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Schirru M, Charef H, Ismaili KE, Fenneteau F, Zugaj D, Tremblay PO, Nekka F. Predicting efficacy assessment of combined treatment of radiotherapy and nivolumab for NSCLC patients through virtual clinical trials using QSP modeling. J Pharmacokinet Pharmacodyn 2024; 51:319-333. [PMID: 38493439 DOI: 10.1007/s10928-024-09903-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/07/2023] [Accepted: 02/05/2024] [Indexed: 03/19/2024]
Abstract
Non-Small Cell Lung Cancer (NSCLC) remains one of the main causes of cancer death worldwide. In the urge of finding an effective approach to treat cancer, enormous therapeutic targets and treatment combinations are explored in clinical studies, which are not only costly, suffer from a shortage of participants, but also unable to explore all prospective therapeutic solutions. Within the evolving therapeutic landscape, the combined use of radiotherapy (RT) and checkpoint inhibitors (ICIs) emerged as a promising avenue. Exploiting the power of quantitative system pharmacology (QSP), we undertook a study to anticipate the therapeutic outcomes of these interventions, aiming to address the limitations of clinical trials. After enhancing a pre-existing QSP platform and accurately replicating clinical data outcomes, we conducted an in-depth study, examining different treatment protocols with nivolumab and RT, both as monotherapy and in combination, by assessing their efficacy through clinical endpoints, namely time to progression (TTP) and duration of response (DOR). As result, the synergy of combined protocols showcased enhanced TTP and extended DOR, suggesting dual advantages of extended response and slowed disease progression with certain combined regimens. Through the lens of QSP modeling, our findings highlight the potential to fine-tune combination therapies for NSCLC, thereby providing pivotal insights for tailoring patient-centric therapeutic interventions.
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Affiliation(s)
- Miriam Schirru
- Laboratoire de recherche en pharmacométrie, Faculté de pharmacie, Université de Montréal, Montreal, Canada.
| | - Hamza Charef
- Laboratoire de recherche en pharmacométrie, Faculté de pharmacie, Université de Montréal, Montreal, Canada
| | - Khalil-Elmehdi Ismaili
- Laboratoire de recherche en pharmacométrie, Faculté de pharmacie, Université de Montréal, Montreal, Canada
| | - Frédérique Fenneteau
- Laboratoire de recherche en pharmacométrie, Faculté de pharmacie, Université de Montréal, Montreal, Canada
| | - Didier Zugaj
- Clinical Pharmacology, Syneos Health, Quebec, Quebec G1P 0A2, Canada
| | | | - Fahima Nekka
- Laboratoire de recherche en pharmacométrie, Faculté de pharmacie, Université de Montréal, Montreal, Canada
- Centre de recherches mathématiques (CRM), Université de Montréal, Montreal, Canada
- Centre for Applied Mathematics in Bioscience and Medicine (CAMBAM), McGill University, Montreal, Canada
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2
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Essongo FE, Mvogo A, Ben-Bolie GH. Dynamics of a diffusive model for cancer stem cells with time delay in microRNA-differentiated cancer cell interactions and radiotherapy effects. Sci Rep 2024; 14:5295. [PMID: 38438408 PMCID: PMC10912232 DOI: 10.1038/s41598-024-55212-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2023] [Accepted: 02/21/2024] [Indexed: 03/06/2024] Open
Abstract
Understand the dynamics of cancer stem cells (CSCs), prevent the non-recurrence of cancers and develop therapeutic strategies to destroy both cancer cells and CSCs remain a challenge topic. In this paper, we study both analytically and numerically the dynamics of CSCs under radiotherapy effects. The dynamical model takes into account the diffusion of cells, the de-differentiation (or plasticity) mechanism of differentiated cancer cells (DCs) and the time delay on the interaction between microRNAs molecules (microRNAs) with DCs. The stability of the model system is studied by using a Hopf bifurcation analysis. We mainly investigate on the critical time delay τ c , that represents the time for DCs to transform into CSCs after the interaction of microRNAs with DCs. Using the system parameters, we calculate the value of τ c for prostate, lung and breast cancers. To confirm the analytical predictions, the numerical simulations are performed and show the formation of spatiotemporal circular patterns. Such patterns have been found as promising diagnostic and therapeutic value in management of cancer and various diseases. The radiotherapy is applied in the particular case of prostate model. We calculate the optimum dose of radiation and determine the probability of avoiding local cancer recurrence after radiotherapy treatment. We find numerically a complete eradication of patterns when the radiotherapy is applied before a time t < τ c . This scenario induces microRNAs to act as suppressors as experimentally observed in prostate cancer. The results obtained in this paper will provide a better concept for the clinicians and oncologists to understand the complex dynamics of CSCs and to design more efficacious therapeutic strategies to prevent the non-recurrence of cancers.
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Affiliation(s)
- Frank Eric Essongo
- Laboratory of Nuclear Physics, Dosimetry and Radiation Protection, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon
| | - Alain Mvogo
- Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon.
| | - Germain Hubert Ben-Bolie
- Laboratory of Nuclear Physics, Dosimetry and Radiation Protection, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon
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3
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Browning AP, Lewin TD, Baker RE, Maini PK, Moros EG, Caudell J, Byrne HM, Enderling H. Predicting Radiotherapy Patient Outcomes with Real-Time Clinical Data Using Mathematical Modelling. Bull Math Biol 2024; 86:19. [PMID: 38238433 PMCID: PMC10796515 DOI: 10.1007/s11538-023-01246-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/01/2023] [Accepted: 12/14/2023] [Indexed: 01/22/2024]
Abstract
Longitudinal tumour volume data from head-and-neck cancer patients show that tumours of comparable pre-treatment size and stage may respond very differently to the same radiotherapy fractionation protocol. Mathematical models are often proposed to predict treatment outcome in this context, and have the potential to guide clinical decision-making and inform personalised fractionation protocols. Hindering effective use of models in this context is the sparsity of clinical measurements juxtaposed with the model complexity required to produce the full range of possible patient responses. In this work, we present a compartment model of tumour volume and tumour composition, which, despite relative simplicity, is capable of producing a wide range of patient responses. We then develop novel statistical methodology and leverage a cohort of existing clinical data to produce a predictive model of both tumour volume progression and the associated level of uncertainty that evolves throughout a patient's course of treatment. To capture inter-patient variability, all model parameters are patient specific, with a bootstrap particle filter-like Bayesian approach developed to model a set of training data as prior knowledge. We validate our approach against a subset of unseen data, and demonstrate both the predictive ability of our trained model and its limitations.
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Affiliation(s)
| | - Thomas D Lewin
- Mathematical Institute, University of Oxford, Oxford, UK
- Roche Pharma Research and Early Development, Roche Innovation Center, Basel, Switzerland
| | - Ruth E Baker
- Mathematical Institute, University of Oxford, Oxford, UK
| | - Philip K Maini
- Mathematical Institute, University of Oxford, Oxford, UK
| | - Eduardo G Moros
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, USA
| | - Jimmy Caudell
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, USA
| | - Helen M Byrne
- Mathematical Institute, University of Oxford, Oxford, UK
| | - Heiko Enderling
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, USA.
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, USA.
- Department of Radiation Oncology, MD Anderson Cancer Center, Houston, TX, USA.
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4
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Olivença DV, Davis JD, Kumbale CM, Zhao CY, Brown SP, McCarty NA, Voit EO. Mathematical models of cystic fibrosis as a systemic disease. WIREs Mech Dis 2023; 15:e1625. [PMID: 37544654 PMCID: PMC10843793 DOI: 10.1002/wsbm.1625] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/16/2022] [Revised: 06/22/2023] [Accepted: 07/06/2023] [Indexed: 08/08/2023]
Abstract
Cystic fibrosis (CF) is widely known as a disease of the lung, even though it is in truth a systemic disease, whose symptoms typically manifest in gastrointestinal dysfunction first. CF ultimately impairs not only the pancreas and intestine but also the lungs, gonads, liver, kidneys, bones, and the cardiovascular system. It is caused by one of several mutations in the gene of the epithelial ion channel protein CFTR. Intense research and improved antimicrobial treatments during the past eight decades have steadily increased the predicted life expectancy of a person with CF (pwCF) from a few weeks to over 50 years. Moreover, several drugs ameliorating the sequelae of the disease have become available in recent years, and notable treatments of the root cause of the disease have recently generated substantial improvements in health for some but not all pwCF. Yet, numerous fundamental questions remain unanswered. Complicating CF, for instance in the lung, is the fact that the associated insufficient chloride secretion typically perturbs the electrochemical balance across epithelia and, in the airways, leads to the accumulation of thick, viscous mucus and mucus plaques that cannot be cleared effectively and provide a rich breeding ground for a spectrum of bacterial and fungal communities. The subsequent infections often become chronic and respond poorly to antibiotic treatments, with outcomes sometimes only weakly correlated with the drug susceptibility of the target pathogen. Furthermore, in contrast to rapidly resolved acute infections with a single target pathogen, chronic infections commonly involve multi-species bacterial communities, called "infection microbiomes," that develop their own ecological and evolutionary dynamics. It is presently impossible to devise mathematical models of CF in its entirety, but it is feasible to design models for many of the distinct drivers of the disease. Building upon these growing yet isolated modeling efforts, we discuss in the following the feasibility of a multi-scale modeling framework, known as template-and-anchor modeling, that allows the gradual integration of refined sub-models with different granularity. The article first reviews the most important biomedical aspects of CF and subsequently describes mathematical modeling approaches that already exist or have the potential to deepen our understanding of the multitude aspects of the disease and their interrelationships. The conceptual ideas behind the approaches proposed here do not only pertain to CF but are translatable to other systemic diseases. This article is categorized under: Congenital Diseases > Computational Models.
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Affiliation(s)
- Daniel V. Olivença
- Center for Engineering Innovation, The University of Texas at Dallas, 800 W. Campbell Road, Richardson, Texas 75080, USA
| | - Jacob D. Davis
- Department of Biomedical Engineering, Georgia Tech and Emory University, Atlanta, Georgia
| | - Carla M. Kumbale
- Department of Biomedical Engineering, Georgia Tech and Emory University, Atlanta, Georgia
| | - Conan Y. Zhao
- Mayo Clinic Alix School of Medicine, Mayo Clinic, Rochester, Minnesota
| | - Samuel P. Brown
- Department of Biological Sciences, Georgia Tech and Emory University, Atlanta, Georgia
| | - Nael A. McCarty
- Department of Pediatrics, Emory University School of Medicine, Atlanta, Georgia
| | - Eberhard O. Voit
- Department of Biomedical Engineering, Georgia Tech and Emory University, Atlanta, Georgia
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Baaz M, Cardilin T, Lignet F, Zimmermann A, El Bawab S, Gabrielsson J, Jirstrand M. Model-based assessment of combination therapies - ranking of radiosensitizing agents in oncology. BMC Cancer 2023; 23:409. [PMID: 37149596 PMCID: PMC10164338 DOI: 10.1186/s12885-023-10899-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/05/2021] [Accepted: 04/27/2023] [Indexed: 05/08/2023] Open
Abstract
BACKGROUND To increase the chances of finding efficacious anticancer drugs, improve development times and reduce costs, it is of interest to rank test compounds based on their potential for human use as early as possible in the drug development process. In this paper, we present a method for ranking radiosensitizers using preclinical data. METHODS We used data from three xenograft mice studies to calibrate a model that accounts for radiation treatment combined with radiosensitizers. A nonlinear mixed effects approach was utilized where between-subject variability and inter-study variability were considered. Using the calibrated model, we ranked three different Ataxia telangiectasia-mutated inhibitors in terms of anticancer activity. The ranking was based on the Tumor Static Exposure (TSE) concept and primarily illustrated through TSE-curves. RESULTS The model described data well and the predicted number of eradicated tumors was in good agreement with experimental data. The efficacy of the radiosensitizers was evaluated for the median individual and the 95% population percentile. Simulations predicted that a total dose of 220 Gy (5 radiation sessions a week for 6 weeks) was required for 95% of tumors to be eradicated when radiation was given alone. When radiation was combined with doses that achieved at least 8 [Formula: see text] of each radiosensitizer in mouse blood, it was predicted that the radiation dose could be decreased to 50, 65, and 100 Gy, respectively, while maintaining 95% eradication. CONCLUSIONS A simulation-based method for calculating TSE-curves was developed, which provides more accurate predictions of tumor eradication than earlier, analytically derived, TSE-curves. The tool we present can potentially be used for radiosensitizer selection before proceeding to subsequent phases of the drug discovery and development process.
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Affiliation(s)
- Marcus Baaz
- Fraunhofer-Chalmers Research Centre for Industrial Mathematics, Gothenburg, Sweden.
- Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden.
| | - Tim Cardilin
- Fraunhofer-Chalmers Research Centre for Industrial Mathematics, Gothenburg, Sweden
| | - Floriane Lignet
- Translational Medicine, Quantitative Pharmacology, Merck Healthcare KGaA, Darmstadt, Germany
| | - Astrid Zimmermann
- Translation Innovation Platform Oncology, Merck Healthcare KGaA, Darmstadt, Germany
| | - Samer El Bawab
- Translational Medicine, Quantitative Pharmacology, Merck Healthcare KGaA, Darmstadt, Germany
- Present Address: Translational Medicine, Servier, Suresnes, France
| | | | - Mats Jirstrand
- Fraunhofer-Chalmers Research Centre for Industrial Mathematics, Gothenburg, Sweden
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Jørgensen ACS, Hill CS, Sturrock M, Tang W, Karamched SR, Gorup D, Lythgoe MF, Parrinello S, Marguerat S, Shahrezaei V. Data-driven spatio-temporal modelling of glioblastoma. ROYAL SOCIETY OPEN SCIENCE 2023; 10:221444. [PMID: 36968241 PMCID: PMC10031411 DOI: 10.1098/rsos.221444] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/08/2022] [Accepted: 02/23/2023] [Indexed: 06/18/2023]
Abstract
Mathematical oncology provides unique and invaluable insights into tumour growth on both the microscopic and macroscopic levels. This review presents state-of-the-art modelling techniques and focuses on their role in understanding glioblastoma, a malignant form of brain cancer. For each approach, we summarize the scope, drawbacks and assets. We highlight the potential clinical applications of each modelling technique and discuss the connections between the mathematical models and the molecular and imaging data used to inform them. By doing so, we aim to prime cancer researchers with current and emerging computational tools for understanding tumour progression. By providing an in-depth picture of the different modelling techniques, we also aim to assist researchers who seek to build and develop their own models and the associated inference frameworks. Our article thus strikes a unique balance. On the one hand, we provide a comprehensive overview of the available modelling techniques and their applications, including key mathematical expressions. On the other hand, the content is accessible to mathematicians and biomedical scientists alike to accommodate the interdisciplinary nature of cancer research.
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Affiliation(s)
| | - Ciaran Scott Hill
- Department of Neurosurgery, The National Hospital for Neurology and Neurosurgery, London WC1N 3BG, UK
- Samantha Dickson Brain Cancer Unit, UCL Cancer Institute, London WC1E 6DD, UK
| | - Marc Sturrock
- Department of Physiology and Medical Physics, Royal College of Surgeons in Ireland, Dublin D02 YN77, Ireland
| | - Wenhao Tang
- Department of Mathematics, Faculty of Natural Sciences, Imperial College London, London SW7 2AZ, UK
| | - Saketh R. Karamched
- Division of Medicine, Centre for Advanced Biomedical Imaging, University College London (UCL), London WC1E 6BT, UK
| | - Dunja Gorup
- Division of Medicine, Centre for Advanced Biomedical Imaging, University College London (UCL), London WC1E 6BT, UK
| | - Mark F. Lythgoe
- Division of Medicine, Centre for Advanced Biomedical Imaging, University College London (UCL), London WC1E 6BT, UK
| | - Simona Parrinello
- Samantha Dickson Brain Cancer Unit, UCL Cancer Institute, London WC1E 6DD, UK
| | - Samuel Marguerat
- Genomics Translational Technology Platform, UCL Cancer Institute, University College London, London WC1E 6DD, UK
| | - Vahid Shahrezaei
- Department of Mathematics, Faculty of Natural Sciences, Imperial College London, London SW7 2AZ, UK
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7
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Valentim CA, Rabi JA, David SA. Cellular-automaton model for tumor growth dynamics: Virtualization of different scenarios. Comput Biol Med 2023; 153:106481. [PMID: 36587567 DOI: 10.1016/j.compbiomed.2022.106481] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/27/2022] [Revised: 12/08/2022] [Accepted: 12/25/2022] [Indexed: 12/29/2022]
Abstract
Mathematical Oncology has emerged as a research field that applies either continuous or discrete models to mathematically describe cancer-related phenomena. Such methods are usually expressed in terms of differential equations, however tumor composition involves specific cellular structure and can demonstrate probabilistic nature, often requiring tailor-made approaches. In this context, cell-based models allow monitoring independent single parameters, which might vary in both time and space. By relying on extant tumor growth models in the literature, this study introduces cellular-automata simulation strategies that admit heterogeneous cell population while capturing both single-cell and cluster-cell behaviors. In this agent-based computational model, tumor cells are limited to follow four possible courses of action, namely: proliferation, migration, apoptosis or quiescence. Despite the apparent simplicity of those actions, the model can represent different complex tumor features depending on parameter settings. This study virtualized five different scenarios, showcasing model capabilities of representing tumor dynamics including alternate dormancy periods, cell death instability and cluster formation. Implementation techniques are also explored together with prospective model expansion towards deterministic features. The proposed stochastic cellular automaton model is able to effectively simulate different scenarios regarding tumor growth effectively, figuring as an interesting tool for in silico modeling, with promising capabilities of expansion to support research in mathematical oncology, thus improving diagnosis tools and/or personalized treatment.
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Affiliation(s)
- Carlos A Valentim
- Department of Biosystems Engineering, University of São Paulo, Pirassununga, Brazil.
| | - José A Rabi
- Department of Biosystems Engineering, University of São Paulo, Pirassununga, Brazil.
| | - Sergio A David
- Department of Biosystems Engineering, University of São Paulo, Pirassununga, Brazil.
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8
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Calatayud J, Jornet M, Mateu J. A phenomenological model for COVID-19 data taking into account neighboring-provinces effect and random noise. STAT NEERL 2022; 77:STAN12278. [PMID: 36247018 PMCID: PMC9538456 DOI: 10.1111/stan.12278] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2022] [Accepted: 09/17/2022] [Indexed: 11/26/2022]
Abstract
We model the incidence of the COVID-19 disease during the first wave of the epidemic in Castilla-Leon (Spain). Within-province dynamics may be governed by a generalized logistic map, but this lacks of spatial structure. To couple the provinces, we relate the daily new infections through a density-independent parameter that entails positive spatial correlation. Pointwise values of the input parameters are fitted by an optimization procedure. To accommodate the significant variability in the daily data, with abruptly increasing and decreasing magnitudes, a random noise is incorporated into the model, whose parameters are calibrated by maximum likelihood estimation. The calculated paths of the stochastic response and the probabilistic regions are in good agreement with the data.
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Affiliation(s)
- Julia Calatayud
- Department of Mathematics Universitat Jaume I Castellón Spain
| | - Marc Jornet
- Department of Mathematics Universitat de València Burjassot Spain
| | - Jorge Mateu
- Department of Mathematics Universitat Jaume I Castellón Spain
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9
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Calatayud J, Jornet M, Mateu J. A stochastic Bayesian bootstrapping model for COVID-19 data. STOCHASTIC ENVIRONMENTAL RESEARCH AND RISK ASSESSMENT : RESEARCH JOURNAL 2022; 36:2907-2917. [PMID: 35035283 PMCID: PMC8749118 DOI: 10.1007/s00477-022-02170-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 01/05/2022] [Indexed: 05/09/2023]
Abstract
We provide a stochastic modeling framework for the incidence of COVID-19 in Castilla-Leon (Spain) for the period March 1, 2020 to February 12, 2021, which encompasses four waves. Each wave is appropriately described by a generalized logistic growth curve. Accordingly, the four waves are modeled through a sum of four generalized logistic growth curves. Pointwise values of the twenty input parameters are fitted by a least-squares optimization procedure. Taking into account the significant variability in the daily reported cases, the input parameters and the errors are regarded as random variables on an abstract probability space. Their probability distributions are inferred from a Bayesian bootstrap procedure. This framework is shown to offer a more accurate estimation of the COVID-19 reported cases than the deterministic formulation.
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Affiliation(s)
- Julia Calatayud
- Departament de Matemàtiques, Universitat Jaume I, 12071 Castellón, Spain
| | - Marc Jornet
- Departament de Matemàtiques, Universitat de València, 46100 Burjassot, Spain
| | - Jorge Mateu
- Departament de Matemàtiques, Universitat Jaume I, 12071 Castellón, Spain
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10
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Khaliq R, Iqbal P, Bhat SA, Sheergojri AR. A fuzzy mathematical model for tumor growth pattern using generalized Hukuhara derivative and its numerical analysis. Appl Soft Comput 2022. [DOI: 10.1016/j.asoc.2022.108467] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
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11
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Zahid MU, Mohsin N, Mohamed ASR, Caudell JJ, Harrison LB, Fuller CD, Moros EG, Enderling H. Forecasting Individual Patient Response to Radiation Therapy in Head and Neck Cancer With a Dynamic Carrying Capacity Model. Int J Radiat Oncol Biol Phys 2021; 111:693-704. [PMID: 34102299 PMCID: PMC8463501 DOI: 10.1016/j.ijrobp.2021.05.132] [Citation(s) in RCA: 35] [Impact Index Per Article: 11.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/13/2020] [Revised: 05/24/2021] [Accepted: 05/28/2021] [Indexed: 12/21/2022]
Abstract
Purpose: To model and predict individual patient responses to radiation therapy. Methods and Materials: We modeled tumor dynamics as logistic growth and the effect of radiation as a reduction in the tumor carrying capacity, motivated by the effect of radiation on the tumor microenvironment. The model was assessed on weekly tumor volume data collected for 2 independent cohorts of patients with head and neck cancer from the H. Lee Moffitt Cancer Center (MCC) and the MD Anderson Cancer Center (MDACC) who received 66 to 70 Gy in standard daily fractions or with accelerated fractionation. To predict response to radiation therapy for individual patients, we developed a new forecasting framework that combined the learned tumor growth rate and carrying capacity reduction fraction (δ) distribution with weekly measurements of tumor volume reduction for a given test patient to estimate δ, which was used to predict patient-specific outcomes. Results: The model fit data from MCC with high accuracy with patient-specific δ and a fixed tumor growth rate across all patients. The model fit data from an independent cohort from MDACC with comparable accuracy using the tumor growth rate learned from the MCC cohort, showing transferability of the growth rate. The forecasting framework predicted patient-specific outcomes with 76% sensitivity and 83% specificity for locoregional control and 68% sensitivity and 85% specificity for disease-free survival with the inclusion of 4 on-treatment tumor volume measurements. Conclusions: These results demonstrate that our simple mathematical model can describe a variety of tumor volume dynamics. Furthermore, combining historically observed patient responses with a few patient-specific tumor volume measurements allowed for the accurate prediction of patient outcomes, which may inform treatment adaptation and personalization.
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Affiliation(s)
- Mohammad U Zahid
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, Florida
| | - Nuverah Mohsin
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, Florida; Dr. Kiran C. Patel College of Allopathic Medicine, Nova Southeastern University, Fort Lauderdale, Florida
| | - Abdallah S R Mohamed
- Department of Radiation Oncology, The University of Texas MD Anderson Cancer Center, Houston, Texas
| | - Jimmy J Caudell
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, Florida
| | - Louis B Harrison
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, Florida
| | - Clifton D Fuller
- Department of Radiation Oncology, The University of Texas MD Anderson Cancer Center, Houston, Texas
| | - Eduardo G Moros
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, Florida
| | - Heiko Enderling
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, Florida; Department of Radiation Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, Florida.
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12
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Cardilin T, Almquist J, Jirstrand M, Zimmermann A, Lignet F, El Bawab S, Gabrielsson J. Exposure-response modeling improves selection of radiation and radiosensitizer combinations. J Pharmacokinet Pharmacodyn 2021; 49:167-178. [PMID: 34623558 PMCID: PMC8940791 DOI: 10.1007/s10928-021-09784-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/30/2021] [Accepted: 09/19/2021] [Indexed: 10/28/2022]
Abstract
A central question in drug discovery is how to select drug candidates from a large number of available compounds. This analysis presents a model-based approach for comparing and ranking combinations of radiation and radiosensitizers. The approach is quantitative and based on the previously-derived Tumor Static Exposure (TSE) concept. Combinations of radiation and radiosensitizers are evaluated based on their ability to induce tumor regression relative to toxicity and other potential costs. The approach is presented in the form of a case study where the objective is to find the most promising candidate out of three radiosensitizing agents. Data from a xenograft study is described using a nonlinear mixed-effects modeling approach and a previously-published tumor model for radiation and radiosensitizing agents. First, the most promising candidate is chosen under the assumption that all compounds are equally toxic. The impact of toxicity in compound selection is then illustrated by assuming that one compound is more toxic than the others, leading to a different choice of candidate.
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Affiliation(s)
- Tim Cardilin
- Fraunhofer-Chalmers Centre, Chalmers Science Park, Gothenburg, Sweden. .,Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden.
| | - Joachim Almquist
- Fraunhofer-Chalmers Centre, Chalmers Science Park, Gothenburg, Sweden.,Clinical Pharmacology and Quantitative Pharmacology, Clinical Pharmacology & Safety Sciences, BioPharmaceuticals R&D, AstraZeneca, Gothenburg, Sweden
| | - Mats Jirstrand
- Fraunhofer-Chalmers Centre, Chalmers Science Park, Gothenburg, Sweden
| | - Astrid Zimmermann
- Translation Innovation Platform Oncology, Merck KGaA, Darmstadt, Germany
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13
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Ellis MA, Dalwadi MP, Ellis MJ, Byrne HM, Waters SL. A Systematically Reduced Mathematical Model for Organoid Expansion. Front Bioeng Biotechnol 2021; 9:670186. [PMID: 34178962 PMCID: PMC8222789 DOI: 10.3389/fbioe.2021.670186] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/20/2021] [Accepted: 05/06/2021] [Indexed: 12/12/2022] Open
Abstract
Organoids are three-dimensional multicellular tissue constructs. When cultured in vitro, they recapitulate the structure, heterogeneity, and function of their in vivo counterparts. As awareness of the multiple uses of organoids has grown, e.g. in drug discovery and personalised medicine, demand has increased for low-cost and efficient methods of producing them in a reproducible manner and at scale. Here we focus on a bioreactor technology for organoid production, which exploits fluid flow to enhance mass transport to and from the organoids. To ensure large numbers of organoids can be grown within the bioreactor in a reproducible manner, nutrient delivery to, and waste product removal from, the organoids must be carefully controlled. We develop a continuum mathematical model to investigate how mass transport within the bioreactor depends on the inlet flow rate and cell seeding density, focusing on the transport of two key metabolites: glucose and lactate. We exploit the thin geometry of the bioreactor to systematically simplify our model. This significantly reduces the computational cost of generating model solutions, and provides insight into the dominant mass transport mechanisms. We test the validity of the reduced models by comparison with simulations of the full model. We then exploit our reduced mathematical model to determine, for a given inlet flow rate and cell seeding density, the evolution of the spatial metabolite distributions throughout the bioreactor. To assess the bioreactor transport characteristics, we introduce metrics quantifying glucose conversion (the ratio between the total amounts of consumed and supplied glucose), the maximum lactate concentration, the proportion of the bioreactor with intolerable lactate concentrations, and the time when intolerable lactate concentrations are first experienced within the bioreactor. We determine the dependence of these metrics on organoid-line characteristics such as proliferation rate and rate of glucose consumption per cell. Finally, for a given organoid line, we determine how the distribution of metabolites and the associated metrics depend on the inlet flow rate. Insights from this study can be used to inform bioreactor operating conditions, ultimately improving the quality and number of bioreactor-expanded organoids.
