Modeling multi-drug chemotherapy: tailoring treatment to individuals

On tumoural growth and treatment under cellular dedifferentiation

Differentiated cancer cells may regain stem cell characteristics; however, the effects of such a cellular dedifferentiation on tumoural growth and treatment are currently understudied. Thus, we here extend a mathematical model of cancer stem cell (CSC) driven tumour growth to also include dedifferentiation. We show that dedifferentiation increases the likelihood of tumorigenesis and the speed of tumoural growth, both modulated by the proliferative potential of the non-stem cancer cells (NSCCs). We demonstrate that dedifferentiation also may lead to treatment evasion, especially when a treatment solely targets CSCs. Conversely, targeting both CSCs and NSCCs in parallel is shown to be more robust to dedifferentiation. Despite dedifferentiation, perturbing CSC-related parameters continues to exert the largest relative effect on tumoural growth; however, we show the existence of synergies between specific CSC- and NSCC-directed treatments which cause superadditive reductions of tumoural growth. Overall, our study demonstrates various effects of dedifferentiation on growth and treatment of tumoural lesions, and we anticipate our results to be helpful in guiding future molecular and clinical research on limiting tumoural growth in vivo.

A Review of Mathematical Models for Tumor Dynamics and Treatment Resistance Evolution of Solid Tumors

Increasing knowledge of intertumor heterogeneity, intratumor heterogeneity, and cancer evolution has improved the understanding of anticancer treatment resistance. A better characterization of cancer evolution and subsequent use of this knowledge for personalized treatment would increase the chance to overcome cancer treatment resistance. Model‐based approaches may help achieve this goal. In this review, we comprehensively summarized mathematical models of tumor dynamics for solid tumors and of drug resistance evolution. Models displayed by ordinary differential equations, algebraic equations, and partial differential equations for characterizing tumor burden dynamics are introduced and discussed. As for tumor resistance evolution, stochastic and deterministic models are introduced and discussed. The results may facilitate a novel model‐based analysis on anticancer treatment response and the occurrence of resistance, which incorporates both tumor dynamics and resistance evolution. The opportunities of a model‐based approach as discussed in this review can be of great benefit for future optimizing and personalizing anticancer treatment.

Practically scheduling hormone therapy for prostate cancer using a mathematical model

Hormone therapy is one of the popular therapeutic methods for prostate cancer. Intermittent androgen suppression (IAS) is the method which stops and resumes hormone therapy repeatedly. The efficacy of IAS differs depending on patients; both the cases have been reported where the relapse of cancer happened and did not happen, for the patients who had undergone IAS. For the patients who cannot avoid the relapse of cancer by IAS, we should delay the relapse of cancer as later as possible. Here we compared some practical methods of determining when to stop and restart hormone therapy for IAS using an existing mathematical model of prostate cancer. The method we suggest is to determine the ratio of on-treatment period and off-treatment period sparsely for each cycle, namely the "sparse search." We also compared the performance of the sparse search with the exhaustive search and the model predictive control. We found that the sparse search can find a good treatment schedule without failure, and the computational cost is not so high compared to the exhaustive method. In addition, we focus on the model predictive control (MPC) method which has been applied to the scheduling of IAS in some existing studies. The MPC is computationary efficient, although it does not always find an optimal schedule in the numerical experiments here. We believe that the MPC method might be also promising because of its reasonable computational costs and its possibility of expanding of the model.

Computational Systems Biology in Cancer: Modeling Methods and Applications

In recent years it has become clear that carcinogenesis is a complex process, both at the molecular and cellular levels. Understanding the origins, growth and spread of cancer, therefore requires an integrated or system-wide approach. Computational systems biology is an emerging sub-discipline in systems biology that utilizes the wealth of data from genomic, proteomic and metabolomic studies to build computer simulations of intra and intercellular processes. Several useful descriptive and predictive models of the origin, growth and spread of cancers have been developed in an effort to better understand the disease and potential therapeutic approaches. In this review we describe and assess the practical and theoretical underpinnings of commonly-used modeling approaches, including ordinary and partial differential equations, petri nets, cellular automata, agent based models and hybrid systems. A number of computer-based formalisms have been implemented to improve the accessibility of the various approaches to researchers whose primary interest lies outside of model development. We discuss several of these and describe how they have led to novel insights into tumor genesis, growth, apoptosis, vascularization and therapy.

