A mathematical model to study the effects of drugs administration on tumor growth dynamics

Formulation and Validation of an Extended Sigmoid Emax Model in Pharmacodynamics

PURPOSE OR OBJECTIVE

Drug concentration-response curves (DRCs) are crucial in pharmacology for assessing the drug effects on biological systems. The widely used sigmoid Emax model, which accounts for response saturation, relies heavily on the effective drug concentration ( E D 50 ). This reliance can lead to validation errors and inaccuracies in model fitting. The Emax model cannot generate multiple DRCs, raising concerns about whether the dataset is fully utilized.

METHODS

This study formulates an extended Emax (eEmax) model designed to overcome these limitations. The eEmax model generates multiple DRCs from a single dataset by using various estimatedα ' s ∈ 0,100 , while keeping E D α fixed, rather than estimating an E D 50 value as in the Emax model.

RESULTS

This model effectively captures a broader range of concentration-response behavior, including non-sigmoidal patterns, thus providing greater flexibility and accuracy compared to the Emax model. Validation using various drug-response data and PKPD frameworks demonstrates the eEmax model's improved accuracy and versatility in handling concentration-response data.

CONCLUSIONS

The eEmax model provides a robust and flexible method for drug concentration-response analysis, facilitating the generation of multiple DRCs from a single dataset and reducing the possibility of validation errors. This model is particularly valuable for its ease of use and its capability to fully utilize datasets, providing its potential in PKPD modeling and drug discovery.

Model-based prediction of effective target exposure for MEN1611 in combination with trastuzumab in HER2-positive advanced or metastatic breast cancer patients

MEN1611 is a novel orally bioavailable PI3K inhibitor currently in clinical development for patients with HER2-positive (HER2+) PI3KCA mutated advanced/metastatic breast cancer (BC) in combination with trastuzumab (TZB). In this work, a translational model-based approach to determine the minimum target exposure of MEN1611 in combination with TZB was applied. First, pharmacokinetic (PK) models for MEN1611 and TZB in mice were developed. Then, in vivo tumor growth inhibition (TGI) data from seven combination studies in mice xenograft models representative of the human HER2+ BC non-responsive to TZB (alterations of the PI3K/AkT/mTOR pathway) were analyzed using a PK-pharmacodynamic (PD) TGI model for co-administration of MEN1611 and TZB. The established PK-PD relationship was used to quantify the minimum effective MEN1611 concentration, as a function of TZB concentration, needed for tumor eradication in xenograft mice. Finally, a range of minimum effective exposures for MEN1611 were extrapolated to patients with BC, considering the typical steady-state TZB plasma levels in patients with BC following three alternative regimens (i.v. 4 mg/kg loading dose +2 mg/kg q1w, i.v. 8 mg/kg loading dose +6 mg/kg q3w or s.c. 600 mg q3w). A threshold of about 2000 ng·h/ml for MEN1611 exposure associated with a high likelihood of effective antitumor activity in a large majority of patients was identified for the 3-weekly and the weekly i.v. schedule for TZB. A slightly lower exposure (i.e., 25% lower) was found for the 3-weekly s.c. schedule. This important outcome confirmed the adequacy of the therapeutic dose administered in the ongoing phase 1b B-PRECISE-01 study in patients with HER2+ PI3KCA mutated advanced/metastatic BC.

A two-stages global sensitivity analysis by using the δ sensitivity index in presence of correlated inputs: application on a tumor growth inhibition model based on the dynamic energy budget theory

Global sensitivity analysis (GSA) evaluates the impact of variability and/or uncertainty of the model parameters on given model outputs. GSA is useful for assessing the quality of Pharmacometric model inference. Indeed, model parameters can be affected by high (estimation) uncertainty due to the sparsity of data. Independence between model parameters is a common assumption of GSA methods. However, ignoring (known) correlations between parameters may alter model predictions and, then, GSA results. To address this issue, a novel two-stages GSA technique based on the δ index, which is well-defined also in presence of correlated parameters, is here proposed. In the first step, statistical dependencies are neglected to identify parameters exerting causal effects. Correlations are introduced in the second step to consider the real distribution of the model output and investigate also the 'indirect' effects due to the correlation structure. The proposed two-stages GSA strategy was applied, as case study, to a preclinical tumor-in-host-growth inhibition model based on the Dynamic Energy Budget theory. The aim is to evaluate the impact of the model parameter estimate uncertainty (including correlations) on key model-derived metrics: the drug threshold concentration for tumor eradication, the tumor volume doubling time and a new index evaluating the drug efficacy-toxicity trade-off. This approach allowed to rank parameters according to their impact on the output, discerning whether a parameter mainly exerts a causal or 'indirect' effect. Thus, it was possible to identify uncertainties that should be necessarily reduced to obtain robust predictions for the outputs of interest.

Replacement, Reduction, and Refinement of Animal Experiments in Anticancer Drug Development: The Contribution of 3D In Vitro Cancer Models in the Drug Efficacy Assessment

In the last decades three-dimensional (3D) in vitro cancer models have been proposed as a bridge between bidimensional (2D) cell cultures and in vivo animal models, the gold standards in the preclinical assessment of anticancer drug efficacy. 3D in vitro cancer models can be generated through a multitude of techniques, from both immortalized cancer cell lines and primary patient-derived tumor tissue. Among them, spheroids and organoids represent the most versatile and promising models, as they faithfully recapitulate the complexity and heterogeneity of human cancers. Although their recent applications include drug screening programs and personalized medicine, 3D in vitro cancer models have not yet been established as preclinical tools for studying anticancer drug efficacy and supporting preclinical-to-clinical translation, which remains mainly based on animal experimentation. In this review, we describe the state-of-the-art of 3D in vitro cancer models for the efficacy evaluation of anticancer agents, focusing on their potential contribution to replace, reduce and refine animal experimentations, highlighting their strength and weakness, and discussing possible perspectives to overcome current challenges.

