CHARACTERIZING UNCERTAINTY FOR MODELING RESPONSE TO TREATMENT Tel-Aviv University Faculty of Exact Sciences Department of Statistics and Operations Research.

CHARACTERIZING UNCERTAINTY FOR MODELING

RESPONSE TO TREATMENT

Tel-Aviv UniversityFaculty of Exact Sciences

Department of Statistics and Operations Research

David M. Steinberg

July 2012UCM 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 assessmentNo

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 modelsC

oll

ect

mo

re d

ata

Kogan et al., Cancer Research, 2012 72(9), pp.2218-2227

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

DC DR

V

Dm

P

Phh

CParPP

RDaR

CRkCDaC

DDkD

DkDkD

DkVpVkD

VnkV

P

PP

RRR

RCCC

RDCCRR

CCRmmlC

mmim

vi

)(

Immunostimulation Immunoinhibition

C R

Math Models for CancerKogan et al., Cancer Research, 2012 72(9), pp.2218-2227

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.

DataBiomarker 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.

1 2 ,( ) ( ; , , )i ij ij i i i jY t g t 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 ModelsStatistical distributions of the parameter estimates.

21

Patient ModelsStatistical 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?

1 2 ,( ) ( ; , , )i ij ij i i i jY t g t P

Fi ~,2

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.