Mathematical Modeling Of Cancer Pdf
PERSONALIZED MEDICINE: A MULTILEVEL APPROACH
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Personalized medicine is becoming an increasing part of modern cancer care.1,2 Patient-specific metrics advise contemporary clinical procedure in terms of vaccine recommendations, screening practice,3 and treatment planning.4 The aim of personalized medicine is to tailor health care specifically to the individual patient, in pursuit of optimal treatment outcome and quality of life. As a strategy, personalized medicine can be highly beneficial in cancer care, because cancer is a disease that presents with high variability across incidences. It is indeed well established that a one-to-fit-all strategy to prevent, diagnose, and treat cancer is a subpar approach.5 Ideally, in line with concepts of personalized medicine, patients should instead be individually evaluated and matched with appropriate cancer care strategies. The personalization of medicine can occur on various levels, as illustrated in Figure 1 . Patient and tumor metrics gathered from macrolevel population data down to microlevel molecular tumor data may aid anticancer decision making in a clinical setting.
FIG 1. (A) Personalized cancer practice on various levels, namely population, patient, and tumor profile. On a population macrolevel, a population can be categorized into various subpopulations (eg, according to age, sex, or smoking habits), which in turn can be evaluated and risk assessed for preventive, detective and corrective oncology practice. On a central, patient-specific level, cancer care may be tailored to fit the needs, lifestyle, and priorities of the patient, in pursuit of medicine that optimizes both treatment outcome and the patient's quality of life. On a tumor-profile level, tumor-specific data can provide information that may contribute toward disease prognoses and intelligent treatment decisions. Narrowing down cancer care personalization to the tumor level allows for a bottom-up approach to personalized tumor treatments. (B) The bench-to-bedside concept depicts the practice of transferring in vitro and in vivo findings from the laboratory to a clinical setting. The contemporary mathematician works on both blackboards and computer keyboards; thus, we analogously, present the phrase blackboard to bedside to describe the action of translating mathematical and computational intelligences to clinical application.
On a population level, a population can be categorized and divided into various subpopulations, which in turn can be evaluated and risk assessed. Certain subpopulations express elevated risks of developing particular cancer types, and likewise certain subpopulations have a predisposition to aggressive disease. This categorization can be determined by inflexible parameters such as age,3,4,6,7genetics,6,8 ethnicity,6 and sex,7 as well as by flexible parameters, such as smoking habits,9 hormonal exposure,8 obesity levels,10 and socioeconomic factors. The combination of wide population data and gathered clinical experience can be used to determine suitable, personalized treatment strategies after tumor detection and also to evaluate the need for cancer screening and vaccine administration.11 For the individual patient to benefit from screenings, screening scheduling should be personalized to enable optimized cancer intervention.9 By deploying mathematical models, which incorporate biologic knowledge and evolutionary concepts, optimized and personalized screening recommendations can be achieved.9 To more competently consider the concerns, quality of life and well-being of a patient, cancer care can be tailored on a patient-specific level. After cancer detection, lifestyle, personal priorities, and economic factors all contribute toward determination of which treatment strategy is the most appropriate for the individual patient.6 On this central, patient-specific level of treatment personalization, the dialogue between patient and clinician is of the essence,6 and it is important that the patient is well informed by the clinician.
In parallel, to keep the clinician as well informed as possible, personalized medicine can be even further detailed and narrowed down to tumor level.12 After tumor detection, disease forecasting and treatment decisions can be informed by tumor-specific data. Because of the high variability of cancer displayed across disease incidences, previous research indicates that tumor prognoses and treatment responses may correlate more with molecular tumor specifics than with larger-scale factors, such as anatomic tumor origin1 or metrics quantified on a patient or population level. Recent advances in biomarker handling,13,14 biopsy techniques, and medical imaging enable tumor assessment15 before and throughout treatments regimes. However, current biopsy procedures may in certain cases be infeasible to perform, and furthermore, tumors are highly evolutive systems that may rapidly and drastically change after a biopsy. Therefore, being able to predict tumor evolution, progression, and treatment response, given tumor-specific input data at an earlier time point, would present an immensely valuable tool in clinical treatment planning. Recently, via rigorous mathematical modeling of cancer tumors, predictive tumor prognosis is being successfully achieved for virtual tumors.16 The current mission at hand is to bridge the gap between virtual and physical tumor control so that preclinical and clinical applications will directly benefit from recent advances made in the field of mathematical and computational oncology.