Fractional transit compartment model for describing drug delayed response to tumors using Mittag-Leffler distribution on age-structured PKPD model

The response of a cell population is often delayed relative to drug injection, and individual cells in a population of cells have a specific age distribution. The application of transit compartment models (TCMs) is a common approach for describing this delay. In this paper, we propose a TCM in which damaged cells caused by a drug are given by a single fractional derivative equation. This model describes the delay as a single equation composed of fractional and ordinary derivatives, instead of a system of ODEs expressed in multiple compartments, applicable to the use of the PK concentration in the model. This model tunes the number of compartments in the existing model and expresses the delay in detail by estimating an appropriate fractional order. We perform model robustness, sensitivity analysis, and change of parameters based on the amount of data. Additionally, we resolve the difficulty in parameter estimation and model simulation using a semigroup property, consisting of a system with a mixture of fractional and ordinary derivatives. This model provides an alternative way to express the delays by estimating an appropriate fractional order without determining the pre-specified number of compartments.

Extended transit compartment model to describe tumor delay using Coxian distribution

The measured response of cell population is often delayed relative to drug injection, and individuals in a population have a specific age distribution. Common approaches for describing the delay are to apply transit compartment models (TCMs). This model reflects that all damaged cells caused by drugs suffer transition processes, resulting in death. In this study, we present an extended TCM using Coxian distribution, one of the phase-type distributions. The cell population attacked by a drug is described via age-structured models. The mortality rate of the damaged cells is expressed by a convolution of drug rate and age density. Then applying to Erlang and Coxian distribution, we derive Erlang TCM, representing the existing model, and Coxian TCMs, reflecting sudden death at all ages. From published data of drug and tumor, delays are compared after parameter estimations in both models. We investigate the dynamical changes according to the number of the compartments. Model robustness and equilibrium analysis are also performed for model validation. Coxian TCM is an extended model considering a realistic case and captures more diverse delays.

A translational model-based approach to inform the choice of the dose in phase 1 oncology trials: the case study of erdafitinib

PURPOSE

Erdafitinib (JNJ-42756493, BALVERSA) is a tyrosine kinase inhibitor indicated for the treatment of advanced urothelial carcinoma. In this work, a translational model-based approach to inform the choice of the doses in phase 1 trials is illustrated.

METHODS

A pharmacokinetic (PK) model was developed to describe the time course of erdafitinib plasma concentrations in mice and rats. Data from multiple xenograft studies in mice and rats were analyzed using the Simeoni tumor growth inhibition (TGI) model. The model parameters were used to derive a range of erdafitinib exposures that might inform the choice of the doses in oncology phase 1 trials. Conversion of exposures to doses was based on preliminary PK assessments from the first-in human (FIH) study.

RESULTS

A one-compartment PK disposition model, with linear absorption and dose-dependent clearance, adequately described the PK data in both mice and rats via an allometric scaling approach. The TGI model was able to describe tumor growth dynamics, providing quantitative measurements of erdafitinib antitumor potency in mice and rats. Based on these estimates, ranges of efficacious unbound concentration were identified for erdafitinib in mice (0.642-5.364 μg/L) and rats (0.782-2.565 μg/L). Based on the FIH data, it was possible to transpose exposures into doses and doses of above 4 mg/day provided erdafitinib exposures associated with significant TGI in animals. The findings were in agreement with the results of the FIH trial, in which the first hints of clinical activities were observed at 6 mg.

CONCLUSION

The successful modeling exercise of erdafitinib preclinical data showed how translational PK-PD modeling might be a tool to help to inform the choice of the doses in FIH studies.

Modeling restoration of gefitinib efficacy by co-administration of MET inhibitors in an EGFR inhibitor-resistant NSCLC xenograft model: A tumor-in-host DEB-based approach

MET receptor tyrosine kinase inhibitors (TKIs) can restore sensitivity to gefitinib, a TKI targeting epidermal growth factor receptor (EGFR), and promote apoptosis in non-small cell lung cancer (NSCLC) models resistant to gefitinib treatment in vitro and in vivo. Several novel MET inhibitors are currently under study in different phases of development. In this work, a novel tumor-in-host modeling approach, based on the Dynamic Energy Budget (DEB) theory, was proposed and successfully applied to the context of poly-targeted combination therapies. The population DEB-based tumor growth inhibition (TGI) model well-described the effect of gefitinib and of two MET inhibitors, capmatinib and S49076, on both tumor growth and host body weight when administered alone or in combination in an NSCLC mice model involving the gefitinib-resistant tumor line HCC827ER1. The introduction of a synergistic effect in the combination DEB-TGI model allowed to capture gefitinib anticancer activity enhanced by the co-administered MET inhibitor, providing also a quantitative evaluation of the synergistic drug interaction. The model-based comparison of the two MET inhibitors highlighted that S49076 exhibited a greater anticancer effect as well as a greater ability in restoring sensitivity to gefitinib than the competitor capmatinib. In summary, the DEB-based tumor-in-host framework proposed here can be applied to routine combination xenograft experiments, providing an assessment of drug interactions and contributing to rank investigated compounds and to select the optimal combinations, based on both tumor and host body weight dynamics. Thus, the combination tumor-in-host DEB-TGI model can be considered a useful tool in the preclinical development and a significant advance toward better characterization of combination therapies.

A Dynamic Energy Budget (DEB) based modeling framework to describe tumor-in-host growth inhibition and cachexia onset during anticancer treatment in in vivo xenograft studies

Cancer anorexia-cachexia syndrome (CACS) is a very severe complication of cancer for which an adequate therapeutic strategy has not yet been defined. Recently, a notable number of new animal models of human CACS has been made available for translational purposes. Under the assumption that tumor-induced adaptations of host metabolism and tumor-host energetic competition play a major role in CACS (together with possible toxicities induced by the anticancer treatment), we developed a new Dynamic Energy Budget (DEB)-based framework, modeling tumor-in-host growth dynamics and cachexia onset in preclinical animal models during anticancer treatments. The tumor-in-host modeling approach was successfully applied on a multitude of in vivo preclinical studies involving different host species, tumor cell lines, type of anticancer agents and experimental settings among which standard xenograft studies. Obtained results strongly suggested the adoption of the tumor-in-host DEB-based approach in the preclinical oncological setting for a joint assessment of drug efficacy and toxicity and for a better design of the experiments. Further applications of the DEB-based approach to the context of poly-targeted combination therapy, anti-cachectic treatments and preclinical to clinical translation are under investigation with extremely encouraging preliminary results.

