Download Characterizing Uncertainty for Modelling Response to

Tel-Aviv University
Faculty of Exact Sciences
Department of Statistics and Operations Research
CHARACTERIZING
UNCERTAINTY FOR MODELING
RESPONSE TO TREATMENT
David M. Steinberg
UCM 2012
July 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
Treated patients.
Patients observed
under Protocol A.
How would they
respond to
Protocol B?
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.
Construct a mathematical model and
a validation criterion
Preparation
Collect more data
Create an updated training data set adding
the recent individual data
Construct a personalized model using the
current data set
Personalization
Compare the current model predictions to
those of previous models
No
Model validation assessment
Yes
Predict treatment outcome and suggest
improved regimens
Prediction
No
Monitoring model accuracy
Yes
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.
Immunostimulation
Immunoinhibition
V
Dm
DC
DR
C
R
Skin
V  ki nv V
D m  ki (V  Vp)  km Dm
D   k D  k D
C
l m
m
CR
C
D R  kCR DC   D DR
C  aC DC  CC  kRCR
R  a D   R
R
R
Lymph node
P
R
P  rP  aP CP
hP
hP  P
Tumor
Math Models for Cancer
Kogan 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.

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:
Yi (tij )  g (tij ;1 , 2i , Pi ) i , j
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.
Patient Models
Statistical distributions of the parameter estimates.
20
Patient Models
Statistical distributions of the parameter estimates;
confidence ellipses for first two parameters.
21
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
Yi (tij )  g (tij ;1 , 2i , Pi ) i , j
 2 ,i ~ F
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.
2.
3.
4.
Sample patients by generating patientspecific parameter vectors.
For each patient, run the model to assess
the expected outcome for this patient
under different protocols of interest.
Characterize population behavior for each
protocol.
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.