MATHEMATICAL AND COMPUTATIONAL ONCOLOGY
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Mathematical and computational oncology has the potential to be especially useful to help personalize clinical cancer practice,17 because it integrates mathematical and computational approaches with traditional bench and clinical experiments. Because of recent advances in imaging techniques, the vast accumulation of experimental and clinical data, and available computational power, in silico studies have gradually been entering the stage of medical research over the past decades.18 Cancer is a highly complex disease, and, although this complexity presents difficulties in model formulation, variable parametrization and implementation, this complexity also infers that there is much biomedical insight to be gained from mathematical models and corresponding in silico experiments. Modeling may unveil new, important information about biologic cancer systems and their submechanisms19 and thus elucidate underlying tumor processes. The advantages of mathematical and computational oncology are multifold. Compared with other types of experiments, in silico experiments are both cheap and quick to produce,17 are highly adaptable,17 and are associated with few ethical concerns. Theories obtained in laboratories or clinics can be tested in silico on virtual tumors before animal testing, so modeling results can be used to guide in vivo experiments.
Today, there exists a wide array of mathematical models that are able to capture various phases of tumor progression and associated mechanisms, such as tumor growth, invasion and metastasis,20-25 angiogenesis,26-29 and treatment responses.30-36 A comprehensive overview of the field may be found in the review article by Lowengrub et al.37 Some of these models have successfully conferred with both in vitro and in vivo experiments or clinical observations38-40; consequently, mathematical tumor modeling is steadily gaining acceptance in the medical community. Recently, several multiscale models have been developed to fully capture the spatiotemporal, multiscale nature of tumor dynamics.41-44 Such models allow for intratumoral cross-scale integration of intracellular, extracellular, and intercellular concepts, which provides comprehensive modeling frameworks to which new biomedical information can easily be added.
Although advances in cancer research are being made in parallel across many disciplines, multidisciplinary collaborations have the potential to accelerate the process of translating cancer research into successful clinical application. To this end, McGuire et al45 provide an implementable pipeline for the interdisciplinary development of cancer therapies. They illustrated how to structure the workflow among clinicians, biologists, and researchers from science, technology, engineering, and mathematics in an optimal, feasible manner. The workflow demonstrates how multidisciplinary research should alternate between performance in parallel and sequentially. It also incorporates refining, iterative processes and an outlined order of operations that act to bring new cancer protocols to the clinic as quickly and safely as possible. The concept and workflow proposed by McGuire et al45 acknowledges, yet transcends, practical limitations, because it allows for collaboration across disciplines, distances, and institutes.
A MULTISCALE MODEL OF SOLID TUMOR DYNAMICS
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To demonstrate the great potential of mathematical tumor models, an established hybrid, multiscale model46 capable of simulating spatiotemporal tumor dynamics under various conditions is presented here in brief. Using this multiscale model, which is implemented with a high performance computational framework, virtual cell populations, or tumors, can be created and various clinical, or preclinical, scenarios can be studied in silico. A handful of selected such scenarios are discussed in the remainder of this paper.
MODEL OVERVIEW AND INTRATUMORAL HETEROGENEITY
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The mathematical model presented here is a multiscale, hybrid cellular automaton in which a cancer cell population was simulated on a three-dimensional lattice. It allows for spatiotemporal dynamics and intratumoral heterogeneity and is summarized in Figure 2 .43,46 The model was implemented in an in-house framework based on C++, and cell maps were visualized using ParaView (Kitware, Clifton Park, NY).47 For a more extensive model description, see previous works by Powathil et al.43,46 On an intracellular level, each cell was modeled individually to integrate subcellular molecular mechanisms and phenotypical variations among cancer cells. Intracellular cell-cycle progression was modeled by a system of ordinary differential equations on the basis of a regulatory molecular network. The model incorporated extracellular regulations, such as oxygen and drug delivery across the lattice, by mechanistic partial differential equations.
