pk/pd modelling using stochastic differential equations 04 27 henrik_madsen_talk.pdf · pk/pd...
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PK/PD Modelling usingStochastic Differential EquationsFMS and DSBS Workshop
Henrik Madsen
DTU Informatics
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Outline
1 Why Stochastic Differential Equations?
2 The grey box modelling concept
3 The Stochastic State Space Model
4 Wrong Error Model gives Wrong Dose
5 Identification, Estimation and Model Validation
6 Identification of Model Structure
7 Software
8 Nonlinear Mixed Effects Models with SDEs
9 Example: Diabetes
10 Systematic Modelling Framework
11 Example: PK/PD Modelling of the HPG Axis
12 Summary
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Problem Scenario
Ordinary differential equation
dA = −KA dt
Y = A + ǫ
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ODE
0 5 10 15 20 25−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
Time
Con
cent
ratio
n −
logs
cale
DataODE
Correlated residuals!!
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Problem Scenario
Stochastic differentialequation
dA = −KA dt + dw
Y = A + e
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ODE vs SDE
0 5 10 15 20 25−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
Time
Con
cent
ratio
n −
logs
cale
DataODE
Correlated residuals
System noise
Measurement noise
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ODE vs SDE
0 5 10 15 20 25−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
Time
Con
cent
ratio
n −
logs
cale
DataODESDE
Correlated residuals
System noise
Measurement noise
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The grey box modelling concept
Combines prior physical knowledge with information in data.
The model is not completely described by physical equations, butequations and the parameters are physically interpretable.
Data
KnowledgePrior
White Grey Black
Physicalknowledge
Deterministicequations
submodelsDetailed
Input−outputrepresentation
Databased
The box in the middle (denoted Grey) may be expressed as therelationship between the amount of prior knowledge and dataavailable.
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Why use grey box modelling?
Prior physical knowledge can be used.
Non-linear and non-stationary models are easily formulated.
Missing data are easily accommodated.
It is possible to estimate state variables that are not measured.
Available physical knowledge and statistical modelling tools iscombined to estimate the parameters of a rather complexdynamical system.
The parameters contain information from the data that can bedirectly interpreted by the scientist.
Fewer parameters→ more power in the statistical tests.
The physical expert and the statistician can collaborate in themodel formulation.
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Stochastic Differential Equations (SDE’s)
ek
The line demonstrates a model prediction, whereas the dotsdenote typical observations.
Notice: Autocorrelated residuals are most often seen- calls for using Stochastic Differential Equations (SDE’s) as an
alternative to Ordinary Differential Equations (ODE’s).
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The continuous-discrete time non-linear stochastic statespace model– The system equation (a set of Itô stochastic differentialeqs.)
The system equation consists of a drift and a diffusion term.
dX t = f (X t ,U t , θθθ) dt + G(X t ,U t , θθθ) dW t , X t0 = X 0
NotationX t ∈ R
n State vectorU t ∈ R
r Known input vectorf Drift termG Diffusion termW t Wiener process of dimension, d , with incre-
mental covariance Qt
θθθ ∈ ΘΘΘ ⊆ Rp Unknown parameter vector
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The observation equation
The observations are in discrete time, functions of state, input, andparameters, and are subject to noise:
Y ti = h(X ti ,U ti , θθθ) + eti
NotationY ti ∈ R
m Observation vectorh Observation functioneti ∈ R
m Gaussian white noise with covariance ΣΣΣti
Observations are available at the time points ti : t1 < . . . < ti < . . . < tNX 0,W t ,eti assumed independent for all (t , ti), t 6= ti
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Advantages of using SDE’s
Provides a decomposition of the total error into process error andmeasurement error.
Facilitates use of statistical tools for model validation.
Provides a systematic framework for pinpointing modeldeficiencies – will be demonstrated later on.
Covariances of system error and measurement error areestimated.
SDE based estimation gives more accurate and reliableparameter estimates than ODE based estimation.
SDEs give more correct (more accurate and realistic) predictionsand simulations.
SDEs give more correct dose (see the following example)
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Example: Glucose/Insulin system
Model predicting theinsulin concentrationhaving glucose asinput
Wrong error modelgives wrong dose!