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Affiliation(s)
- Meredith A. Ellis
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Mohit P. Dalwadi
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Marianne J. Ellis
- Department of Chemical Engineering, University of Bath, Bath, United Kingdom
- Cellesce, Cardiff Medicentre, Heath Park, Cardiff, United Kingdom
| | - Helen M. Byrne
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Sarah L. Waters
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
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Hurtado PJ, Richards C. Building mean field ODE models using the generalized linear chain trick & Markov chain theory. JOURNAL OF BIOLOGICAL DYNAMICS 2021; 15:S248-S272. [PMID: 33847236 DOI: 10.1080/17513758.2021.1912418] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/01/2020] [Accepted: 03/17/2021] [Indexed: 06/12/2023]
Abstract
The well known linear chain trick (LCT) allows modellers to derive mean field ODEs that assume gamma (Erlang) distributed passage times, by transitioning individuals sequentially through a chain of sub-states. The time spent in these sub-states is the sum of k exponentially distributed random variables, and is thus gamma distributed. The generalized linear chain trick (GLCT) extends this technique to the broader phase-type family of distributions, which includes exponential, Erlang, hypoexponential, and Coxian distributions. Phase-type distributions are the family of matrix exponential distributions on [0,∞) that represent the absorption time distributions for finite-state, continuous time Markov chains (CTMCs). Here we review CTMCs and phase-type distributions, then illustrate how to use the GLCT to efficiently build ODE models from underlying stochastic model assumptions. We introduce two novel model families by using the GLCT to generalize the Rosenzweig-MacArthur predator-prey model, and the SEIR model. We illustrate the kinds of complexity that can be captured by such models through multiple examples. We also show the benefits of using a GLCT-based model formulation to speed up the computation of numerical solutions to such models. These results highlight the intuitive nature, and utility, of using the GLCT to derive ODE models from first principles.
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Valentim CA, Rabi JA, David SA. Fractional Mathematical Oncology: On the potential of non-integer order calculus applied to interdisciplinary models. Biosystems 2021; 204:104377. [PMID: 33610556 DOI: 10.1016/j.biosystems.2021.104377] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/23/2020] [Revised: 02/04/2021] [Accepted: 02/04/2021] [Indexed: 12/22/2022]
Abstract
Mathematical Oncology investigates cancer-related phenomena through mathematical models as comprehensive as possible. Accordingly, an interdisciplinary approach involving concepts from biology to materials science can provide a deeper understanding of biological systems pertaining the disease. In this context, fractional calculus (also referred to as non-integer order) is a branch in mathematical analysis whose tools can describe complex phenomena comprising different time and space scales. Fractional-order models may allow a better description and understanding of oncological particularities, potentially contributing to decision-making in areas of interest such as tumor evolution, early diagnosis techniques and personalized treatment therapies. By following a phenomenological (i.e. mechanistic) approach, the present study surveys and explores different aspects of Fractional Mathematical Oncology, reviewing and discussing recent developments in view of their prospective applications.
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Affiliation(s)
- Carlos A Valentim
- Department of Biosystems Engineering, University of São Paulo, Pirassununga Campus, Brazil.
| | - José A Rabi
- Department of Biosystems Engineering, University of São Paulo, Pirassununga Campus, Brazil.
| | - Sergio A David
- Department of Biosystems Engineering, University of São Paulo, Pirassununga Campus, Brazil.
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Bando M, Tsunoyama Y, Suzuki K, Toki H. WAM to SeeSaw model for cancer therapy - overcoming LQM difficulties. Int J Radiat Biol 2020; 97:228-239. [PMID: 33253050 DOI: 10.1080/09553002.2021.1854487] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
Abstract
PURPOSE The assessment of biological effects caused by radiation exposure has been currently carried out with the linear-quadratic (LQ) model as an extension of the linear non-threshold (LNT) model. In this study, we suggest a new mathematical model named as SeaSaw (SS) model, which describes proliferation and cell death effects by taking account of Bergonie-Tribondeau's law in terms of a differential equation in time. We show how this model overcomes the long-standing difficulties of the LQ model. MATERIALS AND METHODS We construct the SS model as an extended Wack-A-Mole (WAM) model by using a differential equation with respect to time in order to express the dynamics of the proliferation effect. A large number of accumulated data of such parameters as α and β in the LQ based models provide us with valuable pieces of information on the corresponding parameter b 1 and the maximum volume V m of the SS model. The dose rate b 1 and the notion of active cell can explain the present data without introduction of β, which is obtained by comparing the SS model with not only the cancer therapy data but also with in vitro experimental data. Numerical calculations are presented to grasp the global features of the SS model. RESULTS The SS model predicts the time dependence of the number of active- and inactive-cells. The SS model clarifies how the effect of radiation depends on the cancer stage at the starting time in the treatment. Further, the time dependence of the tumor volume is calculated by changing individual dose strength, which results in the change of the irradiation duration for the same effect. We can consider continuous irradiation in the SS model with interesting outcome on the time dependence of the tumor volume for various dose rates. Especially by choosing the value of the dose rate to be balanced with the total growth rate, the tumor volume is kept constant. CONCLUSIONS The SS model gives a simple equation to study the situation of clinical radiation therapy and risk estimation of radiation. The radiation parameter extracted from the cancer therapy is close to the value obtained from animal experiment in vitro and in vivo. We expect the SS model leads us to a unified description of radiation therapy and protection and provides a great development in cancer-therapy clinical-planning.
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Affiliation(s)
- Masako Bando
- Research Center for Nuclear Physics, Osaka University, Osaka, Japan
| | - Yuichi Tsunoyama
- Radioisotope Research Center, Agency for Health, Safety and Environment, Kyoto University, Kyoto, Japan
| | - Kazuyo Suzuki
- Preemptive Medicine and Lifestyle-Related Disease Research Center, Kyoto University Hospital, Kyoto University, Kyoto, Japan
| | - Hiroshi Toki
- Research Center for Nuclear Physics, Osaka University, Osaka, Japan
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Jarrett AM, Bloom MJ, Godfrey W, Syed AK, Ekrut DA, Ehrlich LI, Yankeelov TE, Sorace AG. Mathematical modelling of trastuzumab-induced immune response in an in vivo murine model of HER2+ breast cancer. MATHEMATICAL MEDICINE AND BIOLOGY-A JOURNAL OF THE IMA 2020; 36:381-410. [PMID: 30239754 DOI: 10.1093/imammb/dqy014] [Citation(s) in RCA: 21] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/23/2017] [Revised: 06/14/2018] [Accepted: 08/24/2018] [Indexed: 02/06/2023]
Abstract
The goal of this study is to develop an integrated, mathematical-experimental approach for understanding the interactions between the immune system and the effects of trastuzumab on breast cancer that overexpresses the human epidermal growth factor receptor 2 (HER2+). A system of coupled, ordinary differential equations was constructed to describe the temporal changes in tumour growth, along with intratumoural changes in the immune response, vascularity, necrosis and hypoxia. The mathematical model is calibrated with serially acquired experimental data of tumour volume, vascularity, necrosis and hypoxia obtained from either imaging or histology from a murine model of HER2+ breast cancer. Sensitivity analysis shows that model components are sensitive for 12 of 13 parameters, but accounting for uncertainty in the parameter values, model simulations still agree with the experimental data. Given theinitial conditions, the mathematical model predicts an increase in the immune infiltrates over time in the treated animals. Immunofluorescent staining results are presented that validate this prediction by showing an increased co-staining of CD11c and F4/80 (proteins expressed by dendritic cells and/or macrophages) in the total tissue for the treated tumours compared to the controls ($p < 0.03$). We posit that the proposed mathematical-experimental approach can be used to elucidate driving interactions between the trastuzumab-induced responses in the tumour and the immune system that drive the stabilization of vasculature while simultaneously decreasing tumour growth-conclusions revealed by the mathematical model that were not deducible from the experimental data alone.
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Affiliation(s)
- Angela M Jarrett
- Institute for Computational Engineering and Sciences, University of Texas, Austin, TX, USA.,Livestrong Cancer Institutes, University of Texas, Austin, TX, USA
| | - Meghan J Bloom
- Department of Biomedical Engineering, University of Texas, Austin, TX, USA
| | - Wesley Godfrey
- Department of Molecular Biosciences, University of Texas, Austin, TX, USA
| | - Anum K Syed
- Department of Biomedical Engineering, University of Texas, Austin, TX, USA
| | - David A Ekrut
- Institute for Computational Engineering and Sciences, University of Texas, Austin, TX, USA
| | - Lauren I Ehrlich
- Department of Molecular Biosciences, University of Texas, Austin, TX, USA.,Institute for Cellular and Molecular Biology, University of Texas, Austin, TX, USA.,Livestrong Cancer Institutes, University of Texas, Austin, TX, USA
| | - Thomas E Yankeelov
- Institute for Computational Engineering and Sciences, University of Texas, Austin, TX, USA.,Department of Biomedical Engineering, University of Texas, Austin, TX, USA.,Department of Diagnostic Medicine, University of Texas, Austin, TX, USA.,Livestrong Cancer Institutes, University of Texas, Austin, TX, USA
| | - Anna G Sorace
- Department of Biomedical Engineering, University of Texas, Austin, TX, USA.,Department of Diagnostic Medicine, University of Texas, Austin, TX, USA.,Department of Oncology, University of Texas, Austin, TX, USA.,Livestrong Cancer Institutes, University of Texas, Austin, TX, USA
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Investigating the loco-regional control of simultaneous integrated boost intensity-modulated radiotherapy with different radiation fraction sizes for locally advanced non-small-cell lung cancer: clinical outcomes and the application of an extended LQ/TCP model. Radiat Oncol 2020; 15:124. [PMID: 32460796 PMCID: PMC7251706 DOI: 10.1186/s13014-020-01555-x] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/03/2019] [Accepted: 04/27/2020] [Indexed: 12/25/2022] Open
Abstract
Background To investigate the loco-regional progression-free survival (LPFS) of intensity-modulated radiotherapy (IMRT) with different fraction sizes for locally advanced non-small-cell lung cancer (LANSCLC), and to apply a new radiobiological model for tumor control probability (TCP). Methods One hundred and three LANSCLC patients treated with concurrent radiochemotherapy were retrospectively analyzed. Factors potentially predictive of LPFS were assessed in the univariate and multivariate analysis. Patients were divided into group A (2.0 ≤ fraction size<2.2Gy), B (2.2 ≤ fraction size<2.5Gy), and C (2.5 ≤ fraction size≤3.1Gy) according to the tertiles of fraction size. A novel LQRG/TCP model, incorporating four “R”s of radiobiology and Gompertzian tumor growth, was developed to predict LPFS and compared with the classical LQ/TCP model. Results With a median follow-up of 22.1 months, the median LPFS was 23.8 months. Fraction size was independently prognostic of LPFS. The median LPFS of group A, B and C was 13.8, 35.7 months and not reached, respectively. Using the new LQRG/TCP model, the average absolute and relative fitting errors for LPFS were 6.9 and 19.6% for group A, 5.5 and 8.8% for group B, 6.6 and 9.5% for group C, compared with 9.5 and 29.4% for group A, 16.6 and 36.7% for group B, 24.8 and 39.1% for group C using the conventional LQ/TCP model. Conclusions Hypo-fractionated IMRT could be an effective approach for dose intensification in LANSCLC. Compared with conventional LQ model, the LQRG model showed a better performance in predicting follow-up time dependent LPFS.