Pharmacodynamic Modeling of Cell Cycle Effects for Gemcitabine and Trabectedin Combinations in Pancreatic Cancer Cells

Combinations of gemcitabine and trabectedin exert modest synergistic cytotoxic effects on two pancreatic cancer cell lines. Here, systems pharmacodynamic (PD) models that integrate cellular response data and extend a prototype model framework were developed to characterize dynamic changes in cell cycle phases of cancer cell subpopulations in response to gemcitabine and trabectedin as single agents and in combination. Extensive experimental data were obtained for two pancreatic cancer cell lines (MiaPaCa-2 and BxPC-3), including cell proliferation rates over 0-120 h of drug exposure, and the fraction of cells in different cell cycle phases or apoptosis. Cell cycle analysis demonstrated that gemcitabine induced cell cycle arrest in S phase, and trabectedin induced transient cell cycle arrest in S phase that progressed to G₂/M phase. Over time, cells in the control group accumulated in G₀/G₁ phase. Systems cell cycle models were developed based on observed mechanisms and were used to characterize both cell proliferation and cell numbers in the sub G₁, G₀/G₁, S, and G₂/M phases in the control and drug-treated groups. The proposed mathematical models captured well both single and joint effects of gemcitabine and trabectedin. Interaction parameters were applied to quantify unexplainable drug-drug interaction effects on cell cycle arrest in S phase and in inducing apoptosis. The developed models were able to identify and quantify the different underlying interactions between gemcitabine and trabectedin, and captured well our large datasets in the dimensions of time, drug concentrations, and cellular subpopulations.

Dose-Dependent Mutation Rates Determine Optimum Erlotinib Dosing Strategies for EGFR Mutant Non-Small Cell Lung Cancer Patients

BACKGROUND

The advent of targeted therapy for cancer treatment has brought about a paradigm shift in the clinical management of human malignancies. Agents such as erlotinib used for EGFR-mutant non-small cell lung cancer or imatinib for chronic myeloid leukemia, for instance, lead to rapid tumor responses. Unfortunately, however, resistance often emerges and renders these agents ineffective after a variable amount of time. The FDA-approved dosing schedules for these drugs were not designed to optimally prevent the emergence of resistance. To this end, we have previously utilized evolutionary mathematical modeling of treatment responses to elucidate the dosing schedules best able to prevent or delay the onset of resistance. Here we expand on our approaches by taking into account dose-dependent mutation rates at which resistant cells emerge. The relationship between the serum drug concentration and the rate at which resistance mutations arise can lead to non-intuitive results about the best dose administration strategies to prevent or delay the emergence of resistance.

METHODS

We used mathematical modeling, available clinical trial data, and different considerations of the relationship between mutation rate and drug concentration to predict the effectiveness of different dosing strategies.

RESULTS

We designed several distinct measures to interrogate the effects of different treatment dosing strategies and found that a low-dose continuous strategy coupled with high-dose pulses leads to the maximal delay until clinically observable resistance. Furthermore, the response to treatment is robust against different assumptions of the mutation rate as a function of drug concentration.

CONCLUSIONS

For new and existing targeted drugs, our methodology can be employed to compare the effectiveness of different dose administration schedules and investigate the influence of changing mutation rates on outcomes.

A validated mathematical model of tumor growth including tumor-host interaction, cell-mediated immune response and chemotherapy

We consider a dynamical model of cancer growth including three interacting cell populations of tumor cells, healthy host cells and immune effector cells. The tumor-immune and the tumor-host interactions are characterized to reproduce experimental results. A thorough dynamical analysis of the model is carried out, showing its capability to explain theoretical and empirical knowledge about tumor development. A chemotherapy treatment reproducing different experiments is also introduced. We believe that this simple model can serve as a foundation for the development of more complicated and specific cancer models.

