3. Modelling Random effects

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Modelling Random Effects

www.unav.edu/psp

Objectives • To learn the concept, nomenclature, implementation, interpretation, and use of the parameters representing the stochastic elements of a population PK/PD model

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Random effects • Variability • Model for individual parameters • Inter-subject variability • Inter-occasion variability • Other levels of variability • i.e., Inter-study variability

• Residual error • Model for observations • Former intra-individual variability

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Random effects • Variability (model for parameters) • $PK Not a parameter for every single individual Estimate of the variance around the population (θ) estimate

• Residual error (model for DV) • $ERROR Not a parameter for every single observation The estimate of the variance around the predictions ©UNAV-PSP, 2018, all rights reserved.

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Variability (Concept & Nomenclature) • IIV, IPV, ISV, BSV,…. • Quantifies the “discrepancy” between the individual PK parameters and the population (typical) estimates [THETA (θ)] • “Discrepancies” are represented by ETA (η) • For each parameter in the model the set of all “discrepancies” forms a random variable Mean value = 0 (symmetric distribution) 2  Variance= ω 

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Variability (modelling) • Model assumptions • ηs are normally distributed around 0 • Individual parameters cannot be negative • Unimodal distribution of individual parameters

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Variability (modelling) • Exponential model for variability • Taking CL as an example, and for a “n” number of subjects in the population CL1 = THETA(1) x eηCL,1 CL2 = THETA(1) x eηCL,2 ............................ CLn = THETA(1) x eηCL,n

Individual parameters (different across subjects) Typical population parameter (same for all subjects) ©UNAV-PSP, 2018, all rights reserved.

Individual discrepancies with respect the typical value

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Variability (modelling) • Exponential model for variability • Taking CL as an example, and for a “n” number of subjects in the population CL1 = THETA(1) x eηCL,1 CL2 = THETA(1) x eηCL,2 ............................ CLn = THETA(1) x eηCL,n

ω2

0

ηCLn

•ηs are normally distributed around 0 •Individual parameters cannot be negative •Unimodal distribution of individual parameters ©UNAV-PSP, 2018, all rights reserved.

0 THETA(1)

CLn

CL

Variability (implementation) • Usually there is more that one parameter that show inter-individual variability => ω2KA, ω2V, ω2CL.... • ω2KA, ω2V, ω2CL ... are the elements of the variance-covariance matrix Ω

Ω =>

$PK V=THETA(1)*EXP(ETA(1)) ; Model for V1 CL=THETA(2)*EXP(ETA(2)) ; Model for CL

ω2V 0

ω2CL

Ω Diagonal

………………….. $OMEGA 0.1 ; Variance of V $OMEGA 0.1 ; Variance of CL No need to constrain; Units of variance

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Variability (modelling alternatives) • There are alternatives beyond the log-normal distribution of the parameters (exponential model) • • • • •

Semiparametric distributions (Petersson KJF et al., Pharm Res 2007) Mixture models NONPARAMETRIC USE of PRIORS IOV

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Variability (modelling) • Covariance might be present between the diagonal elements of the Ω matrix

$PK V=THETA(1)*EXP(ETA(1)) CL=THETA(2)*EXP(ETA(2))

$PK V=THETA(1)*EXP(ETA(1)) CL=THETA(2)*EXP(ETA(2))

…………………..

…………………..

$OMEGA 0.1 ; Variance of V $OMEGA 0.1 ; Variance of CL

$OMEGA BLOCK(2) 0.1 ; Variance of V 0.01 0.1 ; Variance of CL

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Impact of misspecifications in the Ω matrix

•

Ignoring covariance

•

Detecting covariance

• -2LL • Scatter-matrix plot (η vs η) • Visual predictive check • Mechanistic understanding

Vélez de Mendizábal et al., Pharm Res 2010 ©UNAV-PSP, 2018, all rights reserved.

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$PK TS0=THETA(1)*EXP(ETA(1)) Additive model for KPROL=THETA(2) + ETA(2) inter-subject variability

Relative change from baseline (%)

Additive model

Time

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Logistic model Depot F1

KA ALAG1

Central V

CL IV dose

……… $SUBROUTINES ADVAN2 TRANS2 $PK KA=THETA(1)*EXP(ETA(1)) V=THETA(2)*EXP(ETA(2)) CL=THETA(3)*EXP(ETA(3)) F1 S2=V/1000 ……………

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F1=THETA(4)

$THETA (0, ,1)

Or TVF1=THETA(4) PHI=LOG(TVF1/(1-TVF1)) F1=EXP(PHI+ETA(4))/(1+EXP(PHI+ETA(4))) sd(F1)=THETA(4)*(1-THETA(4))*ω4

Or FF1=THETA(4)*EXP(ETA(4)) F1=FF1/(1+FF1)

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How to interpret estimates of ω2 • To classify the magnitude of the variability the ω2

estimates are often transformed to CV(%) which is much easier to interpret and allows comparisons

• Calculation of CV% from depend on the variability model and involves approximations

