Quantitative mathematical modeling of clinical brain metastasis dynamics in non-small cell lung cancer.

Brain metastases (BMs) are associated with poor prognosis in non-small cell lung cancer (NSCLC), but are only visible when large enough. Therapeutic decisions such as whole brain radiation therapy would benefit from patient-specific predictions of radiologically undetectable BMs. Here, we propose a mathematical modeling approach and use it to analyze clinical data of BM from NSCLC. Primary tumor growth was best described by a gompertzian model for the pre-diagnosis history, followed by a tumor growth inhibition model during treatment. Growth parameters were estimated only from the size at diagnosis and histology, but predicted plausible individual estimates of the tumor age (2.1-5.3 years). Multiple metastatic models were further assessed from fitting either literature data of BM probability (n = 183 patients) or longitudinal measurements of visible BMs in two patients. Among the tested models, the one featuring dormancy was best able to describe the data. It predicted latency phases of 4.4-5.7 months and onset of BMs 14-19 months before diagnosis. This quantitative model paves the way for a computational tool of potential help during therapeutic management.

Mathematical optimisation of the cisplatin plus etoposide combination for managing extensive-stage small-cell lung cancer patients.

BACKGROUND: Small-cell lung cancer (SCLC) represents one of the most aggressive forms of lung cancer. Despite the fair sensitivity of SCLC to chemotherapy and radiotherapy, the current standard treatment regimens have modest survival rates and are associated with potential life-threatening adverse events. Therefore, research into new optimised regimens that increase drug efficacy while respecting toxicity constraints is of primary importance. METHODS: A PK/PD model for the combination of cisplatin and etoposide to treat extensive-stage SCLC patients was generated. The model takes into consideration both the efficacy of the drugs and their haematological toxicity. Using optimisation techniques, the model can be used to propose new regimens. RESULTS: Three new regimens with varying timing for combining cisplatin and etoposide have been generated that respect haematological toxicity constraints and achieve better or similar tumour regression. The proposed regimens are: (1) Protocol OP1: etoposide 80 mg m-2 over 1 h D1, followed by a long infusion 12 h later (over 3 days) of 160 mg m-2 plus cisplatin 80 mg m-2 over 1 h D1, D1-D1 21 days; (2) Protocol OP2: etoposide 80 mg m-2 over 1 h D1, followed by a long infusion 12 h later (over 4 days) of 300 mg m-2 plus cisplatin 100 mg m-2 over 1 h D1, D1-D1 21 days; and (3) Protocol OP3: etoposide 40 mg m-2 over 1 h, followed by a long infusion 6 h later (3 days) of 105 mg m-2 plus cisplatin 50 mg m-2 over 1 h, D1-D1 14 days. CONCLUSIONS: Mathematical modelling can help optimise the design of new cisplatin plus etoposide regimens for managing extensive-stage SCLC patients.

A mathematical model for the administration of temozolomide: comparative analysis of conventional and metronomic chemotherapy regimens.

We propose a mathematical model that takes into account a classical maximum tolerated dose (MTD) chemotherapy regimen (whose primary targets are the tumor cells) as well as a metronomic chemotherapy regimen (whose primary targets are the tumor endothelial cells) for the administration of temozolomide (Temodal(®)) in order to compare the effectiveness of these two types of protocols. The model is built from 4 natural hypotheses: (H1) without treatment the tumor growth follows a Gompertz model, (H2) endothelial cells are more sensitive to temozolomide than cancer cells, (H3) the anti-angiogenic effect blocks tumor growth, and (H4) endothelial cells are more genetically stable than cancer cells and thus less likely to develop resistance to temozolomide. Then, we compared a conventional MTD regimen of 200 mg/m(2) temozolomide J1-J5 every 28 days with a daily metronomic regimen of 85 mg/m(2)/day for cycles of 42 days. Our mathematical model shows that the metronomic regimen induces tumor regression through anti-angiogenic effects while the MTD regimen fails to do so, due to the emergence of temozolomide resistance in cancer cells. Overall, our model is consistent with clinical observations and provides an interesting tool toward the personalization of anticancer treatments, through optimization of dose and schedule of chemotherapy based on individual patient characteristics.

A new model for determining the MTD during phase-I trials in pediatric oncology.

The primary goal of phase I studies in oncology is to determine the MTD (Maximum Tolerated Dose) for a drug. This MTD is determined with respect to an accepted risk (usually 33%) to see a limiting toxity for patients. In this paper we propose a new mathematical model to determine the MTD. An important feature of this model is that the limiting toxicity can be formulated as a combination of several basic graded toxicities such as hematologic or neurological. Another feature is the possibility to take into account several patient covariates to individualize the determination of the MTD. The model is a bayesian model where some prior information has been considered. The model is expected to work better than traditional empirical schemes for determining the MTD because it uses at every step all the available information on patients, and adds some major improvements as compared with existing CRM strategies because it uses whole data made available, including low-grades toxicities. Finally the model has been validated with a retrospective data set on 17 patients from a phase I study on paclitaxel in pediatric oncology. Calculated MTDs for each patient were found to be markedly different than the doses actually given following a traditional dose-escalation methodology. Results suggest that our new model provides a better and safer way to drive dose-escalation in phase-I trials as compared with traditional schemes.

