CHARACTERIZING UNCERTAINTY FOR MODELING RESPONSE TO TREATMENT Tel-Aviv University Faculty of Exact Sciences Department of Statistics and Operations Research David M. Steinberg July 2012 UCM 2012 Sheffield, UK

Based on Joint Work With  Mirit Kagarlitsky, TAU  Zvia Agur, IMBM  Yuri Kogan, IMBM Institute for Medical Bio Mathematics

Overview  Goals  Mathematical models for immunotherapy  Data  Patient and population models  NLME models for separating sources of variance  Protocol assessment  Summary and Conclusions

Goals  Use mathematical models and data to predict outcomes from new treatment protocols in a patient population.  Characterize the variation in response to treatment in a patient population.  Exploit existing trial data to describe the population.  Patient level – use the model to personalize treatment.

Goals Patients observed under Protocol A. How would they respond to Protocol B? Treated patients.

Math Models for Cancer Biomathematics is a science that studies biomedical systems by mathematically analyzing their most crucial relationships. Incorporating biological, pharmacological and medical data within mathematical models of complex physiological and pathological processes, the model can coherently interpret large amounts of diverse information in terms of its clinical consequences. Agur – 2010, Future Medicine

Math Models for Cancer We work with models for immunotherapy treatment of cancer. The models reflect the natural growth of the cancer, the response of the immune system to chemotherapeutic agents, and the consequent effect on the cancer. The models involve compartments and rate constants that govern growth, growth suppression and flows between compartments.

Create an updated training data set adding the recent individual data Model validation assessment No Yes No Yes Construct a personalized model using the current data set Construct a mathematical model and a validation criterion Predict treatment outcome and suggest improved regimens Monitoring model accuracy Preparation Personalization Prediction Compare the current model predictions to those of previous models Collect more data Kogan et al., Cancer Research, (9), pp

Math Models for Cancer Kogan et al. proposed a “success of validation” criterion for the model. The criterion compares data thresholds and asks when sufficient data have been collected to enable accurate prediction of future results. The criterion requires agreement in predictions following three successive observations. The SOV is used to determine a “learning” data set for each subject, from which a personalized treatment regime can be determined.

Math Models for Cancer Our model uses a system of ODE’s to describe vaccination therapy for prostate cancer in terms of interactions of tumor cells, immune cells and vaccine. Assumptions:  Vaccine injection stimulates maturation of dendritic cells.  These become mature antigen-presenting DCs.  Some DCs migrate into lymph nodes.  DCs are exhausted at a given rate and give rise to regulatory DCs.  Antigen-presenting DCs stimulate T-helper cells and activate cytotoxic T lymphocyte (CTL) cells. Some of these cells die or are inactivated by regulatory cells.  Cancer cells grow exponentially at a rate r but are destroyed, with a given efficiency, by CTLs.

Lymph node Skin Tumor DCDC DRDR V DmDm P ImmunostimulationImmunoinhibition CR Math Models for Cancer Kogan et al., Cancer Research, (9), pp

Prediction from the Model  The model tracks tumor size over time.  Expected tumor size can be computed by solving the system of differential equations.  The solution depends on the parameter values and the treatment protocol.  Alternative protocols can be compared for a patient or a population by running the model.

Data Various data sources are available.  Observation of patients.  Direct study of rate constants.  The observational data is not sufficient to estimate all model parameters.  Relevant literature may provide estimates or distributions for some parameters. These may involve “generic” research, not specifically on prostate cancer.

Data  We have data on 38 patients.  The data tracks a biomarker Y over time. The marker should reflect tumor size.  Calibrating the marker to tumor size is subject-specific.

Data Biomarker data for two typical patients, with fitted curves. Time is relative to the start of treatment.

Data Residuals for 16 patients. Plot shows observed/predicted.

Patient Models The general model for a particular patient: Here  1 includes “common” parameters,  2 includes four subject-specific parameters, and  is a random error term. The subject-specific parameters are the tumor growth rate, the CTL killing efficacy and two linear calibration terms. The treatment protocol is specified by P.

Patient Models Distribution of the calibration parameters from nonlinear least squares fits for 40 patients.

Patient Models Distribution of the calibration parameters from nonlinear least squares fits for 40 patients.

20 Patient Models Statistical distributions of the parameter estimates.

21 Patient Models Statistical distributions of the parameter estimates; confidence ellipses for first two parameters.

Patient Models  Substantial variation in parameter values across patients.  High correlations among the parameter values.  The variation could reflect: –Statistical (estimation) uncertainty. –True population heterogeneity.

Population Model  Treat the individual parameters  2 as random effects.  Their distribution describes the heterogeneity of the population.  This generates a nonlinear mixed effects (NLME) model.

NLME Models Common to assume normal distributions. But is this plausible for our application? If not, is there any hope to estimate a more general multivariate density?

NLME Models The covariance matrix for our model is too rich to estimate: 4 variances and 6 covariances. The empirical subject-specific parameter estimates are correlated.

NLME Models Our suggestion: replace the original parameters with the empirical principal components. Assume the new parameters are independent.

NLME Models Model estimation is challenging. Many convergence problems. Work still in progress.

Protocol Assessment Algorithm 1 1. Sample patients by generating patient- specific parameter vectors. 2. For each patient, run the model to assess the expected outcome for this patient under different protocols of interest. 3. Characterize population behavior for each protocol. 4. Use paired data to compare protocols or make a factorial analysis.

Protocol Assessment Paired outcomes are used to compare protocols – how do particular patients succeed on a new protocol versus an old protocol. Marginal outcomes are important to present an overall population picture of protocol success.

Protocol Assessment Algorithm 2 Like Algorithm 1, but in summarizing each patient-protocol pair: 1. Average over a sample of values of the common parameters, reflecting their distribution. 2. For each sampled value of the common parameters, re-analyze the data to estimate the conditional (on the common parameters) distribution of the patient parameters.

Summary & Conclusions  Bio-Mathematical models provide a stronger basis for prediction than empirical models.  They enable us to assess potential treatment protocols that have not been tested in vivo.  It may be difficult to estimate the needed population descriptions.  It is essential to distinguish estimation uncertainty from population heterogeneity.