Impact of Cisplatin Dosing Regimens on Mammary Tumor Growth in an Animal Model

Background

We have recently introduced a modification of the seminal Simeoni model for tumor growth, the modification entailing the incorporation of delay differential equations into its formulation. We found that the modification was competitive with the Simeoni construct in modeling mammary tumor growth under cisplatin treatment in an animal model.

Methods

In our original study, we had two cohorts of animals: untreated, and treatment with bolus injection of cisplatin on day 0. We here explore how modifications in the cisplatin dosing scheme affect tumor growth in our model.

Results

We found that modest fractionation dosing schemes have little ultimate impact on tumor growth. In contrast, metronomic dosing schemes seem quite efficacious, and might yield effective control over tumor progression.

Conclusions

With regard to cisplatin as single agent chemotherapy, a minimum level of drug for a prolonged period of time seems more critical than rapid achievement of a very high dose for a shorter time frame for deterring tumor growth or progression. Exploration of tumor dose schedules with mathematical models can provide valuable insights into potentially effective therapeutic regimens.

Steering the Clinical Translation of Delivery Systems for Drugs and Health Products

Besides the feasibility for industrial scale-up, accelerating the translation from bench to bedside of new technological strategies for controlled delivery and targeting of drugs and other actives relevant for health management, such as medical devices and nutraceuticals, would benefit from an even earlier evaluation in pre-clinical models and clinical settings. At the same time, translational medicine also performs in the opposite direction, incorporating clinical needs and observations into scientific hypotheses and innovative technological proposals. With these aims, the sessions proposed for the 2019 CRS Italy Chapter Workshop will introduce the experience of Italian and worldwide researchers on how to foster the actual work in controlled release and drug delivery towards a reliable pre-clinical and clinical assessment.

Different ODE models of tumor growth can deliver similar results

BACKGROUND

Simeoni and colleagues introduced a compartmental model for tumor growth that has proved quite successful in modeling experimental therapeutic regimens in oncology. The model is based on a system of ordinary differential equations (ODEs), and accommodates a lag in therapeutic action through delay compartments. There is some ambiguity in the appropriate number of delay compartments, which we examine in this note.

METHODS

We devised an explicit delay differential equation model that reflects the main features of the Simeoni ODE model. We evaluated the original Simeoni model and this adaptation with a sample data set of mammary tumor growth in the FVB/N-Tg(MMTVneu)202Mul/J mouse model.

RESULTS

The experimental data evinced tumor growth heterogeneity and inter-individual diversity in response, which could be accommodated statistically through mixed models. We found little difference in goodness of fit between the original Simeoni model and the delay differential equation model relative to the sample data set.

CONCLUSIONS

One should exercise caution if asserting a particular mathematical model uniquely characterizes tumor growth curve data. The Simeoni ODE model of tumor growth is not unique in that alternative models can provide equivalent representations of tumor growth.

Recent advances in physiologically based pharmacokinetic and pharmacodynamic models for anticancer nanomedicines

Nanoparticles (NPs) have distinct pharmacokinetic (PK) properties and can potentially improve the absorption, distribution, metabolism, and elimination (ADME) of small-molecule drugs loaded therein. Owing to the unwanted toxicities of anticancer agents in healthy organs and tissues, their precise delivery to the tumor is an essential requirement. There have been numerous advancements in the development of nanomedicines for cancer therapy. Physiologically based PK (PBPK) models serve as excellent tools for describing and predicting the ADME properties and the efficacy and toxicity of drugs, in combination with pharmacodynamic (PD) models. The recent preliminary application of these modeling approaches to NPs demonstrated their potential benefits in research and development processes relevant to the ADME and pharmacodynamics of NPs and nanomedicines. Here, we comprehensively review the pharmacokinetics of NPs, the developed PBPK models for anticancer NPs, and the developed PD model for anticancer agents.

Combining Multiscale Experimental and Computational Systems Pharmacological Approaches to Overcome Resistance to HER2-targeted Therapy in Breast Cancer

The emergence of human epidermal growth factor receptor type-2 (HER2) therapy resistance in HER2-positive (HER2+) breast cancer (BC) poses a major clinical challenge. The primary mechanisms of resistance include aberrant activation of the HER2 and phosphatidylinositol 3-kinase/mammalian target of rapamycin/AKT8 virus oncogene cellular homolog (PI3K/Akt/mTOR) pathways. The existence of feedback loops in this pathway may engender resistance to targeted therapies such as everolimus, an mTOR inhibitor, resulting in a more aggressive form of refractory HER2+ BC. Here, we hypothesize that a triple and sequential combination therapy of paclitaxel, a potent cytotoxic agent, before concomitant administration of dasatinib, a SRC proto-oncogene nonreceptor tyrosine kinase (Src) family kinase inhibitor, with everolimus, restores sensitivity to treatment in refractory HER2+ BC. This was assessed by a combination of experimental and computational approaches. Quantitative systems pharmacological (QSP), pharmacokinetics (PK), and pharmacodynamics (PD) studies were conducted in static and three-dimensional and dynamic (3DD) cell culture systems using a HER2+ cell line resistant to HER2 therapy, JIMT-1. The dynamic responses in cellular viability and key signaling proteins in the HER2 and PI3K/Akt/mTOR pathways were measured upon treatments with single drugs, combinations, and appropriate controls. A QSP-PK/PD model was developed and used to optimize the sequence and interdose interval of the three agents in the combination. The proposed sequential combination therapy demonstrated strong cytotoxic effects in JIMT-1 cells, and our models predicted the usefulness of this combination over prolonged durations in the 3DD setting. Our combined experimental and QSP-PK/PD modeling approach may serve as a useful screening tool in predicting clinical efficacy of combination therapies in oncology. Nonetheless, further in vivo human xenograft tumor studies are warranted.

