Spatio-Temporal Dynamics of Hypoxia during Radiotherapy

Agent-based modeling in cancer biomedicine: applications and tools for calibration and validation

Computational models are not just appealing because they can simulate and predict the development of biological phenomena across multiple spatial and temporal scales, but also because they can integrate information from well-established in vitro and in vivo models and test new hypotheses in cancer biomedicine. Agent-based models and simulations are especially interesting candidates among computational modeling procedures in cancer research due to the capability to, for instance, recapitulate the dynamics of neoplasia and tumor - host interactions. Yet, the absence of methods to validate the consistency of the results across scales can hinder adoption by turning fine-tuned models into black boxes. This review compiles relevant literature that explores strategies to leverage high-fidelity simulations of multi-scale, or multi-level, cancer models with a focus on verification approached as simulation calibration. We consolidate our review with an outline of modern approaches for agent-based models' validation and provide an ambitious outlook toward rigorous and reliable calibration.

Deciphering oxygen distribution and hypoxia profiles in the tumor microenvironment: a data-driven mechanistic modeling approach

Objective. The distribution of hypoxia within tissues plays a critical role in tumor diagnosis and prognosis. Recognizing the significance of tumor oxygenation and hypoxia gradients, we introduce mathematical frameworks grounded in mechanistic modeling approaches for their quantitative assessment within a tumor microenvironment. By utilizing known blood vasculature, we aim to predict hypoxia levels across different tumor types.Approach. Our approach offers a computational method to measure and predict hypoxia using known blood vasculature. By formulating a reaction-diffusion model for oxygen distribution, we derive the corresponding hypoxia profile.Main results. The framework successfully replicates observed inter- and intra-tumor heterogeneity in experimentally obtained hypoxia profiles across various tumor types (breast, ovarian, pancreatic). Additionally, we propose a data-driven method to deduce partial differential equation models with spatially dependent parameters, which allows us to comprehend the variability of hypoxia profiles within tissues. The versatility of our framework lies in capturing diverse and dynamic behaviors of tumor oxygenation, as well as categorizing states of vascularization based on the dynamics of oxygen molecules, as identified by the model parameters.Significance. The proposed data-informed mechanistic method quantitatively assesses hypoxia in the tumor microenvironment by integrating diverse histopathological data and making predictions across different types of data. The framework provides valuable insights from both modeling and biological perspectives, advancing our comprehension of spatio-temporal dynamics of tumor oxygenation.

Functional impact of oxygen-saturated zeolite nanoparticles on macrophages in the context of glioblastoma: an in vitro and in vivo study

In the context of glioblastoma (GBM), hypoxia and inflammation are two main players of the tumor microenvironment. Hypoxia stimulates various features involves in tumor growth and also maintains a specific environment that favors protumor macrophages. Therefore, targeting hypoxia could potentially restore an anti-tumor M1 phenotype in macrophages. Besides, iron demonstrated its capacity to stimulate the polarization of macrophages towards an M1-like phenotype. In this paper we took advantages of microporous nanoparticles to co-deliver both oxygen and iron to bone marrow derived macrophages (BMDM) enabling the investigation of changes in polarization status and proteomic profiles. The nanoparticles were used in two in vivo models of glioblastoma, specifically, in both immunodeficient and immunocompetent settings. Our in vitro findings revealed that iron doped nanoparticles, saturated with oxygen were deemed safe for macrophages but did not demonstrate the capacity to change the M1 or M2 phenotypes. However, these nanoparticles induced some changes in proteomics pathways. The present study reports on in vivo experimentation that revealed the effects of nanoparticles on the hypoxic fraction, tumor volume, and macrophage phenotype in a GBM model. The findings indicated that the presence of nanoparticles led to a reduction in the hypoxic fraction in one of the GBM models, while no significant changes were observed in the tumor volume or macrophage phenotype. The present data showed that nanoparticles possess the capability of delivering both oxygen and iron to macrophages; though, they do not possess the ability to effectively repolarize M2 macrophages. Such strategies could be used in conjunction with other potent molecules to avoid M1 macrophages to inevitably differentiate to M2 macrophages.

