J Pharmacokinet Pharmacodyn (2014) 41:81–86 DOI 10.1007/s10928-013-9347-8

SHORT REPORT

Parameterisation affects identifiability of population models Vittal Shivva • Julia Korell • Ian G. Tucker Stephen B. Duffull



Received: 1 July 2013 / Accepted: 20 December 2013 / Published online: 31 December 2013 Ó Springer Science+Business Media New York 2013

Abstract Identifiability is an important aspect of model development. In this work, using a simple one compartment population pharmacokinetic model, we show that identifiability of the variances of the random effects parameters are affected by the parameterisation of the fixed effects parameters. Keywords Identifiability  Bateman model  Parameterisation  Population models

Identifiability of models is an important aspect in model based analysis. Recently an informal approach using an information theoretic framework for assessing local identifiability of models was proposed by Shivva et al. [1]. This method provides a framework for consideration of both structural and deterministic identifiability of nonlinear fixed effects and mixed effects models. A model is said to be structurally globally identifiable if all parameters in the model have a unique solution. A model is said to be structurally locally identifiable provided one or more parameters in the model have a finite number of alternative solutions and the remaining parameters have unique solutions. Another way of considering structural local identifiability is if the model is structurally identifiable for a specific (local) set of parameter values and

this is true for all of the finite number of solutions. Finally, a model is said to be structurally unidentifiable if at least one parameter in the model has an infinite number of alternative solutions. Note here the remaining parameters may either have unique solutions or have a finite number of alternative solutions [1]. Deterministic identifiability relates to the precision with which a parameter from a globally or locally identifiable model can be estimated. It is not an absolute condition but rather a condition that relates to the precision of the parameters of interest that a scientist requires. A model that is structurally unidentifiable will also be deterministically unidentifiable. In this work we do not consider global identifiability but rather consider all examples at a specific value of the parameters and hence we distinguish locally identifiable models from unidentifiable models only. We also consider identifiability based on numerical approximations to the nonlinear mixed effects models rather than exact analytical solutions. It has been shown in our previous work (see Shivva et al. [1].) that the parameter F (oral bioavailable fraction) is unidentifiable in the Bateman model (a one compartment first-order input and first-order output model) [2, 3]. The general form of the structural model is shown here: yij ¼

V. Shivva (&)  I. G. Tucker  S. B. Duffull School of Pharmacy, University of Otago, P.O. Box 56, Dunedin 9054, New Zealand e-mail: [email protected] J. Korell Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden

   Di  Fi  kai exp ðCLi =Vi Þ  tij Vi  ðkai  ðCLi =Vi ÞÞ     iid  exp kai  tij þ eij ; eij  N 0; r2 ;

ð1Þ

where yij represents the observed jth response in the ith individual, Di is the dose administered to the ith individual, Fi is the bioavailable fraction following oral administration, kai is the absorption rate constant, Vi is the volume of distribution, CLi is clearance, tij represents the time point of

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Table 1 Empirical set of parameter values used in assessment of identifiability of Bateman mixed effects model Parameter

Mean value (h)

BSV (x2)

CL

4

0.1

k

0.2

0.1

V

20

0.1

ka

1

0.1

F

1

0.1

The vector of parameter values for the ith individual (hi) is given by a covariate model as shown here:   iid hi ¼ g Zi ; hpop expðgi Þ;gi  N ð0; XÞ where X 2 2 3 x11    x21q 6 .. 7 .. ¼ 4 ... . 5 . 2 xq1    x2qq

ð2Þ

the jth observation in the ith individual, eij represents the random error at the jth observation point in the ith individual and r2 represents the residual error variance.

where g is the functional form of the covariate model, Zi is the vector of covariates in ith individual, hpop is the vector of population mean parameter estimates, gi is a q 9 1 vector of random effects in the ith individual, X is the variance covariance matrix representing between subject

Fig. 1 Graphical representation of log |MF| versus log random noise (r2) for the Bateman mixed effects model with parameterisation CL, V, ka and F. In this graph, log |MF| above the abscissa are as represented. Data below the abscissa represent the negative

determinants that do not have log values and are shown for the purpose of displaying discontinuity of the line. a Full model where all parameters are assessed, b fixed effect parameter F fixed, c fixed effect parameter V fixed, d fixed effect parameter CL fixed

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Fig. 2 Graphical representation of log |MF| versus log random noise (r2) for the Bateman mixed effects model with parameterisation CL, V, ka and F with 50 % correlation between random effects CL and V. In this graph, log |MF| above the abscissa are as represented. Data below the abscissa represent the negative determinants that do not

have log values and are shown for the purpose of displaying discontinuity of the line. a Full model where all parameters are assessed, b fixed effect parameter F fixed, c fixed effect parameter F and random effect parameter x2F fixed, d fixed effect parameter F and covariance of random effects x2CL-V fixed

random variability. Diagonal elements of X represent the variances and off-diagonal elements represent the covariances of the random effects. The current study assumed a rich sampling design (n) with time points 0, 0.25, 0.5, 0.75, 1, 2, 4, 8, 12, 18 and 24 h. A single oral dose of 100 mg was assumed in a study population consisting 100 individuals. The local set of parameter values used in the assessment is provided in Table 1. Values for the random noise in assessing identifiability were log(r2) = (-5, -4, -3, -2, -1). The initial parameterisation for assessing the identifiability of the Bateman model was CL, V, ka and F. In this instance, assuming no between subject variability, it was

observed that the model was unidentifiable. Fixing any one of the fixed effects parameters CL, V or F led to an identifiable model. This occurs as the ratios of CL/F and V/ F are identifiable in the model and hence knowledge of any one of these three parameters renders the other two parameters identifiable. For the assessment of the population model, F was fixed and all fixed effects parameters (including F) were assumed to have a random effect. If we assumed that the X matrix was diagonal then all variances of the random effects parameters were identifiable, including the between subject variance of F (Fig. 1). Extending this analysis to a full X matrix (with 50 % correlation between CL and V) resulted in the model being

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J Pharmacokinet Pharmacodyn (2014) 41:81–86

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unidentifiable. Either the variance of F or the covariance between CL and V had to be fixed to make the model identifiable (Fig. 2).

