Investigating Tumour Responses to Combinations of Radiotherapy and Hyperthermia

Abstract

Hyperthermia (HT) is a promising candidate for enhancing the efficacy of radiotherapy (RT), but its use in the clinic has been limited by incomplete understanding of its interactions with RT. In this work, we investigate tumour responses to high temperature HT alone and combined with RT, focussing on how two different mechanisms for growth control may impact tumour sensitivity to these treatments. We extend an existing ordinary differential equation model of tumour growth and RT response to include high HT. In the absence of treatment, this model distinguishes between growth arrest due to nutrient insufficiency and competition for space, and exhibits three growth regimes: nutrient limited (NL), space limited (SL) and bistable (BS), where both mechanisms for growth arrest coexist. We construct three virtual tumour populations corresponding to the NL, SL and BS regimes and, for each population, we identify the treatment (RT, HT or RT + HT) and dosing regimen that maximise the reduction in tumour burden at the treatment end-point. We thus generate experimentally testable predictions that may explain highly variable experimental and clinical responses to RT and HT and assist patient-specific treatment design.

Keywords: Growth arrest mechanisms; Hyperthermia; ODE system; Radiotherapy.

Fig. 1

Schematic showing the interactions between T, $T_{\text{S}}$, $T_{\text{R}}$ and $T_{\text{H}}$ cells, the vasculature, V, and the oxygen concentration, c, in response to high HT (purple) and RT (orange) in the model (2)–(7).

Fig. 2

Given a high HT treatment of duration ${\delta }_{\text{H}}=60\text{min}$ starting at time $t_{\text{H}}=0$, we plot ${\mu }^{\ast }(t)$, defined in Eq. (13), for $t\in [0,7]\text{days}$. The parameters ${\mu }_0={\mu }_0^{\ast }\tau$, with $\tau =1\text{min}$, $k_2$ and ${\mu }_{\mathrm{\Lambda }}$ are fixed to the values stated in Table 2, $k_1=100\tilde{\beta }$ and $\tilde{\beta }\in \{0.001,0.005,0.01\}$. There is no significant difference in the time evolution of ${\mu }^{\ast }$ as $k_1$ varies. Thus, in this parameter regime, the inhibition of DNA repair is treatment-independent.

Fig. 3

Scatter plots showing the steady state tumour volume, $T^{\ast }$, and logarithm of the oxygen concentration, $ln(c^{\ast })$, in the absence of treatment for each $(q_3,q_1)$ pair and fixed value of V(0) used to generate the virtual NL, SL and BS tumours. We show the NL steady states for the BS cohort; its SL steady states are qualitatively the same as those for the SL cohort, except $T^{\ast }:=1-V(0)=0.99725$ and $ln(c^{\ast })\in [-4.6,-0.55]$.

Fig. 4

Schematic showing the RT, HT and $\text{RT}+\text{HT}$ fractionation schedules whose efficacy we compare when combining a conventional RT schedule comprising $5\times 2\text{Gy}$ for $N_{\text{wks}}=8$ weeks and a conventional HT schedule with $\tilde{\beta }=0.005$. Yellow lightning symbols represent a RT fraction and red flame symbols represent a high HT fraction.

Fig. 5

Schematic showing our two-step process for comparing RT, HT and $\text{RT}+\text{HT}$ treatments on a toy dataset generated by taking a subset of the simulation results for a particular tumour in the SL cohort. The orange boxes highlight the best treatment, for comparable RT, HT and $\text{RT}+\text{HT}$ schedules, and the red box highlights the best treatment and dosing regimen for this tumour.

Fig. 6

Violin plots of the distributions of ${\mathrm{\Delta }}_{\text{viable}}^{\text{R}}$ and ${\mathrm{\Delta }}_{\text{total}}^{\text{R}}$ following conventional RT for tumours in the a NL and SL cohorts, and b BS cohort. SL tumours respond well to RT as they experience reductions in viable and total tumour volumes, while NL tumours have more limited responses, with smaller reductions in viable volume and increases in total volume. In the BS cohort, RT has a deleterious effect. Figures a and b were reproduced from Colson et al. (2023).