Predicting the Tumour Response to Radiation by Modelling the Five Rs of Radiotherapy Using PET Images

Despite the intensive use of radiotherapy in clinical practice, its effectiveness depends on several factors. Several studies showed that the tumour response to radiation differs from one patient to another. The non-uniform response of the tumour is mainly caused by multiple interactions between the tumour microenvironment and healthy cells. To understand these interactions, five major biologic concepts called the "5 Rs" have emerged. These concepts include reoxygenation, DNA damage repair, cell cycle redistribution, cellular radiosensitivity and cellular repopulation. In this study, we used a multi-scale model, which included the five Rs of radiotherapy, to predict the effects of radiation on tumour growth. In this model, the oxygen level was varied in both time and space. When radiotherapy was given, the sensitivity of cells depending on their location in the cell cycle was taken in account. This model also considered the repair of cells by giving a different probability of survival after radiation for tumour and normal cells. Here, we developed four fractionation protocol schemes. We used simulated and positron emission tomography (PET) imaging with the hypoxia tracer 18F-flortanidazole (18F-HX4) images as input data of our model. In addition, tumour control probability curves were simulated. The result showed the evolution of tumours and normal cells. The increase in the cell number after radiation was seen in both normal and malignant cells, which proves that repopulation was included in this model. The proposed model predicts the tumour response to radiation and forms the basis for a more patient-specific clinical tool where related biological data will be included.

Lattice or Oxygen-Guided Radiotherapy: What If They Converge? Possible Future Directions in the Era of Immunotherapy

Palliative radiotherapy has a great role in the treatment of large tumor masses. However, treating a bulky disease could be difficult, especially in critical anatomical areas. In daily clinical practice, short course hypofractionated radiotherapy is delivered in order to control the symptomatic disease. Radiation fields generally encompass the entire tumor mass, which is homogeneously irradiated. Recent technological advances enable delivering a higher radiation dose in small areas within a large mass. This goal, previously achieved thanks to the GRID approach, is now achievable using the newest concept of LATTICE radiotherapy (LT-RT). This kind of treatment allows exploiting various radiation effects, such as bystander and abscopal effects. These events may be enhanced by the concomitant use of immunotherapy, with the latter being ever more successfully delivered in cancer patients. Moreover, a critical issue in the treatment of large masses is the inhomogeneous intratumoral distribution of well-oxygenated and hypo-oxygenated areas. It is well known that hypoxic areas are more resistant to the killing effect of radiation, hence the need to target them with higher aggressive doses. This concept introduces the "oxygen-guided radiation therapy" (OGRT), which means looking for suitable hypoxic markers to implement in PET/CT and Magnetic Resonance Imaging. Future treatment strategies are likely to involve combinations of LT-RT, OGRT, and immunotherapy. In this paper, we review the radiobiological rationale behind a potential benefit of LT-RT and OGRT, and we summarize the results reported in the few clinical trials published so far regarding these issues. Lastly, we suggest what future perspectives may emerge by combining immunotherapy with LT-RT/OGRT.

Phenotypic variation modulates the growth dynamics and response to radiotherapy of solid tumours under normoxia and hypoxia

In cancer, treatment failure and disease recurrence have been associated with small subpopulations of cancer cells with a stem-like phenotype. In this paper, we develop and investigate a phenotype-structured model of solid tumour growth in which cells are structured by a stemness level, which varies continuously between stem-like and terminally differentiated behaviours. Cell evolution is driven by proliferation and death, as well as advection and diffusion with respect to the stemness structure variable. Here, the magnitude and sign of the advective flux are allowed to vary with the oxygen level. We use the model to investigate how the environment, in particular oxygen levels, affects the tumour's population dynamics and composition, and its response to radiotherapy. We use a combination of numerical and analytical techniques to quantify how under physiological oxygen levels the cells evolve to a differentiated phenotype and under low oxygen level (i.e., hypoxia) they de-differentiate. Under normoxia, the proportion of cancer stem cells is typically negligible and the tumour may ultimately become extinct whereas under hypoxia cancer stem cells comprise a dominant proportion of the tumour volume, enhancing radio-resistance and favouring the tumour's long-term survival. We then investigate how such phenotypic heterogeneity impacts the tumour's response to treatment with radiotherapy under normoxia and hypoxia. Of particular interest is establishing how the presence of radio-resistant cancer stem cells can facilitate a tumour's regrowth following radiotherapy. We also use the model to show how radiation-induced changes in tumour oxygen levels can give rise to complex re-growth dynamics. For example, transient periods of hypoxia induced by damage to tumour blood vessels may rescue the cancer cell population from extinction and drive secondary regrowth.

