Elsevier

Mathematical Biosciences

Volume 329, November 2020, 108456
Mathematical Biosciences

Original Research Article
A computational study of combination HIFU–chemotherapy as a potential means of overcoming cancer drug resistance

https://doi.org/10.1016/j.mbs.2020.108456Get rights and content

Highlights

  • A PDE model is proposed to study the effect of hyperthermia–chemotherapy treatment.

  • Efficacy of the combined treatment depends on the temperature at the focal point.

  • Hyperthermia can delay the re-formation of an aggressive tumor.

  • Increasing the exposure time to hyperthermia makes the combined treatment efficient.

  • The effect of hyperthermia is more pronounced at high concentrations of drugs.

Abstract

The application of local hyperthermia, particularly in conjunction with other treatment strategies (like chemotherapy and radiotherapy) has been known to be a useful means of enhancing tumor treatment outcomes. However, to our knowledge, there has been no mathematical model designed to capture the impact of the combination of hyperthermia and chemotherapies on tumor growth and control. In this study, we propose a nonlinear Partial Differential Equation (PDE) model which describes the tumor response to chemotherapy, and use the model to study the effects of hyperthermia on the response of prototypical tumor to the generic chemotherapeutic agent. Ultrasound energy is delivered to the tumor through High Intensity Focused Ultrasound (HIFU), as a noninvasive technique to elevate the tumor temperature in a controlled manner. The proposed tumor growth model is coupled with the nonlinear density dependent Westervelt and Penne’s bio-heat equations, used to calculate the net delivered energy and temperature of the tumor and its surrounding normal tissue. The tumor is assumed to be composed of two species: drug-sensitive and drug-resistant. The central assumption underlying our model is that the drug-resistant species is converted to a drug-sensitive type when the tumor temperature is elevated above a certain threshold temperature. The ”in silico” results obtained, confirm that hyperthermia can result in less aggressive tumor development and emphasize the importance of designing an optimized thermal dose strategy. Furthermore, our results suggest that increasing the length of the on/off cycle of the transducer is an efficient approach to treatment scheduling in the sense of optimizing tumor eradication.

Introduction

The challenges posed by cancer to human health are formidable and have preoccupied numerous researchers over the past several decades. Different approaches have been proposed to eradicate tumors and prevent tumor metastasis, and these include surgery, radiotherapy, chemotherapy, hormonal therapy, and (more recently) immunotherapy. All these approaches have their associated side effects and limitations and their success depends on numerous factors such as the cancer type, the stage at which cancer is diagnosed and patient age and gender [1], [2], [3], [4]. However, even if the first administration of these strategies is successful, patients have a high probability of relapse if they are treated simply by these approaches. High Intensity Focused Ultrasound (HIFU) is an emerging treatment strategy that has been used effectively for cancer treatment. Its primary usage is to destroy tumor colonies in a minimally invasive manner. For this purpose, the temperature of a tumor region lying within the focal area of a transducer is elevated to 65 C or higher and ablation occurs. In addition to complete necrosis, HIFU can relieve the tumor associated pain, which is not well controlled by pharmacological agents. Furthermore, it can eradicate tumor cells, control malignant proliferation, and prevent metastasis [4], [5]. However, HIFU alone is not an efficient treatment approach for some types of cancers and relapse can occur [6], [7]. Moreover, controlling and monitoring the temperature of surrounding normal tissue presents unique challenges, especially in tissue environments containing bone. Besides tumor ablation and all the other associated advantages of using HIFU in cancer treatment, HIFU is a type of local external hyperthermia treatment (a controlled temperature rise) that can be used in combination therapies for example with chemotherapy and/or radiotherapy to increase therapeutic effects [8], [9], [10], [11]. In local hyperthermia, the temperature of a solid tumor is elevated above the normal body temperature to achieve certain therapeutic effects. In most cases, the threshold temperature that activates various physiological responses such as the release of Heat Shock Proteins (a specific group of proteins that chaperone denatured proteins and thereby prevent formation of toxic protein aggregates [12]), and improved tumor perfusion [8] is 42 °C. It has been reported that tumors are more susceptible to hyperthermia than normal tissues because of their faster cell division, increased hypoxia, low pH, and limited temperature regulation due to poor fluid transfer [13]. Therefore, hyperthermia does not cause serious injury to adjoining normal tissues as a result of the elevated temperatures. Several mechanisms have been proposed by which hyperthermia either sensitizes the tumors to therapeutic interventions or strengthens the efficiency of other strategies (more specifically chemotherapy). These mechanisms include but are not limited to enhanced inhibition of clonogenic cell growth in vitro at higher temperatures [8], sensitization of cells to DNA damaging agents, interference with the cell cycle, DNA and protein synthesis [10], [14], improvement of anti-tumor immunity [15], [16] and sensitization of cells exposed to chemotherapy [17]. Hyperthermia can also enhance the delivery of drugs to targeted regions without exceeding the maximum tolerated dose and without the associated complications of toxicity in healthy tissues [18], [19]. Some positive effects of hyperthermia on drug transport include increased drug diffusion, and a change in the cell density and shape that may result in enhanced drug transport [20]. Nevertheless, these responses to thermal stress may cause negative effects. For example the increase in perfusion in tumors can enhance drug removal from the tumor [20] or the associated release of heat shock proteins may play a role in the development of thermotolerance [12]. Among all the aforementioned mechanisms, heat-induced repair of DNA damage has been extensively studied. However, how hyperthermia sensitizes cells to DNA damaging agents still has not been fully elucidated [10].