SDEs give morereliable estimates ofcovariate effects
J.B. Møller et.al.: Predictive performance for population models using stochastic differential equations applied on data from an oral glucose
tolerance test, Journal of PK/PD, 2009
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Methods for Identification, Estimation and Model Validation
Model Identification: See the next slide. Parameter Estimation:
− (Approx.) Maximum Likelihood Methods− Estimation Functions− Prediction Error Methods
Model Validation:− Test whether the estimated model describes the data.− Autocorrelation functions – or Lag Dependent Functions.− Other classical methods ...
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Identification of Model Structure
The diffusion term gives information for pinpointing modeldeficiencies.
Assume that we based on ’large’ values of relevant diffusionterm(s) suspect r ∈ θθθ to be a function of the states, input or time.
Then consider the extended state space model :
dX t = f (X t ,U t , θθθ) dt + G(X t ,U t , θθθ) dW t , X t0 = X 0
drt = dW ∗t
Y ti = h(X ti ,U ti , θθθ) + eti
(1)
which corresponds to a random walk description of rt .
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Identification of Model Structure
Do we observe a significiant reduction of the relevant diffusionterm(s)?
In that case calculate the smoothed state estimate rt |N (use forinstance the software tool CTSM - see the next slide).
Plot rt |N versus the states, inputs and time.
Identify a possible functional relationship.
Build that functional relationship into the stochastic state spacemodel.
Estimate the model parameters and evaluate the improvement –using e.g. likelihood ratio tests.
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Continuous Time Stochastic Modelling (CTSM)
The parameter estimation is performed by using the softwareCTSM.
The software has been developed at IMM.
Download from: www.imm.dtu.dk/ctsm
The program analyses the model equations to determine thesymbolic names of the parameters to fix and which to estimate.
The program returns the uncertainty of the parameter estimatesas well.
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Continuous Time Stochastic Modelling (CTSM)– Linear case
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Continuous Time Stochastic Modelling (CTSM)– Non-linear case
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Population Modelling – Introduction
Data originating from several population members/subjects
Identical experiments
More data - better estimates of parameters and uncertainties.
Software: Population Stochastic Modelling (PSM)
Software download: www.imm.dtu.dk/psm
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Population Modelling – Introduction
Data originating from several population members/subjects
Identical experiments
More data - better estimates of parameters and uncertainties.
Software: Population Stochastic Modelling (PSM)
Software download: www.imm.dtu.dk/psm
Nonlinear mixed effects model with SDEs
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Population Modelling – Stages
A population model consists of 2 stages
1st stage models the process variation within a single populationmember/subject
2nd stage models the variation in parameters between populationmembers/subjects like:
φφφi = g(θθθ,Z i) · exp(ηηηi)
ηηηi ∈ N(0,ΩΩΩ)
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Population – Parameter estimation
Parameter estimation using likelihood theory
Single member/subject - likelihood based on product ofconditional densities for each time series of length ni (called p1
below).
Population likelihood is a combination of the random effects ηand the single member likelihoods
L(θθθ|YNni ) ∝
N∏
i=1
∫p1(Yini |φφφi)p2(ηηηi |ΩΩΩ)dηηηi
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Diabetes in figures
150.000 diagnosed with diabetes in Denmark− Treatment costs 2.5B kr./year− 150.000 unaware of their diabetes condition
171 millions diagnosed world wide− Expected to reach 366 million by 2030
Increased risk for heart diseases, blindness, nerve damage andkidney damage
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Diabetes physiology
Insulin is secreted from the Pancreas− Extracted by the liver− Half-life approx. 5 min.
C-peptide is co-secreted in equimolar amounts− Not extracted by the liver− Half-life approx. 30 min.
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Data – 24H study
12 type 2 diabetic patients Three standardized meals
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ISR estimate by deconvolution
C-peptide (the observation) is modelled with a 2-compartmentmodel
ISR modelled as a random walk (the third state in x)
dx =
−(k1 + ke) k2 1
k1 −k2 00 0 0
x dt + diag
00σISR
dωωω
y = C1 + ǫ
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ISR estimate by deconvolution
Smoothed estimate of ISR for individual 1 and 2.
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Systematic Modelling Framework
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Final model
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Individual profile
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Population profile
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Summary
By using stochastic differential equations for PK/PD modelling wehave
better predictions and simulations
systematic methods for model development
methods for finding the best model (LR-tests, etc.)
statistical methods for model validation and structure modification
more accurate estimates of the effects of covariates
more accurate estimates of the dose
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