•

Its important therefore to specify the expression used to compute the CV, to allow others to reproduce our results

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How to interpret estimates of ω2 •

Exponential model (log-normal distribution)

$PK V=THETA(1)*EXP(ETA(1)) CL=THETA(2)*EXP(ETA(2)) $OMEGA 0.1 ; Variance of V (ω2) 0.1 ; Variance of CL (ω2)

Two expressions for CV(%) 𝑖𝑖 𝐶𝐶𝐶𝐶 % =

𝑖𝑖𝑖𝑖 𝐶𝐶𝐶𝐶 % =

𝜔𝜔 2 × 100 2

𝑒𝑒 𝜔𝜔 − 1 × 100

Both give similar results with low estimates of ω2 (

Covariance ≠ 0

σ2FCO 0

Σ Diagonal

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Σ =>

σ2

M

σ2FCO σ2FCO, M σ2M Σ Full

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Residual error (implementation) $ERROR Y = F + EPS(1) ………………….. $SIGMA 0.1 ; Variance of ε; σ2 No need to constrain; Units of variance

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• Y, observation • F, model predictions (θ, η) • EPS, ε Required NMTRAN terms

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Residual error (modeling) • As in the case of IIV several type of models are possible Additive model

Proportional model

Cp Cp

o

o

o

o

o o

o o

Combined model

o

o

o o

o

o o

o o

o

εs are independent from the magnitude of the prediction

εs have a magnitude proportional to prediction

Y = F + EPS(1)

Y=F+F*EPS(1)

𝝈𝝈𝟐𝟐 (ng/mL)

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𝝈𝝈𝟐𝟐 X 100 [CV(%)]

εs are proportional to predictions at high values & independent at low values

Y=F+F*EPS(1)+EPS(2) 𝝈𝝈𝟐𝟐 X 100 (CV(%)

𝝈𝝈𝟐𝟐 (ng/mL)

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Residual variability • Most used expression for the $ERROR block $ERROR IPRED=F

W= THETA(X)

; Additive model; σ=THETA(X)

Y=IPRED+W*EPS(1)

W= F* THETA(Y)

; proportional model; σ=THETA(Y)

IWRES=(DV-IPRED)/W $SIGMA 1 FIX

Since SIGMA is fixed to 1 Variance = 1 SD = sqrt(variance)=1 ©UNAV-PSP, 2018, all rights reserved.

W=SQRT(THETA(Y)**2*F**2+ THETA(X)**2) Proportional

Additive

The estimates of THETA(X) and THETA(Y) are in units of SD and CV/100%, respectively THETA(X) & THETA(Y) constrained to be non negative

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Overview statistical model

±ω p

0

±σ

ηp,i

εij

0

Parameter logC θp

Observation tij ©UNAV-PSP, 2018, all rights reserved.

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MU referencing • EM methods work more efficiently if the arithmetic association between thetas with the etas and individual parameters is supply • The association of one or more thetas with one eta(x) must be given through a variable called MU_x TVCL= THETA(1) CL=TVCL*EXP(ETA(1))

TVCL= THETA(2)*(WT/70)^THETA(5) CL=TVCL*EXP(ETA(1))

linear relationship between MU and ETA MU_1=LOG(THETA(1)) CL=EXP(MU_1 + ETA(1))

MU_1=LOG(THETA(2))+THETA(5)*LOG(WT/70) CL=EXP(MU_1 + ETA(1))

Even better, linear relationship between THETAS and MUs MU_1=THETA(1) CL=EXP(MU_1 + ETA(1)) ©UNAV-PSP, 2018, all rights reserved.

MU_1=THETA(2)+THETA(5)*LOG(WT/70) CL=EXP(MU_1 + ETA(1)) 27

MU referencing • Some basic rules: • One ETA can only be associated with one MU (same number) • CL= EXP(MU_1 + ETA(1) + ETA(2)) • CL=EXP(MU_1 + MU_2 + ETA(2)) • CL = EXP(MU_1 + MU_1*ETA(1))

• Thetas that appear in a MU cannot be used in subsequent lines MU_1= THETA(5) KTR=EXP(MU_1 + ETA(1)) MTT=4/THETA(5)

MU_1= THETA(5) KTR=EXP(MU_1 + ETA(1)) MTT=4/KTR

• Thetas with no random effects cannot be MU modeled  alternative, fix its omega to a low value 0.0225 (15%) • Time varying covariates cannot be MU modeled ©UNAV-PSP, 2018, all rights reserved.

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Residual error (additional considerations) $PK V=THETA(1)*EXP(ETA(1)) CL=THETA(2)*EXP(ETA(2))

$ERROR IPRED=F W=SQRT(THETA(3)**2+THETA(4)**2*IPRED**2) *EXP(ETA(3)) Y=IPRED+W*EPS(1)

………. $OMEGA 0.1 $OMEGA 0.1 $OMEGA 0.1 ©UNAV-PSP, 2018, all rights reserved.

