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Stochastic Differential Equations in NONMEM®: Implementation, Application, and Comparison with Ordinary Differential Equations

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Purpose

The objective of the present analysis was to explore the use of stochastic differential equations (SDEs) in population pharmacokinetic/pharmacodynamic (PK/PD) modeling.

Methods

The intra-individual variability in nonlinear mixed-effects models based on SDEs is decomposed into two types of noise: a measurement and a system noise term. The measurement noise represents uncorrelated error due to, for example, assay error while the system noise accounts for structural misspecifications, approximations of the dynamical model, and true random physiological fluctuations. Since the system noise accounts for model misspecifications, the SDEs provide a diagnostic tool for model appropriateness. The focus of the article is on the implementation of the Extended Kalman Filter (EKF) in NONMEM® for parameter estimation in SDE models.

Results

Various applications of SDEs in population PK/PD modeling are illustrated through a systematic model development example using clinical PK data of the gonadotropin releasing hormone (GnRH) antagonist degarelix. The dynamic noise estimates were used to track variations in model parameters and systematically build an absorption model for subcutaneously administered degarelix.

Conclusions

The EKF-based algorithm was successfully implemented in NONMEM for parameter estimation in population PK/PD models described by systems of SDEs. The example indicated that it was possible to pinpoint structural model deficiencies, and that valuable information may be obtained by tracking unexplained variations in parameters.

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Abbreviations

EKF:

Extended Kalman Filter

FOCE:

first-order conditional estimation

GnRH:

gonadotropin releasing hormone

LLOQ:

lower limit of quantification

ODE:

ordinary differential equation

PK/PD:

pharmacokinetics/pharmacodynamics

RSE:

relative standard error

SC:

subcutaneous

SDE:

stochastic differential equation. Variables:

e :

measurement error

ε:

residual error

η :

inter-individual variability

K :

Kalman gain

Ω:

inter-individual covariance

P :

state covariance

φ :

individual parameter

R :

output prediction covariance

Σ:

measurement error covariance

σ w :

diffusion term

t :

time

θ :

population mean parameter

w :

Wiener process

x :

state variable

y :

measurement

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Acknowledgments

The authors wish to acknowledge the help of Professor Stuart Beal (UCSF), who instructed us on how to implement the EKF algorithm in NONMEM. This work was financially supported by Ferring Pharmaceuticals A/S and Center for Information Technology, Denmark.

Author information

Authors and Affiliations

  1. Experimental Medicine, Ferring Pharmaceuticals A/S, DK-2300, Copenhagen S, Denmark

    Christoffer W. Tornøe & Henrik Agersø

  2. Informatics and Mathematical Modelling, Technical University of Denmark, DK-2800, Lyngby, Denmark

    Christoffer W. Tornøe, Rune V. Overgaard, Henrik A. Nielsen & Henrik Madsen

  3. Department of Pharmaceutical Biosciences, Division of Pharmacokinetics and Drug Therapy, Uppsala University, Box 591, 751 24, Uppsala, Sweden

    Christoffer W. Tornøe & E. Niclas Jonsson

Authors
  1. Christoffer W. Tornøe
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  2. Rune V. Overgaard
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  3. Henrik Agersø
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  4. Henrik A. Nielsen
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  5. Henrik Madsen
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  6. E. Niclas Jonsson
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Corresponding author

Correspondence to Christoffer W. Tornøe.

Appendix 1

Appendix 1

NONMEM SDE Control Stream (See Table I)

1

$PROBLEM SDE 1-COMP MODEL WITH 2 ABS COMPONENT

65

A7 = A7

$INPUT ID HOUR TIME AMT CMT DV EVID MDV

A8 = A8

$DATA SDE.dta IGNORE = @

A9 = A9

$SUBROUTINE ADVANCE TOL 6 DP

ENDIF

5

$MODEL COMP = (ABS1) COMP = (CENT)

; EKF state update equations

COMP = (P11) COMP = (P12) COMP = (P13)

70

IF(EVID.EQ.3) THEN

COMP = (P22) COMP = (P23) COMP = (P33)

;Output cov. R_j∣j−1 = C P_j∣j−1C^T + Sig Sig^T

RVAR = A9/(A3*A3) + SIG**2

$THETA (0,5.50)(0,100)(0,30.0)(0,900)(0,0.10,1)

;Kalman Gain K_j = P_j∣j−1 C^T R_j∣j−1^−1

10

 (0,0.70,1)(0.0.20)(0,1)(0,250)(0,1)

K1 = A6/(A3*RVAR)

$OMEGA 0.25 0.25 0.25 0.25

75

K2 = A8/(A3*RVAR)