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19
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Farayola MF, Shafie S, Mohd Siam F, Khan I. Numerical simulation of normal and cancer cells' populations with fractional derivative under radiotherapy. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2020; 187:105202. [PMID: 31835107 DOI: 10.1016/j.cmpb.2019.105202] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/30/2019] [Revised: 11/08/2019] [Accepted: 11/11/2019] [Indexed: 06/10/2023]
Abstract
Background This paper presents a numerical simulation of normal and cancer cells' population dynamics during radiotherapy. The model used for the simulation was the improved cancer treatment model with radiotherapy. The model simulated the population changes during a fractionated cancer treatment process. The results gave the final populations of the cells, which provided the final volumes of the tumor and normal cells. Method The improved model was obtained by integrating the previous cancer treatment model with the Caputo fractional derivative. In addition, the cells' population decay due to radiation was accounted for by coupling the linear-quadratic model into the improved model. The simulation of the treatment process was done with numerical variables, numerical parameters, and radiation parameters. The numerical variables include the populations of the cells and the time of treatment. The numerical parameters were the model factors which included the proliferation rates of cells, competition coefficients of cells, and perturbation constant for normal cells. The radiation parameters were clinical data based on the treatment procedure. The numerical parameters were obtained from the previous literature while the numerical variables and radiation parameters, which were clinical data, were obtained from reported data of four cancer patients treated with radiotherapy. The four cancer patients had tumor volumes of 28.4 cm3, 18.8 cm3, 30.6 cm3, and 12.6 cm3 and were treated with different treatment plans and a fractionated dose of 1.8 Gy each. The initial populations of cells were obtained by using the tumor volumes. The computer simulations were done with MATLAB. Results The final volumes of the tumors, from the results of the simulations, were 5.67 cm3, 4.36 cm3, 5.74 cm3, and 6.15 cm3 while the normal cells' volumes were 28.17 cm3, 18.68 cm3, 30.34 cm3, and 12.54 cm3. The powers of the derivatives were 0.16774, 0.16557, 0.16835, and 0.16. A variance-based sensitivity analysis was done to corroborate the model with the clinical data. The result showed that the most sensitive factors were the power of the derivative and the cancer cells' proliferation rate. Conclusion The model provided information concerning the status of treatments and can also predict outcomes of other treatment plans.
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Affiliation(s)
- Musiliu Folarin Farayola
- Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Johor Bahru, Johor, Malaysia
| | - Sharidan Shafie
- Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Johor Bahru, Johor, Malaysia.
| | - Fuaada Mohd Siam
- Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Johor Bahru, Johor, Malaysia.
| | - Ilyas Khan
- Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam.
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Glioma invasion and its interplay with nervous tissue and therapy: A multiscale model. J Theor Biol 2020; 486:110088. [DOI: 10.1016/j.jtbi.2019.110088] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/04/2019] [Revised: 09/23/2019] [Accepted: 11/18/2019] [Indexed: 01/05/2023]
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21
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Lewin TD, Byrne HM, Maini PK, Caudell JJ, Moros EG, Enderling H. The importance of dead material within a tumour on the dynamics in response to radiotherapy. ACTA ACUST UNITED AC 2020; 65:015007. [DOI: 10.1088/1361-6560/ab4c27] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
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Hormuth DA, Jarrett AM, Yankeelov TE. Forecasting tumor and vasculature response dynamics to radiation therapy via image based mathematical modeling. Radiat Oncol 2020; 15:4. [PMID: 31898514 PMCID: PMC6941255 DOI: 10.1186/s13014-019-1446-2] [Citation(s) in RCA: 19] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/16/2019] [Accepted: 12/12/2019] [Indexed: 12/11/2022] Open
Abstract
BACKGROUND Intra-and inter-tumoral heterogeneity in growth dynamics and vascularity influence tumor response to radiation therapy. Quantitative imaging techniques capture these dynamics non-invasively, and these data can initialize and constrain predictive models of response on an individual basis. METHODS We have developed a family of 10 biologically-based mathematical models describing the spatiotemporal dynamics of tumor volume fraction, blood volume fraction, and response to radiation therapy. To evaluate this family of models, rats (n = 13) with C6 gliomas were imaged with magnetic resonance imaging (MRI) three times before, and four times following a single fraction of 20 Gy or 40 Gy whole brain irradiation. The first five 3D time series data of tumor volume fraction, estimated from diffusion-weighted (DW-) MRI, and blood volume fraction, estimated from dynamic contrast-enhanced (DCE-) MRI, were used to calibrate tumor-specific model parameters. The most parsimonious and well calibrated of the 10 models, selected using the Akaike information criterion, was then utilized to predict future growth and response at the final two imaging time points. Model predictions were compared at the global level (percent error in tumor volume, and Dice coefficient) as well as at the local or voxel level (concordance correlation coefficient). RESULT The selected model resulted in < 12% error in tumor volume predictions, strong spatial agreement between predicted and observed tumor volumes (Dice coefficient > 0.74), and high level of agreement at the voxel level between the predicted and observed tumor volume fraction and blood volume fraction (concordance correlation coefficient > 0.77 and > 0.65, respectively). CONCLUSIONS This study demonstrates that serial quantitative MRI data collected before and following radiation therapy can be used to accurately predict tumor and vasculature response with a biologically-based mathematical model that is calibrated on an individual basis. To the best of our knowledge, this is the first effort to characterize the tumor and vasculature response to radiation therapy temporally and spatially using imaging-driven mathematical models.
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Affiliation(s)
- David A Hormuth
- Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 E. 24th Street, POB 4.102, 1 University Station (C0200), Austin, TX, USA.
- Livestrong Cancer Institutes, The University of Texas at Austin, Austin, TX, USA.
| | - Angela M Jarrett
- Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 E. 24th Street, POB 4.102, 1 University Station (C0200), Austin, TX, USA
- Livestrong Cancer Institutes, The University of Texas at Austin, Austin, TX, USA
| | - Thomas E Yankeelov
- Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 E. 24th Street, POB 4.102, 1 University Station (C0200), Austin, TX, USA
- Livestrong Cancer Institutes, The University of Texas at Austin, Austin, TX, USA
- Departments of Biomedical Engineering, The University of Texas at Austin, Austin, TX, USA
- Departments of Diagnostic Medicine, The University of Texas at Austin, Austin, TX, USA
- Departments of Oncology, The University of Texas at Austin, Austin, TX, USA
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Mathematical Models of Cancer Cell Plasticity. JOURNAL OF ONCOLOGY 2019; 2019:2403483. [PMID: 31814825 PMCID: PMC6877945 DOI: 10.1155/2019/2403483] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/02/2019] [Revised: 08/20/2019] [Accepted: 09/17/2019] [Indexed: 01/29/2023]
Abstract
Quantitative modelling is increasingly important in cancer research, helping to integrate myriad diverse experimental data into coherent pictures of the disease and able to discriminate between competing hypotheses or suggest specific experimental lines of enquiry and new approaches to therapy. Here, we review a diverse set of mathematical models of cancer cell plasticity (a process in which, through genetic and epigenetic changes, cancer cells survive in hostile environments and migrate to more favourable environments, respectively), tumour growth, and invasion. Quantitative models can help to elucidate the complex biological mechanisms of cancer cell plasticity. In this review, we discuss models of plasticity, tumour progression, and metastasis under three broadly conceived mathematical modelling techniques: discrete, continuum, and hybrid, each with advantages and disadvantages. An emerging theme from the predictions of many of these models is that cell escape from the tumour microenvironment (TME) is encouraged by a combination of physiological stress locally (e.g., hypoxia), external stresses (e.g., the presence of immune cells), and interactions with the extracellular matrix. We also discuss the value of mathematical modelling for understanding cancer more generally.
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Brady R, Enderling H. Mathematical Models of Cancer: When to Predict Novel Therapies, and When Not to. Bull Math Biol 2019; 81:3722-3731. [PMID: 31338741 PMCID: PMC6764933 DOI: 10.1007/s11538-019-00640-x] [Citation(s) in RCA: 79] [Impact Index Per Article: 15.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/23/2019] [Accepted: 07/02/2019] [Indexed: 12/27/2022]
Abstract
The number of publications on mathematical modeling of cancer is growing at an exponential rate, according to PubMed records, provided by the US National Library of Medicine and the National Institutes of Health. Seminal papers have initiated and promoted mathematical modeling of cancer and have helped define the field of mathematical oncology (Norton and Simon in J Natl Cancer Inst 58:1735-1741, 1977; Norton in Can Res 48:7067-7071, 1988; Hahnfeldt et al. in Can Res 59:4770-4775, 1999; Anderson et al. in Comput Math Methods Med 2:129-154, 2000. https://doi.org/10.1080/10273660008833042 ; Michor et al. in Nature 435:1267-1270, 2005. https://doi.org/10.1038/nature03669 ; Anderson et al. in Cell 127:905-915, 2006. https://doi.org/10.1016/j.cell.2006.09.042 ; Benzekry et al. in PLoS Comput Biol 10:e1003800, 2014. https://doi.org/10.1371/journal.pcbi.1003800 ). Following the introduction of undergraduate and graduate programs in mathematical biology, we have begun to see curricula developing with specific and exclusive focus on mathematical oncology. In 2018, 218 articles on mathematical modeling of cancer were published in various journals, including not only traditional modeling journals like the Bulletin of Mathematical Biology and the Journal of Theoretical Biology, but also publications in renowned science, biology, and cancer journals with tremendous impact in the cancer field (Cell, Cancer Research, Clinical Cancer Research, Cancer Discovery, Scientific Reports, PNAS, PLoS Biology, Nature Communications, eLife, etc). This shows the breadth of cancer models that are being developed for multiple purposes. While some models are phenomenological in nature following a bottom-up approach, other models are more top-down data-driven. Here, we discuss the emerging trend in mathematical oncology publications to predict novel, optimal, sometimes even patient-specific treatments, and propose a convention when to use a model to predict novel treatments and, probably more importantly, when not to.
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Affiliation(s)
- Renee Brady
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL, 33647, USA
| | - Heiko Enderling
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL, 33647, USA.
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Tumor Clearance Analysis on a Cancer Chemo-Immunotherapy Mathematical Model. Bull Math Biol 2019; 81:4144-4173. [PMID: 31264136 DOI: 10.1007/s11538-019-00636-7] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2018] [Accepted: 06/20/2019] [Indexed: 01/08/2023]
Abstract
Mathematical models may allow us to improve our knowledge on tumor evolution and to better comprehend the dynamics between cancer, the immune system and the application of treatments such as chemotherapy and immunotherapy in both short and long term. In this paper, we solve the tumor clearance problem for a six-dimensional mathematical model that describes tumor evolution under immune response and chemo-immunotherapy treatments. First, by means of the localization of compact invariant sets method, we determine lower and upper bounds for all cells populations considered by the model and we use these results to establish sufficient conditions for the existence of a bounded positively invariant domain in the nonnegative orthant by applying LaSalle's invariance principle. Then, by exploiting a candidate Lyapunov function we determine sufficient conditions on the chemotherapy treatment to ensure tumor clearance. Further, we investigate the local stability of the tumor-free equilibrium point and compute conditions for asymptotic stability and tumor persistence. All conditions are given by inequalities in terms of the system parameters, and we perform numerical simulations with different values on the chemotherapy treatment to illustrate our results. Finally, we discuss the biological implications of our work.
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Li QW, Qiu B, Wang B, Zhang J, Chen L, Zhou Y, Qin JK, Guo SP, Xie WH, Hui ZG, Liang Y, Guo JY, Wang H, Zhu M, Shen WT, Duan LY, Chen LK, Zhang L, Long H, Wang YM, Liu H. Comparison of hyper- and hypofractionated radiation schemes with IMRT technique in small cell lung cancer: Clinical outcomes and the introduction of extended LQ and TCP models. Radiother Oncol 2019; 136:98-105. [PMID: 31015136 DOI: 10.1016/j.radonc.2019.03.035] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/03/2018] [Revised: 03/27/2019] [Accepted: 03/31/2019] [Indexed: 11/16/2022]
Abstract
PURPOSE To evaluate the outcomes of 45 Gy/15 fractions/once-daily and 45 Gy/30 fractions/twice-daily radiation schemes utilizing intensity-modulated radiation therapy (IMRT) in extensive stage small cell lung cancer (SCLC), and to build up a new radiobiological model for tumor control probability (TCP) considering multiple biological effects. METHODS Fifty-eight consecutive patients diagnosed with extensive stage SCLC, treated with chemotherapy and chest irradiation, were retrospectively reviewed. Thirty-seven received hyperfractionated IMRT (Hyper-IMRT, 45 Gy/30 fractions/twice-daily) and 21 received hypofractionated IMRT (Hypo-IMRT, 45 Gy/15 fractions/once-daily). Local progression-free survival (LPFS) and overall survival (OS) were calculated and compared. An extended linear-quadratic (LQ) model, LQRG, incorporating cell repair, redistribution, reoxygenation, regrowth and Gompertzian tumor growth was created based on the clinical data. The TCP model was reformulated to predict LPFS. The classical LQ and TCP models were compared with the new models. Akaike information criterion (AIC) was used to assess the quality of the models. RESULTS The 2-year LPFS (34.1% vs 27.9%, p = 0.44) and OS (76.9% vs 76.9%, p = 0.26) were similar between Hyper- and Hypo-IMRT patients. According to the LQRG model, the α/β calculated was 9.2 (95% confidence interval: 8.7-9.9) Gy after optimization. The average absolute and relative fitting errors for LPFS were 9.1% and 18.7% for Hyper-IMRT, and 8.8% and 16.2% for Hypo-IMRT of the new TCP model, compared with 29.1% and 62.3% for Hyper-IMRT, and 30.7% and 65.3% for Hypo-IMRT of the classical model. CONCLUSIONS Hypo- and Hyper-IMRT resulted in comparable local control in the chest irradiation of extensive stage SCLC. The LQRG model has better performance in predicting the TCP (or LPFS) of the two schemes.