Mathematical modeling of cancer progression and response to chemotherapy

The complex, constantly evolving and multifaceted nature of cancer has made it difficult to identify unique molecular and pathophysiological signatures for each disease variant, consequently hindering development of effective therapies. Mathematical modeling and computer simulation are tools that can provide a robust framework to better understand cancer progression and response to chemotherapy. Successful therapeutic agents must overcome biological barriers occurring at multiple space and time scales and still reach targets at sufficient concentrations. A multiscale computer simulator founded on the integration of experimental data and mathematical models can provide valuable insights into these processes and establish a technology platform for analyzing the effectiveness of chemotherapeutic drugs, with the potential to cost-effectively and efficiently screen drug candidates during the drug-development process.

Engineered cell-based therapies: a vanguard of design-driven medicine

Engineered cell-based therapies are uniquely capable of performing sophisticated therapeutic functions in vivo, and this strategy is yielding promising clinical benefits for treating cancer. In this review, we discuss key opportunities and challenges for engineering customized cellular functions using cell-based therapy for cancer as a representative case study. We examine the historical development of chimeric antigen receptor (CAR) therapies as an illustration of the engineering design cycle. We also consider the potential roles that the complementary disciplines of systems biology and synthetic biology may play in realizing safe and effective treatments for a broad range of patients and diseases. In particular, we discuss how systems biology may facilitate both fundamental research and clinical translation, and we describe how the emerging field of synthetic biology is providing novel modalities for building customized cellular functions to overcome existing clinical barriers. Together, these approaches provide a powerful set of conceptual and experimental tools for transforming information into understanding, and for translating understanding into novel therapeutics to establish a new framework for design-driven medicine.

Pharmacodynamic modeling of cell cycle and apoptotic effects of gemcitabine on pancreatic adenocarcinoma cells

PURPOSE

The standard of care for treating patients with pancreatic adenocarcinomas includes gemcitabine (2',2'-difluorodeoxycytidine). Gemcitabine primarily elicits its response by stalling the DNA replication forks of cells in the S phase of the cell cycle. To provide a quantitative framework for characterizing the cell cycle and apoptotic effects of gemcitabine, we developed a pharmacodynamic model in which the activation of cell cycle checkpoints or cell death is dependent on gemcitabine exposure.

METHODS

Three pancreatic adenocarcinoma cell lines (AsPC-1, BxPC-3, and MiaPaca-2) were exposed to varying concentrations (0-100,000 ng/mL) of gemcitabine over a period of 96 h in order to quantify proliferation kinetics and cell distributions among the cell cycle phases. The model assumes that the drug can inhibit cycle-phase transitioning in each of the 3 phases (G1, S, and G2/M) and can cause apoptosis of cells in G1 and G2/M phases. Fitting was performed using the ADAPT5 program.

RESULTS

The time course of gemcitabine effects was well described by the model, and parameters were estimated with good precision. Model predictions and experimental data show that gemcitabine induces cell cycle arrest in the S phase at low concentrations, whereas higher concentrations induce arrest in all cell cycle phases. Furthermore, apoptotic effects of gemcitabine appear to be minimal and take place at later time points.

CONCLUSION

The pharmacodynamic model developed provides a quantitative, mechanistic interpretation of gemcitabine efficacy in 3 pancreatic cancer cell lines, and provides useful insights for rational selection of chemotherapeutic agents for combination therapy.

In silico modelling of treatment-induced tumour cell kill: developments and advances

Mathematical and stochastic computer (in silico) models of tumour growth and treatment response of the past and current eras are presented, outlining the aims of the models, model methodology, the key parameters used to describe the tumour system, and treatment modality applied, as well as reported outcomes from simulations. Fractionated radiotherapy, chemotherapy, and combined therapies are reviewed, providing a comprehensive overview of the modelling literature for current modellers and radiobiologists to ignite the interest of other computational scientists and health professionals of the ever evolving and clinically relevant field of tumour modelling.

