Mathematical models in drug development

Transcription

1 Summary Mathematical modelling of tumor growth inhibition for the development of anticancer drugs Giuseppe De Nicolao Department of Computer Science and Systems Theory University of Pavia Italy Tumor growth inhibition (TGI) studies The pharmacokinetic-pharmacodynamic model Results: discovery candidates Benefits & Further developments Oncology drugs and extrapolation to patients Drug development Mathematical models in drug development It takes years Average cost: 9 ML Dollar Only drugs out of pay back the costs Attrition (rate of): Fraction of discarded candidates Facts: Only % of Phase candidates is eventually registered: Attrition = 9% Phase II anticancer drugs: Attrition > % Pase III: Attrition = % (anticancer: 9%) Main causes of attrition in clinics: % lack of efficacy % excessive toxicity Fact: Approval of NCE (New Chemical Entities) is at a historical minimum Moral: pharma companies can maintain their growth rate only by reducing attrition What to do? predict efficacy and toxicity in discovery and/or preclinical studies Improve predictivity of animal models (also by means of mathematical modeling and simulation)

2 Computational methods for reducing attrition Computational methods for reducing attrition ADMET prediction Systems biology Systems Biology ADME / ADMET property prediction PK/PD modelling PK / PD modelling Systems biology Cellular models, mathematical models of diseases, virtual patient Methodologies: system theory, dynamical models Firms: Entelos (99), Beyond Genomics (), Bioseek (), Gene Network International (), etc... Investments: (source: Nature Biotechnology, October ) Entelos: ML $ Beyond Genomics: ML $ Bioseek:.ML $ Gene Network International:. ML $ ADME / ADMET property prediction Prediction of ADMET properties (Absorbtion, Distribution, Metabolism, Excretion and Toxicity) from molecular features. Methodologies: multivariate statistics, neural networks, machine learning, data mining. Firms: Umetrics, Tripos, Spotfire, Molecular Discovery, Entelos, SimulationsPlus, Shrodinger, Biorad, Inpharmatica, Accelryce, Compudrug, Leadscope, Lhasa, Lion Bioscience, Logichem, MDL Information Systems, Multicase, etc. PK / PD modelling Prediction of pharmacokinetic (PK) and pharmacodynamic (PD) properties from in vivo/in vitro experiment Firms: Bayer Technology Services, Pharsight, Globomax, Medeval, Cyprotex, Optimata, Simcyp, etc... Investments: (source: Nature Biotechnology, October ) Optimata (999): ML $ In silico technologies will enable drug manufacturers to accelerate the selection process, reduce the cost of preclinical and clinical studies and increase their overall chances of success. We estimate that they could collectively save at least $ ML and two to three years per drug PwC Report Pharma - Silicon Rally: The race to e-r&d by ~ % of pharmaceutical R&D expenditure will be on computer simulation and modelling, a figure set to rise to % by A brave new world of drug development, Curr. Drug Disc

3 TGI studies: the experimental setting Tumor growth inhibition (TGI) studies Experiments on animals Human tumor cells inoculated into athymic mice Tumor dimensions measured by caliper Control Treated Aim: assess drug effectiveness TGI studies: the experimental setting Efficacy is then reported as percentage of decrease of the average tumor weight of treated animals in comparison to the average of the control group. 9 mg/kg bid x days mg/kg mg/kg control qd tid x days x day TW in control group =.9 g TW in treated group =. g Efficacy reported as: (.9-.)/.9=% mg/kg bid x days Uncertainty mg/kg qd on x the days definition of the optimal mg/kg tid time x day at which the efficacy control should be evaluated. For example: the qd schedule is less active up to days in comparison to the bid schedule but then it appears the most efficacious one. The pharmacokineticpharmacodynamic model The PK-PD approach The ideal situation: having a PK/PD model, linking the dosing regimen to the tumor growth dynamics.in this way it is possible to predict the response of tumor growth dynamics at different regimens mg/kg TID x day 9 9 mg/kg QD x days 9 PHA- (exp): observed tumor weights Building the PK-PD model Strategy: Model of tumor growth in control animals (unperturbed growth) Model of tumor growth in treated animals, including the effect of the anticancer agent (perturbed growth) 9 9