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Affiliation(s)
- Qi-Wen Li
- Department of Radiation Oncology, State Key Laboratory of Oncology in South China, Collaborative Innovation Center for Cancer Medicine, Sun Yat-sen University Cancer Center, Guangzhou, China
| | - Bo Qiu
- Department of Radiation Oncology, State Key Laboratory of Oncology in South China, Collaborative Innovation Center for Cancer Medicine, Sun Yat-sen University Cancer Center, Guangzhou, China
| | - Bin Wang
- Department of Radiation Oncology, State Key Laboratory of Oncology in South China, Collaborative Innovation Center for Cancer Medicine, Sun Yat-sen University Cancer Center, Guangzhou, China
| | - Jun Zhang
- Department of Radiation Oncology, State Key Laboratory of Oncology in South China, Collaborative Innovation Center for Cancer Medicine, Sun Yat-sen University Cancer Center, Guangzhou, China
| | - Li Chen
- Department of Radiation Oncology, State Key Laboratory of Oncology in South China, Collaborative Innovation Center for Cancer Medicine, Sun Yat-sen University Cancer Center, Guangzhou, China
| | - Yin Zhou
- Evidance Medical Technologies Inc., Suzhou, China
| | | | - Su-Ping Guo
- Department of Radiation Oncology, State Key Laboratory of Oncology in South China, Collaborative Innovation Center for Cancer Medicine, Sun Yat-sen University Cancer Center, Guangzhou, China
| | - Wei-Hao Xie
- Department of Radiation Oncology, State Key Laboratory of Oncology in South China, Collaborative Innovation Center for Cancer Medicine, Sun Yat-sen University Cancer Center, Guangzhou, China
| | - Zhou-Guang Hui
- Department of Radiation Oncology, National Cancer Center/Cancer Hospital, Chinese Academy of Medical Sciences and Peking Union Medical College, Beijing, China
| | - Ying Liang
- Department of Medical Oncology, State Key Laboratory of Oncology in South China, Collaborative Innovation Center for Cancer Medicine, Sun Yat-sen University Cancer Center, Guangzhou, China
| | - Jin-Yu Guo
- Department of Radiation Oncology, State Key Laboratory of Oncology in South China, Collaborative Innovation Center for Cancer Medicine, Sun Yat-sen University Cancer Center, Guangzhou, China
| | - He Wang
- Homology Medical Technologies Inc., Suzhou, China
| | - Meng Zhu
- Homology Medical Technologies Inc., Suzhou, China
| | - Wen-Tong Shen
- Department of Radiation Oncology, Ruijin Hospital, Shanghai Jiaotong University School of Medicine, Shanghai, China
| | - Long-Yan Duan
- Department of Radiation Oncology, Ruijin Hospital, Shanghai Jiaotong University School of Medicine, Shanghai, China
| | - Li-Kun Chen
- Department of Medical Oncology, State Key Laboratory of Oncology in South China, Collaborative Innovation Center for Cancer Medicine, Sun Yat-sen University Cancer Center, Guangzhou, China
| | - Li Zhang
- Department of Medical Oncology, State Key Laboratory of Oncology in South China, Collaborative Innovation Center for Cancer Medicine, Sun Yat-sen University Cancer Center, Guangzhou, China
| | - Hao Long
- Department of Thoracic Surgery, State Key Laboratory of Oncology in South China, Collaborative Innovation Center for Cancer Medicine, Sun Yat-sen University Cancer Center, Guangzhou, China
| | - Yi-Ming Wang
- Department of Oncology, The First Affiliated Hospital of Jinan University, Guangzhou, China
| | - Hui Liu
- Department of Radiation Oncology, State Key Laboratory of Oncology in South China, Collaborative Innovation Center for Cancer Medicine, Sun Yat-sen University Cancer Center, Guangzhou, China.
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Cardilin T, Almquist J, Jirstrand M, Zimmermann A, Lignet F, El Bawab S, Gabrielsson J. Modeling long-term tumor growth and kill after combinations of radiation and radiosensitizing agents. Cancer Chemother Pharmacol 2019; 83:1159-1173. [PMID: 30976845 PMCID: PMC6499765 DOI: 10.1007/s00280-019-03829-y] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2018] [Accepted: 04/01/2019] [Indexed: 11/30/2022]
Abstract
PURPOSE Radiation therapy, whether given alone or in combination with chemical agents, is one of the cornerstones of oncology. We develop a quantitative model that describes tumor growth during and after treatment with radiation and radiosensitizing agents. The model also describes long-term treatment effects including tumor regrowth and eradication. METHODS We challenge the model with data from a xenograft study using a clinically relevant administration schedule and use a mixed-effects approach for model-fitting. We use the calibrated model to predict exposure combinations that result in tumor eradication using Tumor Static Exposure (TSE). RESULTS The model is able to adequately describe data from all treatment groups, with the parameter estimates taking biologically reasonable values. Using TSE, we predict the total radiation dose necessary for tumor eradication to be 110 Gy, which is reduced to 80 or 30 Gy with co-administration of 25 or 100 mg kg-1 of a radiosensitizer. TSE is also explored via a heat map of different growth and shrinkage rates. Finally, we discuss the translational potential of the model and TSE concept to humans. CONCLUSIONS The new model is capable of describing different tumor dynamics including tumor eradication and tumor regrowth with different rates, and can be calibrated using data from standard xenograft experiments. TSE and related concepts can be used to predict tumor shrinkage and eradication, and have the potential to guide new experiments and support translations from animals to humans.
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Affiliation(s)
- Tim Cardilin
- Fraunhofer-Chalmers Centre, Chalmers Science Park, Gothenburg, Sweden. .,Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden.
| | - Joachim Almquist
- Fraunhofer-Chalmers Centre, Chalmers Science Park, Gothenburg, Sweden
| | - Mats Jirstrand
- Fraunhofer-Chalmers Centre, Chalmers Science Park, Gothenburg, Sweden
| | - Astrid Zimmermann
- Translation Innovation Platform Oncology, Merck Healthcare KGaA, Darmstadt, Germany
| | - Floriane Lignet
- Translational Medicine, Quantitative Pharmacology, Merck Healthcare KGaA, Darmstadt, Germany
| | - Samer El Bawab
- Translational Medicine, Quantitative Pharmacology, Merck Healthcare KGaA, Darmstadt, Germany
| | - Johan Gabrielsson
- Department of Biomedical Sciences and Veterinary Public Health, Swedish University of Agricultural Sciences, Uppsala, Sweden
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Jarrett AM, Lima EABF, Hormuth DA, McKenna MT, Feng X, Ekrut DA, Resende ACM, Brock A, Yankeelov TE. Mathematical models of tumor cell proliferation: A review of the literature. Expert Rev Anticancer Ther 2018; 18:1271-1286. [PMID: 30252552 PMCID: PMC6295418 DOI: 10.1080/14737140.2018.1527689] [Citation(s) in RCA: 59] [Impact Index Per Article: 9.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
Abstract
INTRODUCTION A defining hallmark of cancer is aberrant cell proliferation. Efforts to understand the generative properties of cancer cells span all biological scales: from genetic deviations and alterations of metabolic pathways to physical stresses due to overcrowding, as well as the effects of therapeutics and the immune system. While these factors have long been studied in the laboratory, mathematical and computational techniques are being increasingly applied to help understand and forecast tumor growth and treatment response. Advantages of mathematical modeling of proliferation include the ability to simulate and predict the spatiotemporal development of tumors across multiple experimental scales. Central to proliferation modeling is the incorporation of available biological data and validation with experimental data. Areas covered: We present an overview of past and current mathematical strategies directed at understanding tumor cell proliferation. We identify areas for mathematical development as motivated by available experimental and clinical evidence, with a particular emphasis on emerging, non-invasive imaging technologies. Expert commentary: The data required to legitimize mathematical models are often difficult or (currently) impossible to obtain. We suggest areas for further investigation to establish mathematical models that more effectively utilize available data to make informed predictions on tumor cell proliferation.
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Affiliation(s)
- Angela M Jarrett
- a Institute for Computational Engineering and Sciences , The University of Texas at Austin , Austin , USA
- b Livestrong Cancer Institutes , The University of Texas at Austin , Austin , USA
| | - Ernesto A B F Lima
- a Institute for Computational Engineering and Sciences , The University of Texas at Austin , Austin , USA
| | - David A Hormuth
- a Institute for Computational Engineering and Sciences , The University of Texas at Austin , Austin , USA
- b Livestrong Cancer Institutes , The University of Texas at Austin , Austin , USA
| | - Matthew T McKenna
- c Department of Biomedical Engineering , Vanderbilt University , Nashville , USA
| | - Xinzeng Feng
- a Institute for Computational Engineering and Sciences , The University of Texas at Austin , Austin , USA
| | - David A Ekrut
- a Institute for Computational Engineering and Sciences , The University of Texas at Austin , Austin , USA
| | - Anna Claudia M Resende
- a Institute for Computational Engineering and Sciences , The University of Texas at Austin , Austin , USA
- d Department of Computational Modeling , National Laboratory for Scientific Computing , Petrópolis , Brazil
| | - Amy Brock
- b Livestrong Cancer Institutes , The University of Texas at Austin , Austin , USA
- e Department of Biomedical Engineering , The University of Texas at Austin , Austin , USA
| | - Thomas E Yankeelov
- a Institute for Computational Engineering and Sciences , The University of Texas at Austin , Austin , USA
- b Livestrong Cancer Institutes , The University of Texas at Austin , Austin , USA
- e Department of Biomedical Engineering , The University of Texas at Austin , Austin , USA
- f Department of Diagnostic Medicine , The University of Texas at Austin , Austin , USA
- g Department of Oncology , The University of Texas at Austin , Austin , USA
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Henscheid N, Clarkson E, Myers KJ, Barrett HH. Physiological random processes in precision cancer therapy. PLoS One 2018; 13:e0199823. [PMID: 29958271 PMCID: PMC6025881 DOI: 10.1371/journal.pone.0199823] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/26/2018] [Accepted: 06/14/2018] [Indexed: 02/07/2023] Open
Abstract
Many different physiological processes affect the growth of malignant lesions and their response to therapy. Each of these processes is spatially and genetically heterogeneous; dynamically evolving in time; controlled by many other physiological processes, and intrinsically random and unpredictable. The objective of this paper is to show that all of these properties of cancer physiology can be treated in a unified, mathematically rigorous way via the theory of random processes. We treat each physiological process as a random function of position and time within a tumor, defining the joint statistics of such functions via the infinite-dimensional characteristic functional. The theory is illustrated by analyzing several models of drug delivery and response of a tumor to therapy. To apply the methodology to precision cancer therapy, we use maximum-likelihood estimation with Emission Computed Tomography (ECT) data to estimate unknown patient-specific physiological parameters, ultimately demonstrating how to predict the probability of tumor control for an individual patient undergoing a proposed therapeutic regimen.
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Affiliation(s)
- Nick Henscheid
- Center for Gamma-Ray Imaging, University of Arizona, Tucson, AZ, United States of America
- Program in Applied Mathematics, University of Arizona, Tucson, AZ, United States of America
| | - Eric Clarkson
- Center for Gamma-Ray Imaging, University of Arizona, Tucson, AZ, United States of America
- Program in Applied Mathematics, University of Arizona, Tucson, AZ, United States of America
- Department of Medical Imaging, University of Arizona, Tucson, AZ, United States of America
- College of Optical Sciences, University of Arizona, Tucson, AZ, United States of America
| | - Kyle J. Myers
- Center for Devices and Radiological Health, Food and Drug Administration, Silver Spring, MD, United States of America
| | - Harrison H. Barrett
- Center for Gamma-Ray Imaging, University of Arizona, Tucson, AZ, United States of America
- Program in Applied Mathematics, University of Arizona, Tucson, AZ, United States of America
- Department of Medical Imaging, University of Arizona, Tucson, AZ, United States of America
- College of Optical Sciences, University of Arizona, Tucson, AZ, United States of America
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Walker R, Poleszczuk J, Pilon-Thomas S, Kim S, Anderson AARA, Czerniecki BJ, Harrison LB, Moros EG, Enderling H. Immune interconnectivity of anatomically distant tumors as a potential mediator of systemic responses to local therapy. Sci Rep 2018; 8:9474. [PMID: 29930290 PMCID: PMC6013469 DOI: 10.1038/s41598-018-27718-1] [Citation(s) in RCA: 27] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2017] [Accepted: 06/07/2018] [Indexed: 12/14/2022] Open
Abstract
Complex interactions occur between tumor and host immune system at each site in the metastatic setting, the outcome of which can determine behavior ranging from dormancy to rapid growth. An additional layer of complexity arises from the understanding that cytotoxic T cells can traffic through the host circulatory system. Coupling mathematical models of local tumor-immune dynamics and systemic T cell trafficking allows us to simulate the evolution of tumor and immune cell populations in anatomically distant sites following local therapy and thus computationally evaluate immune interconnectivity. Results suggest that the presence of a secondary site may either inhibit or promote growth of the primary, depending on the capacity for immune recruitment of each tumor and the resulting systemic redistribution of T cells. Treatment such as surgical resection and radiotherapy can be simulated to estimate both the decrease in tumor volume at the local treatment-targeted site, and the change in overall tumor burden and tumor growth trajectories across all sites. Qualitatively similar responses of distant tumors to local therapy (positive and negative abscopal effects) to those reported in the clinical setting were observed. Such findings may facilitate an improved understanding of general disease kinetics in the metastatic setting: if metastatic sites are interconnected through the immune system, truly local therapy does not exist.