The dynamics of drug resistance: a mathematical perspective

Resistance to chemotherapy is a key impediment to successful cancer treatment that has been intensively studied for the last three decades. Several central mechanisms have been identified as contributing to the resistance. In the case of multidrug resistance (MDR), the cell becomes resistant to a variety of structurally and mechanistically unrelated drugs in addition to the drug initially administered. Mathematical models of drug resistance have dealt with many of the known aspects of this field, such as pharmacologic sanctuary and location/diffusion resistance, intrinsic resistance, induced resistance and acquired resistance. In addition, there are mathematical models that take into account the kinetic/phase resistance, and models that investigate intracellular mechanisms based on specific biological functions (such as ABC transporters, apoptosis and repair mechanisms). This review covers aspects of MDR that have been mathematically studied, and explains how, from a methodological perspective, mathematics can be used to study drug resistance. We discuss quantitative approaches of mathematical analysis, and demonstrate how mathematics can be used in combination with other experimental and clinical tools. We emphasize the potential benefits of integrating analytical and mathematical methods into future clinical and experimental studies of drug resistance.

Tumor growth instability and its implications for chemotherapy

Optimal delivery of chemotherapy intensity is dependent on host- and tumor-specific characteristics. In this article, the chemotherapy late intensity schedule is revised to account for tumor growth instability, where a small tumor cell fraction emerges that exhibits a higher proliferation rate than the parent strain. Modeling this instability as simplified two-population dynamics, we find that: (a) if this instability precedes the onset of treatment, the slope of the linear increase of the drug concentration for the standard "Norton-Simon late intensity schedule" changes and the initial value of the dose strongly depends on the ratio of the two tumor cell populations and on their distinct growth rates; and (b) if the instability trails the initial treatment, the effective chemotherapeutic drug concentration changes as well. Both cases point toward testable potential refinements of the Norton-Simon late intensity schedule.

Integrating cell-cycle progression, drug penetration and energy metabolism to identify improved cancer therapeutic strategies

The effectiveness of chemotherapeutic drugs in tumors is reduced by multiple effects including drug diffusion and variable susceptibility of local cell populations. We hypothesized that quantifying the interactions between drugs and tumor microenvironments could be used to identify more effective anti-cancer strategies. To test this hypothesis we created a mathematical model that integrated intracellular metabolism, nutrient and drug diffusion, cell-cycle progression, cellular drug effects, and drug pharmacokinetics. To our knowledge, this is the first model that combines these elements and has coupled them to experimentally derived parameters. Drug cytotoxicity was assumed to be cell-cycle phase specific, and progression through the cell cycle was assumed to be dependent on ATP generation. The model consisted of a coupled set of nonlinear partial differential, ordinary differential and algebraic equations with an outer free boundary, which was solved using orthogonal collocation on a moving grid of finite elements. Model simulations showed the existence of an optimum drug diffusion coefficient: a low diffusivity prevents effective penetration before the drug is cleared from the blood and a high diffusivity limits drug retention. This result suggests that increasing the molecular weight of the anti-cancer drug paclitaxel from 854 to approximately 20,000 by nanoparticle conjugation would improve its efficacy. The simulations also showed that fast growing tumors are less responsive to therapy than are slower tumors with more quiescent cells, demonstrating the competing effects of regrowth and cytotoxicity. The therapeutic implications of the simulation results are that (1) monolayer cultures are inadequate for accurately determining therapeutic effects in vitro, (2) decreasing the diffusivity of paclitaxel could increase its efficacy, and (3) measuring the proliferation fraction in tumors could enhance the prediction of therapeutic efficacy.

Integration of modeling and simulation into hospital-based decision support systems guiding pediatric pharmacotherapy

Background

Decision analysis in hospital-based settings is becoming more common place. The application of modeling and simulation approaches has likewise become more prevalent in order to support decision analytics. With respect to clinical decision making at the level of the patient, modeling and simulation approaches have been used to study and forecast treatment options, examine and rate caregiver performance and assign resources (staffing, beds, patient throughput). There us a great need to facilitate pharmacotherapeutic decision making in pediatrics given the often limited data available to guide dosing and manage patient response. We have employed nonlinear mixed effect models and Bayesian forecasting algorithms coupled with data summary and visualization tools to create drug-specific decision support systems that utilize individualized patient data from our electronic medical records systems.

