Optimizing drug regimens in cancer chemotherapy by an efficacy-toxicity mathematical model

Combination Chemotherapy Optimization with Discrete Dosing

Chemotherapy drug administration is a complex problem that often requires expensive clinical trials to evaluate potential regimens; one way to alleviate this burden and better inform future trials is to build reliable models for drug administration. This paper presents a mixed-integer program for combination chemotherapy (utilization of multiple drugs) optimization that incorporates various important operational constraints and, besides dose and concentration limits, controls treatment toxicity based on its effect on the count of white blood cells. To address the uncertainty of tumor heterogeneity, we also propose chance constraints that guarantee reaching an operable tumor size with a high probability in a neoadjuvant setting. We present analytical results pertinent to the accuracy of the model in representing biological processes of chemotherapy and establish its potential for clinical applications through a numerical study of breast cancer.

Model-based assessment of combination therapies - ranking of radiosensitizing agents in oncology

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.

Design, synthesis, and biological evaluation of novel bioactive thalidomide analogs as anticancer immunomodulatory agents

Cancer is still a dangerous disease with a high mortality rate all over the world. In our attempt to develop potential anticancer candidates, new quinazoline and phthalazine based compounds were designed and synthesized. The new derivatives were built in line with the pharmacophoric features of thalidomide. The new derivatives as well as thalidomide were examined against three cancer cell lines, namely: hepatocellular carcinoma (HepG-2), breast cancer (MCF-7) and prostate cancer (PC3). Then the effects on the expression levels of caspase-8, VEGF, NF-κB P65, and TNF-α in HepG-2 cells were evaluated. The biological data revealed the high importance of phthalazine based compounds (24a-c), which were far better than thalidomide with regard to the antiproliferative activity. 24b showed IC₅₀ of 2.51, 5.80 and 4.11 μg mL^-1 compared to 11.26, 14.58, and 16.87 μg mL^-1 for thalidomide against the three cell lines respectively. 24b raised caspase-8 level by about 7 folds, compared to 8 folds reported for thalidomide. Also, VEGF level in HepG-2 cells treated with 24b was 185.3 pg mL^-1, compared to 432.5 pg mL^-1 in control cells. Furthermore, the immunomodulatory properties were proven to 24b, which reduced TNF-α level by approximately half. At the same time, NF-κB P65 level in HepG-2 cells treated with 24b was 76.5 pg mL^-1 compared to 278.1 and 110.5 pg mL^-1 measured for control cells and thalidomide treated HepG-2 cells respectively. Moreover, an in vitro viability study against Vero non-cancerous cell line was investigated and the results reflected a high safety profile of all tested compounds. This work suggests 24b as a promising lead compound for development of new immunomodulatory anticancer agents.

Singlet Oxygen Activatable Prodrugs of Paclitaxel, SN-38, MMC, and CA4: Non-mitochondria Targeted Prodrugs

We established a light-activatable prodrug strategy that produces the combination effect of photodynamic therapy (PDT) and site-specific chemotherapy. Prodrugs are activated by singlet oxygen (SO), generated from PS and visible or near IR light, in either intra- or inter-molecular manner. The goal of this study is to evaluate cytotoxic effects of non-mitochondria targeted prodrugs of a number of anticancer drugs with different mechanisms of action. They were tested in both 2D and 3D in vitro conditions via inter-molecular activation of prodrugs by SO generated in mitochondria by protoporphyrin IX-PDT (PpIX-PDT). Prodrugs of anticancer drugs (paclitaxel, SN-38, combrestatin A4, and mitomycin C) were synthesized using facile and high yielding reactions. Non-mitochondria targeted prodrugs showed limited dark toxicity while all of them showed greatly enhanced phototoxicity compared to PpIX-PDT in the 2D culture model. Prodrugs generated up to about 95% cell killing at 2.5 μM when administered with hexyl-aminolevulinate (HAL) to produce Protoporphyrin IX in cancer cells in both 2D monolayer and 3D spheroids model. The data demonstrate that mitochondria-targeting of prodrugs is not fully essential for our inter-molecular activation prodrug strategy. The prodrug strategy also worked for anticancer drugs with diverse MOAs.

