Modeling the effect of intratumoral heterogeneity of radiosensitivity on tumor response over the course of fractionated radiation therapy

Abstract

Background: Standard radiobiology theory of radiation response assumes a uniform innate radiosensitivity of tumors. However, experimental data show that there is significant intratumoral heterogeneity of radiosensitivity. Therefore, a model with heterogeneity was developed and tested using existing experimental data to show the potential effects from the presence of an intratumoral distribution of radiosensitivity on radiation therapy response over a protracted radiation therapy treatment course.

Methods: The standard radiation response curve was modified to account for a distribution of radiosensitivity, and for variations in the repopulation rates of the tumor cell subpopulations. Experimental data from the literature were incorporated to determine the boundaries of the model. The proposed model was then used to show the changes in radiosensitivity of the tumor during treatment, and the effects of fraction size, α/β ratio and variation of the repopulation rates of tumor cells.

Results: In the presence of an intratumoral distribution of radiosensitivity, there is rapid selection of radiation-resistant cells over a course of fractionated radiation therapy. Standard treatment fractionation regimes result in the near-complete replacement of the initial population of sensitive cells with a population of more resistant cells. Further, as treatment progresses, the tumor becomes more resistant to further radiation treatment, making each fractional dose less efficacious. A wider initial distribution induces increased radiation resistance. Hypofractionation is more efficient in a heterogeneous tumor, with increased cell kill for biologically equivalent doses, while inducing less resistance. The model also shows that a higher growth rate in resistant cells can account for the accelerated repopulation that is seen during the clinical treatment of patients.

Conclusions: Modeling of tumor cell survival with radiosensitivity heterogeneity alters the predicted tumor response, and explains the induction of radiation resistance by radiation treatment, the development of accelerated repopulation, and the potential beneficial effects of hypofractionation. Tumor response to treatment may be better predicted by assaying for the distribution of radiosensitivity, or the extreme of the radiosensitivity, rather than measuring the initial, general radiation sensitivity of the untreated tumor.

Keywords: Accelerated repopulation; Fractionated radiotherapy; Intratumoral radiosensitivity heterogeneity; Linear-quadratic model; Radiation resistance.

Fig. 1

Model fitting of in vitro measurements in [30] of the change in the radiosensitivity of OE33 esophageal adenocarcinoma cell culture after exposure to fractionated radiation. a Representation of pre-treatment intratumoral distribution of α and β parameters. The sum of cell subset percentages is equal to 100%. b SF₂ distribution with respect to α and β parameters. c Model fitting of tumor cell survival curves, and (d) the corresponding pre- and post-treatment α and β distributions. Colormap in (d) represents the normalized densities of each tumor cell subsets before and after treatment, where the red arrow pointing from right to left represents the evolution of α and β from pre- to post-treatment values. Pre-treatment parameters in Eq. (3) were α_c = 0.40 Gy^− 1, β_c = 0.02 Gy^− 2, σ_α = 6.5 × 10^− 2 Gy^− 1 and σ_β = 3.5 × 10^− 3 Gy^− 2

Fig. 2

Model fitting of in vitro measurements in [31] of the change in the radiosensitivity of LNCaP, PC3, and Du145 prostate cancer cell cultures after exposure to fractionated radiation. Pre-treatment parameters in Eq. (3) were (a) α_c = 0.43 Gy^− 1, (b) α_c = 0.35 Gy^− 1 and (c) α_c = 0.30 Gy^− 1 with β_c = 0.02 Gy^− 2, σ_α = 1.0 × 10^− 1 Gy^− 1 and σ_β = 3.5 × 10^− 3 Gy^− 2

Fig. 3

Variation of radiosensitivity during fractionated radiation therapy of a tumor as in Fig. 1. a Survival of tumor cell subsets characterized by different SF₂ values in response to a standard fractionation scheme of 70 Gy in 35 daily fractions at 2.0 Gy/day with weekend interruptions. b Intratumoral composition of radiosensitivity for different tumor cell subsets before, during and after treatment. c-e Pre- and post-treatment α and β distributions after 15, 25 and 35 fractions. Colormap represents the normalized densities of tumor cell subsets before treatment and after 15, 25 and 35 fractions. The red arrows pointing from right to left represents the evolution of α and β from pre- to post-treatment values. f-g Variation on the mean values and standard deviations of α/β ratio and SF₂ within the tumor during fractionated radiation therapy