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Affiliation(s)
- Rachel Walker
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL, USA
| | - Jan Poleszczuk
- Nalecz Institute of Biocybernetics and Biomedical Engineering, Polish Academy of Sciences, Warsaw, Poland
| | - Shari Pilon-Thomas
- Department of Immunology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL, USA
| | - Sungjune Kim
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL, USA
| | - Alexander A R A Anderson
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL, USA
| | - Brian J Czerniecki
- Department of Breast Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL, USA
| | - Louis B Harrison
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL, USA
| | - Eduardo G Moros
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL, USA
| | - Heiko Enderling
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL, USA. .,Department of Radiation Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL, USA.
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The Evolution of Tumour Composition During Fractionated Radiotherapy: Implications for Outcome. Bull Math Biol 2018; 80:1207-1235. [PMID: 29488054 DOI: 10.1007/s11538-018-0391-9] [Citation(s) in RCA: 30] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2017] [Accepted: 01/08/2018] [Indexed: 01/29/2023]
Abstract
Current protocols for delivering radiotherapy are based primarily on tumour stage and nodal and metastases status, even though it is well known that tumours and their microenvironments are highly heterogeneous. It is well established that the local oxygen tension plays an important role in radiation-induced cell death, with hypoxic tumour regions responding poorly to irradiation. Therefore, to improve radiation response, it is important to understand more fully the spatiotemporal distribution of oxygen within a growing tumour before and during fractionated radiation. To this end, we have extended a spatially resolved mathematical model of tumour growth, first proposed by Greenspan (Stud Appl Math 51:317-340, 1972), to investigate the effects of oxygen heterogeneity on radiation-induced cell death. In more detail, cell death due to radiation at each location in the tumour, as determined by the well-known linear-quadratic model, is assumed also to depend on the local oxygen concentration. The oxygen concentration is governed by a reaction-diffusion equation that is coupled to an integro-differential equation that determines the size of the assumed spherically symmetric tumour. We combine numerical and analytical techniques to investigate radiation response of tumours with different intratumoral oxygen distribution profiles. Model simulations reveal a rapid transient increase in hypoxia upon regrowth of the tumour spheroid post-irradiation. We investigate the response to different radiation fractionation schedules and identify a tumour-specific relationship between inter-fraction time and dose per fraction to achieve cure. The rich dynamics exhibited by the model suggest that spatial heterogeneity may be important for predicting tumour response to radiotherapy for clinical applications.
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Cardilin T, Almquist J, Jirstrand M, Zimmermann A, El Bawab S, Gabrielsson J. Model-Based Evaluation of Radiation and Radiosensitizing Agents in Oncology. CPT-PHARMACOMETRICS & SYSTEMS PHARMACOLOGY 2017; 7:51-58. [PMID: 29218836 PMCID: PMC5784742 DOI: 10.1002/psp4.12268] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/18/2017] [Revised: 11/08/2017] [Accepted: 11/08/2017] [Indexed: 12/25/2022]
Abstract
Radiotherapy is one of the major therapy forms in oncology, and combination therapies involving radiation and chemical compounds can yield highly effective tumor eradication. In this paper, we develop a tumor growth inhibition model for combination therapy with radiation and radiosensitizing agents. Moreover, we extend previous analyses of drug combinations by introducing the tumor static exposure (TSE) curve. The TSE curve for radiation and radiosensitizer visualizes exposure combinations sufficient for tumor regression. The model and TSE analysis are then tested on xenograft data. The calibrated model indicates that the highest dose of combination therapy increases the time until tumor regrowth 10-fold. The TSE curve shows that with an average radiosensitizer concentration of 1.0 μg/mL the radiation dose can be decreased from 2.2 Gy to 0.7 Gy. Finally, we successfully predict the effect of a clinically relevant treatment schedule, which contributes to validating both the model and the TSE concept.
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Affiliation(s)
- Tim Cardilin
- Fraunhofer-Chalmers Centre, Chalmers Science Park, Gothenburg, Sweden.,Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden
| | - Joachim Almquist
- Fraunhofer-Chalmers Centre, Chalmers Science Park, Gothenburg, Sweden.,Department of Biology and Biological Engineering, Chalmers University of Technology, Gothenburg, Sweden
| | - Mats Jirstrand
- Fraunhofer-Chalmers Centre, Chalmers Science Park, Gothenburg, Sweden
| | | | - Samer El Bawab
- Global Early Development - Quantitative Pharmacology and Drug Disposition, Quantitative Pharmacology, Merck, Darmstadt, Germany
| | - Johan Gabrielsson
- Department of Biomedical Sciences and Veterinary Public Health, Swedish University of Agricultural Sciences, Uppsala, Sweden
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A branching process model of heterogeneous DNA damages caused by radiotherapy in in vitro cell cultures. Math Biosci 2017; 294:100-109. [PMID: 29054768 DOI: 10.1016/j.mbs.2017.09.006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2017] [Revised: 07/21/2017] [Accepted: 09/23/2017] [Indexed: 11/22/2022]
Abstract
This paper deals with the dynamic modeling and simulation of cell damage heterogeneity and associated mutant cell phenotypes in the therapeutic responses of cancer cell populations submitted to a radiotherapy session during in vitro assays. Each cell is described by a finite number of phenotypic states with possible transitions between them. The population dynamics is then given by an age-dependent multi-type branching process. From this representation, we obtain formulas for the average size of the global survival population as well as the one of subpopulations associated with 10 mutation phenotypes. The proposed model has been implemented into Matlab© and the numerical results corroborate the ability of the model to reproduce four major types of cell responses: delayed growth, anti-proliferative, cytostatic and cytotoxic.
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Geng C, Paganetti H, Grassberger C. Prediction of Treatment Response for Combined Chemo- and Radiation Therapy for Non-Small Cell Lung Cancer Patients Using a Bio-Mathematical Model. Sci Rep 2017; 7:13542. [PMID: 29051600 PMCID: PMC5648928 DOI: 10.1038/s41598-017-13646-z] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2017] [Accepted: 09/19/2017] [Indexed: 12/13/2022] Open
Abstract
The goal of this work was to develop a mathematical model to predict Kaplan-Meier survival curves for chemotherapy combined with radiation in Non-Small Cell Lung Cancer patients for use in clinical trial design. The Gompertz model was used to describe tumor growth, radiation effect was simulated by the linear-quadratic model with an α/β-ratio of 10, and chemotherapy effect was based on the log-cell kill model. To account for repopulation during treatment, we considered two independent methods: 1) kickoff-repopulation using exponential growth with a decreased volume doubling time, or 2) Gompertz-repopulation using the gradually accelerating growth rate with tumor shrinkage. The input parameters were independently estimated by fitting to the SEER database for untreated tumors, RTOG-8808 for radiation only, and RTOG-9410 for sequential chemo-radiation. Applying the model, the benefit from concurrent chemo-radiation comparing to sequential for stage III patients was predicted to be a 6.6% and 6.2% improvement in overall survival for 3 and 5-years respectively, comparing well to the 5.3% and 4.5% observed in RTOG-9410. In summary, a mathematical model was developed to model tumor growth over extended periods of time, and can be used for the optimization of combined chemo-radiation scheduling and sequencing.
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Affiliation(s)
- Changran Geng
- Department of Radiation Oncology, Massachusetts General Hospital, Harvard Medical School, 30 Fruit Street, Boston, MA, 02114, USA.
- Department of Nuclear Science and Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People's Republic of China.
| | - Harald Paganetti
- Department of Radiation Oncology, Massachusetts General Hospital, Harvard Medical School, 30 Fruit Street, Boston, MA, 02114, USA
| | - Clemens Grassberger
- Department of Radiation Oncology, Massachusetts General Hospital, Harvard Medical School, 30 Fruit Street, Boston, MA, 02114, USA.
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An Optimal Control Approach for the Treatment of Solid Tumors with Angiogenesis Inhibitors. MATHEMATICS 2017. [DOI: 10.3390/math5040049] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
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36
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Ingalls B, Duncker B, Kim D, McConkey B. Systems Level Modeling of the Cell Cycle Using Budding Yeast. Cancer Inform 2017. [DOI: 10.1177/117693510700300020] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022] Open
Abstract
Proteins involved in the regulation of the cell cycle are highly conserved across all eukaryotes, and so a relatively simple eukaryote such as yeast can provide insight into a variety of cell cycle perturbations including those that occur in human cancer. To date, the budding yeast Saccharomyces cerevisiae has provided the largest amount of experimental and modeling data on the progression of the cell cycle, making it a logical choice for in-depth studies of this process. Moreover, the advent of methods for collection of high-throughput genome, transcriptome, and proteome data has provided a means to collect and precisely quantify simultaneous cell cycle gene transcript and protein levels, permitting modeling of the cell cycle on the systems level. With the appropriate mathematical framework and sufficient and accurate data on cell cycle components, it should be possible to create a model of the cell cycle that not only effectively describes its operation, but can also predict responses to perturbations such as variation in protein levels and responses to external stimuli including targeted inhibition by drugs. In this review, we summarize existing data on the yeast cell cycle, proteomics technologies for quantifying cell cycle proteins, and the mathematical frameworks that can integrate this data into representative and effective models. Systems level modeling of the cell cycle will require the integration of high-quality data with the appropriate mathematical framework, which can currently be attained through the combination of dynamic modeling based on proteomics data and using yeast as a model organism.
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Affiliation(s)
- B.P. Ingalls
- Department of Applied Mathematics, University of Waterloo
| | | | - D.R. Kim
- Department of Biology, University of Waterloo
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Mechanistic Model for Cancer Growth and Response to Chemotherapy. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2017; 2017:3676295. [PMID: 28928794 PMCID: PMC5591976 DOI: 10.1155/2017/3676295] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/09/2017] [Accepted: 07/19/2017] [Indexed: 12/28/2022]
Abstract
Cancer treatment has developed over the years; however not all patients respond to this treatment, and therefore further research is needed. In this paper, we employ mathematical modeling to understand the behavior of cancer and its interaction with therapy. We study a drug delivery and drug-cell interaction model along with cell proliferation. Due to the fact that cancer cells grow when there are enough nutrients and oxygen, proliferation can be a barrier against a response to therapy. To understand the effect of this factor, we perform numerical simulations of the model for different values of the parameters with a continuous delivery of the drug. The numerical results showed that cancer dies after short apoptotic cycles if the cancer is highly vascularized or if the penetration of the drug is high. This suggests promoting angiogenesis or perfusion of the drug. This result is similar to the situation where proliferation is not considered since the constant release of drug overcomes the growth of the cells and thus the effect of proliferation can be neglected.
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38
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Caudell JJ, Torres-Roca JF, Gillies RJ, Enderling H, Kim S, Rishi A, Moros EG, Harrison LB. The future of personalised radiotherapy for head and neck cancer. Lancet Oncol 2017; 18:e266-e273. [PMID: 28456586 PMCID: PMC7771279 DOI: 10.1016/s1470-2045(17)30252-8] [Citation(s) in RCA: 142] [Impact Index Per Article: 20.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/07/2016] [Revised: 01/30/2017] [Accepted: 01/31/2017] [Indexed: 12/25/2022]
Abstract
Radiotherapy has long been the mainstay of treatment for patients with head and neck cancer and has traditionally involved a stage-dependent strategy whereby all patients with the same TNM stage receive the same therapy. We believe there is a substantial opportunity to improve radiotherapy delivery beyond just technological and anatomical precision. In this Series paper, we explore several new ideas that could improve understanding of the phenotypic and genotypic differences that exist between patients and their tumours. We discuss how exploiting these differences and taking advantage of precision medicine tools-such as genomics, radiomics, and mathematical modelling-could open new doors to personalised radiotherapy adaptation and treatment. We propose a new treatment shift that moves away from an era of empirical dosing and fractionation to an era focused on the development of evidence to guide personalisation and biological adaptation of radiotherapy. We believe these approaches offer the potential to improve outcomes and reduce toxicity.
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Affiliation(s)
- Jimmy J Caudell
- Department of Radiation Oncology, Moffitt Cancer Center and Research Institute, Tampa, FL, USA
| | - Javier F Torres-Roca
- Department of Radiation Oncology, Moffitt Cancer Center and Research Institute, Tampa, FL, USA
| | - Robert J Gillies
- Department of Cancer Imaging and Metabolism, Moffitt Cancer Center and Research Institute, Tampa, FL, USA
| | - Heiko Enderling
- Department of Integrated Mathematical Oncology, Moffitt Cancer Center and Research Institute, Tampa, FL, USA
| | - Sungjune Kim
- Department of Radiation Oncology, Moffitt Cancer Center and Research Institute, Tampa, FL, USA
| | - Anupam Rishi
- Department of Radiation Oncology, Moffitt Cancer Center and Research Institute, Tampa, FL, USA
| | - Eduardo G Moros
- Department of Radiation Oncology, Moffitt Cancer Center and Research Institute, Tampa, FL, USA
| | - Louis B Harrison
- Department of Radiation Oncology, Moffitt Cancer Center and Research Institute, Tampa, FL, USA.