A Model-Based Framework to Identify Optimal Administration Protocols for Immunotherapies in Castration-Resistance Prostate Cancer

Prostate cancer (PCa) is one of the most frequent cancer in male population. Androgen deprivation therapy is the first-line strategy for the metastatic stage of the disease, but, inevitably, PCa develops resistance to castration (CRPC), becoming incurable. In recent years, clinical trials are testing the efficacy of anti-CTLA4 on CRPC. However, this tumor seems to be resistant to immunotherapies that are very effective in other types of cancers, and, so far, only the dendritic cell vaccine sipuleucel-T has been approved. In this work, we employ a mathematical model of CRPC to determine the optimal administration protocol of ipilimumab, a particular anti-CTLA4, as single treatment or in combination with the sipuleucel-T, by considering both the effect on tumor population and the drug toxicity. To this end, we first introduce a dose-depending function of toxicity, estimated from experimental data, then we define two different optimization problems. We show the results obtained by imposing different constraints, and how these change by varying drug efficacy. Our results suggest administration of high-doses for a brief period, which is predicted to be more efficient than solutions with prolonged low-doses. The model also highlights a synergy between ipilimumab and sipuleucel-T, which leads to a better tumor control with lower doses of ipilimumab. Finally, tumor eradication is also conceivable, but it depends on patient-specific parameters.

Improving cancer treatments via dynamical biophysical models

Despite significant advances in oncological research, cancer nowadays remains one of the main causes of mortality and morbidity worldwide. New treatment techniques, as a rule, have limited efficacy, target only a narrow range of oncological diseases, and have limited availability to the general public due their high cost. An important goal in oncology is thus the modification of the types of antitumor therapy and their combinations, that are already introduced into clinical practice, with the goal of increasing the overall treatment efficacy. One option to achieve this goal is optimization of the schedules of drugs administration or performing other medical actions. Several factors complicate such tasks: the adverse effects of treatments on healthy cell populations, which must be kept tolerable; the emergence of drug resistance due to the intrinsic plasticity of heterogeneous cancer cell populations; the interplay between different types of therapies administered simultaneously. Mathematical modeling, in which a tumor and its microenvironment are considered as a single complex system, can address this complexity and can indicate potentially effective protocols, that would require experimental verification. In this review, we consider classical methods, current trends and future prospects in the field of mathematical modeling of tumor growth and treatment. In particular, methods of treatment optimization are discussed with several examples of specific problems related to different types of treatment.

A review on numerical modeling for magnetic nanoparticle hyperthermia: Progress and challenges

Recent progress in nanotechnology has advanced the development of magnetic nanoparticle (MNP) hyperthermia as a potential therapeutic platform for treating diseases. Due to the challenges in reliably predicting the spatiotemporal distribution of temperature in the living tissue during the therapy of MNP hyperthermia, critical for ensuring the safety as well as efficacy of the therapy, the development of effective and reliable numerical models is warranted. This article provides a comprehensive review on the various mathematical methods for determining specific loss power (SLP), a parameter used to quantify the heat generation capability of MNPs, as well as bio-heat models for predicting heat transfer phenomena and temperature distribution in living tissue upon the application of MNP hyperthermia. This article also discusses potential applications of the bio-heat models of MNP hyperthermia for therapeutic purposes, particularly for cancer treatment, along with their limitations that could be overcome.

Optimization of Cancer Treatment in the Frequency Domain

Thorough exploration of alternative dosing frequencies is often not performed in conventional pharmacometrics approaches. Quantitative systems pharmacology (QSP) can provide novel insights into optimal dosing regimen and drug behaviors which could add a new dimension to the design of novel treatments. However, methods for such an approach are currently lacking. Recently, we illustrated the utility of frequency-domain response analysis (FdRA), an analytical method used in control engineering, using several generic pharmacokinetic-pharmacodynamic case studies. While FdRA is not applicable to models harboring ever increasing variables such as those describing tumor growth, studying such models in the frequency domain provides valuable insight into optimal dosing frequencies. Through the analysis of three distinct tumor growth models (cell cycle-specific, metronomic, and acquired resistance), we demonstrate the application of a simulation-based analysis in the frequency domain to optimize cancer treatments. We study the response of tumor growth to dosing frequencies while simultaneously examining treatment safety, and found for all three models that above a certain dosing frequency, tumor size is insensitive to an increase in dosing frequency, e.g., for the cell cycle-specific model, one dose per 3 days, and an hourly dose yield the same reduction of tumor size to 3% of the initial size after 1 year of treatment. Additionally, we explore the effect of drug elimination rate changes on the tumor growth response. In summary, we show that the frequency-domain view of three models of tumor growth dynamics can help in optimizing drug dosing regimen to improve treatment success.