Methods

Pharmacokinetic and pharmacodynamic nonlinear mixed-effect models of specific drugs are generated based on historical data in relevant pediatric populations or from adults when no pediatric data is available. These models are re-executed with individual patient data allowing for patient-specific guidance via a Bayesian forecasting approach. The models are called and executed in an interactive manner through our web-based dashboard environment which interfaces to the hospital's electronic medical records system.

Results

The methotrexate dashboard utilizes a two-compartment, population-based, PK mixed-effect model to project patient response to specific dosing events. Projected plasma concentrations are viewable against protocol-specific nomograms to provide dosing guidance for potential rescue therapy with leucovorin. These data are also viewable against common biomarkers used to assess patient safety (e.g., vital signs and plasma creatinine levels). As additional data become available via therapeutic drug monitoring, the model is re-executed and projections are revised.

Conclusion

The management of pediatric pharmacotherapy can be greatly enhanced via the immediate feedback provided by decision analytics which incorporate the current, best-available knowledge pertaining to dose-exposure and exposure-response relationships, especially for narrow therapeutic agents that are difficult to manage.

Estimation of likely cancer cure using first- and second-order product densities of population balance models

The objective of chemotherapy is to eradicate all cancerous cells. However, due to the stochastic behavior of cells, the elimination of all cancerous cells must be discussed probabilistically. We hypothesize, and demonstrate in the results, that the mean and standard deviation of a cancer cell population, derived through the probabilistic interpretation of population balance equations, are sufficient to estimate the likelihood of cancer eradication. Our analysis of a binary cell division model reveals that an expected cancer population that is six standard deviations less than one cell provides a good estimate for the treatment durations that nearly ensures treatment successes. This approximation is evaluated and tested on two other physiologically likely scenarios: variable patient response to chemotherapy and the presence of a dormant population. We find that early identification of individual patient susceptibility to the chemotherapeutic agent is extremely important to all patients as treatment adjustments for non-responders greatly enhances their likelihood of cure while responders need not be subjected to needlessly harsh treatments. Presence of a dormant population increases both the required treatment duration and population variability, but the same estimation method holds. This work is a step toward using stochastic models for a quantitative evaluation of chemotherapy.

Potential impact of advanced clinical information technology on cancer care in 2015

New clinical information technologies now sporadically available will soon be in routine clinical use, bringing many changes to all phases of the cancer care continuum. For example, new technologies such as: (1) The next generation Internet; (2) Real-time clinical decision support systems; (3) Off-line, population-based systems; (4) Large, integrated, individual patient-level phenotypic and genotypic databases with intelligent data mining capabilities; (5) Wireless, invasive and non-invasive physiologic monitoring devices; (6) Natural Language Processing (NLP) systems; and (7) Mathematical models of complex biological systems all have the potential to impact significantly the provision of cancer care throughout its continuum. While new information management and communication techniques and technologies will reduce many of the inefficiencies and inaccuracies of our present systems, there will be an equal, and potentially far more dangerous, set of unintended consequences. Informatics investigators, cancer specialists, and health system administrators must focus on the study of what is working and what is not, as well as, on development and testing of the new clinical information management and communication technologies, if we are to be ready for the future.

A spatiotemporal, patient individualized simulation model of solid tumor response to chemotherapy in vivo: the paradigm of glioblastoma multiforme treated by temozolomide

A novel four-dimensional, patient-specific Monte Carlo simulation model of solid tumor response to chemotherapeutic treatment in vivo is presented. The special case of glioblastoma multiforme treated by temozolomide is addressed as a simulation paradigm. Nevertheless, a considerable number of the involved algorithms are generally applicable. The model is based on the patient's imaging, histopathologic and genetic data. For a given drug administration schedule lying within acceptable toxicity boundaries, the concentration of the prodrug and its metabolites within the tumor is calculated as a function of time based on the drug pharamacokinetics. A discretization mesh is superimposed upon the anatomical region of interest and within each geometrical cell of the mesh the most prominent biological "laws" (cell cycling, necrosis, apoptosis, mechanical restictions, etc.) are applied. The biological cell fates are predicted based on the drug pharmacodynamics. The outcome of the simulation is a prediction of the spatiotemporal activity of the entire tumor and is virtual reality visualized. A good qualitative agreement of the model's predictions with clinical experience supports the applicability of the approach. The proposed model primarily aims at providing a platform for performing patient individualized in silico experiments as a means of chemotherapeutic treatment optimization.