An interface model for dosage adjustment connects hematotoxicity to pharmacokinetics.

When modeling is required to describe pharmacokinetics and pharmacodynamics simultaneously, it is difficult to link time-concentration profiles and drug effects. When patients are under chemotherapy, despite the huge amount of blood monitoring numerations, there is a lack of exposure variables to describe hematotoxicity linked with the circulating drug blood levels. We developed an interface model that transforms circulating pharmacokinetic concentrations to adequate exposures, destined to be inputs of the pharmacodynamic process. The model is materialized by a nonlinear differential equation involving three parameters. The relevance of the interface model for dosage adjustment is illustrated by numerous simulations. In particular, the interface model is incorporated into a complex system including pharmacokinetics and neutropenia induced by docetaxel and by cisplatin. Emphasis is placed on the sensitivity of neutropenia with respect to the variations of the drug amount. This complex system including pharmacokinetic, interface, and pharmacodynamic hematotoxicity models is an interesting tool for analysis of hematotoxicity induced by anticancer agents. The model could be a new basis for further improvements aimed at incorporating new experimental features.

New adaptive method for phase I trials in oncology.

Assessment of the maximum tolerated dose for a small sample of patients is the objective of phase I trials in oncology. We propose a new adaptive approach performing differentiation of each type of toxicity and their grades, definition of the dose-limiting toxicity as a logical combination of toxicity events, modeling of the risk of toxic events as a function of the drug dose, and integration of individual covariates. An application study with retrospective data from a phase I Taxol pediatric trial revealed age as an influential covariate, allowing individualization of a patient's maximum tolerated dose. A simulation study illustrates the practice of sequential inclusion of patients, of constraints in the dose-finding process and of trial stopping rules. The approach presented here allows quick maximum tolerated dose assessment with a small sample of patients and assists the design of phase I trials in oncology.

Optimizing drug regimens in cancer chemotherapy: a simulation study using a PK-PD model.

In cancer chemotherapy, it is important to design treatment strategies for drug protocols that ensure a desired rate of tumor cell kill without overdosing the host. Mathematical modeling was used for optimization in which we minimize the end value of the tumor cells while limiting toxicity by always maintaining the white blood cell count beyond a limit. The optimal solution for this is a mixture of an initial bolus application of drug followed by no drug and then continuous infusion that keeps the normal cell population at its lower limit while decreasing the tumor cell population.

Information tools for exploratory data analysis in population pharmacokinetics.

For a group of individuals, population pharmacokinetic studies describe the interindividual variability through a statistical distribution. These studies conducted during the drug development serve as a useful marker of the safety of the drug, provide information that might be decisive for future experiments and, in a clinical context, help establish guidelines for optimal use in each patient. As complementary tools to the existing statistical and graphical techniques for population pharmacokinetic data analysis, indexes derived from information theory were used to select the most appropriate modelfor the statistical distribution, to detect atypical individuals, and to screen influential covariates. The rationale for using these indexes is shown using simulated and real data.

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

In cancer chemotherapy, it is important to design treatment strategies that ensure a desired rate of tumor cell kill without unacceptable toxicity. To optimize treatment, we used a mathematical model describing the pharmacokinetics of anticancer drugs, antitumor efficacy, and drug toxicity. This model was associated with constraints on the allowed plasma concentrations, drug exposure, and leukopenia. Given a schedule of drug administrations, the mathematical model optimized the drug doses that can minimize the tumor burden while limiting toxicity at the level of the white blood cells. The main result is that the optimal drug administration is an initial high-dose chemotherapy up to saturation of constraints associated with normal cell toxicity and a maintenance continuous infusion at a moderate rate. Data related to etoposide investigations were used in a feasibility study. Simulations with the optimized protocol showed better performances than usual clinical protocols. Model-based optimal drug doses provide for greater cytoreduction, while limiting the risk of unacceptable toxicity.

Dosage regimen calculations with optimal control theory.

In clinical pharmacokinetics, dosage regimen calculation involves determination of either: (1) the drug amount to be administered according to a set time schedule, or (2) the time schedule to be used for appropriate drug amounts. The goal is to guarantee that the time profile of circulating drug levels is between the thresholds of toxicity and efficacy. For the first case, solutions are obtained by using the property of linearity when it holds. Herein, we present several results concerning the second case. The optimal control theory allows determination of switching times between the minimal and maximal input rates in order to ensure the fastest transition from an initial state to the therapeutic levels. Fundamental results are reported and the approach is developed for drugs administered by intravenous infusion. By means of the phase trajectories, general graphical rules are presented to design the optimal control and to determine the reachable areas. A numerical example is given, comparisons of the method with others are attempted and potential developments are pointed out.