Optimizing drug development in oncology by clinical trial simulation: Why and how?

In therapeutic research, the safety and efficacy of pharmaceutical products are necessarily tested on humans via clinical trials after an extensive and expensive preclinical development period. Methodologies such as computer modeling and clinical trial simulation (CTS) might represent a valuable option to reduce animal and human assays. The relevance of these methods is well recognized in pharmacokinetics and pharmacodynamics from the preclinical phase to postmarketing. However, they are barely used and are poorly regarded for drug approval, despite Food and Drug Administration and European Medicines Agency recommendations. The generalization of CTS could be greatly facilitated by the availability of software for modeling biological systems, by clinical trial studies and hospital databases. Data sharing and data merging raise legal, policy and technical issues that will need to be addressed. Development of future molecules will have to use CTS for faster development and thus enable better patient management. Drug activity modeling coupled with disease modeling, optimal use of medical data and increased computing speed should allow this leap forward. The realization of CTS requires not only bioinformatics tools to allow interconnection and global integration of all clinical data but also a universal legal framework to protect the privacy of every patient. While recognizing that CTS can never replace 'real-life' trials, they should be implemented in future drug development schemes to provide quantitative support for decision-making. This in silico medicine opens the way to the P4 medicine: predictive, preventive, personalized and participatory.

How to mathematically optimize drug regimens using optimal control

This article gives an overview of a technique called optimal control, which is used to optimize real-world quantities represented by mathematical models. I include background information about the historical development of the technique and applications in a variety of fields. The main focus here is the application to diseases and therapies, particularly the optimization of combination therapies, and I highlight several such examples. I also describe the basic theory of optimal control, and illustrate each of the steps with an example that optimizes the doses in a combination regimen for leukemia. References are provided for more complex cases. The article is aimed at modelers working in drug development, who have not used optimal control previously. My goal is to make this technique more accessible in the biopharma community.

Frequency-Domain Response Analysis for Quantitative Systems Pharmacology Models

Drug dosing regimen can significantly impact drug effect and, thus, the success of treatments. Nevertheless, trial and error is still the most commonly used method by conventional pharmacometric approaches to optimize dosing regimen. In this tutorial, we utilize four distinct classes of quantitative systems pharmacology models to introduce frequency-domain response analysis, a method widely used in electrical and control engineering that allows the analytical optimization of drug treatment regimen from the dynamics of the model.

Revisiting Dosing Regimen Using Pharmacokinetic/Pharmacodynamic Mathematical Modeling: Densification and Intensification of Combination Cancer Therapy

Controlling effects of drugs administered in combination is particularly challenging with a densified regimen because of life-threatening hematological toxicities. We have developed a mathematical model to optimize drug dosing regimens and to redesign the dose intensification-dose escalation process, using densified cycles of combined anticancer drugs. A generic mathematical model was developed to describe the main components of the real process, including pharmacokinetics, safety and efficacy pharmacodynamics, and non-hematological toxicity risk. This model allowed for computing the distribution of the total drug amount of each drug in combination, for each escalation dose level, in order to minimize the average tumor mass for each cycle. This was achieved while complying with absolute neutrophil count clinical constraints and without exceeding a fixed risk of non-hematological dose-limiting toxicity. The innovative part of this work was the development of densifying and intensifying designs in a unified procedure. This model enabled us to determine the appropriate regimen in a pilot phase I/II study in metastatic breast patients for a 2-week-cycle treatment of docetaxel plus epirubicin doublet, and to propose a new dose-ranging process. In addition to the present application, this method can be further used to achieve optimization of any combination therapy, thus improving the efficacy versus toxicity balance of such a regimen.

