Mathematical modeling of radiotherapy: impact of model selection on estimating minimum radiation dose for tumor control

Abstract

Introduction: Radiation therapy (RT) is one of the most common anticancer therapies. Yet, current radiation oncology practice does not adapt RT dose for individual patients, despite wide interpatient variability in radiosensitivity and accompanying treatment response. We have previously shown that mechanistic mathematical modeling of tumor volume dynamics can simulate volumetric response to RT for individual patients and estimation personalized RT dose for optimal tumor volume reduction. However, understanding the implications of the choice of the underlying RT response model is critical when calculating personalized RT dose.

Methods: In this study, we evaluate the mathematical implications and biological effects of 2 models of RT response on dose personalization: (1) cytotoxicity to cancer cells that lead to direct tumor volume reduction (DVR) and (2) radiation responses to the tumor microenvironment that lead to tumor carrying capacity reduction (CCR) and subsequent tumor shrinkage. Tumor growth was simulated as logistic growth with pre-treatment dynamics being described in the proliferation saturation index (PSI). The effect of RT was simulated according to each respective model for a standard schedule of fractionated RT with 2 Gy weekday fractions. Parameter sweeps were evaluated for the intrinsic tumor growth rate and the radiosensitivity parameter for both models to observe the qualitative impact of each model parameter. We then calculated the minimum RT dose required for locoregional tumor control (LRC) across all combinations of the full range of radiosensitvity and proliferation saturation values.

Results: Both models estimate that patients with higher radiosensitivity will require a lower RT dose to achieve LRC. However, the two models make opposite estimates on the impact of PSI on the minimum RT dose for LRC: the DVR model estimates that tumors with higher PSI values will require a higher RT dose to achieve LRC, while the CCR model estimates that higher PSI values will require a lower RT dose to achieve LRC.

Discussion: Ultimately, these results show the importance of understanding which model best describes tumor growth and treatment response in a particular setting, before using any such model to make estimates for personalized treatment recommendations.

Keywords: mathematical modeling; model comparison; oncology; personalized oncology; radiotherapy.

Figure 1

Study Overview and Hypothesis. The primary objective of this study is to assess how the selection of the underlying mathematical model of tumor volume dynamics affects estimates for optimal RT dose. This is premised on the idea that even when the same observations of tumor volume changes (both due to off-treatment growth and regression due to treatment effect) are used as inputs for differing models of response to RT there may be different estimates of the optimal RT dose.

Figure 2

Tumor dynamics models. (A) Simulated example of tumor growth modeled as logistic growth. The blue curve shows tumor volume, V, over time and the orange line the tumor carrying capacity, K (B) Simulated examples of response to RT, which is modeled by either direct tumor volume reduction (DVR, left) or tumor carrying capacity reduction (CCR, right). Timing of RT fractions, simulated as a standard fractionated RT course with weekday fractions, is shown by the arrows above each plot.

Figure 3

Effect of radiation sensitivity parameter in DVR and CCR models. (A) Tumor volume trajectories simulated using the DVR model with values of α ∈ (0,0.2) Gy^-1, where larger α values lead to greater decrease in tumor volume. (B) Tumor volume trajectories simulated using the CCR model with values of δ ∈ (0,0.2), where larger δ values lead to greater decrease in tumor volume. For both models, λ = 0.1 day^-1 and PSI = 0.9, and the value of the respective radiation sensitivity parameters are indicated by the color bar. All simulations have an arbitrary initial tumor volume with a fractionated RT regimen, where treatment is applied every weekday for a total of five weeks of treatment.

Figure 4

Effect of intrinsic tumor growth rate, λ, in DVR and CCR models. (A) Tumor volume trajectories simulated using the DVR model with α = 0.1 Gy^-1 and λ ∈ (0,0.1) day^-1. Lower λ values lead to greater net reduction in tumor volume at the end of the treatment course. (B) Tumor volume trajectories simulated using the CCR model with δ = 0.1 and λ ∈ (0,0.1) day^-1. Higher λ values lead to greater net reduction in tumor volume at the end of the treatment course. The inset shows the initial phase of simulated RT, where the tumor volume remains above the carrying capacity and lower λ still results in lower tumor volumes. For all simulations PSI = 0.7, and the values of λ are indicated by the color bar. All simulations have an arbitrary initial tumor volume with a fractionated RT regimen, where treatment is applied every weekday for a total of five weeks of treatment.

Figure 5

Minimum cumulative dose (D_min) for LRC estimates for DVR and CCR models. (A, E) Sample volume trajectories for representative parameter pairs across the parameter range, where the bold symbols indicate the location on the heatmap in (B, F). Green curves are the tumor volume plotted as function of cumulative dose, which increases linearly with treatment time; horizontal dashed line indicates the 32.2% volume reduction cutoff used to calculate D_min; vertical red dashed line indicates the calculated D_min with the specific value of D_min indicated on the x-axis. Patient-specific parameters for each simulation are indicated on the corresponding plots. For all simulations, λ = 0.07 day^-1. (B, F) Heatmaps of D_min over the clinically relevant range for the radiosensitivity parameter (α or δ) and PSI. All simulations have an arbitrary initial tumor volume with a fractionated RT regimen, where treatment is applied every weekday for a total of five weeks of treatment. Black curves indicate “iso-dose” levels with the number of RT fractions required for the indicated dose. White areas indicate parameter regions where sufficient volume reduction was not achieved in the 8 weeks of simulated RT. (C, G) Plots of the radiosensitivity parameters (α or δ) against D_min for PSI = 0.7, 0.8, 0.9. Colored dots are data points sampled from the heatmaps in (B, F); the corresponding solid lines are exponential fits to the data (fitted coefficients in SI). (D, H) Plots of PSI against D_min for 3 different values of the radiosensitivity parameters. Colored dots are data points sampled from the heatmaps in (B, F); the corresponding solid lines are exponential fits to the data (fitted coefficients in SI Tables 1 – 4 ).