Modeling tumor growth inhibition and toxicity outcome after administration of anticancer agents in xenograft mice: A Dynamic Energy Budget (DEB) approach

Host features, such as cell proliferation rates, caloric intake, metabolism and energetic conditions, significantly influence tumor growth; at the same time, tumor growth may have a dramatic impact on the host conditions. For example, in clinics, at certain stages of the tumor growth, cachexia (body weight reduction) may become so relevant to be considered as responsible for around 20% of cancer deaths. Unfortunately, anticancer therapies may also contribute to the development of cachexia due to reduced food intake (anorexia), commonly observed during the treatment periods. For this reason, cachexia is considered one of the major toxicity findings to be evaluated also in preclinical studies. However, although various pharmacokinetic-pharmacodynamic (PK-PD) tumor growth inhibition (TGI) models are currently available, the mathematical modeling of cachexia onset and TGI after an anticancer administration in preclinical experiments is still an open issue. To cope with this, a new PK-PD model, based on a set of tumor-host interaction rules taken from Dynamic Energy Budget (DEB) theory and a set of drug tumor inhibition equations taken from the well-known Simeoni TGI model, was developed. The model is able to describe the body weight reduction, splitting the cachexia directly induced by tumor and that caused by the drug treatment under study. It was tested in typical preclinical studies, essentially designed for efficacy evaluation and routinely performed as a part of the industrial drug development plans. For the first time, both the dynamics of tumor and host growth could be predicted in xenograft mice untreated or treated with different anticancer agents and following different schedules. The model code is freely available for downloading at http://repository.ddmore.eu (model number DDMODEL00000274).

Sequential Therapeutic Response Modeling for Tumor Treatment Using Computational Hybrid Control Systems Approach

OBJECTIVE

Tumorigenesis is due to uncontrolled cell division arising from mutations and alterations in the proliferative controls of the cell population. The fight against tumor growth and development has often relied on combination therapy that has been acclaimed as one of the main standards of care in cancer therapeutics and prevention of drug-related resistances. The toxicity of the combinatorial drugs raises a significant concern whenever patients take two or more drugs concurrently at the maximum tolerated dose. A promising solution in tumor treatment involves the administration of the drugs in an alternating or sequential fashion rather than a simultaneous manner. In this paper, we investigate how feasible such an approach is from a mathematical perspective and propose a switched hybrid control systems framework.

METHODS

We explore the response of tumor cells dynamics to sequential drugs administration with the aid of a time-dependent switching strategy. A transit compartmentalized model is employed to describe the tumor cells progression to death.

RESULTS

The design of the time-based drug switching logic ensures the proliferating tumor cells are repressed.

CONCLUSIONS

Simulation results are provided using the tumor growth dynamics with sequential drugs intake to demonstrate the effectiveness of the proposed method in reducing the tumor size.

SIGNIFICANCE

This paper is the first attempt to provide a switched hybrid control systems framework on sequential drug administration to biomedical researchers and clinicians.

Time-Dependent Effects in Algae for Chemicals with Different Adverse Outcome Pathways: A Novel Approach

Chemicals affect unicellular algae as a result of toxicokinetic and toxicodynamic processes. The internal concentration of chemicals in algae cells typically reaches equilibrium within minutes, while damage cumulatively increases over hours. The time gap between the steady state of internal exposure and damage development is thus suspected to span up to hours, mainly due to toxicodynamic processes. The quantification of rate-limited toxicodynamic processes, aggregated as a progressive effect from an initiating molecular event through biological key events toward the adverse outcome on algae growth inhibition, might discriminate between different adverse outcome pathways (AOPs). To support our hypothesis, we selected six chemicals according to different physicochemical properties and three distinctly dissimilar AOPs. The time courses of internal concentrations were linked to the observed affected Scenedesmus vacuolatus growth using toxicokinetic-toxicodynamic modeling. Effects on cell growth were explained by effect progression and not by the time to reach internal equilibrium concentration. Effect progression rates ranged over 6 orders of magnitude for all chemicals but varied by less than 1 order of magnitude within similar AOP (photosystem II inhibitors > reactive chemicals > lipid biosynthesis inhibitors), meaning that inhibitors of photosystem II advance an effect toward algae growth fastest compared to reactive chemicals and inhibitors of lipid biosynthesis.

Optimal Design for Informative Protocols in Xenograft Tumor Growth Inhibition Experiments in Mice

Tumor growth inhibition (TGI) models are increasingly used during preclinical drug development in oncology for the in vivo evaluation of antitumor effect. Tumor sizes are measured in xenografted mice, often only during and shortly after treatment, thus preventing correct identification of some TGI model parameters. Our aims were (i) to evaluate the importance of including measurements during tumor regrowth and (ii) to investigate the proportions of mice included in each arm. For these purposes, optimal design theory based on the Fisher information matrix implemented in PFIM4.0 was applied. Published xenograft experiments, involving different drugs, schedules, and cell lines, were used to help optimize experimental settings and parameters using the Simeoni TGI model. For each experiment, a two-arm design, i.e., control versus treatment, was optimized with or without the constraint of not sampling during tumor regrowth, i.e., "short" and "long" studies, respectively. In long studies, measurements could be taken up to 6 g of tumor weight, whereas in short studies the experiment was stopped 3 days after the end of treatment. Predicted relative standard errors were smaller in long studies than in corresponding short studies. Some optimal measurement times were located in the regrowth phase, highlighting the importance of continuing the experiment after the end of treatment. In the four-arm designs, the results showed that the proportions of control and treated mice can differ. To conclude, making measurements during tumor regrowth should become a general rule for informative preclinical studies in oncology, especially when a delayed drug effect is suspected.

Improvement of Parameter Estimations in Tumor Growth Inhibition Models on Xenografted Animals: a Novel Method to Handle the Interval Censoring Caused by Measurement of Smaller Tumors

The purpose of this study was to explore the interval censoring induced by caliper measurements on smaller tumors during tumor growth experiments in preclinical studies and to show its impact on parameter estimations. A new approach, the so-called interval-M3 method, is proposed to specifically handle this type of data. Thereby, the interval-M3 method was challenged with different methods (including classical methods for handling below quantification limit values) using Stochastic Simulation and Estimation process to take into account the censoring. In this way, 1000 datasets were simulated under the design of a typical of tumor growth study in xenografted mice, and then, each method was used for parameter estimation on the simulated datasets. Relative bias and relative root mean square error (relative RMSE) were consequently computed for comparison purpose. By not considering the censoring, parameter estimations appeared to be biased and particularly the cytotoxic effect parameter, k 2 , which is the parameter of interest to characterize the efficacy of a compound in oncology. The best performance was noted with the interval-M3 method which properly takes into account the interval censoring induced by caliper measurement, giving overall unbiased estimations for all parameters and especially for the antitumor effect parameter (relative bias = 0.49%, and relative RMSE = 4.06%).