Nanosized zeolites as a gas delivery platform in a glioblastoma model

Approaches able to counteract, at least temporarily, hypoxia, a well-known factor of resistance to treatment in solid tumors are highly desirable. Herein, we report the use of nanosized zeolite crystals as hyperoxic/hypercapnic gas carriers for glioblastoma. First, the non-toxic profile of nanosized zeolite crystals in living animals (mice, rats and non-human primates) and in various cell types is presented. Second, the ability of the nanosized zeolites to act as a vasoactive agent for a targeted re-oxygenation of the tumor after intravenous injection is shown. As attested by an MRI protocol, the zeolites were able to increase oxygenation and blood volume specifically within the brain tumor whilst no changes in the healthy-non tumoral brain-were observed. The first proof of concept for the use of metal-containing nanosized zeolites as a tool for vectorization of hyperoxic/hypercapnic gases in glioblastoma is revealed.

A minimal modeling framework of radiation and immune system synergy to assist radiotherapy planning

Recent evidence indicates the ability of radiotherapy to induce local and systemic tumor-specific immune responses as a result of immunogenic cell death. However, fractionation regimes routinely used in clinical practice typically ignore the synergy between radiation and the immune system, and instead attempt to completely eradicate tumors by the direct lethal effect of radiation on cancer cells. This paradigm is expected to change in the near future due to the potential benefits of considering radiation-induced antitumor immunity during treatment planning. Towards this goal, we propose a minimal modeling framework based on key aspects of the tumor-immune system interplay to simulate the effects of radiation on tumors and the immunological consequences of radiotherapy. The impacts of tumor-associated vasculature and intratumoral oxygen-mediated heterogeneity on treatment outcomes are ininvestigated. The model provides estimates of the minimum radiation doses required for tumor eradication given a certain number of treatment fractions. Moreover, estimates of treatment duration for disease control given predetermined fractional radiation doses can be also obtained. Although theoretical in nature, this study motivates the development and establishment of immune-based decision-support tools in radiotherapy planning.

Stochastic multicellular modeling of x-ray irradiation, DNA damage induction, DNA free-end misrejoining and cell death

The repair or misrepair of DNA double-strand breaks (DSBs) largely determines whether a cell will survive radiation insult or die. A new computational model of multicellular, track structure-based and pO₂-dependent radiation-induced cell death was developed and used to investigate the contribution to cell killing by the mechanism of DNA free-end misrejoining for low-LET radiation. A simulated tumor of 1224 squamous cells was irradiated with 6 MV x-rays using the Monte Carlo toolkit Geant4 with low-energy Geant4-DNA physics and chemistry modules up to a uniform dose of 1 Gy. DNA damage including DSBs were simulated from ionizations, excitations and hydroxyl radical interactions along track segments through cell nuclei, with a higher cellular pO₂ enhancing the conversion of DNA radicals to strand breaks. DNA free-ends produced by complex DSBs (cDSBs) were able to misrejoin and produce exchange-type chromosome aberrations, some of which were asymmetric and lethal. A sensitivity analysis was performed and conditions of full oxia and anoxia were simulated. The linear component of cell killing from misrejoining was consistently small compared to values in the literature for the linear component of cell killing for head and neck squamous cell carcinoma (HNSCC). This indicated that misrejoinings involving DSBs from the same x-ray (including all associated secondary electrons) were rare and that other mechanisms (e.g. unrejoined ends) may be important. Ignoring the contribution by the indirect effect toward DNA damage caused the DSB yield to drop to a third of its original value and the cDSB yield to drop to a tenth of its original value. Track structure-based cell killing was simulated in all 135306 viable cells of a 1 mm³ hypoxic HNSCC tumor for a uniform dose of 1 Gy.

Three-dimensional tumor growth in time-varying chemical fields: a modeling framework and theoretical study

Background

Contemporary biological observations have revealed a large variety of mechanisms acting during the expansion of a tumor. However, there are still many qualitative and quantitative aspects of the phenomenon that remain largely unknown. In this context, mathematical and computational modeling appears as an invaluable tool providing the means for conducting in silico experiments, which are cheaper and less tedious than real laboratory experiments.