4 The PK-PD model: unperturbed growth The PK-PD model: unperturbed growth Tumor growth has two phases: exponential and linear $ "W (t) W (t) # W * & W = GF(W (t)) = % & W (t) > W * ' " W(t) A, A, A, A, linear A, CRI HTC, 9 HTC, 9! exp W = W* = " / W (t ) L time " "!! (." + % & +,, / W (t ) )) # " * #$ '& - W () = L! The PK-PD model: unperturbed growth Results: individual fittings of control animals The PK-PD model: perturbed growth Modeling the drug tumor cell interaction The tumor growth in treated animals follows the same law of the control animals minus a loss due to the effect of the drug, according to the following scheme: PK k Z Z (t ) = GF " K! c( t )! Z (t ) cycling cells " / W (t ) W =!! (." + % & +,, / W (t ) )) # &' - " * #$ W () = L The PK-PD model: perturbed growth Modeling the damage and the delay of the cell death. Since the death of tumor cells is not immediate with respect to the drug treatment, a delay in the time of death has to be introduced. A transit compartment model is used for describing this feature. The PK-PD model: the general scheme The complete PK/PD model assumes that all the cells (cycling plus the damaged cells) contribute to the weight of the tumor and that all the cells entered in the motality chain go irreversibly to death. PK control animals: In control animals: damaged cells W ( t ) = Z () cycling cells treated animals: In treated animals: Mortality chain damaged cells Z Z K K! c (t ) Z K Z K Z K cell death W (t ) = Z () + Z () + Z () + Z () Zn K Z death Z n+ K K "! Z (t ) " K! c (t )! Z (t )! (." + %! & +,,! W ( t ) )) # & - " * #$ ' Z (t ) = K! c ( t )! Z ( t ) " K! Z ( t ) Z = This can be written as a system of differential equations: Z (t ) = K! Z ( t ) " K! Z ( t ) Z ( ) = L Z ( ) = Z ( ) = Z (t ) = K! Z ( t ) " K! Z ( t ) Z ( ) =

5 The PK-PD model: The PK-PD model: the drug-specific parameters the drug-specific parameters High K values give sharp and immediate response. Low K values give delayed and smoothed response. Dependence of the tumor growth kinetics on K: it is a multiplicative factor representing the drug potency. Control K =. K =. K =. K =. Dependence of the distribution of probability of death on K. K (days -).... time [days] The PK-PD model: secondary parameters Derived parameters: the threshold concentration (CT) According to the model, it is possible to define a threshold concentration: CT = /k such that, if CSS / CT! complete tumor regression. PHA- exp: simulated tumor weights Mean (days ) The PK-PD model: simultaneous fittings Since in the model, the perturbed growth collapses into the unperturbed one in the absence of treatment, average data can be used for simultaneous modeling of control and treated groups, thus: Making an efficient use of all the information PHA- (exp): observed and predicted tumor weights Giving robustness to the estimates 9 C(t) < CT. Observed mg/kg bid x days Predicted mg/kg bid x days control Observed mg/kg qd x days mg/day x days C(t) / CT mg/day x days Predicted mg/kg qd x days Observed mg/kg tid x day mg/day x days. Predicted mg/kg tid x day Predicted control Observed control Treatment start Treatment end The PK-PD model: simultaneous fittings Meaning of a good simultaneous fitting Predictive power of the model Results: discovery candidates 9% (days)

6 obs mg/kg qdxd pred mg/kg qdxd obs control pred control Example : Project AR Exploring the response of the tumor at different doses and schedules, from IV bolus to infusion. Exp. : Exp. Another : compound was tested mg/kg after i.v. (tidxd) bolus qdx with different mg/kg doses bidxd+stopxd+bidxd and schedules. PHA-9 (exp): observed and predicted tumor weights mg/kg qdxd Based on the derived parameter C T and the PK parameters of the compound, the dose to be given by infusion for five days able to give a significant tumor regression was calculated and the corresponding tumor growth was predicted. PHA-9 (exp): observed and predicted tumor weights IV infusion obs mg/kg (tid x d) qd x pred mg/kg (tid x d) qd x obs mg/kg bidxd+stopxd+bidxd pred mg/kg bidxd+stopxd+bidxd obs control pred control For the development of the project it was of interest testing the efficacy of a long-term infusion. 9 Time (days) Prediction? 9 Time (days) Based on the derived parameter C T and the PK parameters of the compound, the dose to be given by infusion for five days able to give a significant tumor regression was calculated and the corresponding tumor growth was predicted. Example : Project AU Comparing the antitumor activity of two compounds; due to their different toxicological profiles different doses were administered. IV infusion 9 Time (days) Prediction Conclusion: the compound maintains the efficacy also after infusion, excellent predictability of the response by the model 9 Time (days) Experiment A Dose Volume Route Scheduling Tot mice TGI drug X mg/kg ml/kg % Control animals iv < bid Group iv < bid Group iv < bid PK samples from Group Experiment B Dose Volume Route Scheduling Tot mice TGI drug Y mg/kg ml/kg % Control animals iv < bid Group iv < bid PK samples from Group Using the PK parameters previously obtained, the plasma concentrations of the drug X predicted after daily IV bolus at the doses of and mg/kg bid were derived and included in the PK/PD model. TheTGI model was then applied to the tumor growth curves observed in the efficacy experiment. Using the PK parameters previously obtained, the plasma concentrations of the drug Y predicted after daily IV bolus at the doses of mg/kg bid were derived and included in the PK/PD model. TheTGI model was then applied to the tumor growth curves observed in the efficacy experiment parameter value CV% K /h.99. K /µm/h. 9.! /h. 9.! g/h.. L g.. Ct µm. Time (hr) F Observed F Predicted F Observed F Predicted F Observed F Predicted parameter value CV% K /h.. K /µm/h..! /h..! g/h.. L g.. Ct µm. h F Observed F Predicted F Observed F Predicted Conclusion: Drug Y is much more potent in comparison to drug X (one order of magnitude,. vs.)