Physiologically based pharmacokinetic and pharmacodynamic modeling in cancer drug development: status, potential and gaps

INTRODUCTION

Modeling and simulation have become important means of answering questions relevant to the development of a drug, making it possible to assess risks early and to reduce costs. Physiologically based pharmacokinetic and pharmacodynamic (PBPK/PD) models contribute to a comprehensive understanding of the drug, covering specific questions from early discovery through lifecycle management stages. As for other disease areas, in oncology, PBPK and PD models are important topics that remain to be addressed.

AREAS COVERED

This review describes current PBPK and PD approaches, their applicability in drug development in general and specifically in the area of oncology. It discusses the current status and then focuses on key challenges and the potential for future use. It provides cases in which modeling currently cannot answer the questions and assesses the requirements to close gaps for PBPK/PD in oncology.

EXPERT OPINION

PBPK/PD models have led to improvements in identifying risks and reducing costs during the drug development process. Nevertheless, there is a lot of potential, where more rigorous integration of biological knowledge and specific experimental design would result in a more comprehensive biological picture. Ideally, such approaches would reveal the extent to which preclinical work can be extrapolated to clinical settings, thus enabling reliable prediction and, ultimately, reducing failed trials in clinical oncology.

An analytical model for nanoparticles concentration resulting from infusion into poroelastic brain tissue

We consider the infusion of a diluted suspension of nanoparticles (NPs) into poroelastic brain tissue, in view of relevant biomedical applications such as intratumoral thermotherapy. Indeed, the high impact of the related pathologies motivates the development of advanced therapeutic approaches, whose design also benefits from theoretical models. This study provides an analytical expression for the time-dependent NPs concentration during the infusion into poroelastic brain tissue, which also accounts for particle binding onto cells (by recalling relevant results from the colloid filtration theory). Our model is computationally inexpensive and, compared to fully numerical approaches, permits to explicitly elucidate the role of the involved physical aspects (tissue poroelasticity, infusion parameters, NPs physico-chemical properties, NP-tissue interactions underlying binding). We also present illustrative results based on parameters taken from the literature, by considering clinically relevant ranges for the infusion parameters. Moreover, we thoroughly assess the model working assumptions besides discussing its limitations. While not laying any claims of generality, our model can be used to support the development of more ambitious numerical approaches, towards the preliminary design of novel therapies based on NPs infusion into brain tissue.

Biomarker- versus drug-driven tumor growth inhibition models: an equivalence analysis

The mathematical modeling of tumor xenograft experiments following the dosing of antitumor drugs has received much attention in the last decade. Biomarker data can further provide useful insights on the pathological processes and be used for translational purposes in the early clinical development. Therefore, it is of particular interest the development of integrated pharmacokinetic-pharmacodynamic (PK-PD) models encompassing drug, biomarker and tumor-size data. This paper investigates the reciprocal consistency of three types of models: drug-to-tumor, such as established drug-driven tumor growth inhibition (TGI) models, drug-to-biomarker, e.g. indirect response models, and biomarker-to-tumor, e.g. the more recent biomarker-driven TGI models. In particular, this paper derives a mathematical relationship that guarantees the steady-state equivalence of the cascade of drug-to-biomarker and biomarker-to-tumor models with a drug-to-tumor TGI model. Using the Simeoni TGI model as a reference, conditions for steady-state equivalence are worked out and used to derive a new biomarker-driven model. Simulated and real data are used to show that in realistic cases the steady-state equivalence extends also to transient responses. The possibility of predicting the drug-to-tumor potency of a new candidate drug based only on biomarker response is discussed.

Mathematical models for absorption and efficacy of ovarian cancer treatments

The creation of personal and individualized anti-cancer treatments has been a major goal in the progression of cancer discovery as evident by the continuous research efforts in genetics and population based PK/PD studies. In this paper we use our clinical decision support tool, called ChemoDSS, to evaluate the effectiveness of three treatments recommended by the NCCN guidelines for ovarian cancer using pre-clinical data from the literature. In particular, we analyze the treatments of PC (i.e., Paclitaxel and Cispaltin), DC (i.e., Docetaxel and Carboplatin), and PBC (i.e., Paclitaxel, Bevacizumab, and Carboplatin). Our in silico analysis of the ovarian cancer treatments shows that PC was the most effective regimen for treating ovarian cancer compared to DC and PBC, which is consistent with literature findings. We demonstrate that we can successfully evaluate the effectiveness of the selected ovarian cancer treatment regimens using ChemoDSS.

Modeling of human tumor xenografts and dose rationale in oncology

Xenograft models are commonly used in oncology drug development. Although there are discussions about their ability to generate meaningful data for the translation from animal to humans, it appears that better data quality and better design of the preclinical experiments, together with appropriate data analysis approaches could make these data more informative for clinical development. An approach based on mathematical modeling is necessary to derive experiment-independent parameters which can be linked with clinically relevant endpoints. Moreover, the inclusion of biomarkers as predictors of efficacy is a key step towards a more general mechanism-based strategy.

Optimal Intervention in Markovian Gene Regulatory Networks With Random-Length Therapeutic Response to Antitumor Drug

The most effective cancer treatments are the ones that prolong patients' lives while offering a reasonable quality of life during and after treatment. The treatments must also carry out their actions rapidly and with high efficiency such that a very large percentage of tumor cells die or shift into a state where they stop proliferating. Due to biological and microenvironmental variabilities within tumor cells, the action period of an administered drug can vary among a population of patients. In this paper, based on a recently proposed model for tumor growth inhibition, we first probabilistically characterize the variability of the length of drug action. Then, we present a methodology to devise optimal intervention strategies for any Markovian genetic regulatory network governing the tumor when the antitumor drug has a random-length duration of action.