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Klamka J, Maurer H, Swierniak A. Local controllability and optimal control for a model of combined anticancer therapy with control delays. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2017; 14:195-216. [PMID: 27879128 DOI: 10.3934/mbe.2017013] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We study some control properties of a class of two-compartmental models of response to anticancer treatment which combines anti-angiogenic and cytotoxic drugs and take into account multiple control delays. We formulate sufficient local controllability conditions for semilinear systems resulting from these models. The control delays are related to PK/PD effects and some clinical recommendations, e.g., normalization of the vascular network. The optimized protocols of the combined therapy for the model, considered as solutions to an optimal control problem with delays in control, are found using necessary conditions of optimality and numerical computations. Our numerical approach uses dicretization and nonlinear programming methods as well as the direct optimization of switching times. The structural sensitivity of the considered control properties and optimal solutions is also discussed.
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Affiliation(s)
- Jerzy Klamka
- Silesian University of Technology, Department of Automatic Control, Akademicka 16, 44101 Gliwice, Poland.
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Oroji A, Omar M, Yarahmadian S. An Îto stochastic differential equations model for the dynamics of the MCF-7 breast cancer cell line treated by radiotherapy. J Theor Biol 2016; 407:128-137. [DOI: 10.1016/j.jtbi.2016.07.035] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/06/2015] [Revised: 05/05/2016] [Accepted: 07/20/2016] [Indexed: 01/07/2023]
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Cardilin T, Almquist J, Jirstrand M, Sostelly A, Amendt C, El Bawab S, Gabrielsson J. Tumor Static Concentration Curves in Combination Therapy. AAPS JOURNAL 2016; 19:456-467. [PMID: 27681102 DOI: 10.1208/s12248-016-9991-1] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/24/2016] [Accepted: 09/09/2016] [Indexed: 11/30/2022]
Abstract
Combination therapies are widely accepted as a cornerstone for treatment of different cancer types. A tumor growth inhibition (TGI) model is developed for combinations of cetuximab and cisplatin obtained from xenograft mice. Unlike traditional TGI models, both natural cell growth and cell death are considered explicitly. The growth rate was estimated to 0.006 h-1 and the natural cell death to 0.0039 h-1 resulting in a tumor doubling time of 14 days. The tumor static concentrations (TSC) are predicted for each individual compound. When the compounds are given as single-agents, the required concentrations were computed to be 506 μg · mL-1 and 56 ng · mL-1 for cetuximab and cisplatin, respectively. A TSC curve is constructed for different combinations of the two drugs, which separates concentration combinations into regions of tumor shrinkage and tumor growth. The more concave the TSC curve is, the lower is the total exposure to test compounds necessary to achieve tumor regression. The TSC curve for cetuximab and cisplatin showed weak concavity. TSC values and TSC curves were estimated that predict tumor regression for 95% of the population by taking between-subject variability into account. The TSC concept is further discussed for different concentration-effect relationships and for combinations of three or more compounds.
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Affiliation(s)
- Tim Cardilin
- Fraunhofer-Chalmers Centre, Chalmers Science Park, Gothenburg, Sweden. .,Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden.
| | - Joachim Almquist
- Fraunhofer-Chalmers Centre, Chalmers Science Park, Gothenburg, Sweden.,Department of Biology and Biological Engineering, Chalmers University of Technology, Gothenburg, Sweden
| | - Mats Jirstrand
- Fraunhofer-Chalmers Centre, Chalmers Science Park, Gothenburg, Sweden
| | - Alexandre Sostelly
- Global Early Development-Quantitative Pharmacology and Drug Disposition, Quantitative Pharmacology, Merck, Darmstadt, Germany.,Pharmaceutical Research and Early Development, Hoffmann-Le Roche, Basel, Switzerland
| | | | - Samer El Bawab
- Global Early Development-Quantitative Pharmacology and Drug Disposition, Quantitative Pharmacology, Merck, Darmstadt, Germany
| | - Johan Gabrielsson
- Department of Biomedical Sciences and Veterinary Public Health, Swedish University of Agricultural Sciences, Uppsala, Sweden
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Ballhausen H, Belka C. In the linear quadratic model, the Poisson approximation and the Zaider-Minerbo formula agree on the ranking of tumor control probabilities, up to a critical cell birth rate. Int J Radiat Biol 2016; 93:279-285. [PMID: 27658488 DOI: 10.1080/09553002.2017.1238527] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
Abstract
PURPOSE To provide a rule for the agreement or disagreement of the Poisson approximation (PA) and the Zaider-Minerbo formula (ZM) on the ranking of treatment alternatives in terms of tumor control probability (TCP) in the linear quadratic model. MATERIALS AND METHODS A general criterion involving a critical cell birth rate was formally derived. For demonstration, the criterion was applied to a distinct radiobiological model of fast growing head and neck tumors and a respective range of 22 conventional and nonconventional head and neck schedules. RESULTS There is a critical cell birth rate bcrit below which PA and ZM agree on which one out of two alternative treatment schemes with single-cell survival curves S'(t) and S''(t) offers better TCP: [Formula: see text] For cell birth rates b above this critical cell birth rate, PA and ZM disagree if and only if b >bcrit > 0. In case of the exemplary head and neck schedules, out of 231 possible combinations, only 16 or 7% were found where PA and ZM disagreed. In all 231 cases the prediction of the criterion was numerically confirmed, and cell birth rates at crossovers between schedules matched the calculated critical cell birth rates. CONCLUSIONS TCP estimated by PA and ZM almost never numerically coincide. Still, in many cases both formulas at least agree about which one out of two alternative fractionation schemes offers better TCP. In case of fast growing tumors featuring a high cell birth rate, however, ZM may suggest a re-evaluation of treatment options.
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Affiliation(s)
- Hendrik Ballhausen
- a Department of Radiation Oncology , University Hospitals of LMU Munich , Munich , Germany
| | - Claus Belka
- a Department of Radiation Oncology , University Hospitals of LMU Munich , Munich , Germany
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Hatzikirou H, Alfonso JCL, Mühle S, Stern C, Weiss S, Meyer-Hermann M. Cancer therapeutic potential of combinatorial immuno- and vasomodulatory interventions. J R Soc Interface 2016; 12:rsif.2015.0439. [PMID: 26510827 DOI: 10.1098/rsif.2015.0439] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/23/2023] Open
Abstract
Currently, most of the basic mechanisms governing tumour-immune system interactions, in combination with modulations of tumour-associated vasculature, are far from being completely understood. Here, we propose a mathematical model of vascularized tumour growth, where the main novelty is the modelling of the interplay between functional tumour vasculature and effector cell recruitment dynamics. Parameters are calibrated on the basis of different in vivo immunocompromised Rag1(-/-) and wild-type (WT) BALB/c murine tumour growth experiments. The model analysis supports that tumour vasculature normalization can be a plausible and effective strategy to treat cancer when combined with appropriate immunostimulations. We find that improved levels of functional tumour vasculature, potentially mediated by normalization or stress alleviation strategies, can provide beneficial outcomes in terms of tumour burden reduction and growth control. Normalization of tumour blood vessels opens a therapeutic window of opportunity to augment the antitumour immune responses, as well as to reduce intratumoral immunosuppression and induced hypoxia due to vascular abnormalities. The potential success of normalizing tumour-associated vasculature closely depends on the effector cell recruitment dynamics and tumour sizes. Furthermore, an arbitrary increase in the initial effector cell concentration does not necessarily imply better tumour control. We evidence the existence of an optimal concentration range of effector cells for tumour shrinkage. Based on these findings, we suggest a theory-driven therapeutic proposal that optimally combines immuno- and vasomodulatory interventions.
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Affiliation(s)
- H Hatzikirou
- Center for Advancing Electronics, Technische Universität Dresden, 01062 Dresden, Germany Center for Information Services and High Performance Computing, Technische Universität Dresden, 01062 Dresden, Germany Department of Systems Immunology and Braunschweig Integrated Centre of Systems Biology, Helmholtz Center for Infectious Research, Inhoffenstrasse 7, 38124 Braunschweig, Germany
| | - J C L Alfonso
- Center for Advancing Electronics, Technische Universität Dresden, 01062 Dresden, Germany Center for Information Services and High Performance Computing, Technische Universität Dresden, 01062 Dresden, Germany
| | - S Mühle
- Molecular Immunology, Helmholtz Center for Infectious Research, Inhoffenstrasse 7, 38124 Braunschweig, Germany
| | - C Stern
- Molecular Immunology, Helmholtz Center for Infectious Research, Inhoffenstrasse 7, 38124 Braunschweig, Germany
| | - S Weiss
- Molecular Immunology, Helmholtz Center for Infectious Research, Inhoffenstrasse 7, 38124 Braunschweig, Germany Institute of Immunology, Medical School Hannover, Carl-Neuberg-Strasse 1, 30625 Hannover, Germany
| | - M Meyer-Hermann
- Department of Systems Immunology and Braunschweig Integrated Centre of Systems Biology, Helmholtz Center for Infectious Research, Inhoffenstrasse 7, 38124 Braunschweig, Germany Institute for Biochemistry, Biotechnology and Bioinformatics, Technische Universität Braunschweig, 38106 Braunschweig, Germany
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Yang J, Zhao TJ, Yuan CQ, Xie JH, Hao FF. A nonlinear competitive model of the prostate tumor growth under intermittent androgen suppression. J Theor Biol 2016; 404:66-72. [PMID: 27259386 DOI: 10.1016/j.jtbi.2016.05.033] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2015] [Revised: 05/22/2016] [Accepted: 05/23/2016] [Indexed: 11/19/2022]
Abstract
Hormone suppression has been the primary modality of treatment for prostate cancer. However long-term androgen deprivation may induce androgen-independent (AI) recurrence. Intermittent androgen suppression (IAS) is a potential way to delay or avoid the AI relapse. Mathematical models of tumor growth and treatment are simple while they are capable of capturing the essence of complicated interactions. Game theory models have analyzed that tumor cells can enhance their fitness by adopting genetically determined survival strategies. In this paper, we consider the survival strategies as the competitive advantage of tumor cells and propose a new model to mimic the prostate tumor growth in IAS therapy. Then we investigate the competition effect in tumor development by numerical simulations. The results indicate that successfully IAS-controlled states can be achieved even though the net growth rate of AI cells is positive for any androgen level. There is crucial difference between the previous models and the new one in the phase diagram of successful and unsuccessful tumor control by IAS administration, which means that the suggestions from the models for medication can be different. Furthermore we introduce quadratic logistic terms to the competition model to simulate the tumor growth in the environment with a finite carrying capacity considering the nutrients or inhibitors. The simulations show that the tumor growth can reach an equilibrium state or an oscillatory state with the net growth rate of AI cells being androgen independent. Our results suggest that the competition and the restraint of a limited environment can enhance the possibility of relapse prevention.
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Affiliation(s)
- Jing Yang
- Institute of Biophysics, School of Science, Hebei University of Technology, Tianjin 300401, China
| | - Tong-Jun Zhao
- Institute of Biophysics, School of Science, Hebei University of Technology, Tianjin 300401, China.
| | - Chang-Qing Yuan
- Institute of Biophysics, School of Science, Hebei University of Technology, Tianjin 300401, China
| | - Jing-Hui Xie
- Institute of Biophysics, School of Science, Hebei University of Technology, Tianjin 300401, China
| | - Fang-Fang Hao
- Institute of Biophysics, School of Science, Hebei University of Technology, Tianjin 300401, China
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Abstract
The controlling factors that underlie the growth of tumors have often been hard to identify because of the presence in this system of a large number of intracellular biochemical parameters. Here, we propose a simplifying framework to identify the key physical parameters that govern the early growth of tumors. We model growth by means of branching processes where cells of different types can divide and differentiate. First, using this process that has only one controlling parameter, we study a one cell type model and compute the probability for tumor survival and the time of tumor extinction. Second, we show that when cell death and cell division are perfectly balanced, stochastic effects dominate the growth dynamics and the system exhibits a near-critical behavior that resembles a second-order phase transition. We show, in this near-critical regime, that the time interval before tumor extinction is power-law distributed. Finally, we apply this branching formalism to infer, from experimental growth data, the number of different cell types present in the observed tumor.
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Affiliation(s)
- Guillaume Remy
- FAS Center for Systems Biology, Department of Molecular and Cellular Biology, School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, United States of America. ICFP Physics Master Program, Ecole Normale Superieure de Paris, 45 Rue d'Ulm, Paris 75005, France
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Argyri KD, Dionysiou DD, Misichroni FD, Stamatakos GS. Numerical simulation of vascular tumour growth under antiangiogenic treatment: addressing the paradigm of single-agent bevacizumab therapy with the use of experimental data. Biol Direct 2016; 11:12. [PMID: 27005569 PMCID: PMC4804544 DOI: 10.1186/s13062-016-0114-9] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/13/2016] [Accepted: 03/14/2016] [Indexed: 11/30/2022] Open
Abstract
Background Antiangiogenic agents have been recently added to the oncological armamentarium with bevacizumab probably being the most popular representative in current clinical practice. The elucidation of the mode of action of these agents is a prerequisite for personalized prediction of antiangiogenic treatment response and selection of patients who may benefit from this kind of therapy. To this end, having used as a basis a preexisting continuous vascular tumour growth model which addresses the targeted nature of antiangiogenic treatment, we present a paper characterized by the following three features. First, the integration of a two-compartmental bevacizumab specific pharmacokinetic module into the core of the aforementioned preexisting model. Second, its mathematical modification in order to reproduce the asymptotic behaviour of tumour volume in the theoretical case of a total destruction of tumour neovasculature. Third, the exploitation of a range of published animal datasets pertaining to antitumour efficacy of bevacizumab on various tumour types (breast, lung, head and neck, colon). Results Results for both the unperturbed growth and the treatment module reveal qualitative similarities with experimental observations establishing the biologically acceptable behaviour of the model. The dynamics of the untreated tumour has been studied via a parameter analysis, revealing the role of each relevant input parameter to tumour evolution. The combined effect of endogenous proangiogenic and antiangiogenic factors on the angiogenic potential of a tumour is also studied, in order to capture the dynamics of molecular competition between the two key-players of tumoural angiogenesis. The adopted methodology also allows accounting for the newly recognized direct antitumour effect of the specific agent. Conclusions Interesting observations have been made, suggesting a potential size-dependent tumour response to different treatment modalities and determining the relative timing of cytotoxic versus antiangiogenic agents administration. Insight into the comparative effectiveness of different antiangiogenic treatment strategies is revealed. The results of a series of in vivo experiments in mice bearing diverse types of tumours (breast, lung, head and neck, colon) and treated with bevacizumab are successfully reproduced, supporting thus the validity of the underlying model. Reviewers This article was reviewed by L. Hanin, T. Radivoyevitch and L. Edler.