A Review of Modeling Approaches to Predict Drug Response in Clinical Oncology

Model-based approaches have emerged as important tools for quantitatively understanding temporal relationships between drug dose, concentration, and effect over the course of treatment, and have now become central to optimal drug development and tailored drug treatment. In oncology, the therapeutic index of a chemotherapeutic drug is typically narrow and a full dose-response relationship is not available, often because of treatment failure. Noting the benefits of model-based approaches and the low therapeutic index of oncology drugs, in recent years, modeling approaches have been increasingly used to streamline oncologic drug development through early identification and quantification of dose-response relationships. With this background, this report reviews publications that used model-based approaches to evaluate drug treatment outcome variables in oncology therapeutics, ranging from tumor size dynamics to tumor/biomarker time courses and survival response.

Revisiting dosing regimen using PK/PD modeling: the MODEL1 phase I/II trial of docetaxel plus epirubicin in metastatic breast cancer patients

The MODEL1 trial is the first model-driven phase I/II dose-escalation study of densified docetaxel plus epirubicin administration in metastatic breast cancer patients, a regimen previously known to induce unacceptable life-threatening toxicities. The primary objective was to determine the maximum tolerated dose of this densified regimen. Study of the efficacy was a secondary objective. Her2-negative, hormone-resistant metastatic breast cancer patients were treated with escalating doses of docetaxel plus epirubicin every 2 weeks for six cycles with granulocyte colony stimulating factor support. A total of 16 patients were treated with total doses ranging from 85 to 110 mg of docetaxel plus epirubicin per cycle. Dose escalation was controlled by a non-hematological toxicity model. Dose densification was guided by a model of neutrophil kinetics, able to optimize docetaxel plus epirubicin dosing with respect to pre-defined acceptable levels of hematological toxicity while ensuring maximal efficacy. The densified treatment was safe since hematological toxicity was much lower compared to previous findings, and other adverse events were consistent with those observed with this regimen. The maximal tolerated dose was 100 mg given every 2 weeks. The response rate was 45 %; median progression-free survival was 10.4 months, whereas 54.6 months of median overall survival was achieved. The optimized docetaxel plus epirubicin dosing regimen led to fewer toxicities associated with higher efficacy as compared with standard or empirical densified dosing. This study suggests that model-driven dosage adjustment can lead to improved efficacy-toxicity balance in patients with cancer when several anticancer drugs are combined.

Mathematical modeling for novel cancer drug discovery and development

INTRODUCTION

Mathematical modeling enables: the in silico classification of cancers, the prediction of disease outcomes, optimization of therapy, identification of promising drug targets and prediction of resistance to anticancer drugs. In silico pre-screened drug targets can be validated by a small number of carefully selected experiments.

AREAS COVERED

This review discusses the basics of mathematical modeling in cancer drug discovery and development. The topics include in silico discovery of novel molecular drug targets, optimization of immunotherapies, personalized medicine and guiding preclinical and clinical trials. Breast cancer has been used to demonstrate the applications of mathematical modeling in cancer diagnostics, the identification of high-risk population, cancer screening strategies, prediction of tumor growth and guiding cancer treatment.

EXPERT OPINION

Mathematical models are the key components of the toolkit used in the fight against cancer. The combinatorial complexity of new drugs discovery is enormous, making systematic drug discovery, by experimentation, alone difficult if not impossible. The biggest challenges include seamless integration of growing data, information and knowledge, and making them available for a multiplicity of analyses. Mathematical models are essential for bringing cancer drug discovery into the era of Omics, Big Data and personalized medicine.

Maximum tolerated dose versus metronomic scheduling in the treatment of metastatic cancers

Although optimal control theory has been used for the theoretical study of anti-cancerous drugs scheduling optimization, with the aim of reducing the primary tumor volume, the effect on metastases is often ignored. Here, we use a previously published model for metastatic development to define an optimal control problem at the scale of the entire organism of the patient. In silico study of the impact of different scheduling strategies for anti-angiogenic and cytotoxic agents (either in monotherapy or in combination) is performed to compare a low-dose, continuous, metronomic administration scheme with a more classical maximum tolerated dose schedule. Simulation results reveal differences between primary tumor reduction and control of metastases but overall suggest use of the metronomic protocol.

