Evaluating the biologically effective dose (BED) concept using a dynamic tumor simulation model.

PURPOSE: To evaluate three different formulae for calculating the biologically effective dose (BED) by use of a multipopulation reaction-diffusion simulation to determine whether these formulae produce equivalent effects for different treatment regimes. METHODS: The standard BED formula, BEDs , was updated to account both for spacial nonuniformity in dose and for cellular regrowth between fractions, by creating two new formulae: BEDϕ and BEDϕT . These BED formulae were used to calculate dose per fraction values for two, three, and five fraction treatments and to compare the tumor volumes of those treatments to those of a single fraction. A spherical tumor model based on the reaction-diffusion equation was used to calculate the final volume of each tumor 185 days after the delivery of the first fraction. The percent difference in volume between single-fraction and multiple-fraction treatments was used as a measure to test the accuracy of each BED formula. RESULTS: Percent differences in volume between single- and multiple-fraction treatment regimes varied up to approximately 18.5% if the dose per fraction was calculated using BEDs but the delivered dose was nonuniform. Proper application of spacial nonuniformity in dose and tumor regrowth correction factors modified the dose per fraction values by no more than 5%, but resulted in the improvement of simulated tumor volumes down to around 2% or lower difference in volume. CONCLUSIONS: Treatment regimes with the same BED value should have the same effect. However, small changes in the dose per fraction delivered in multiple-fraction treatments can have a large effect on the tumor volume of a treatment when the dose is delivered nonuniformly or when tumor regrowth between fractions is ignored. Inclusion of these correction factors is important for the underlying assumption that treatments with equal BED will have equal effects on the clinically observed tumor volume.

A mathematical model of tumor growth and its response to single irradiation.

BACKGROUND: Mathematical modeling of biological processes is widely used to enhance quantitative understanding of bio-medical phenomena. This quantitative knowledge can be applied in both clinical and experimental settings. Recently, many investigators began studying mathematical models of tumor response to radiation therapy. We developed a simple mathematical model to simulate the growth of tumor volume and its response to a single fraction of high dose irradiation. The modelling study may provide clinicians important insights on radiation therapy strategies through identification of biological factors significantly influencing the treatment effectiveness. METHODS: We made several key assumptions of the model. Tumor volume is composed of proliferating (or dividing) cancer cells and non-dividing (or dead) cells. Tumor growth rate (or tumor volume doubling time) is proportional to the ratio of the volumes of tumor vasculature and the tumor. The vascular volume grows slower than the tumor by introducing the vascular growth retardation factor, θ. Upon irradiation, the proliferating cells gradually die over a fixed time period after irradiation. Dead cells are cleared away with cell clearance time. The model was applied to simulate pre-treatment growth and post-treatment radiation response of rat rhabdomyosarcoma tumors and metastatic brain tumors of five patients who were treated with Gamma Knife stereotactic radiosurgery (GKSRS). RESULTS: By selecting appropriate model parameters, we showed the temporal variation of the tumors for both the rat experiment and the clinical GKSRS cases could be easily replicated by the simple model. Additionally, the application of our model to the GKSRS cases showed that the α-value, which is an indicator of radiation sensitivity in the LQ model, and the value of θ could be predictors of the post-treatment volume change. CONCLUSIONS: The proposed model was successful in representing both the animal experimental data and the clinically observed tumor volume changes. We showed that the model can be used to find the potential biological parameters, which may be able to predict the treatment outcome. However, there is a large statistical uncertainty of the result due to the small sample size. Therefore, a future clinical study with a larger number of patients is needed to confirm the finding.