FIG 2. Fundamental mathematical components of the multiscale model. Cell-cycle dynamics are modeled by a system of ordinary differential equations composed of six dependent variables, specifically five protein concentrations and cell mass. Oxygen dynamics obey a mechanistic partial differential equation (PDE). Various treatment regulations are incorporated in the model. Chemotherapy dynamics are modeled by a mechanistic PDE, and radiotherapy response follows the linear-quadratic model. The cellular survival probability S(x, t) after radiation depends on oxygenation status and cell-cycle progression. See Powathil et al48,49 for additional details.
Many cancers are derived from one cancerous seeding cell, which by detection time has produced a tumor with subclonal diversity that displays a few dominant subclones.50 It has been observed that cells collected from the same tumor may display different subclonal50 and spatiotemporal features influenced by intracellular, extracellular,51 and intercellular mechanisms. Consequently, a multitude of tumor metrics will vary within a tumor mass,14 and this diversity may not be captured by current diagnosis tools.50 However, in silico experiments provide a platform on which to conveniently study implications of spatiotemporal heterogeneities within a tumor, which thus allows us to study what is not empirically feasible by other methods.
IMPLICATIONS OF INTRATUMORAL OXYGEN PROFILES ON RADIOTHERAPY OUTCOMES
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Hypoxic cancer cells express reduced sensitivity to anticancer treatments, such as radiotherapy and some chemotherapeutic drugs.52 Thus, hypoxia has an adverse effect on treatment delivery and may significantly affect clinical outcome. Consequently, it is important to consider spatiotemporal oxygen dynamics during study, or implementation, of treatment delivery.49 Accordingly, a previous in silico study by Enderling et al53 deployed an agent-based model, which accounted for variations in stemness among cancer cells, to emphasize the impact that cancer cell morphology and spatiotemporal heterogeneity has on radiotherapeutic response and hence on success and treatment optimization. Another in silico experiment that considered the hierarchy of tumor-initiating cells was presented by Scott et al54; they investigated phenotypic tumor-infiltrating cell features and extrinsic effects on tumor growth using a hybrid cellular automaton. A recent, spatially resolved mathematical study by Lewin et al55 demonstrated how radiotherapy-induced cell kill alters the intratumoral cell and oxygen composition and, consequently, how a certain tumor's radiotherapy response will vary across a series of administered fractions. In their study, avascular spherical symmetrical tumor growth was modeled by an integrodifferential equation that was coupled with a reaction-diffusion equation that governed oxygen distribution. Radiotherapy response was modeled using the well-established linear-quadratic (LQ) model.
In the presented modeling framework, oxygen dynamics and corresponding consequences were incorporated on multiple scales. Oxygenation status affects a cell's proliferation rate; furthermore, cellular response to radiotherapy is considered to be a function of oxygenation status, cell-cycle advancement, and radiation dosage.31,46 Figure 3 illustrates how two different virtual tumors (tumor A and tumor B) with different oxygen profiles evolved and responded to radiotherapy. Both tumors stemmed from three identically located seeding cancer cells and were simulated on equisized, three-dimensional lattices. On the lattice that contains tumor A, oxygen was produced on all extracellular lattice points that were not occupied or enclosed by cancer cells. On the second lattice, which correspondingly contains tumor B, oxygen was produced only in one octant of the lattice, as demonstrated in Figure 3 . It can be seen that tumor A was significantly more oxygenated than tumor B; consequently, the in silico experiment demonstrated a few key differences between the dynamics of the two tumors. First, the well-oxygenated tumor proliferated faster than the poorly oxygenated tumor, because hypoxia impeded cell-cycle progression. Second, the two tumors displayed different distributions of cellular oxygen levels. Last, radiotherapy was more effective in the well-oxygenated tumor than in the poorly oxygenated tumor. These results demonstrate how useful it can be to consider a detailed tumor profile before treatment planning starts.