Reparameterisation of the model to k, V, ka and F (k = elimination rate constant), assuming no between subject variability revealed that fixing either V or F led to an identifiable model (here, it is the ratio V/F that is identifiable). When considering the population model (again all fixed effects parameters were assumed to have a random effect) and assuming a diagonal X matrix, we see that the between subject variability of F is no longer identifiable. Fixing either V or F and the random effect of either of the between subject variance of V or F led to an identifiable model (Fig. 3). When this analysis was extended to include a full X matrix (with 50 % correlation

Fig. 4 Graphical representation of log |MF| versus log random noise (r2) for the Bateman mixed effects model with parameterisation k, V, ka and F with 50 % correlation between random effects k and V. In this graph, log |MF| above the abscissa are as represented. Data below the abscissa represent the negative determinants that do not have log

values and are shown for the purpose of displaying discontinuity of the line. a Full model where all parameters are assessed, b fixed effect parameter F fixed, c fixed effect parameter F and random effect parameter x2F fixed, d fixed effect parameter F and covariance of random effects x2k-V fixed

b Fig. 3 Graphical representation of log |MF| versus log random noise

(r2) for the Bateman mixed effects model with parameterisation k, V, ka and F. In this graph, log |MF| above the abscissa are as represented. Data below the abscissa represent the negative determinants that do not have log values and are shown for the purpose of displaying discontinuity of the line. a Fixed effect parameter F fixed, b fixed effect parameter V fixed, c fixed effect parameter F and random effect parameter x2F fixed, d fixed effect parameter V and random effect parameter x2V fixed, e fixed effect parameter F and random effect parameter x2V fixed, f fixed effect parameter V and random effect parameter x2F fixed

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between k and V) we see no further differences in identifiability. The covariance of k and V was identifiable and the between subject variance of F remained unidentifiable (Fig. 4). The results obtained for the reparameterisation (diagonal X matrix) of the Bateman population model are now comparable to the Dost population model [4] as presented in our previous work (Shivva et al. [1]). This indicates that parameterisation of the fixed effects parameters of the Bateman model impacts on the identifiability of the variance of the random effects models. It is reasonable to expect that the identifiability of more complex PK models will also not be invariant to the choice of parameterisation. It is seen in this work with a diagonal X matrix that reparameterisation of a simple one compartment model with first-order input and output from CL, V, ka and F to k, V, ka and F affects the identifiability of the between subject variance on F. Assessment with a full X matrix showed that when the parameterisation is CL, V, ka and F, then the covariance between CL and V provides a locally equivalent solution to the between subject variance of F. Either the variance of F or the covariance between CL and V, but not both, can be estimated with this parameterisation. However, this issue does not hold in the reparameterised model (k, V, ka and F) with covariance between k and V, though k is a derived rate constant of CL and V and V/F is an identifiable parameter combination in the structural model. Both fixed and variance parameter of F cannot be estimated irrespective of whether covariance between k and V is fixed or estimated in this parameterisation. Generalisation of this work to other models (or different parameter sets of the examined model) should be considered with caution. The assumptions and limitations here are (1) that we only consider identifiability for a local set of parameter values, (2) that random effects parameters behave inconsistently with respect to their fixed effects partners and therefore no rule of thumb would seem possible, (3) we use a first-order approximation, plus an assumption that no correlation exists between fixed and random effects when computing the population Fisher

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information matrix (MF), which could negatively affect our ability to compute a positive |MF| even if the model is locally identifiable and (4) we assess convergence (hence identifiability) by showing a continuous log–log relationship over a (non-infinite) range of values of the residual variance. Despite these assumptions and limitations it is reasonable to take the conservative approach that parameterisation may affect the identifiability of variance parameters for mixed effects models and that this effect is likely to hold across a reasonable range of parameter values. Care should be taken when considering parameterisation of mixed effects models with respect to the influence this may have on local identifiability of the random effects models in a population analysis. This becomes an important aspect in population analysis with complex PK and PKPD models as the associated influence on local identifiability of random effects parameters with fixed effects parameterisation is not necessarily predictable a priori. Acknowledgments Vittal Shivva was supported by University of Otago Postgraduate Scholarship. Conflict of interest

The authors declared no conflict of interest.

References 1. Shivva V, Korell J, Tucker IG, Duffull S (2013) An approach for identifiability of population pharmacokinetic–pharmacodynamic models. CPT Pharmacometrics Syst Pharmacol 2(e49):1–9. doi:10. 1038/psp.2013.25 2. Bateman H (1910) Solution of a system of differential equations occurring in the theory of radioactive transformations. Proc Camb Philos Soc 15:423–427 3. Garrett ER (1994) The Bateman function revisited: a critical reevaluation of the quantitative expressions to characterize concentrations in the one compartment body model as a function of time with first-order invasion and first-order elimination. J Pharmacokinet Biopharm 22(2):103–128. doi:10.1007/bf02353538 4. Dost FH (1968) Grundlagen der Pharmakokinetik, vol 2. G. Thieme, Stuttgart, pp 38–47

Parameterisation affects identifiability of population models.

Identifiability is an important aspect of model development. In this work, using a simple one compartment population pharmacokinetic model, we show th...
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