Several mathematical models at different levels of complexity have been proposed to describe the homogeneous and heterogeneous development of tumor colonies [21], [22], [23], [24], [25], [26]. The focus of these studies is on using drugs as a single modal treatment strategy and on identifying crucial parameters that influence their efficacy. For this purpose, systems of coupled spatio-temporal or ordinary differential equations are used to capture the spatial and temporal behavior of tumor colonies and drug concentrations where, without loss of generality, a few simplifying assumptions are made. Authors in these studies assume that the evolution of the cell populations is influenced by a constant diffusion rate and as a result of proliferation, or that the populations evolve only through proliferation. Another assumption that is made in all cancer growth modeling (to the best of our knowledge) is that there are no voids within the tumor i.e. summation of tumor volume fractions remains constant. If we consider tumors to be composed of a group of multi-species cells that grow and diffuse, then the constraint that all volume fractions should sum to unity implies movement of cells away from the initial tumor mass. This puts a severe limitation on the ability of these models to predict the spatial interaction between different species of tumor cells. Furthermore, the models (under this assumption) are not able to describe the hollowing phenomena that may occur as a consequence of nutrient limitations or cell lysis.

There are some computational studies that investigate the ability of HIFU to eradicate tumors. These studies use the Rayleigh–Sommerfeld integral or Westervelt equation to simulate the effects of acoustic wave propagation and the Pennes’ bio-heat transfer equation to calculate the temperature field in solid tumors and the surrounding microenvironment [27], [28], [29]. The main objectives of these studies are to find a way to ablate the tumor colonies without inducing serious damage to the surrounding normal tissue and to decrease the computational cost of the applied numerical algorithms.

Although there are several clinical, experimental, and computational studies that examine the effects of hyperthermia and chemotherapy individually (as single modal treatment strategies) on tumor development [4], [15], [18], [30], [31], to the best of our knowledge, no attempt has been made so far to introduce a mathematical model that describes the impact of applying combined hyperthermia–chemotherapy treatment on tumor growth and control. Our main objective in this study is to fill this gap in the literature and relax the assumptions made in previous tumor growth models. For this purpose, we introduce a mathematical model in which the tumor is assumed to be composed of two different cancer species with different characteristics, which determine specific susceptibility to medical treatment. We will use nonlinear reaction–diffusion equations to describe the development of tumor colonies and these equations will be coupled with a semi-linear equation for drug resource. To predict the temperature field, we use the Pennes’ bio-heat equation and to determine the net acoustic energy distribution produced by the ultrasound propagation we use the nonlinear density dependent Westervelt equation. Note that the model that we introduce and the parameter values that we use for our simulation results are for a generic tumor and drug and we do not focus on any specific tumor-type or chemotherapeutic agent. However, the results obtained, provide insight into the potential efficacy of a hypothermia–chemotherapy combination treatment approach and lay the foundation for further investigation and future work (both experimental and theoretical).

Section snippets

Governing equation

Our model consists of two coupled sets of equations which are coupled together, which can be solved separately due to a separation of time scales. The first set includes the density dependent Westervelt equation that describes the nonlinear propagation of acoustic wave [32] denoted by: ρ1ρp=1c022pt2δc043pt3βρ0c042p2t2In Eq. (1), p[Pa] is the acoustic pressure, ρ[g/m3] and c0[m/s] are the density and sound velocity of the acoustic medium respectively, β is a nonlinear coefficient, δ is

Existence of solution

In this study the model is formulated in terms of a system of nonlinear equations in bounded domains ΩχRn, χ=P,T,M,n{2,3}. Due to the singularity in the diffusion coefficient of tumor species and for the biological and physical interpretation of the model, it is important to show that all dependent variables are nonnegative and that the tumor density is bounded above by unity. Nonlinear diffusion effects make it difficult to show that solutions of the PDE system (2) have these properties. To

Illustrative simulations

The objective in this section is to illustrate how hyperthermia might lead to the transformation of drug-resistant to drug-sensitive species and consequently a more effective eradication procedure and to find an optimum simulation set-up (in terms of achieving maximum removal of tumor) for the following sections. For this purpose we first discuss in detail a typical simulation, illustrated in Fig. 2, Fig. 3. Two treatment strategies are considered: a single modal treatment approach where only

Conclusion

Our primary objective in this paper was to introduce a spatio-temporal model to explore the effects of combining hyperthermia and chemotherapy on tumor growth through computer simulations. Our model incorporates two types of tumor species (drug-resistant and drug-sensitive), as well as drug concentration, with the governing equations given by Penne’s bio-heat equation, and a nonlinear wave equation for the acoustic waves emitted from a transducer (as the energy source for temperature

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research was supported in part by the Natural Science and Engineering Research Council of Canada (NSERC) with a Discovery Grant awarded to SS. We also thank the University of Waterloo for providing financial support for MG as a postdoctoral fellow.

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