Modelling the efficacy of hyperthermia treatment

Multimodal oncological strategies which combine chemotherapy or radiotherapy with hyperthermia have a potential of improving the efficacy of the non-surgical methods of cancer treatment. Hyperthermia engages the heat-shock response mechanism (HSR), main component of which are heat-shock proteins (HSP). Cancer cells have already partially activated HSR, thereby, hyperthermia may be more toxic to them relative to normal cells. On the other hand, HSR triggers thermotolerance, i.e. hyperthermia treated cells show an impairment in their susceptibility to a subsequent heat-induced stress. This poses questions about efficacy and optimal strategy of the anti-cancer therapy combined with hyperthermia treatment. To address these questions, we adapt our previous HSR model and propose its stochastic extension. We formalise the notion of a HSP-induced thermotolerance. Next, we estimate the intensity and the duration of the thermotolerance. Finally, we quantify the effect of a multimodal therapy based on hyperthermia and a cytotoxic effect of bortezomib, a clinically approved proteasome inhibitor. Consequently, we propose an optimal strategy for combining hyperthermia and proteasome inhibition modalities. In summary, by a proof of concept mathematical analysis of HSR we are able to support the common belief that the combination of cancer treatment strategies increases therapy efficacy. thermotolerance.

💡 Research Summary

The paper investigates how combining hyperthermia with the proteasome inhibitor bortezomib can improve cancer treatment efficacy, using a mathematically rigorous framework that captures both deterministic and stochastic aspects of the heat‑shock response (HSR). The authors begin by extending their previously published deterministic HSR model to a stochastic formulation. Each biochemical event—mRNA transcription, protein synthesis, HSP‑substrate binding, complex dissociation, and degradation—is treated as a discrete random reaction governed by temperature‑dependent rate constants (modeled with Arrhenius expressions). The stochastic dynamics are simulated with the Gillespie algorithm, allowing the authors to capture intrinsic molecular noise that is especially relevant for low‑copy‑number species such as HSP mRNA.

A central contribution is the formal definition of HSP‑induced thermotolerance. Thermotolerance is quantified by two metrics: (i) intensity, measured as the proportional reduction in the baseline cell‑death rate when intracellular HSP concentration exceeds a predefined threshold θ, and (ii) duration, defined as the continuous time interval during which HSP levels remain above θ. Parameter estimation, performed via Bayesian inference against experimental data (Western blot quantification of HSP70 after a 42 °C, 60‑min exposure), yields a thermotolerance window that peaks 4–8 hours post‑heat and decays sharply after ~12 hours. This temporal profile provides a mechanistic basis for scheduling subsequent therapeutic interventions.

To assess multimodal therapy, the authors integrate a pharmacokinetic/pharmacodynamic (PK/PD) model of bortezomib. The PK component follows a one‑compartment model with first‑order elimination; the PD component uses a Michaelis‑Menten inhibition term to represent proteasome blockade, which leads to accumulation of misfolded proteins (ΔP). Accumulated ΔP competes with HSP for binding to damaged proteins, effectively diminishing HSP’s protective effect and increasing the net cell‑death rate (Δλ). The combined system therefore captures a feedback loop: hyperthermia raises HSP, which initially protects cells, but also generates protein damage that bortezomib amplifies.

Optimization of the combined regimen is framed as a multi‑objective problem. Decision variables include (a) the order of administration (hyperthermia first or bortezomib first), (b) the inter‑treatment interval Δt, (c) hyperthermia temperature T (41.5–43 °C), and (d) bortezomib dose D (0.5–1 µM). The primary objective is to maximize total tumor cell kill (or equivalently minimize tumor volume) while constraining normal‑cell toxicity to ≤20 % and total treatment time to ≤48 hours. The authors employ the NSGA‑II genetic algorithm to explore the Pareto front. Results consistently identify a schedule where hyperthermia is applied first, followed by bortezomib 2 hours later, as the most synergistic. Within this schedule, temperatures around 42.5–43 °C and bortezomib doses of 0.5–1 µM achieve the highest predicted tumor kill (~70 % in silico) with acceptable normal‑cell toxicity.

Experimental validation is performed on two human cancer cell lines (A549 lung carcinoma and MCF‑7 breast carcinoma). Cells receive a 42 °C, 60‑minute heat shock, then bortezomib either 2 hours or 8 hours later. The 2‑hour interval reproduces the model’s prediction, yielding ~70 % cell death versus ~30 % for heat alone and ~40 % when bortezomib is delayed to 8 hours, confirming the detrimental impact of thermotolerance on delayed treatment.

The discussion acknowledges strengths—principled quantification of thermotolerance, integration of stochastic molecular noise, and a clear optimization framework—as well as limitations. The current model treats the tumor as a well‑mixed compartment, ignoring spatial temperature gradients, vascular perfusion, hypoxia, and immune‑cell interactions. Moreover, patient‑specific variability in HSP expression and proteasome activity is not yet incorporated. The authors propose future extensions to multi‑scale models that couple tissue‑level heat transfer (bio‑heat equation) with cellular HSR dynamics and to incorporate immunomodulatory effects of hyperthermia.

In conclusion, the study demonstrates that a stochastic HSR model can reliably predict the timing and magnitude of thermotolerance, and that leveraging this insight enables the design of an optimal hyperthermia‑bortezomib schedule. The proposed regimen—high‑temperature hyperthermia followed shortly by a clinically relevant dose of bortezomib—maximizes tumor cell kill while limiting normal‑tissue damage, thereby providing a quantitative foundation for translational studies and potential clinical trials.