Results

This paper aims at developing an extensible and computationally efficient framework for in silico modeling of tumor growth in a 3-dimensional, inhomogeneous and time-varying chemical environment. The resulting model consists of a set of mathematically derived and algorithmically defined operators, each one addressing the effects of a particular biological mechanism on the state of the system. These operators may be extended or re-adjusted, in case a different set of starting assumptions or a different simulation scenario needs to be considered.

Conclusion

In silico modeling provides an alternative means for testing hypotheses and simulating scenarios for which exact biological knowledge remains elusive. However, finer tuning of pertinent methods presupposes qualitative and quantitative enrichment of available biological evidence. Validation in a strict sense would further require comprehensive, case-specific simulations and detailed comparisons with biomedical observations.

Hybrid modeling frameworks of tumor development and treatment

Tumors are complex multicellular heterogeneous systems comprised of components that interact with and modify one another. Tumor development depends on multiple factors: intrinsic, such as genetic mutations, altered signaling pathways, or variable receptor expression; and extrinsic, such as differences in nutrient supply, crosstalk with stromal or immune cells, or variable composition of the surrounding extracellular matrix. Tumors are also characterized by high cellular heterogeneity and dynamically changing tumor microenvironments. The complexity increases when this multiscale, multicomponent system is perturbed by anticancer treatments. Modeling such complex systems and predicting how tumors will respond to therapies require mathematical models that can handle various types of information and combine diverse theoretical methods on multiple temporal and spatial scales, that is, hybrid models. In this update, we discuss the progress that has been achieved during the last 10 years in the area of the hybrid modeling of tumors. The classical definition of hybrid models refers to the coupling of discrete descriptions of cells with continuous descriptions of microenvironmental factors. To reflect on the direction that the modeling field has taken, we propose extending the definition of hybrid models to include of coupling two or more different mathematical frameworks. Thus, in addition to discussing recent advances in discrete/continuous modeling, we also discuss how these two mathematical descriptions can be coupled with theoretical frameworks of optimal control, optimization, fluid dynamics, game theory, and machine learning. All these methods will be illustrated with applications to tumor development and various anticancer treatments.

This article is characterized under:

Analytical and Computational Methods > Computational Methods

Translational, Genomic, and Systems Medicine > Therapeutic Methods

Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models

Approaches to combat hypoxia in cancer therapy and the potential for in silico models in their evaluation

AIM

The negative impact of tumour hypoxia on cancer treatment outcome has been long-known, yet there has been little success combating it. This paper investigates the potential role of in silico modelling to help test emerging hypoxia-targeting treatments in cancer therapy.

METHODS

A Medline search was undertaken on the current landscape of in silico models that simulate cancer therapy and evaluate their ability to test hypoxia-targeting treatments. Techniques and treatments to combat tumour hypoxia and their current challenges are also presented.

RESULTS

Hypoxia-targeting treatments include tumour reoxygenation, hypoxic cell radiosensitization with nitroimidazoles, hypoxia-activated prodrugs and molecular targeting. Their main challenges are toxicity and not achieving adequate delivery to hypoxic regions of the tumour. There is promising research toward combining two or more of these techniques. Different types of in silico therapy models have been developed ranging from temporal to spatial and from stochastic to deterministic models. Numerous models have compared the effectiveness of different radiotherapy fractionation schedules for controlling hypoxic tumours. Similarly, models could help identify and optimize new treatments for overcoming hypoxia that utilize novel hypoxia-targeting technology.

CONCLUSION

Current therapy models should attempt to incorporate more sophisticated modelling of tumour angiogenesis/vasculature and vessel perfusion in order to become more useful for testing hypoxia-targeting treatments, which typically rely upon the tumour vasculature for delivery of additional oxygen, (pro)drugs and nanoparticles.

Towards personalized computational oncology: from spatial models of tumour spheroids, to organoids, to tissues

A main goal of mathematical and computational oncology is to develop quantitative tools to determine the most effective therapies for each individual patient. This involves predicting the right drug to be administered at the right time and at the right dose. Such an approach is known as precision medicine. Mathematical modelling can play an invaluable role in the development of such therapeutic strategies, since it allows for relatively fast, efficient and inexpensive simulations of a large number of treatment schedules in order to find the most effective. This review is a survey of mathematical models that explicitly take into account the spatial architecture of three-dimensional tumours and address tumour development, progression and response to treatments. In particular, we discuss models of epithelial acini, multicellular spheroids, normal and tumour spheroids and organoids, and multi-component tissues. Our intent is to showcase how these in silico models can be applied to patient-specific data to assess which therapeutic strategies will be the most efficient. We also present the concept of virtual clinical trials that integrate standard-of-care patient data, medical imaging, organ-on-chip experiments and computational models to determine personalized medical treatment strategies.