7 dose_(mg/kg)=, mouse= time_(hr) Observed Predicted Example : Project C Comparing two compounds, drug X given IV and drug Y given orally. Experiment A Dose Volume Route Scheduling Tot mice TGI drug X mg/kg ml/kg % Control animals iv < daily Group iv < daily PK samples from Group Experiment B Dose Volume Route Scheduling Tot mice TGI drug Y mg/kg ml/kg % Control animals oral < bid Group oral < bid 9 Group oral < bid 9 Group oral < bid PK samples from Group, Observed and predicted tumor growth curves after obtained in A tumor bearing female mice after IV bolus administration of drug X given at the dose of mg/kg/day for days. L g.! /day.! g/day. K /day.9 K /µm/day.9 Ct µm. days F Observed F Predicted F Observed F Predicted PK/PD analysis of drug Y given orally at the doses of,, twice a day for consecutive days. L (g).! (/day).! (g/day). K (/day).9 K (/µm/day).9 CT (µm). Benefits & further developments F Observed Conclusion: F Predicted F Observed F Predicted F Observed F Predicted F Observed F Predicted drug X time (hr) drug Y (the oral compound) has a potency similar or even higher compared to The PK-PD model: achievements Simple model with few parameters. Identifiable and physiologically relevant model parameters. Estimates of drug potency can be obtained independently from dose levels and schedules (ranking of compounds). Applicable to different cell lines. Prediction of tumor growth kinetics at different schedules. Savings in animals, time and resourses. The PK-PD model: references Simeoni M, Magni P, Cammia C, De Nicolao G, Croci V, Pesenti E, Germani M, Poggesi I, Rocchetti M. Predictive pharmacokinetic-pharmacodynamic modeling of tumor growth kinetics in xenograft models after administrations of anticancer agents. Cancer Research. : 9-. Rocchetti M, Poggesi I, Germani M, Fiorentini F, Pellizzoni C, Zugnoni P, Pesenti E, Simeoni M, De Nicolao G. A PK-PD model for predicting tumor growth inhibition in mice: a useful tool in oncology drug development. Basic & Clinical Pharmacology and Toxicolog., 9: -. Magni P, Simeoni M, Poggesi I, Rocchetti M, De Nicolao G. A mathematical model to study the effects of drugs administration on tumor growth dynamics. Mathematical Bioscience., :,.

8 The PK-PD model: what is going on. Oncology drugs and extrapolation to patients The typical question made at the end of the presentation: How does this translate to humans? Making predictions from animals to humans Facing the dilemma of oncology drug failures in the clinic. Kola I, Landis J. Can the pharmaceutical industry reduce attrition rates? Nature Rev Drug Discov., - (). About % of Investigational New Drug (IND) applications for new molecular entities submitted to the US Food and Drug Administration (FDA) progress beyond the investigational phase. The success rate is even lower in oncology (~%). The problematic issues that underlie the low rate of approval of new oncological drugs include the lack of preclinical systems (both in vitro assays and in vivo animal models) that can accurately predict the efficacy and toxicity of new agents,,, The PK-PD model: from animals to humans Relationship of doses of anticancer drugs in humans (cumulative doses given in -week cycles, midpoint of range) vs. C T xcl h Dose (mg/m, -week) f -fluorouracil d irinotecan a b c paclitaxel g.. C T xcl h (mg/m /h) e r=.99 The PK-PD model: from animals to humans Retrospective PK-PD analysis of drug D, whose development was interrupted due to absence of clinical benefit at the dose adopted in the phase II clinical trials of mg/m qwk. Observed and model-fitted tumor growth curves obtained in nude mice given i.v. either the vehicle (!) or drug D ( mg/kg " or. mg/kg #, given qdx from Day ).... Time (hr) C T =. ng/ml The PK-PD model: from animals to humans Dose (mg/m, -week) Actual dose from phase I trial g f -fluorouracil d irinotecan a b c paclitaxel.. C T xcl h (mg/m /h) e C T xcl h =. mg/m /h

9 Acknowledgments Maurizio Rocchetti (Nerviano Medical Sciences) Monica Simeoni and Italo Poggesi (currently GlaxoSmithKline) Paolo Magni (University of Pavia)