Optimization of vascular-targeting drugs in a computational model of tumor growth

A biophysical tool is introduced that seeks to provide a theoretical basis for helping drug design teams assess the most promising drug targets and design optimal treatment strategies. The tool is grounded in a previously validated computational model of the feedback that occurs between a growing tumor and the evolving vasculature. In this paper, the model is particularly used to explore the therapeutic effectiveness of two drugs that target the tumor vasculature: angiogenesis inhibitors (AIs) and vascular disrupting agents (VDAs). Using sensitivity analyses, the impact of VDA dosing parameters is explored, as is the effects of administering a VDA with an AI. Further, a stochastic optimization scheme is utilized to identify an optimal dosing schedule for treatment with an AI and a chemotherapeutic. The treatment regimen identified can successfully halt simulated tumor growth, even after the cessation of therapy.

Computational modeling of tumor response to vascular-targeting therapies--part I: validation

Mathematical modeling techniques have been widely employed to understand how cancer grows, and, more recently, such approaches have been used to understand how cancer can be controlled. In this manuscript, a previously validated hybrid cellular automaton model of tumor growth in a vascularized environment is used to study the antitumor activity of several vascular-targeting compounds of known efficacy. In particular, this model is used to test the antitumor activity of a clinically used angiogenesis inhibitor (both in isolation, and with a cytotoxic chemotherapeutic) and a vascular disrupting agent currently undergoing clinical trial testing. I demonstrate that the mathematical model can make predictions in agreement with preclinical/clinical data and can also be used to gain more insight into these treatment protocols. The results presented herein suggest that vascular-targeting agents, as currently administered, cannot lead to cancer eradication, although a highly efficacious agent may lead to long-term cancer control.

Evaluation of α2-integrin expression as a biomarker for tumor growth inhibition for the investigational integrin inhibitor E7820 in preclinical and clinical studies

E7820 is an orally active inhibitor of α(2)-integrin mRNA expression, currently tested in phases I and II. We aimed to evaluate what levels of inhibition of integrin expression are needed to achieve tumor stasis in mice, and to compare this to the level of inhibition achieved in humans. Tumor growth inhibition was measured in mice bearing a pancreatic KP-1 tumor, dosed at 12.5-200 mg/kg over 21 days. In the phase I study, E7820 was administered daily for 28 days over a range of 0-200 mg, followed by a 7-day washout period. PK-PD models were developed in NONMEM. α(2)-Integrin expression measured on platelets, corresponding to tumor stasis at t = 21 in 50% and 90% of the mice (I(int,50), I(int,90)) were calculated. It was evaluated if these levels of inhibition could be achieved in patients at tolerable doses. One hundred nineteen α(2)-Integrin measurements and 210 tumor size measurements were available from mice. The relationship between PK and α(2)-integrin expression was modeled using an indirect-effect model, subsequently linked to an exponential tumor growth model. I(inh,50) and I(inh,90) were 14.7% (RSE 7%) and 17.9% (RSE 8%). Four hundred sixty two α(2)-integrin measurements were available from 29 patients. Using the schedule of 100 mg qd (MTD), α(2)-integrin expression was inhibited more strongly than the I(int,50) and I(int,90) in greater than 95% and greater than 50% of patients, respectively. Moderate inhibition of α(2)-integrin expression corresponded to tumor stasis in mice, and similar levels could be reached in patients with the dose level of 100 mg qd.

Model-based prediction of phase III overall survival in colorectal cancer on the basis of phase II tumor dynamics

PURPOSE

We developed a drug-disease simulation model to predict antitumor response and overall survival in phase III studies from longitudinal tumor size data in phase II trials.

METHODS

We developed a longitudinal exposure-response tumor-growth inhibition (TGI) model of drug effect (and resistance) using phase II data of capecitabine (n = 34) and historical phase III data of fluorouracil (FU; n = 252) in colorectal cancer (CRC); and we developed a parametric survival model that related change in tumor size and patient characteristics to survival time using historical phase III data (n = 245). The models were validated in simulation of antitumor response and survival in an independent phase III study (n = 1,000 replicates) of capecitabine versus FU in CRC.