Accurate, robust and harmonized implementation of morpho-functional imaging in treatment planning for personalized radiotherapy

In this work we present a methodology able to use harmonized PET/CT imaging in dose painting by number (DPBN) approach by means of a robust and accurate treatment planning system. Image processing and treatment planning were performed by using a Matlab-based platform, called CARMEN, in which a full Monte Carlo simulation is included. Linear programming formulation was developed for a voxel-by-voxel robust optimization and a specific direct aperture optimization was designed for an efficient adaptive radiotherapy implementation. DPBN approach with our methodology was tested to reduce the uncertainties associated with both, the absolute value and the relative value of the information in the functional image. For the same H&N case, a single robust treatment was planned for dose prescription maps corresponding to standardized uptake value distributions from two different image reconstruction protocols: One to fulfill EARL accreditation for harmonization of [¹⁸F]FDG PET/CT image, and the other one to use the highest available spatial resolution. Also, a robust treatment was planned to fulfill dose prescription maps corresponding to both approaches, the dose painting by contour based on volumes and our voxel-by-voxel DPBN. Adaptive planning was also carried out to check the suitability of our proposal.

Different plans showed robustness to cover a range of scenarios for implementation of harmonizing strategies by using the highest available resolution. Also, robustness associated to discretization level of dose prescription according to the use of contours or numbers was achieved. All plans showed excellent quality index histogram and quality factors below 2%. Efficient solution for adaptive radiotherapy based directly on changes in functional image was obtained. We proved that by using voxel-by-voxel DPBN approach it is possible to overcome typical drawbacks linked to PET/CT images, providing to the clinical specialist confidence enough for routinely implementation of functional imaging for personalized radiotherapy.

An agent-based model for drug-radiation interactions in the tumour microenvironment: Hypoxia-activated prodrug SN30000 in multicellular tumour spheroids

Multicellular tumour spheroids capture many characteristics of human tumour microenvironments, including hypoxia, and represent an experimentally tractable in vitro model for studying interactions between radiotherapy and anticancer drugs. However, interpreting spheroid data is challenging because of limited ability to observe cell fate within spheroids dynamically. To overcome this limitation, we have developed a hybrid continuum/agent-based model (ABM) for HCT116 tumour spheroids, parameterised using experimental models (monolayers and multilayers) in which reaction and diffusion can be measured directly. In the ABM, cell fate is simulated as a function of local oxygen, glucose and drug concentrations, determined by solving diffusion equations and intracellular reactions. The model is lattice-based, with cells occupying discrete locations on a 3D grid embedded within a coarser grid that encompasses the culture medium; separate solvers are employed for each grid. The generated concentration fields account for depletion in the medium and specify concentration-time profiles within the spheroid. Cell growth and survival are determined by intracellular oxygen and glucose concentrations, the latter based on direct measurement of glucose diffusion/reaction (in multilayers) for the first time. The ABM reproduces known features of spheroids including overall growth rate, its oxygen and glucose dependence, peripheral cell proliferation, central hypoxia and necrosis. We extended the ABM to describe in detail the hypoxia-dependent interaction between ionising radiation and a hypoxia-activated prodrug (SN30000), again using experimentally determined parameters; the model accurately simulated clonogenic cell killing in spheroids, while inclusion of reversible cell cycle delay was required to account for the marked spheroid growth delay after combined radiation and SN30000. This ABM of spheroid growth and response exemplifies the utility of integrating computational and experimental tools for investigating radiation/drug interactions, and highlights the critical importance of understanding oxygen, glucose and drug concentration gradients in interpreting activity of therapeutic agents in spheroid models.