$Sigma 1 FIX

K3 = A9/(A3*RVAR)

;State update eq. x_j∣j = x_j∣−1+K_j (y_j−y_j∣j−1)

$PK

A1UP = A1+K1*(OBS-LOG(A3/V1))

15

 CL = THETA(1)

A2UP = A2+K2*(OBS-LOG(A3/V1))

V1 = THETA(2)

80

A3UP = A3+K3*(OBS-LOG(A3/V1))

HL1 = THETA(3)*EXP(ETA(1))

;State cov. update eq.P_j∣j = P_j∣j−1K_j R_j∣j−1 K_j^T

HL2 = THETA(4)*EXP(ETA(2)) + HL1

P1UP = A4-K1*RVAR*K1

 KA1 = LOG(2)/HL1

P2UP = A5-K1*RVAR*K2

20

KA2 = LOG(2)/HL2

P3UP = A6-K1*RVAR*K3

TFR = THETA(5)

85

P4UP = A7-K2*RVAR*K2

 RHO = LOG(TFR/(1-TFR))

P5UP = A8-K2*RVAR*K3

 FRAC = EXP(RHO+ETA(3))/(1+EXP(RHO+ETA(3)))

P6UP = A9-K3*RVAR*K3

TBIO = 1

ENDIF

25

 IF(CONC. EQ. 20)TBIO = THETA(6)

;Update states

BIO = TBIO8EXP(ETA(4))

90

IF (A_0FLG.EQ. 1) THEN

SIG = THETA (7)

A_0(1)A1UP

SGW1 = THETA (8)

A_0 (2) = A2UP

SGW2 = THETA (9)

A_0 (3) = A3UP

30

SGW3 = THETA (10)

A_0 (4) = P1UP

F1 = FRAC*BIO

95

A_0 (5) = P2UP

F2 = (1-FRAC)*BIO

A_0 (6) = P3UP

;Initialize state and state covariance equations

A_0 (7) = P4UP

IF (NEWIND.NE.2) THEN

A_0 (8) = P5UP

35

AHT1 = 0

A_0 (9) = P6UP

AHT2 = 0

100

ENDIF

AHT3 = 0

 

PHT1 = 0

$DES

PHT2 = 0

;State predicton eq. dx_t∣j/dt = g(x_t∣j,d,phi)

40

PHT3 = 0

DADT(1) = −KA1*A(1)

PHT4 = 0

105

DADT(2) = −KA2*A(2)

PHT5 = 0

DADT(3) = −KA1*A(1) + KA2*A(2) - CL/V1*A(3)

PHT6 = 0

;State cov. dP_t∣j/dt = A_t P_t∣j+P_t∣j A_t^T+SGW SGW^T SGW^T

ENDIF

DADT (4) = −2*KA1*A(4) + SGW1*SGW1

45

;Store observations for EKF update

DADT (5) = −(KA1+KA2)*A(5)

IF (EVID.EQ.0) OBS = DV

110

DADT (6) = −(KA1+CL/V1)*A(6) + KA1*A(4) + KA2*A(5)

;Store one-step predictions for EKF update

DADT (7) = −2*KA2*A(7) + SGW2*SGW2

IF (EVID.NE.3) THEN

DADT (8) = −(KA2+CL/V1)*A(8) + KA1*(5) + KA2*A(7)

50

A1 = A(1)

DADT (9) = 2*KA1*A(6)+2*KA2*A(8)−2*CL/V1*A(9)+SGW3*SGW3

A2 = A(2)

 

A3 = A(3)

115

$ERROR (OBS ONLY)

A4 = A(4)

 IF (ICALL.EQ.4) THEN

A5 = A(5)

  IF (DV.NE.0) Y = LOG(DV)

A6 = A(6)

  RETURN

55

A7 = A(7)

 ENDIF

A8 = A(8)

120

IPRED = LOG(A(3)/V1)

A9 = A(9)

W = SQRT(A(9)/(A(3)*A(3))+SIG**2)

ELSE

IRES = DV – IPRED

A1 = A1

IWRES = IRES/W

60

A2 = A2

Y = IPRED + W*EPS(1)

A3 = A3

125

$SIM (1)

A4 = A4

$EST METH=1 INTER

A5 = A5

$COV

A6 = A6

 

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Tornøe, C.W., Overgaard, R.V., Agersø, H. et al. Stochastic Differential Equations in NONMEM®: Implementation, Application, and Comparison with Ordinary Differential Equations. Pharm Res 22, 1247–1258 (2005). https://doi.org/10.1007/s11095-005-5269-5

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  • DOI: https://doi.org/10.1007/s11095-005-5269-5

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