clinical studies. To achieve the same level of tumor control in a hypoxic tumor region relative to well oxygenated tumor regions, the delivered dose should, in principle, be escalated by a factor equal to the oxygen enhancement ratio (OER). The theoretically required hypoxia-targeted dose escalation could be as large as 100% because it has been estimated that hypoxic tumor regions may have an OER = 2 for conventional fractionation. However, our results indicate that clinically acceptable values of the TCP would require much lower hypoxia-targeted dose escalation (<50%) when the effects of total and hypoxic tumor volumes are taken into account. CONCLUSIONS: The reported studies and models suggest that the effect of total tumor volume on the TCP is negligible for oxygenated head and neck and non-small cell lung tumors treated with conventional fractionation. According to our simulations, the volume effects in the TCP observed in clinical studies are defined primarily by the hypoxic volume. This information can be useful for the analysis of treatment outcomes and the dose escalation to target tumor hypoxia.

Ill-posed problem and regularization in reconstruction of radiobiological parameters from serial tumor imaging data.

The main objective of this article is to improve the stability of reconstruction algorithms for estimation of radiobiological parameters using serial tumor imaging data acquired during radiation therapy. Serial images of tumor response to radiation therapy represent a complex summation of several exponential processes as treatment induced cell inactivation, tumor growth rates, and the rate of cell loss. Accurate assessment of treatment response would require separation of these processes because they define radiobiological determinants of treatment response and, correspondingly, tumor control probability. However, the estimation of radiobiological parameters using imaging data can be considered an inverse ill-posed problem because a sum of several exponentials would produce the Fredholm integral equation of the first kind which is ill posed. Therefore, the stability of reconstruction of radiobiological parameters presents a problem even for the simplest models of tumor response. To study stability of the parameter reconstruction problem, we used a set of serial CT imaging data for head and neck cancer and a simplest case of a two-level cell population model of tumor response. Inverse reconstruction was performed using a simulated annealing algorithm to minimize a least squared objective function. Results show that the reconstructed values of cell surviving fractions and cell doubling time exhibit significant nonphysical fluctuations if no stabilization algorithms are applied. However, after applying a stabilization algorithm based on variational regularization, the reconstruction produces statistical distributions for survival fractions and doubling time that are comparable to published in vitro data. This algorithm is an advance over our previous work where only cell surviving fractions were reconstructed. We conclude that variational regularization allows for an increase in the number of free parameters in our model which enables development of more-advanced parameter reconstruction algorithms.