FIG 3. Graphs that illustrate the implication of intratumoral oxygen profiles; cell death by radiation is visualized as instantaneous for clarity. (A) At 0 hours, three seeding cancer cells are planted on a three- dimensional lattice. These cells develop into a tumor that, at 700 hours, is exposed to radiotherapy. (B) Oxygen is produced on all extracellular lattice points in lattice A and on all extracellular lattice points in one octant only in lattice B. (C) Cell maps at 500 and 700 hours (before radiotherapy) and at 710 hours (after radiotherapy) are provided, and colors correspond to cellular oxygenation levels. (D) Graph of tumor growth over time and shrinkage in response to radiotherapy at 700 hours. (E) Intratumoral oxygen distribution at time of radiotherapy administration.
CLINICAL RELEVANCE OF RADIATION BYSTANDER EFFECTS
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At low radiation doses, the direct effects of delivered radiation are considered relatively low; consequently, the nontargeted effects of radiation play an important role in the determination of cellular radiation response.48,56,57 In most cases of clinical radiation delivery, the majority of normal tissue adjacent to the targeted tumor is exposed to low-dose radiation in an effort to maximize the dose delivered to the tumor but minimize the damage to surrounding normal structures.57 Hence, it is important to study the impact of nontargeted effects in clinical radiation therapy and their role in radiation effectiveness and potential radiation-induced carcinogenesis.57
Examples of such nontargeted effects include the phenomena known as bystander effects, in which signals produced by irradiated cells influence the behavior of nonirradiated cells.48,56,58 It is difficult to study these effects independently of direct effects, and, moreover, mediators and targets for bystander signals are poorly understood even after several experimental studies.49,57,59-64 Recently, several modeling attempts have been made to study and understand radiation bystander effects.65,66
In one of these attempts, Powathil et al66 used a multiscale mathematical modeling framework (Fig 2 ) to study the impact of radiation and radiation-induced bystander effects on both normal and tumor cells. Here, bystander effects are considered to be produced by bystander signals that diffuse through the medium/microenvironment.48,59 Radiated and nonirradiated cells that are exposed to these signals are assumed to respond probabilistically, in which varying outcomes, such as cell death, repair delay, and mutation, depend on localized signal intensity. Figures 4A and 4B show the survival fraction of the cells after radiation, with and without bystander-induced cell kill, for two cases of dose delivery.66 In Figure 4A ,66,67 tumor cells are fully exposed to the given radiation dose, but the surrounding normal cells receive gradients of the dose. In Figure 4B , tumor cells are fully exposed to the given radiation dose, but the surrounding normal cells are spared completely. The plots show that, at low doses (< 0.5 Gy), bystander effects contribute to a higher cell kill than do direct effects. This indicates that bystander effects might play a role in low-dose radiation hypersensitivity, as observed in multiple experimental studies, such as one shown in Figure 4C (which compares curves for the LQ model and the LQ model with bystander effects). These in silico results are also confirmed with experiments by Fernandez et al.59
FIG 4. Plots that show radiation bystander effects and the difference in cellular survival fraction when bystander responses are considered. (A) Tumor cells are fully exposed to the given radiation dose, whereas the surrounding normal cells receive a gradient of the dose. (B) Tumor cells are exposed to the fully given radiation dose, whereas the surrounding normal cells are spared completely. (C) Experimental result: survival of asynchronous T98G human glioma cells irradiated with 240 kVp x-rays, measured using the cell-sort protocol. Used with Elsevier permission from Joiner et al35 and adapted from Powathil et al.58
DRUG RESISTANCE AND DRUG RESPONSE
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After chemotherapy, tumor recurrence is a prevalent clinical problem, and reappearing cancers are often observed to be increasingly drug resistant. Cancer cells may possess, acquire, and communicate drug-resistant traits, which enable them to survive under otherwise lethal conditions, after exposure to chemotherapeutic drugs. The efficacy of chemotherapy treatments is significantly influenced by drug-resistant traits expressed by the targeted tumor.