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Affiliation(s)
- Katerina D Argyri
- In Silico Oncology and In Silico Medicine Group, Laboratory of Microwaves and Fiber Optics, Institute of Communication and Computer Systems, School of Electrical and Computer Engineering, National Technical University of Athens, 9 Iroon Polytechniou, Zografos, GR 157 80, Athens, Greece
| | - Dimitra D Dionysiou
- In Silico Oncology and In Silico Medicine Group, Laboratory of Microwaves and Fiber Optics, Institute of Communication and Computer Systems, School of Electrical and Computer Engineering, National Technical University of Athens, 9 Iroon Polytechniou, Zografos, GR 157 80, Athens, Greece
| | - Fay D Misichroni
- In Silico Oncology and In Silico Medicine Group, Laboratory of Microwaves and Fiber Optics, Institute of Communication and Computer Systems, School of Electrical and Computer Engineering, National Technical University of Athens, 9 Iroon Polytechniou, Zografos, GR 157 80, Athens, Greece
| | - Georgios S Stamatakos
- In Silico Oncology and In Silico Medicine Group, Laboratory of Microwaves and Fiber Optics, Institute of Communication and Computer Systems, School of Electrical and Computer Engineering, National Technical University of Athens, 9 Iroon Polytechniou, Zografos, GR 157 80, Athens, Greece.
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Wilson S, Tod M, Ouerdani A, Emde A, Yarden Y, Adda Berkane A, Kassour S, Wei MX, Freyer G, You B, Grenier E, Ribba B. Modeling and predicting optimal treatment scheduling between the antiangiogenic drug sunitinib and irinotecan in preclinical settings. CPT-PHARMACOMETRICS & SYSTEMS PHARMACOLOGY 2015; 4:720-7. [PMID: 26904386 PMCID: PMC4759705 DOI: 10.1002/psp4.12045] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/30/2015] [Accepted: 09/27/2015] [Indexed: 12/17/2022]
Abstract
We present a system of nonlinear ordinary differential equations used to quantify the complex dynamics of the interactions between tumor growth, vasculature generation, and antiangiogenic treatment. The primary dataset consists of longitudinal tumor size measurements (1,371 total observations) in 105 colorectal tumor‐bearing mice. Mice received single or combination administration of sunitinib, an antiangiogenic agent, and/or irinotecan, a cytotoxic agent. Depending on the dataset, parameter estimation was performed either using a mixed‐effect approach or by nonlinear least squares. Through a log‐likelihood ratio test, we conclude that there is a potential synergistic interaction between sunitinib when administered in combination with irinotecan in preclinical settings. Model simulations were then compared to data from a follow‐up preclinical experiment. We conclude that the model has predictive value in identifying the therapeutic window in which the timing between the administrations of these two drugs is most effective.
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Affiliation(s)
- S Wilson
- Inria, project-team Numed, Ecole Normale Supérieure de Lyon Lyon France
| | - M Tod
- EMR 3738, Ciblage Thérapeutique en Oncologie, Faculté de Médecine et de Maïeutique Lyon-Sud, Université Lyon 1 Oullins France
| | - A Ouerdani
- Inria, project-team Numed, Ecole Normale Supérieure de Lyon Lyon France
| | - A Emde
- Department of Biological Regulation Weizmann Institute of Science Rehovot Israel
| | - Y Yarden
- Department of Biological Regulation Weizmann Institute of Science Rehovot Israel
| | - A Adda Berkane
- CellVax Laboratory facility Bâtiment Marcenac, Ecole Nationale Vétérinaire d'Alfort Maisons Alfort France
| | - S Kassour
- CellVax Laboratory facility Bâtiment Marcenac, Ecole Nationale Vétérinaire d'Alfort Maisons Alfort France
| | - M X Wei
- CellVax Laboratory facility Bâtiment Marcenac, Ecole Nationale Vétérinaire d'Alfort Maisons Alfort France
| | - G Freyer
- EMR 3738, Ciblage Thérapeutique en Oncologie, Faculté de Médecine et de Maïeutique Lyon-Sud, Université Lyon 1 Oullins France
| | - B You
- EMR 3738, Ciblage Thérapeutique en Oncologie, Faculté de Médecine et de Maïeutique Lyon-Sud, Université Lyon 1 Oullins France
| | - E Grenier
- Inria, project-team Numed, Ecole Normale Supérieure de Lyon Lyon France
| | - B Ribba
- Inria, project-team Numed, Ecole Normale Supérieure de Lyon Lyon France
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Tomezak M, Abbadie C, Lartigau E, Cleri F. A biophysical model of cell evolution after cytotoxic treatments: Damage, repair and cell response. J Theor Biol 2015; 389:146-58. [PMID: 26549470 DOI: 10.1016/j.jtbi.2015.10.017] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/19/2015] [Revised: 10/26/2015] [Accepted: 10/28/2015] [Indexed: 10/22/2022]
Abstract
We present a theoretical agent-based model of cell evolution under the action of cytotoxic treatments, such as radiotherapy or chemotherapy. The major features of cell cycle and proliferation, cell damage and repair, and chemical diffusion are included. Cell evolution is based on a discrete Markov chain, with cells stepping along a sequence of discrete internal states from 'normal' to 'inactive'. Probabilistic laws are introduced for each type of event a cell can undergo during its life: duplication, arrest, senescence, damage, reparation, or death. We adjust the model parameters on a series of cell irradiation experiments, carried out in a clinical LINAC, in which the damage and repair kinetics of single- and double-strand breaks are followed. Two showcase applications of the model are then presented. In the first one, we reconstruct the cell survival curves from a number of published low- and high-dose irradiation experiments. We reobtain a very good description of the data without assuming the well-known linear-quadratic model, but instead including a variable DSB repair probability. The repair capability of the model spontaneously saturates to an exponential decay at increasingly high doses. As a second test, we attempt to simulate the two extreme possibilities of the so-called 'bystander' effect in radiotherapy: the 'local' effect versus a 'global' effect, respectively activated by the short-range or long-range diffusion of some factor, presumably secreted by the irradiated cells. Even with an oversimplified simulation, we could demonstrate a sizeable difference in the proliferation rate of non-irradiated cells, the proliferation acceleration being much larger for the global than the local effect, for relatively small fractions of irradiated cells in the colony.
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Affiliation(s)
- M Tomezak
- Institut d׳Electronique Microelectronique et Nanotechnologie (IEMN), UMR Cnrs 8520, 59652 Villeneuve d׳Ascq, France; CNRS, Institut Pasteur de Lille, UMR Cnrs 8161 Mechanisms of Tumorigenesis and Targeted Therapies, 59000 Lille, France
| | - C Abbadie
- CNRS, Institut Pasteur de Lille, UMR Cnrs 8161 Mechanisms of Tumorigenesis and Targeted Therapies, 59000 Lille, France; Université de Lille I, Sciences et Technologies, 59650 Villeneuve d׳Ascq, France
| | - E Lartigau
- Centre de Lutte contre le Cancer "Oscar Lambret", 59000 Lille, France
| | - F Cleri
- Institut d׳Electronique Microelectronique et Nanotechnologie (IEMN), UMR Cnrs 8520, 59652 Villeneuve d׳Ascq, France; Université de Lille I, Sciences et Technologies, 59650 Villeneuve d׳Ascq, France.
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Talkington A, Durrett R. Estimating Tumor Growth Rates In Vivo. Bull Math Biol 2015; 77:1934-54. [PMID: 26481497 PMCID: PMC4764475 DOI: 10.1007/s11538-015-0110-8] [Citation(s) in RCA: 74] [Impact Index Per Article: 8.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2014] [Accepted: 09/23/2015] [Indexed: 12/31/2022]
Abstract
In this paper, we develop methods for inferring tumor growth rates from the observation of tumor volumes at two time points. We fit power law, exponential, Gompertz, and Spratt’s generalized logistic model to five data sets. Though the data sets are small and there are biases due to the way the samples were ascertained, there is a clear sign of exponential growth for the breast and liver cancers, and a 2/3’s power law (surface growth) for the two neurological cancers.
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Rockne RC, Trister AD, Jacobs J, Hawkins-Daarud AJ, Neal ML, Hendrickson K, Mrugala MM, Rockhill JK, Kinahan P, Krohn KA, Swanson KR. A patient-specific computational model of hypoxia-modulated radiation resistance in glioblastoma using 18F-FMISO-PET. J R Soc Interface 2015; 12:rsif.2014.1174. [PMID: 25540239 PMCID: PMC4305419 DOI: 10.1098/rsif.2014.1174] [Citation(s) in RCA: 58] [Impact Index Per Article: 6.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022] Open
Abstract
Glioblastoma multiforme (GBM) is a highly invasive primary brain tumour that has poor prognosis despite aggressive treatment. A hallmark of these tumours is diffuse invasion into the surrounding brain, necessitating a multi-modal treatment approach, including surgery, radiation and chemotherapy. We have previously demonstrated the ability of our model to predict radiographic response immediately following radiation therapy in individual GBM patients using a simplified geometry of the brain and theoretical radiation dose. Using only two pre-treatment magnetic resonance imaging scans, we calculate net rates of proliferation and invasion as well as radiation sensitivity for a patient's disease. Here, we present the application of our clinically targeted modelling approach to a single glioblastoma patient as a demonstration of our method. We apply our model in the full three-dimensional architecture of the brain to quantify the effects of regional resistance to radiation owing to hypoxia in vivo determined by [(18)F]-fluoromisonidazole positron emission tomography (FMISO-PET) and the patient-specific three-dimensional radiation treatment plan. Incorporation of hypoxia into our model with FMISO-PET increases the model-data agreement by an order of magnitude. This improvement was robust to our definition of hypoxia or the degree of radiation resistance quantified with the FMISO-PET image and our computational model, respectively. This work demonstrates a useful application of patient-specific modelling in personalized medicine and how mathematical modelling has the potential to unify multi-modality imaging and radiation treatment planning.
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Affiliation(s)
- Russell C Rockne
- Department of Neurological Surgery, Northwestern University and Feinberg School of Medicine, 676 N Saint Clair Street, Suite 1300, Chicago, IL 60611, USA Northwestern Brain Tumor Institute, Northwestern University, 675 N Saint Clair Street, Suite 2100, Chicago, IL 60611, USA,
| | - Andrew D Trister
- Department of Radiation Oncology, University of Washington, School of Medicine, 1959 NE Pacific Street, Seattle, WA 98195, USA
| | - Joshua Jacobs
- Department of Neurological Surgery, Northwestern University and Feinberg School of Medicine, 676 N Saint Clair Street, Suite 1300, Chicago, IL 60611, USA Northwestern Brain Tumor Institute, Northwestern University, 675 N Saint Clair Street, Suite 2100, Chicago, IL 60611, USA
| | - Andrea J Hawkins-Daarud
- Department of Neurological Surgery, Northwestern University and Feinberg School of Medicine, 676 N Saint Clair Street, Suite 1300, Chicago, IL 60611, USA Northwestern Brain Tumor Institute, Northwestern University, 675 N Saint Clair Street, Suite 2100, Chicago, IL 60611, USA
| | - Maxwell L Neal
- Department of Pathology, University of Washington, School of Medicine, 1959 NE Pacific Street, Seattle, WA 98195, USA
| | - Kristi Hendrickson
- Department of Radiation Oncology, University of Washington, School of Medicine, 1959 NE Pacific Street, Seattle, WA 98195, USA
| | - Maciej M Mrugala
- Department of Neurology, University of Washington, School of Medicine, 1959 NE Pacific Street, Seattle, WA 98195, USA
| | - Jason K Rockhill
- Department of Radiation Oncology, University of Washington, School of Medicine, 1959 NE Pacific Street, Seattle, WA 98195, USA
| | - Paul Kinahan
- Department of Radiology, University of Washington, School of Medicine, 1959 NE Pacific Street, Seattle, WA 98195, USA
| | - Kenneth A Krohn
- Department of Radiation Oncology, University of Washington, School of Medicine, 1959 NE Pacific Street, Seattle, WA 98195, USA Department of Radiology, University of Washington, School of Medicine, 1959 NE Pacific Street, Seattle, WA 98195, USA
| | - Kristin R Swanson
- Department of Neurological Surgery, Northwestern University and Feinberg School of Medicine, 676 N Saint Clair Street, Suite 1300, Chicago, IL 60611, USA Northwestern Brain Tumor Institute, Northwestern University, 675 N Saint Clair Street, Suite 2100, Chicago, IL 60611, USA
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