RESULTS

The TGI model provided a good fit of longitudinal tumor size data. A lognormal distribution best described the survival time, and baseline tumor size and change in tumor size from baseline at week 7 were predictors (P < .00001). Predicted change of tumor size and survival time distributions in the phase III study for both capecitabine and FU were consistent with observed values, for example, 431 days (90% prediction interval, 362 to 514 days) versus 401 days observed for survival in the capecitabine arm. A modest survival improvement of 39 days (90% prediction interval, -21 to 110 days) versus 35 days observed was predicted for capecitabine.

CONCLUSION

The modeling framework successfully predicted survival in a phase III trial on the basis of capecitabine phase II data in CRC. It is a useful tool to support end-of-phase II decisions and design of phase III studies.

Elucidation of relationship between tumor size and survival in non-small-cell lung cancer patients can aid early decision making in clinical drug development

Four non-small-cell lung cancer (NSCLC) registration trials were utilized to develop models linking survival to risk factors and changes in tumor size during treatment. The purpose was to leverage existing quantitative knowledge to facilitate future development of anti-NSCLC drugs. Eleven risk factors were screened using a Cox model. A mixed exponential decay and linear growth model was utilized for modeling tumor size. Survival times were described in a parametric model. Eastern Cooperative Oncology Group (ECOG) score and baseline tumor size were consistent prognostic factors of survival. Tumor size was well described by the mixed model. The parametric survival model includes ECOG score, baseline tumor size, and week 8 tumor size change as predictors of survival duration. The change in tumor size at week 8 allows early assessment of the activity of an experimental regimen. The survival model and the tumor model will be beneficial for early screening of candidate drugs, simulating NSCLC trials, and optimizing trial designs.

Mathematical and numerical analysis for a model of growing metastatic tumors

In cancer diseases, the appearance of metastases is a very pejorative forecast. Chemotherapies are systemic treatments which aim at the elimination of the micrometastases produced by a primitive tumour. The efficiency of chemotherapies closely depends on the protocols of administration. Mathematical modeling is an invaluable tool to help in evaluating the best treatment strategy. Iwata et al. [K. Iwata, K. Kawasaki, N. Shigesad, A dynamical model for the growth and size distribution of multiple metastatic tumors, J. Theor. Biol. 203 (2000) 177.] proposed a partial differential equation (PDE) that describes the metastatic evolution of an untreated tumour. In this article, we conducted a thorough mathematical analysis of this model. Particularly, we provide an explicit formula for the growth rate parameter, as well as a numerical resolution of this PDE. By increasing our understanding of the existing model, this work is crucial for further extension and refinement of the model. It settles down the framework necessary for the consideration of drugs administration effects on tumour development.

Cell cycle checkpoint models for cellular pharmacology of paclitaxel and platinum drugs

A pharmacokinetic-pharmacodynamic mathematical model is developed for cellular pharmacology of chemotherapeutic drugs for which the decisive step towards cell death occurs at a point in the cell cycle, presumably corresponding to a cell cycle checkpoint. For each cell, the model assumes a threshold level of some intracellular species at that checkpoint, beyond which the cell dies. The threshold level is assumed to have a log-normal distribution in the cell population. The kinetics of formation of the lethal intracellular species depends on the drug, and on the cellular pharmacokinetics and binding kinetics of the cell. Specific models are developed for paclitaxel and for platinum drugs (cisplatin, oxaliplatin and carboplatin). In the case of paclitaxel, two separate mechanisms of cell death necessitate a model that accounts for two checkpoints, with different intracellular species. The model was tested on a number of in vitro cytotoxicity data sets for these drugs, and found overall to give significantly better fits than previously proposed cellular pharmacodynamic models. It provides an explanation for the asymptotic convergence of dose-response curves as exposure time becomes long.