A predictive pharmacokinetic-pharmacodynamic model of tumor growth kinetics in xenograft mice after administration of anticancer agents given in combination

Purpose

In clinical oncology, combination treatments are widely used and increasingly preferred over single drug administrations. A better characterization of the interaction between drug effects and the selection of synergistic combinations represent an open challenge in drug development process. To this aim, preclinical studies are routinely performed, even if they are only qualitatively analyzed due to the lack of generally applicable mathematical models.

Methods

This paper presents a new pharmacokinetic–pharmacodynamic model that, starting from the well-known single agent Simeoni TGI model, is able to describe tumor growth in xenograft mice after the co-administration of two anticancer agents. Due to the drug action, tumor cells are divided in two groups: damaged and not damaged ones. The damaging rate has two terms proportional to drug concentrations (as in the single drug administration model) and one interaction term proportional to their product. Six of the eight pharmacodynamic parameters assume the same value as in the corresponding single drug models. Only one parameter summarizes the interaction, and it can be used to compute two important indexes that are a clear way to score the synergistic/antagonistic interaction among drug effects.

Results

The model was successfully applied to four new compounds co-administered with four drugs already available on the market for the treatment of three different tumor cell lines. It also provided reliable predictions of different combination regimens in which the same drugs were administered at different doses/schedules.

Conclusions

A good and quantitative measurement of the intensity and nature of interaction between drug effects, as well as the capability to correctly predict new combination arms, suggest the use of this generally applicable model for supporting the experiment optimal design and the prioritization of different therapies.

Electronic supplementary material

The online version of this article (doi:10.1007/s00280-013-2208-8) contains supplementary material, which is available to authorized users.

Systems pharmacological analysis of paclitaxel-mediated tumor priming that enhances nanocarrier deposition and efficacy

Paclitaxel (PAC)-mediated apoptosis decompresses and primes tumors for enhanced deposition of nanoparticulate agents such as pegylated liposomal doxorubicin (DXR). A quantitative pharmacokinetic/pharmacodynamic (PK/PD) approach was developed to analyze efficacy and identify optima for PAC combined with sterically stabilized liposome (SSL)-DXR. Using data extracted from diverse literature sources, Cremophor-paclitaxel (Taxol(®)) PK was described by a carrier-mediated dispositional model and SSL-DXR PK was described by a two-compartment model with first-order drug release. A hybrid-physiologic, well-stirred model with partition coefficients (Kp) captured intratumor concentrations. Apoptotic responses driving tumor priming were modeled using nonlinear, time-dependent transduction functions. The tumor growth model used net first-order growth and death rate constants, and two transit compartments that captured the temporal displacement of tumor exposure versus effect, and apoptotic signals from each agent were used to drive cytotoxic effects of the combination. The final model captured plasma and intratumor PK data, apoptosis induction profiles, and tumor growth for all treatments/sequences. A feedback loop representing PAC-induced apoptosis effects on Kp(_DXR) enabled the model to capture tumor-priming effects. Simulations to explore time- and sequence-dependent effects of priming indicated that PAC priming increased K(p_DXR) 3-fold. The intratumor concentrations producing maximal and half-maximal effects were 18 and 7.2 μg/ml for PAC, and 17.6 and 14.3 μg/ml for SSL-DXR. The duration of drug-induced apoptosis was 27.4 h for PAC and 15.8 h for SSL-DXR. Simulations suggested that PAC administered 24 h before peak priming could increase efficacy 2.5-fold over experimentally reported results. The quantitative approach developed in this article is applicable for evaluating tumor-priming strategies using diverse agents.

A Systems Biology Approach in Therapeutic Response Study for Different Dosing Regimens-a Modeling Study of Drug Effects on Tumor Growth using Hybrid Systems

Motivated by the frustration of translation of research advances in the molecular and cellular biology of cancer into treatment, this study calls for cross-disciplinary efforts and proposes a methodology of incorporating drug pharmacology information into drug therapeutic response modeling using a computational systems biology approach. The objectives are two fold. The first one is to involve effective mathematical modeling in the drug development stage to incorporate preclinical and clinical data in order to decrease costs of drug development and increase pipeline productivity, since it is extremely expensive and difficult to get the optimal compromise of dosage and schedule through empirical testing. The second objective is to provide valuable suggestions to adjust individual drug dosing regimens to improve therapeutic effects considering most anticancer agents have wide inter-individual pharmacokinetic variability and a narrow therapeutic index. A dynamic hybrid systems model is proposed to study drug antitumor effect from the perspective of tumor growth dynamics, specifically the dosing and schedule of the periodic drug intake, and a drug’s pharmacokinetics and pharmacodynamics information are linked together in the proposed model using a state-space approach. It is proved analytically that there exists an optimal drug dosage and interval administration point, and demonstrated through simulation study.

TGI-Simulator: a visual tool to support the preclinical phase of the drug discovery process by assessing in silico the effect of an anticancer drug

This paper presents TGI-Simulator, a software tool designed to show, through a 2-D graphical animation, the simulated time effect of an anticancer drug on a tumor mass by exploiting the well-known Tumor Growth Inhibition (TGI) model published by Simeoni et al. [1]. Simeoni TGI model is a mathematical model routinely used by pharma companies and researchers during the drug development process. The application is based on a Java graphical user interface (GUI) including a self installing differential equation solver implemented in Matlab together with an optimization algorithm that performs model identification via Weighted Least Squares (WLS). However, it can graphically show also the simulation results obtained within other scientific software tools, if they are preventively stored into a suitable ASCII file. The tool would be a valid support also for researchers with no specific skills in scientific calculations and in pharmacokinetic and pharmacodynamic modeling but daily involved in pharma companies drug development processes at different levels. The availability of a movie with a temporal varying 2-D iconographic representation is an original instrument to communicate results and learn Simeoni TGI model and its potential application in preclinical studies.