Studies in 3D cultures, notably multicellular tumour spheroids that mimic many features of solid tumours, have great potential for speeding up anticancer drug development. However the increased complexity of 3D cultures makes interpretation of experiments more difficult. We have developed a hybrid continuum/agent-based mathematical model, validated by experiments, to aid interpretation of spheroid experiments in developing drugs designed to eliminate radiation-resistant hypoxic cells. This model includes key features of the tumour microenvironment including oxygen and glucose transport and regions of hypoxia where the cells are resistant to radiation, but sensitive to hypoxia-activated prodrugs such as SN30000. This enables us to predict the growth and cell response in untreated spheroids and compare the results to spheroids treated with radiation and SN30000. We demonstrate good prediction of cellular responses in spheroids treated with radiation and SN30000 and good agreement with spheroid regrowth after treatment when additional effects of cellular growth delay are added. This demonstrates that the modelling approach has potential to improve interpretation of experimental investigations of drug and radiation combinations.

Integration of PET/MR Hybrid Imaging into Radiation Therapy Treatment

Hybrid PET/MR imaging is in early development for treatment planning. This article briefly reviews research and clinical applications of PET/MR imaging in radiation oncology. With improvements in workflow, more specific tracers, and fast and robust acquisition protocols, PET/MR imaging will play an increasingly important role in better target delineation for treatment planning and have clear advantages in the evaluation of tumor response and in a better understanding of tumor heterogeneity. With advances in treatment delivery and the potential of integrating PET/MR imaging with research on radiomics for radiation oncology, quantitative and physiologic information could lead to more precise and personalized RT.

Predicting Patient-Specific Radiotherapy Protocols Based on Mathematical Model Choice for Proliferation Saturation Index

Radiation is commonly used in cancer treatment. Over 50% of all cancer patients will undergo radiotherapy (RT) as part of cancer care. Scientific advances in RT have primarily focused on the physical characteristics of treatment including beam quality and delivery. Only recently have inroads been made into utilizing tumor biology and radiobiology to design more appropriate RT protocols. Tumors are composites of proliferating and growth-arrested cells, and overall response depends on their respective proportions at irradiation. Prokopiou et al. (Radiat Oncol 10:159, 2015) developed the concept of the proliferation saturation index (PSI) to augment the clinical decision process associated with RT. This framework is based on the application of the logistic equation to pre-treatment imaging data in order to estimate a patient-specific tumor carrying capacity, which is then used to recommend a specific RT protocol. It is unclear, however, how dependent clinical recommendations are on the underlying tumor growth law. We discuss a PSI framework with a generalized logistic equation that can capture kinetics of different well-known growth laws including logistic and Gompertzian growth. Estimation of model parameters on the basis of clinical data revealed that the generalized logistic model can describe data equally well for a wide range of the generalized logistic exponent value. Clinical recommendations based on the calculated PSI, however, are strongly dependent on the specific growth law assumed. Our analysis suggests that the PSI framework may best be utilized in clinical practice when the underlying tumor growth law is known, or when sufficiently many tumor growth models suggest similar fractionation protocols.

Stochastic multi-scale models of competition within heterogeneous cellular populations: Simulation methods and mean-field analysis

We propose a modelling framework to analyse the stochastic behaviour of heterogeneous, multi-scale cellular populations. We illustrate our methodology with a particular example in which we study a population with an oxygen-regulated proliferation rate. Our formulation is based on an age-dependent stochastic process. Cells within the population are characterised by their age (i.e. time elapsed since they were born). The age-dependent (oxygen-regulated) birth rate is given by a stochastic model of oxygen-dependent cell cycle progression. Once the birth rate is determined, we formulate an age-dependent birth-and-death process, which dictates the time evolution of the cell population. The population is under a feedback loop which controls its steady state size (carrying capacity): cells consume oxygen which in turn fuels cell proliferation. We show that our stochastic model of cell cycle progression allows for heterogeneity within the cell population induced by stochastic effects. Such heterogeneous behaviour is reflected in variations in the proliferation rate. Within this set-up, we have established three main results. First, we have shown that the age to the G1/S transition, which essentially determines the birth rate, exhibits a remarkably simple scaling behaviour. Besides the fact that this simple behaviour emerges from a rather complex model, this allows for a huge simplification of our numerical methodology. A further result is the observation that heterogeneous populations undergo an internal process of quasi-neutral competition. Finally, we investigated the effects of cell-cycle-phase dependent therapies (such as radiation therapy) on heterogeneous populations. In particular, we have studied the case in which the population contains a quiescent sub-population. Our mean-field analysis and numerical simulations confirm that, if the survival fraction of the therapy is too high, rescue of the quiescent population occurs. This gives rise to emergence of resistance to therapy since the rescued population is less sensitive to therapy.