Modeling oxaliplatin drug delivery to circadian rhythms in drug metabolism and host tolerance

To make possible the design of optimal (circadian and other period) time-scheduled regimens for cytotoxic drug delivery by intravenous infusion, a pharmacokinetic-pharmacodynamic (PK-PD, with circadian periodic drug dynamics) model of chemotherapy on a population of tumor cells and its tolerance by a population of fast renewing healthy cells is presented. The application chosen for identification of the model parameters is the treatment by oxaliplatin of Glasgow osteosarcoma, a murine tumor, and the healthy cell population is the jejunal mucosa, which is the main target of oxaliplatin toxicity in mice. The model shows the advantage of a periodic time-scheduled regimen, compared to the conventional continuous constant infusion of the same daily dose, when the biological time of peak infusion is correctly chosen. Furthermore, it is well adapted to using mathematical optimization methods of drug infusion flow, choosing tumor population minimization as the objective function and healthy tissue preservation as a constraint. Such a constraint is in clinical settings tunable by physicians by taking into account the patient's state of health.

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.

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.

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.

Two-mechanism peak concentration model for cellular pharmacodynamics of Doxorubicin

A mathematical model is presented for the cellular uptake and cytotoxicity of the anticancer drug doxorubicin. The model assumes sigmoidal, Hill-type dependence of cell survival on drug-induced damage. Experimental evidence indicates distinct intracellular and extracellular mechanisms of doxorubicin cytotoxicity. Drug-induced damage is therefore expressed as the sum of two terms, representing the peak values over time of concentrations of intracellular and extracellular drugs. Dependence of cell kill on peak values of concentration rather than on an integral over time is consistent with observations that dose-response curves for doxorubicin converge to a single curve as exposure time is increased. Drug uptake by cells is assumed to include both saturable and unsaturable components, consistent with experimental data. Overall, the model provides better fits to in vitro cytotoxicity data than previous models. It shows how saturation of cellular uptake or binding with concentration can result in plateaus in the dose-response curve at high concentrations and short exposure, as observed experimentally in some cases. The model provides a unified framework for analyzing doxorubicin cellular pharmacokinetic and pharmacodynamic data, and can be applied in mathematical models for tumor response and treatment optimization.

Predictive pharmacokinetic-pharmacodynamic modeling of tumor growth kinetics in xenograft models after administration of anticancer agents

The available mathematical models describing tumor growth and the effect of anticancer treatments on tumors in animals are of limited use within the drug industry. A simple and effective model would allow applying quantitative thinking to the preclinical development of oncology drugs. In this article, a minimal pharmacokinetic-pharmacodynamic model is presented, based on a system of ordinary differential equations that link the dosing regimen of a compound to the tumor growth in animal models. The growth of tumors in nontreated animals is described by an exponential growth followed by a linear growth. In treated animals, the tumor growth rate is decreased by a factor proportional to both drug concentration and number of proliferating tumor cells. A transit compartmental system is used to model the process of cell death, which occurs at later times. The parameters of the pharmacodynamic model are related to the growth characteristics of the tumor, to the drug potency, and to the kinetics of the tumor cell death. Therefore, such parameters can be used for ranking compounds based on their potency and for evaluating potential differences in the tumor cell death process. The model was extensively tested on discovery candidates and known anticancer drugs. It fitted well the experimental data, providing reliable parameter estimates. On the basis of the parameters estimated in a first experiment, the model successfully predicted the response of tumors exposed to drugs given at different dose levels and/or schedules. It is, thus, possible to use the model prospectively, optimizing the design of new experiments.

Mechanistic models for myelosuppression

As myelosuppression is the dose-limiting toxicity for most chemotherapeutic drugs, modelers attempt to find relationships between drug and toxicity to optimize treatment. Mechanistic models, i.e. models based on physiology and pharmacology, are preferable over empirical models, as prior information can be utilized and as they generally are more reliable for extrapolations. To account for different dosing-regimens and possible schedule-dependent effects, the whole concentration-time profile should be used as input into the pharmacokinetic-pharmacodynamic model. It is also of importance to model the whole time course of myelosuppression to be able to predict both the degree and duration of toxicity as well as consecutive courses of therapy. A handful of (semi)-mechanistic pharmacokinetic-pharmacodynamic models with the above properties have been developed and are reviewed. Ideally, a model of myelosuppression should separate drug-specific parameters from system related parameters to be applicable across drugs and useful under different clinical settings. Introduction of mechanistic models of myelosuppression in the design and evaluation of clinical trials can guide in the decision of optimal sampling times, contribute to knowledge of optimal doses and treatment regimens at an earlier time point and identify sub-groups of patients at a high risk of myelosuppression.