A model of vascular tumour growth in mice combining longitudinal tumour size data with histological biomarkers

Optimising the delivery of antiangiogenic drugs requires the development of drug-disease models of vascular tumour growth that incorporate histological data indicative of cytostatic action. In this study, we formulated a model to analyse the dynamics of tumour progression in nude mice xenografted with HT29 or HCT116 colorectal cancer cells. In 30 mice, tumour size was periodically measured, and percentages of hypoxic and necrotic tissue were assessed using immunohistochemistry techniques on tumour samples after euthanasia. The simultaneous analysis of histological data together with longitudinal tumour size data prompted the development of a semi-mechanistic model integrating random effects of parameters. In this model, the peripheral non-hypoxic tissue proliferates according to a generalised-logistic equation where the maximal tumour size is represented by a variable called 'carrying capacity'. The ratio of the whole tumour size to the carrying capacity was used to define the hypoxic stress. As this stress increases, non-hypoxic tissue turns hypoxic. Hypoxic tissue does not stop proliferating, but hypoxia constitutes a transient stage before the tissue becomes necrotic. As the tumour grows, the carrying capacity increases owing to the process of angiogenesis. The model is shown to correctly predict tumour growth dynamics as well as percentages of necrotic and hypoxic tissues within the tumour. We show how the model can be used as a theoretical tool to investigate the effects of antiangiogenic treatments on tumour growth. This model provides a tool to analyse tumour size data in combination with histological biomarkers such as the percentages of hypoxic and necrotic tissue and is shown to be useful for gaining insight into the effects of antiangiogenic drugs on tumour growth and composition.

Modeling the efficacy of trastuzumab-DM1, an antibody drug conjugate, in mice

Trastuzumab-DM1 (T-DM1) is a novel antibody-drug conjugate under investigation for the treatment of human epidermal growth factor receptor 2 (HER2)-positive metastatic breast cancer. One challenge in oncologic drug development is determining the optimal dose and treatment schedule. A novel dose regimen-finding strategy was developed for T-DM1 using experimental data and pharmacokinetic/pharmacodynamic modeling. To characterize the disposition of T-DM1, pharmacokinetic studies were conducted in athymic nude and beige nude mice. The pharmacokinetics of T-DM1 were described well by a two-compartment model. Tumor response data were obtained from single-dose, multiple-dose and time-dose-fractionation studies of T-DM1 in animal models of HER2-positive breast cancer, specifically engineered to be insensitive to trastuzumab. A sequential population-based pharmacokinetic/pharmacodynamic modeling approach was developed to describe the anti-tumor activity of T-DM1. A cell-cycle-phase nonspecific tumor cell kill model incorporating transit compartments captured well the features of tumor growth and the activity of T-DM1. Key findings of the model were that tumor cell growth rate played a significant role in the sensitivity of tumors to T-DM1; anti-tumor activity was schedule independent; and tumor response was linked to the ratio of exposure to a concentration required for tumor stasis.

Modelling the genesis and treatment of cancer: the potential role of physiologically based pharmacodynamics

Physiologically based modelling of pharmacodynamics/toxicodynamics requires an a priori knowledge on the underlying mechanisms causing toxicity or causing the disease. In the context of cancer, the objective of the expert meeting was to discuss the molecular understanding of the disease, modelling approaches used so far to describe the process, preclinical models of cancer treatment and to evaluate modelling approaches developed based on improved knowledge. Molecular events in cancerogenesis can be detected using 'omics' technology, a tool applied in experimental carcinogenesis, but also for diagnostics and prognosis. The molecular understanding forms the basis for new drugs, for example targeting protein kinases specifically expressed in cancer. At present, empirical preclinical models of tumour growth are in great use as the development of physiological models is cost and resource intensive. Although a major challenge in PKPD modelling in oncology patients is the complexity of the system, based in part on preclinical models, successful models have been constructed describing the mechanism of action and providing a tool to establish levels of biomarker associated with efficacy and assisting in defining biologically effective dose range selection for first dose in man. To follow the concentration in the tumour compartment enables to link kinetics and dynamics. In order to obtain a reliable model of tumour growth dynamics and drug effects, specific aspects of the modelling of the concentration-effect relationship in cancer treatment that need to be accounted for include: the physiological/circadian rhythms of the cell cycle; the treatment with combinations and the need to optimally choose appropriate combinations of the multiple agents to study; and the schedule dependence of the response in the clinical situation.

Comparison of two pharmacodynamic transduction models for the analysis of tumor therapeutic responses in model systems

Semi-mechanistic pharmacodynamic (PD) models that capture tumor responses to anticancer agents with fidelity can provide valuable insights that could aid in the optimization of dosing regimens and the development of drug delivery strategies. This study evaluated the utility and potential interchangeability of two transduction-type PD models: a cell distribution model (CDM) and a signal distribution model (SDM). The evaluation was performed by simulating dense and sparse tumor response data with one model and analyzing it using the other. Performance was scored by visual inspection and precision of parameter estimation. Capture of tumor response data was also evaluated for a liposomal formulation of paclitaxel in the paclitaxel-resistant murine Colon-26 model. A suitable PK model was developed by simultaneous fitting of literature data for paclitaxel formulations in mice. Analysis of the simulated tumor response data revealed that the SDM was more flexible in describing delayed drug effects upon tumor volume progression. Dense and sparse data simulated using the CDM were fit very well by the SDM, but under some conditions, data simulated using the SDM were fitted poorly by the CDM. Although both models described the dose-dependent therapeutic responses of Colon-26 tumors, the fit by the SDM contained less bias. The CDM and SDM are both useful transduction models that recapitulate, with fidelity, delayed drug effects upon tumor growth. However, they are mechanistically distinct and not interchangeable. Both fit some types of tumor growth data well, but the SDM appeared more robust, particularly where experimental data are sparse.

Preclinical pharmacokinetic/pharmacodynamic models to predict synergistic effects of co-administered anti-cancer agents

PURPOSE

Pharmacokinetic/pharmacodynamic (PK/PD) models have been shown to be useful in predicting tumor growth rates in mouse xenografts. We applied novel PK/PD models to the published anticancer combination therapies of tumor growth inhibition to simulate synergistic changes in tumor growth rates. The parameters from the PK/PD model were further used to estimate clinical doses of the combination.