Fig. 7

For the NL and SL cohorts, we show how the efficacy of RT depends on the values of $q_1$ and $q_3$. We stratify tumour responses into five classes: negative, limited, moderate, positive and strongly positive, as described in Sect. 3.3.

Fig. 8

Violin plots of the distributions of ${\mathrm{\Delta }}_{\text{V}}^{\text{H}}$, ${\mathrm{\Delta }}_{\text{viable}}^{\text{H}}$ and ${\mathrm{\Delta }}_{\text{total}}^{\text{H}}$ following conventional high HT in the NL, SL and BS cohorts. All but one NL tumour respond positively to treatment, with typically greater reductions in tumour cell and vascular volumes in the NL and BS cohorts than in the SL cohort.

Fig. 9

For the NL and SL cohorts, we show how the efficacy of conventional high HT depends on the values of $q_1$ and $q_3$. Tumour responses are stratified into five classes: negative, limited, moderate, positive and strongly positive, as described in Sect. 3.3. $A_1-C_1$ are representative NL tumours corresponding to $(q_3,q_1)$ sets $(2.72\times {10}^{-1},5.03\times {10}^{-1}),(4.01\times {10}^{-2},7.78)$ and (9.94, 7.60), respectively, while $A_2-C_2$ are representative SL tumours corresponding to $(q_3,q_1)$ sets $(2.10\times {10}^{-1},1.51\times {10}^{-1})$, $(1.43\times {10}^{-1},2.14)$ and (7.61, 2.14), respectively.

Fig. 10

Schematic showing the relationships between the undamaged tumour volume, T, HT-damaged tumour volume, $T_{\text{H}}$, vascular volume, V, and oxygen concentration, c, in model (14)–(19). When HT is applied, T cells are lethally damaged, becoming $T_{\text{H}}$ cells. HT also kills V, while $T_{\text{H}}$ cells induce vascular regrowth via angiogenesis; the balance of these two processes determines whether V decreases or increases during treatment. Increases (decreases) in V lead to an increase (decrease) in the supply of c to the tumour and, thus, an increase (decrease) in tumour cell proliferation and decrease (increase) in hypoxic cell death. The balance between changes in oxygen supply by V and oxygen consumption by T determine whether c increases or decreases during treatment, which, in turn, determines the extent of regrowth of T during treatment.

Fig. 11

For high HT ($\tilde{\beta }=0.005$), we solve Eqs. (14)–(19) for $t\in (0,8.064\times {10}^4]$ subject to initial conditions (22). We fix $V(0)=0.0005$ and $(q_3,q_1)$ as indicated by the points $A_1$-$C_1$ in Fig. 9. Low values of $q_1$ and $q_3$ ($A_1$) lead to worse treatment outcomes than high values of $q_1$ and/or $q_3$ ($B_1$, $C_1$) as they are associated with greater accumulation of dead material, HT-induced angiogenesis and tumour cell regrowth.

Fig. 12

Schematics showing the biological processes that explain positive and negative responses to high HT in the a NL and b SL cohorts and how they depend on the values of $q_1$ and $q_3$.

Fig. 13

For high HT ($\tilde{\beta }=0.005$), we solve Eqs. (14)–(19) for $t\in (0,8.064\times {10}^4]$ subject to initial conditions (22). We fix $V(0)=0.005$ and $(q_3,q_1)$ as indicated by the points $A_2-C_2$ in Fig. 9. Low values of $q_1$ ($A_2$) lead to worse outcomes than high values of $q_1$ ($B_2$, $C_2$) as higher oxygen levels promote greater tumour regrowth and dead cell accumulation.

Fig. 14

For the NL, BS and SL virtual cohorts, we show how the distributions of ${\mathrm{\Delta }}_{\text{V}}^{\text{H}}$, ${\mathrm{\Delta }}_{\text{viable}}^{\text{H}}$ and ${\mathrm{\Delta }}_{\text{total}}^{\text{H}}$ change as $\tilde{\beta }\in \{0.001,0.005,0.01\}$ varies (for fixed $N_{\text{wks}}=8$). The values of ${\mathrm{\Delta }}_{\text{V}}^{\text{H}}$, ${\mathrm{\Delta }}_{\text{viable}}^{\text{H}}$ and ${\mathrm{\Delta }}_{\text{total}}^{\text{H}}$ typically decrease as $\tilde{\beta }$ increases.