Tumor dynamics in cancer cell populations that display drug-resistant traits, derived from inheritance or mutations, which are spontaneous, drug-induced or communicated via exosomes, can be studied in silico.68 The coexistence of drug-resistant and drug-sensitive cells yields a competition between subpopulations, and the evolution and ecology of cancer cell populations obey Darwinian rules.69 It has been demonstrated that drug-resistant subpopulations are more fit to survive in the presence of drugs, whereas drug-sensitive cells are more fit to survive in the absence of drugs. This is in accordance with the argument that cells may indeed acquire drug resistance as a survival endeavor; however, by doing so, other cellular driving mechanisms are compromised.69 To study these concepts in silico, we here investigated tumor growth and chemotherapy response of a tumor composed of two subpopulations, one of which was sensitive to drugs and one of which was drug resistant. The drug-resistant subpopulation was immune to drug effects but had reduced proliferation ability, specifically a cell-cycle length of roughly twice that of the sensitive subpopulation. At the start of the in silico experiment, one drug-sensitive and one drug-resistant cell were placed on the lattice. As is clearly demonstrated in Figure 5 , the sensitive subpopulation dominated in size before drug administration because of its higher proliferation rate. Drugs, specifically cisplatin, were administered at 700 and 800 hours in a single or double dosage. The results demonstrated how the choice of drug dosage influences the outcome of chemotherapy. Although a higher drug dosage killed more cancerous cells, it also altered the composition of sensitive-to-resistant cells in favor of the drug-resistant subpopulation. Repeated use of such chemotherapy administration would quickly render the tumor uncontrollable by chemotherapy. This conceptually shows that suboptimal chemotherapy administration may contribute toward drug resistance. In other words, chemotherapy that does not kill a tumor may make it stronger.68
FIG 5. Graphs that illustrate drug resistance and drug response, in which cells are visualized as euthanized instantaneously for clarity. (A) At 0 hours, one sensitive and one drug resistant cell is placed on the lattice and drugs are instantaneously administered at 700 and 800 hours. (B) Number of cells over time for a single (1) and double (2) drug dosage. (C) Cell maps at 400, 700, 800, and 900 hours.
Traditionally, the goal of clinical chemotherapy has been to reduce the tumor size as much as possible with minimal toxicity. However, Hamis et al68 recently demonstrated that chemotherapy treatments, in terms of dosage administration and scheduling, should be tumor specific to simultaneously suppress tumor size and drug-resistant tumor features.68 A treatment strategy backed up by mathematical modeling is adaptive chemotherapy,69 which takes advantage of the competition between drug-sensitive and drug-resistant subpopulations in an effort to improve treatment outcomes. This promising, and mathematically validated, strategy does however imply breaking free from the clinically traditional set-dosage, set-schedule practice. This shows that mathematical models, such as the one described here, can be useful to motivate, develop, and guide personalized treatment protocols informed by available data.
HYPOXIA-ACTIVATED PRODRUGS
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Because of recognized implications of hypoxia, multiple strategies to combat hypoxia have been explored.70 Hypoxia-activated prodrugs (HAPs) present a means to not only combat, but better yet, exploit hypoxia in anticancer therapies.52 HAPs comprise bioreductive prodrugs that reduce to active cytotoxins upon reaching hypoxic regions.64,71,72 Because tumors, as a rule, express levels of hypoxia of higher severity than do other relevant tissues in the host system, tumors can be targeted. HAPs may thus operate as trojan horses, which are essentially harmless until they are converted in targets—in other words, hypoxic tumors regions. Combination treatments of HAPs and ionizing radiation have produced optimistic preclinical results,72 but, despite the conceptual promise of HAPs, clinical trials of HAPs have produced mixed outcomes. Mathematical modeling can help elucidate the (somewhat disheartening) results of clinical HAP trials.
Previous mathematical studies have already contributed to the overall understanding of HAPs. Work by Foehrenbacher et al73 quantified bystander effects elicited by the HAP PR-104 using a Green's function method, in customized form, and pharmacokinetic/pharmacodynamic modeling. Similar mathematical concepts were used in a later study to compare class I HAPs with class II HAPs and to determine optimal properties for class II HAPs.74 TH-302 is a HAP that has been tested for an array of cancers in clinical trials. Using a stochastic model, Lindsay et al75 studied TH-302–erlotinib monotherapies and combination therapies to conclude that a combination therapy of the two drugs impedes the uprising of drug resistance. HAPs bioreduce to activated form under hypoxic conditions, so a previous study by Wojtkowiak et al76 investigated, and conceptually validated, the strategy of deliberately exacerbating intratumoral hypoxia using exogenous pyruvate to amplify TH-302 activity. Their study combined metabolic profiling and electron paramagnetic resonance imaging with mathematical modeling, in which HAP dynamics was modeled using reaction-diffusion/convection equations coupled with fluid-structure interactions.