Analysis of resonance chemotherapy in leukemia treatment via multi-staged population balance models

An age-structured population balance model that explicitly models cell cycle phases is developed to investigate the effects of cell cycle specific (CCS) drugs. In particular, the benefits of timing CCS drug treatments in resonance chemotherapy are predicted and measured directly in vitro before evaluating likely in vivo scenarios. The phase transition rates are measured in vitro for the HL60 leukemia cell line and are used to predict the transient phase dynamics after exposure to the S phase specific drug, camptothecin. The phase oscillations predicted by the model are observed experimentally and the timing of a second camptothecin pulse is shown to significantly alter the overall treatment effectiveness. To explore the feasibility of designing resonance chemotherapeutic treatments to preferentially eliminate one cell type over another, Jurkat and HL60 leukemia cells are exposed to the same dual-pulse camptothecin treatment regimen. With the model framework validated for simplified cases, the model is used to extrapolate the effectiveness of resonance chemotherapy considering in vivo effects such as quiescence, drug metabolism, drug properties, and transport considerations that were not included in the in vitro experiments. While resonance chemotherapy is intuitive and looks promising in vitro, when in vivo considerations are included in the model, the phenomenon is dampened and the window of applicability becomes narrower.

Simulating chemotherapeutic schemes in the individualized treatment context: the paradigm of glioblastoma multiforme treated by temozolomide in vivo

A novel patient individualized, spatiotemporal Monte Carlo simulation model of tumor response to chemotherapeutic schemes in vivo is presented. Treatment of glioblastoma multiforme by temozolomide is considered as a paradigm. The model is based on the patient's imaging, histopathologic and genetic data. A discretization mesh is superimposed upon the anatomical region of interest and within each geometrical cell of the mesh the most prominent biological "laws" (cell cycling, apoptosis, etc.) in conjunction with pharmacokinetics and pharmacodynamics information are applied. A good qualitative agreement of the model's predictions with clinical experience supports the applicability of the approach to chemotherapy optimization.

The application of mathematical modelling to aspects of adjuvant chemotherapy scheduling

In this paper simple models for tumour growth incorporating age-structured cell cycle dynamics are considered in the presence of two non-cross-resistant S-phase specific chemotherapeutic drugs. According to the seminal work of Goldie and Coldman, if one cannot deliver two cell cycle phase non-specific, non-cross-resistant drugs simultaneously, for example due to toxicity, and both drugs are identical apart from resistance, one should alternate their delivery as rapidly as possible. However consider S-phase specific drugs. One might speculate that, for example, alternating the two drugs at intervals of T, where T is the mean cell cycle time, is better than alternating the drugs at intervals of T/2, as the latter strategy allows the possibility of a cell cycle sanctuary. Such speculation implicitly requires a sufficiently low variance of the cell cycle time, and hence it is not clear if such reasoning prevents a generalisation of the results of Goldie and Coldman. This question is addressed in this paper via a detailed modelling investigation, as motivated by suggestions for future colorectal adjuvant chemotherapy trials and developments in hepatic arterial infusion technology. It is shown that the cell cycle distribution of the resistant cell populations is strongly influenced by the chemotherapy schedule. The consequences of this can be dramatic, and can lead to chemotherapy failure at resonant chemotherapy timings, especially for a small standard deviation of the cell cycle time. The novel aspects of this observation are highlighted compared to other models in the literature exhibiting resonant behaviour in the timing of a periodic chemotherapy protocol. The above investigation also results in the principal prediction of this paper that reducing the drug alternation time to approximately a few hours, if possible, can result in substantial improvements in predicted chemotherapy outcomes. Critically, such improvements are not predicted by the Goldie Coldman model or other chemotherapy scheduling models in the literature.