Fig. 4

Comparison of treatment response of tumors characterized by same distribution parameters in Eq. (3) but centered at different α and β values. a-c Pre- and post-treatment intratumoral radiosensitivity distributions catered at different α and β combinations compared to the pre-treatment distribution in Fig. 3, dashed ellipses distribution 1 (D1), after a standard fractionation scheme of 70 Gy in 35 daily fractions at 2.0 Gy/day with weekend interruptions. Colormap represents the normalized densities of tumor cell subsets before and after treatment. The red arrows pointing from right to left represents the evolution of α and β from pre- to post-treatment values. d-f Intratumoral composition of radiosensitivity for different cell subsets in distribution 2 (D2) in (a), distribution 3 (D3) in (b) and distribution 4 (D4) in (c) before, during and after treatment. Shift in the mean values of (g) SF₂ and (h) α/β ratio during treatment with respect to the pre-treatment values. Pre-treatment parameters in Eq. (3) were α_c = 0.40 Gy^− 1 and β_c = 0.04 Gy^− 2 (D2 in a), α_c = 0.25 Gy^− 1 and β_c = 0.02 Gy^− 2 (D3 in b), and α_c = 0.33 Gy^− 1 and β_c = 0.033 Gy^− 2 (D4 in c) with σ_α = 6.5 × 10^− 2 Gy^− 1 and σ_β = 3.5 × 10^− 3 Gy^− 2

Fig. 5

Comparison of treatment response of tumors characterized by different distribution widths in Eq. (3) and centered at same α and β parameters. a-b Pre- and post-treatment α and β distributions of different radiosensitivity heterogeneities (widths) compared to the pre-treatment distribution 4 (D4) in Fig. 4c (dashed ellipses) after a standard fractionation scheme of 70 Gy in 35 daily fractions at 2.0 Gy/day with weekend interruptions. Colormap represents the normalized densities of tumor cell subsets before and after treatment. The red arrows pointing from right to left represents the evolution of α and β from pre- to post-treatment values. c Tumor cell survival in response to fractionated radiotherapy. Shift in the mean values of (d) SF₂ and (e) α/β ratio during treatment with respect to the pre-treatment values. f-g Intratumoral composition of radiosensitivity for different cell subsets in distribution 5 (D5) in (a) and distribution 6 (D6) in (b) before, during and after treatment. Pre-treatment parameters in Eq. (3) were σ_α = 4.5 × 10^− 2 Gy^− 1 (D5 in a) and σ_α = 8.5 × 10^− 2 Gy^− 1 (D6 in b) with σ_β = 3.5 × 10^− 3 Gy^− 2, α_c = 0.33 Gy^− 1 and β_c = 0.033 Gy^− 2

Fig. 6

Variation of intratumoral radiosensitivity during BED equivalent fractionated radiation therapy. a Survival of a tumor to different BED-equivalent radiation therapy fractionation schemes at 2.0Gy/day, 2.4Gy/day, 3.0Gy/day, 4.2Gy/day and 7.0Gy/day in 25, 20, 15, 10 and 5 fractions with 5 consecutive treatments per 7-day week. b Intratumoral composition of radiosensitivity for different tumor cell subsets before and after treatments at 2.0Gy/day and 7.0Gy/day in 25 and 5 fractions with 5 consecutive treatments per 7-day week. c Intratumoral distribution of radiosensitivity parameters α and β before and after treatments. The red arrows pointing from right to left represents the evolution of α and β from pre- to post-treatment values. Shift in the mean values of (d) SF₂ and (e) α/β ratio within the tumor at the end of BED-equivalent fractionation schemes. Pre-treatment parameters in Eq. (3) were α_c = 0.33 Gy^− 1 and β_c = 0.033 Gy^− 2 with σ_α = 6.5 × 10^− 2 Gy^− 1 and σ_β = 3.5 × 10^− 3 Gy^− 2 corresponding to the distribution 4 (D4) in Fig. 4c

Fig. 7

Radiotherapy response of tumors with more rapidly and slowly growing resistant cells. a, c Non-uniform distribution of daily repopulation rates, (a, b) rapidly and (c, d) slowly growing resistant cells, within a tumor as in Fig. 4c. Parameters in Eq. (4) were μ = 0.3 and θ = 5.5. b, d Tumor cell survival to BED-equivalent fractionations at 2.0 Gy/day in 25 fractions and 3.0 Gy/day in 15 fractions both with weekend interruptions. Results were obtained for a uniform repopulation of 15% and the non-uniform repopulation distributions as in (a, c)