Here, we extended the multiscale framework summarized in Figure 2 to study HAP dynamics and appropriate tumor conditions for HAP success. In the model, HAPs were assumed to follow similar dynamics as that of chemotherapy, in which HAPs bioreduced to activated form (ie, AHAPs) after they were on a sufficiently hypoxic lattice point. Cells that were hypoxic enough to activate the prodrug are called activators, and other cancer cells are bystander cells. Both activators and bystander cells may be euthanized by the DNA linking effect of AHAPs. Figure 6 illustrates an in silico experiment in which two tumor spheroids, tumor A and tumor B, are subjected to HAPs. Both tumors consisted of 20% activator cells. In tumor A, the activator cells were concentrically located in the center of the tumor spheroid, but the activator cells in tumor B were scattered in clusters. The results indicate that, despite the fact that tumor A and tumor B contained the same number of activator and bystander cells, HAPs were more effective in tumor B than in tumor A. This demonstrates that not only the hypoxic tumor fraction but also the spatiotemporal distribution of hypoxic cells may influence HAP efficacy. Tumor-specific treatment personalization is, again, of the essence, because the results demonstrate that the successfulness of HAP treatments strongly correlates with tumor-specific features. This indicates that, in a clinical setting, patients might need to be carefully selected for HAP treatments via sufficiently comprehensive tumor assessments.
FIG 6. The effect of hypoxia-activated prodrugs, in which cell death is visualized as instantaneous for clarity. (A) Number, and total, of activator and bystander cells during hypoxia-activated prodrug treatment of tumors A and B. (B) Cell maps of tumors A and B at the time of treatment (T), 1.5 hours after the treatment (T + 1.5), and 3 hours after the treatment (T + 3). AHAP, activated form of hypoxia-activated prodrugs.
DISCUSSION
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Mathematical modeling has chronologically tailed clinical implementation of tumor treatment strategies. Historically, this time lag is validated in the early era of modern cancer care practice, which preceded advanced technology. However, with current imaging and biopsy technologies; sophisticated in vitro and in vivo laboratories; accumulating data from experiments and clinics; available computational power18; and biologic, medical, and mathematical knowledge, mathematical oncology today constitutes an up-to-date complement to traditional cancer research. Modeling has the potential to both optimize currently available anticancer protocols and contemporaneously aid preclinical developments of new anticancer therapies. Thus, the time lag between clinical applications and mathematical modeling is conceptually being eliminated. However, to comprehensively transfer insights from blackboard to bedside, actualized collaboration between clinicians and mathematicians, as well as between biologists and experimentalists, is key.77
In this article, we demonstrated how mathematical tumor models can be used to study pertinent treatment scenarios in silico. Once validated, data driven, predictive mathematical models can eventually help plan personalized treatment protocols by acting as in silico test bases. For a more complete translation of intelligence from blackboard-to-bedside, there exists a need to integrate tumor-host interactions mathematically. However, cell population models, such as those described here, constitute a crucial stepping stone to realization of a bottom-up approach toward personalized medicine.
© 2019 by American Society of Clinical Oncology
Conception and design: All authors
Data analysis and interpretation: All authors
Collection and assembly of data: All authors
Manuscript writing: All authors
Final approval of manuscript: All authors
Accountable for all aspects of work: All authors
AUTHORS' DISCLOSURES OF POTENTIAL CONFLICTS OF INTEREST
The following represents disclosure information provided by authors of this manuscript. All relationships are considered compensated. Relationships are self-held unless noted. I = Immediate Family Member, Inst = My Institution. Relationships may not relate to the subject matter of this manuscript. For more information about ASCO's conflict of interest policy, please refer to www.asco.org/rwc or ascopubs.org/jco/site/ifc.
No potential conflicts of interest were reported.
ACKNOWLEDGMENT
We thank Swansea University Research Studentship for support.
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Mathematical Modeling Of Cancer Pdf
Source: https://ascopubs.org/doi/full/10.1200/CCI.18.00068
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