METHODS

A PK/PD model was built that linked the dosing regimen of a compound to the inhibition of tumor growth in mouse xenograft models. Two subsequent PK/PD models were developed to simulate the published tumor growth profiles of combination treatments. Model I predicts the tumor growth curve assuming that the effect of two anticancer drugs, AZD7762 and irinotecan, is synergistic when given in combination. Model II predicts the tumor growth curve assuming that the effect of co-administering flavopiridol and irinotecan is maximally synergistic when dosed at an optimal interval.

RESULTS

Model I was able to account for the synergistic effects of AZD7762 following the administration of irinotecan. When Model II was applied to the antitumor activity of irinotecan and flavopiridol combination therapy, the modeling was able to reproduce the optimal dosing interval between administrations of the compounds. Furthermore, Model II was able to estimate the biologically active dose of flavopiridol recommended for phase II studies.

CONCLUSIONS

The timing of clinical combination therapy doses is often selected empirically. PK/PD models provide a theoretical structure useful in the design of the optimal clinical dose, frequency of administration and the optimal timing of administration between anticancer agents to maximize tumor suppression.

Phenomenological modeling of tumor diameter growth based on a mixed effects model

Over the last few years, taking advantage of the linear kinetics of the tumor growth during the steady-state phase, tumor diameter-based rather than tumor volume-based models have been developed for the phenomenological modeling of tumor growth. In this study, we propose a new tumor diameter growth model characterizing early, late and steady-state treatment effects. Model parameters consist of growth rhythms, growth delays and time constants and are meaningful for biologists. Biological experiments provide in vivo longitudinal data. The latter are analyzed using a mixed effects model based on the new diameter growth function, to take into account inter-mouse variability and treatment factors. The relevance of the tumor growth mixed model is firstly assessed by analyzing the effects of three therapeutic strategies for cancer treatment (radiotherapy, concomitant radiochemotherapy and photodynamic therapy) administered on mice. Then, effects of the radiochemotherapy treatment duration are estimated within the mixed model. The results highlight the model suitability for analyzing therapeutic efficiency, comparing treatment responses and optimizing, when used in combination with optimal experiment design, anti-cancer treatment modalities.

Modeling of tumor growth and anticancer effects of combination therapy

Combination therapies are widely used in the treatment of patients with cancer. Selecting synergistic combination strategies is a great challenge during early drug development. Here, we present a pharmacokinetic/pharmacodynamic (PK/PD) model with a smooth nonlinear growth function to characterize and quantify anticancer effect of combination therapies using time-dependent data. To describe the pharmacological effect of combination therapy, an interaction term was introduced into a semi-mechanistic anticancer PK/PD model. This approach enables testing of a pharmacological hypothesis with respect to an anticipated pharmacological synergy of drug combinations, such as an assumed pharmacological synergy of complementary inhibition of a particular signaling pathway. The model was applied to three real data sets derived from preclinical screening experiments using xenograft mice. The suggested model fitted well the observed data from mono- to combination-therapy and allowed physiologically meaningful interpretation of the experiments. The tested drug combinations were assessed for their ability to act as synergistic modulators of tumor growth inhibition by the interaction parameter psi. The presented approach has practical implications because it can be applied to standard xenograft experiments and may assist in the selection of both optimal drug combinations and administration schedules. The unique feature of the presented approach is the ability to characterize the nature of combined drug interaction as well as to quantify the intensity of such interactions by assessing the time course of combined drug effect.

A minimal model of tumor growth inhibition

The preclinical development of antitumor drugs greatly benefits from the availability of models capable of predicting tumor growth as a function of the drug administration schedule. For being of practical use, such models should be simple enough to be identifiable from standard experiments conducted on animals. In the present paper, a stochastic model is derived from a set of minimal assumptions formulated at cellular level. Tumor cells are divided in two groups: proliferating and nonproliferating. The probability that a proliferating cell generates a new cell is a function of the tumor weight. The probability that a proliferating cell becomes nonproliferating is a function of the plasma drug concentration. The time-to-death of a nonproliferating cell is a random variable whose distribution reflects the nondeterministic delay between drug action and cell death. The evolution of the expected value of tumor weight obeys two differential equations (an ordinary and a partial differential one), whereas the variance is negligible. Therefore, the tumor growth dynamics can be well approximated by the deterministic evolution of its expected value. The tumor growth inhibition model, which is a lumped parameter model that in the last few years has been successfully applied to several antitumor drugs, is shown to be a special case of the minimal model presented here.

A model-based approach to the in vitro evaluation of anticancer activity

PURPOSE

The use of in vitro screening tests for characterizing the activity of anticancer agents is a standard practice in oncology research and development. In these studies, human A2780 ovarian carcinoma cells cultured in plates are exposed to different concentrations of the compounds for different periods of time. Their anticancer activity is then quantified in terms of EC(50) comparing the number of metabolically active cells present in the treated and the control arms at specified time points. The major concern of this methodology is the observed dependency of the EC(50) on the experimental design in terms of duration of exposure. This dependency could affect the efficacy ranking of the compounds, causing possible biases especially in the screening phase, when compound selection is the primary purpose of the in vitro analysis. To overcome this problem, the applicability of a modeling approach to these in vitro studies was evaluated.

METHODS

The model, consisting of a system of ordinary differential equations, represents the growth of tumor cells using a few identifiable and biologically relevant parameters related to cell proliferation dynamics and drug action. In particular, the potency of the compounds can be measured by a unique and drug-specific parameter that is essentially independent of drug concentration and exposure time. Parameter values were estimated using weighted nonlinear least squares.

RESULTS

The model was able to adequately describe the growth of tumor cells at different experimental conditions. The approach was validated both on commercial drugs and discovery candidate compounds. In addition, from this model the relationship between EC(50) and the exposure time was derived in an analytic form.

CONCLUSIONS

The proposed approach provides a new tool for predicting and/or simulating cell responses to different treatments with useful indications for optimizing in vitro experimental designs. The estimated potency parameter values obtained from different compounds can be used for an immediate ranking of anticancer activity.