Fig. 15

Scatter plots showing the HT dosing regimen (defined in the table) which maximises $|{\mathrm{\Delta }}_{\text{viable}}^{\text{H}}+{\mathrm{\Delta }}_{\text{total}}^{\text{H}}|$ for each $(q_3,q_1)$ pair used to generate the NL, BS and SL virtual cohorts. The best dosing regimen depends on the efficacy of high HT: when HT response is weaker, larger HT fractions applied over a shorter time period typically maximise treatment efficacy (purple), whereas, when HT response is stronger, lower HT fractions applied over a longer time period are typically best (orange).

Fig. 16

We solve Eqs. (14)–(19) subject to initial conditions (22). We fix $V(0)=0.0005$, and $(q_3,q_1)$ as indicated by points $A_1$ and $C_1$ in Fig. 9. In both cases, we fix the thermal dose (left) $\tilde{\beta }=0.001$ or (right) $\tilde{\beta }=0.01$, and simulate high HT for (left) 80 weeks or (right) 8 weeks. A larger thermal dose is necessary for $A_1$ to achieve a sustained reduction in tumour burden. In contrast, a lower thermal dose applied over a longer period yields a more gradual, but larger, reduction in tumour burden for $C_1$.

Fig. 17

For the $(q_3,q_1)$ pairs used to generate the SL virtual cohort, scatter plot a shows the best treatment and scatter plot b shows the values of $|{\mathrm{\Delta }}_{\text{total}}^{\text{R}}+{\mathrm{\Delta }}_{\text{viable}}^{\text{R}}|-|{\mathrm{\Delta }}_{\text{total}}^{\text{H}}+{\mathrm{\Delta }}_{\text{viable}}^{\text{H}}|$ for comparable RT and HT protocols (recall Fig. 4) corresponding to the optimal schedules in Fig. 18. Tumours that respond much better to RT than HT typically respond best to RT, and vice versa, while tumours with comparable responses to RT and HT alone respond best to combined treatment.

Fig. 18

Scatter plots showing the best dosing regimen (defined in the table) for the $(q_3,q_1)$ pairs used to generate the SL virtual cohort. When HT alone is best (see Fig. 17), a longer type 1 regimen with lower HT fractions is typically recommended for tumours that have a stronger response to HT alone and weaker response to RT alone. A shorter type 2 regimen with higher HT fractions is recommended for tumours with a weaker response to HT and stronger response to RT. When $\text{RT}+\text{HT}$ is best (see Fig. 17), tumours with low values of $q_3$ respond best to the type 2 regimen which includes lower RT fractions, while tumours with high values of $q_3$ respond best to a longer type 3 regimen combining low HT fractions and high RT fractions. When RT alone is best (see Fig. 17), high RT fractions applied at high frequency maximise treatment efficacy.

Fig. 19

For RT ($R=0.5$, $N_{\text{frac}}=1$) and high HT ($\tilde{\beta }=0.005$) schedules, alone and combined, we solve Eqs. (14)–(19) for $t\in (0,1.6128\times {10}^5]$ subject to initial conditions (22). We set $(V(0),q_3,q_1)=(0.005,8.81,3.74\times {10}^{-1})$. This SL tumour has a limited response to high HT and a positive response to RT (compared to the rest of the SL cohort). Combined treatment is most effective as RT and HT act synergistically, with RT significantly enhancing HT effects.

Fig. 20

For conventional RT and HT schedules, alone and combined, we solve Eqs. (14)–(19) for $t\in (0,8.064\times {10}^4]$ subject to initial conditions (22). We set $(V(0),q_3,q_1)=(0.005,9.53,3.21\times {10}^{-2})$. This SL tumour responds positively to RT, but has a limited response to high HT. During combined treatment, HT-induced vascular damage reduces oxygen levels and RT cell kill rates. Combined with fast tumour regrowth between fractions, this slows the reduction in viable volume, enabling net growth of the viable volume. As a result, RT alone is the best treatment.