Lecture 9 NonParametric PKPD

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Lecture 9Non-Parametric PK/PD
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"NON-PARAMETRIC" APPROACH
SEMI-PARAMETRIC" APPROACH
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Fundamental Assumption
Connecting the observed concentrations captures
the shape of the PK vs. time curve
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2. Curve Representation Smoothing
2.1. Goal Curve that looks like "perfect"
data Dilemma If follow data too closely, may
believe noise (large variance) If smooth too
much (e.g., to straight line), may ignore signal
(large local bias).
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General Concept Convolution

• In linear systems theory, you can take any
  input and decompose it into a series of boluses

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General Concept Convolution

• Each of these boluses is then treated to the
  usual disposition of drug

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General Concept Convolution

• You then add up all of the concentrations that
  resulted from each bolus to get the overall
  profile

This is really just a statement of the
superposition principle
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Consider an input with two boluses

• A bolus is given at 0 and 5 minutes
• The resulting concentration is the sum of the
  response to each bolus

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We can generalize the PK function as a unit
disposition function

• The fate of a unit of drug injected into the
  system as a bolus
• e.g.,
• More generally, we can call the fate of a unit
  function the UDF(t)
• Engineers will recognizes this as the IRF or
  impulse response function
• When you estimate PK, you are estimating the
  parameters of the UDF

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We can write for anyseries of N boluses
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The input function I(t)

• Let I(t) be the amount of a bolus at time t, with
  a time resolution of 1 second
• We can then generalize the prior equation as

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The Convolution Integral

• To make this smooth, rather than have a bolus
  every second, we can treat the input as a
  continuous function
• Or, more generally
• This is what you have been doing in all the
  linear PK analysis

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I(t) can be any function
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D(t) can be any non-saturable function
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Usual Simulation Problem isConvolution

• Calculate Cp(t) given I(t), D(t)

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However, the typical PK estimation problem is
actually DEconvolution

• Given Cp(t) and I(t), calculate D(t) (i.e.,
  calculate the parameters of the UDF

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Controlled release problems are also DEconvolution

• Given Cp(t) and D(t), calculate I(t) (i.e.,
  calculate the release rate to give the desired PK
  profile

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"NONPARAMETRIC" PK/PD
Apply nonparametric methods to exploratory
analysis of PK/PD data.
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Typical experiment showing hysteresis
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Add an effect site
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Add an effect site
Input
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Add an effect site
Disposition
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For the Effect Site
I(t
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The input is the plasma concentration over time
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For the Effect Site
The disposition function is the first order delay
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So, non-parametric PK/PD meansfitting ke0 so
that
So, how do you do this?
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Various Approaches
Divide it into infusions
Divide it into polygons connect the dots
Approximate data using splines, local linear
regression to exponential functions, smoothing
functions, etc
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Once you know Ce, then fitCe to Effect

• Standard models Sigmoidal, Biphasic, power,
  linear, etc.
• Or, use no model at all just graph out the
  relationship!

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How to implement a simpleconnect-the-dots

• Use linear when function is increasing
• Use log-linear when function is decreasing

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Connect the dotsIncreasing concentrations
Compute Ce(t) as convolution of above with
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Connect the dotsDecreasing concentrations
Compute Ce(t) as convolution of above with
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Connect the dotsEasy to implement

• Excel example
• NONMEM Example
• Direct collapse the hysteresis loop
• ke0obj.exe
• Enter time CP data pairs
• Enter time effect data pairs
• Estimates keo that minimizes the area within the
  hysteresis loop

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Excel Example
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NONMEM Code
INPUT ID TIME EFFDV CP PRED E0THETA(1)
Baseline
EMAXTHETA(2) Max Effect
C50THETA(3)EXP(ETA(1)) C50, only parameter
with interindividual variability GAMTHETA(4)
Gamma - don't add an ETA
to this - very hard to fit KE0THETA(5)EXP(ETA
(2)) Ke0 IF (TIME.EQ.0) THEN CE, PTIME, PCP,
PCE 0 DT1TIME-PTIME IF(CP.GE.PCP) THEN
SLOPE (CP-PCP)/DT1 DELTADT1SLOPE(KE0PCP-S
LOPE)(1-EXP(-KE0DT1))/KE0 ELSE SLOPE
(LOG(CP)-LOG(PCP))/DT1 DELTAPCPKE0/(KE0SLOPE
)(EXP(DT1SLOPE)-EXP(-KE0DT1)) ENDIF CE
PCEEXP(-KE0DT)DELTA Full sigmoidal Emax PD
model with simple additive error Y
E0(EMAX-E0)CEGAM/(CEGAMC50GAM)
EPS(1) Variables for NONMEM to remember at the
next iteration PTIME TIME Prior Time PCP CP
Prior plasma concentration PCE CE
Prior effect site concentration
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ExamplePropofol and EEG
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Time
Ce first attempt
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ExamplePropofol and EEG
Time
Ce revised model
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Main Caveat

• You must have a reasonable idea of the shape of
  the concentration vs. time curve

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You are given these data to fit
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But it was actually a study with target
controlled IV drug delivery
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Key Points aboutNonparametric PK/PD

• If you know Cp(t) and effect(t), you can estimate
  keo and Ce(t)
• Assumes the equilibration between the plasma and
  the effect site is first order
• Assumes that the relationship between Ce and
  effect is independent of time, and dose
• Need reasonably high resolution measures of
  concentration and effect
• Enough to connect the dots
• Makes no assumptions about the shape of either
  the PK model or the PD model

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PowerPoint Files

• Go to anesthesia.stanford.edu, then navigate to
  online lectures, then to Advanced PK/PD for
• These lecture notes
• Excel spreadsheet example
• NONMEM PRED routine
• ke0obj.c and ke0obj.exe

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PowerPoint Files

• Go to anesthesia.stanford.edu, then navigate to
  online lectures, then to Advanced PK/PD for
• These lecture notes
• Excel spreadsheet example
• NONMEM PRED routine
• ke0obj.c and ke0obj.exe