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Data transformation • Sometime data transformation is required • Residual distribution is not normal or centered around 0 • Logarithmic transformation of the data (transform both sides) • No additional parameter

• Box-cox transformation (similar to the case of the inter-patient variability) • Addition of an extra parameter

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Data logarithmically transformed • • • •

Dataset modifications ID TIME AMT DV=DRODP DV=LNDV An additional column is incorporated to the dataset $ERROR • IPRED=LOG(F) ; what if predictions are equal to 0 (i.e., dosing records) • Y=IPRED+W*EPS(1)

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Data logarithmically transformed • • • •

Dataset modifications ID TIME AMT DV=DROP DV=LNDV An additional column is incorporated to the dataset $ERROR (ONLY OBSERVATION) • IPRED=LOG(F) ; what if predictions are equal to 0 (i.e., dosing records) • Y=IPRED+W*EPS(1)

• In general when data are log-transformed an additive error model is enough • The expression for a combined error model with log-transformed data is: • SQRT( THETA(X)**2/IPRED**2+THETA(Y)**2)

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Inter- Occasion Variability (IOV) • Intra-patient variability • Concept • Random change in parameters between different study periods/occasions • Can be used to explore presence of time dependencies • Non random changes • Disease progression effects

• Data requirements • More than one sample/measurement per occasion • Otherwise un-distinguishable from residual error

Karlsson and Sheiner JPB 1993 ©UNAV-PSP, 2018, all rights reserved.

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Inter- Occasion Variability (IOV) IOV

CLn θCL CLn

IIV

Occasion ©UNAV-PSP, 2018, all rights reserved.

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Inter- Occasion Variability (IOV) Time dependent PK // Disease Progression

CLn θCL CLn Occasion ©UNAV-PSP, 2018, all rights reserved.

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Inter- Occasion Variability (Coding) $PK OCC1=0 OCC2=0 IF(OCC.EQ.1) OCC1=1 IF(OCC.EQ.2) OCC2=1 IOV1=OCC1*ETA(3)+OCC2*ETA(4) TVCL=THETA(1) CL=TVCL*EXP(ETA(1)+IOV1) TVV=THETA(2) IOV2=OCC1*ETA(5)+OCC2*ETA(6) V=TVV*EXP(ETA(2)+IOV2)

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$OMEGA BLOCK(1) 0.1 ; Var_eta3 $OMEGA BLOCK(1) SAME ; Var_eta4 $OMEGA BLOCK(1) 0.1 ; Var_eta5 $OMEGA BLOCK(1) SAME ; Var_eta6 • Distributions of ETA3 and ETA4 have the same variance • Otherwise non random variations

• Individual η3 and η4 values neither are the same nor are correlated •

Otherwise non random variations

• Same for ETA5 and ETA6

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Inter- Occasion Variability (Coding) $ABBR REPLACE ETA(OCC_CL)=ETA(3,4) $ABBR REPLACE ETA(OCC_V)=ETA(5,6) ; when OCC=1, eta(3) to be used for CL and eta(5) for V ; when OCC=2, eta(4) and eta(6) $PK CL=TVCL*EXP(ETA(OCC_CL)) V =TVV *EXP(ETA(OCC_V))

$OMEGA BLOCK(1) 0.1 ; Var_eta3 $OMEGA BLOCK(1) SAME ; Var_eta4 $OMEGA BLOCK(1) 0.1 ; Var_eta5 $OMEGA BLOCK(1) SAME ; Var_eta6

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Inter- Occasion Variability (Coding) $ABBR REPLACE ETA(OCC_CL)=ETA(3,4) $ABBR REPLACE ETA(OCC_V)=ETA(5,6) ; when OCC=1, eta(3) to be used for CL and eta(5) for V; ; when OCC=2, eta(4) and eta(6) $PK CL=TVCL*EXP(ETA(1)+ETA(OCC_CL)) V =TVV *EXP(ETA(2)+ETA(OCC_V)) $OMEGA BLOCK(2) 0.1 ; Var_eta1 0.01 0.1 ; Var eta2 $OMEGA BLOCK(2) 0.1 ; Var_eta3 0.01 0.1 ; Var_eta4 $OMEGA BLOCK(2) SAME ; Var_eta5 & eta6 ©UNAV-PSP, 2018, all rights reserved.

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Inter- Occasion Variability (Dataset) ID 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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TIME 0 0.25 0.5 1 1.5 2 4 6 8 12 24 336 336.25 336.5 337 337.5 338 340 342 344 348 360

TAD 0 0.25 0.5 1 1.5 2 4 6 8 12 24 0 0.25 0.5 1 1.5 2 4 6 8 12 24

AMT 10 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0

DV . C1a C2a C3a C4a C5a C6a C7a C8a C9a C10a . C1b C2b C3b C4b C5b C6b C7b C8b C9b C10b

OCC 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2

EVID 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0

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Inter- Occasion Variability • Bias • Increase in residual error • • • • • •

What if more than one occasion and some with just one sample How to bin occasions Sometime IOV gretar than IIV, Others no IIV and just IOV High IOV is not good Tool to suggest complex behavior

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