STRUCTURAL IDENTIFIABILITY OF PARALLEL PHARMACOKINETIC EXPERIMENTS AS CONSTRAINED SYSTEMS

STRUCTURAL IDENTIFIABILITY OF PARALLEL PHARMACOKINETIC EXPERIMENTS AS CONSTRAINED SYSTEMS S.Y. Amy Cheung1, James W.T. Yates2, Leon Aarons1 1

School of Pharmacy and Pharmaceutical Sciences, University of Manchester, Oxford Road, U.K. 2 Discovery DMPK, AstraZeneca, Alderley Park, Macclesfield, U.K.

Abstract: Pharmacokinetic analysis in humans using compartmental models is restricted with respect to the estimation of parameter values. This is because the experimenter usually is only able to apply inputs and observations in a very small number of compartments in the system. This has implications for the structural identifiability of such systems and consequently limits the complexity and mechanistic relevance of the models that may be applied to such experiments. A number of strategies are presented whereby models are rendered globally identifiable by considering a series of experiments in parallel. Examples are taken from the pharmacokinetic literature and analysed using this parallel experiment methodology. Copyright © 2006 IFAC Keywords: Algebraic approaches, Constrained parameters, Controllability, First-order systems, Identifiability, Parameter identification, Parameterization, Pharmacokinetic data, State-space models.

1.

INTRODUCTION

Pharmacokinetics is the study of the absorption, distribution, metabolism and elimination of a therapeutic agent in the body. There are a number of ‘classical’ compartmental models for such purposes (Wagner 1993), however any introduction of greater complexity or physiological relevance can result in problems for parameter estimation. This is because in humans it is usually only practical to apply inputs to and observe the compartment representing the blood. Structural identifiability (Bellman and Åström 1970; Anderson 1983) is the property of a model that there is sufficient information in the experimental inputoutput design to uniquely identify the unconstrained model parameters. Testing a pharmacokinetic model for structural identifiability is an important aspect of experimental design (Saccomani and Cobelli 1993). It is an a priori test because the system structure determines it and as such can be performed prior to experimentation. Modeller and experimenter should always follow this order of model analysis because strictly speaking without this unique ‘behaviour to parameter value’ correspondence any characterisation

of the system by parameter values might be invalid. This is, of course, dependent upon the application in hand and structural identifiability analysis can indirectly indicate which parameter values may be uniquely identified and the nature of any indeterminacy. A non-unique correspondence between observed dynamics and model parameter values presents problems for the experimenter. Not least of these is that the experiment would have been designed to investigate the kinetics of the real system, but interpretation of the data by the model might give more than one explanation of the behaviour via estimated parameter values. This restriction limits the scope and application of more mechanistic pharmacokinetic models. Here a mathematical analysis is presented of methods that may render models globally identifiable. These methods are to be found implicitly in the pharmacokinetic literature (Moghadaminia et al 2003; Nelson and Schaldemose 1959), however the authors believe this is the first time that such experimental arrangements have been analysed formally.

1. Parallel experiments as formal model structures are discussed as well as the implications for structural identifiability. Three case studies are presented that demonstrate the concepts discussed.

2. 3.

p ∈ P is globally identifiable if and only if [ p] = { p} , that is to say a single parameter set is associated with the observed behaviour. p ∈ P is locally identifiable if [p] is countably infinite2. p ∈ P is unidentifiable otherwise.

2. STRUCTURAL IDENTIFIABILITY ANALYSIS Consider a parameter vector p that parameterises the linear compartmental system x
(t ) = A( p ) x(t ) + B( p )u (t ) x (0 ) = x 0 ( p )

(1)

y (t ) = C ( p) x(t )

2.1 Similarity Transformation Method

xi

where in the case of pharmacokinetic modelling,

is the quantity of drug in compartment i, and y is the drug concentration in the observed compartments. The function u(t) corresponds to some input to the system (for example an intravenous bolus dose). The entries of the matrices A, B and C are dependent upon the parameter vector p. Let U be the set of continuous inputs/doses (to which u(t) in (1) belongs) and Y be the set of the associated continuous responses (to which y(t) belongs). Then define the behaviour of the compartmental system Σ p to be Σ p :U → Y ,

(2)

a map from time continuous inputs to time continuous outputs subject to the system specification (1). This is the input-output behaviour of the system viewed as a ‘black box’. For a given experimental design there is a corresponding fixed input function, u (t ) for 0 < t ≤ T . This will result in a particular observed output y p (t ) = Σ p (u (t )) .

Thus, as a structural property, the model σ is globally (locally) structurally identifiable if all p ∈ P are globally (locally) identifiable. Otherwise the model is said to be unidentifiable.

The similarity transformation method (Walter 1982) provides a method for exhaustively searching for all models that give the same input-output behaviour. The method of similarity transformation is based upon finding an isometry (Vajda and Rabitz 1989) between systems of the form (1). In the case of linear systems, given two systems ~ ~ ~ ( A, B, C ) and ( A, B , C ) then if the following conditions (1-3) are satisfied then the systems have equivalent input-output behaviour. 1. 2. 3.

~ ~ ~

For two systems ( A, B, C ) and ( A, B , C ) to be similar, there must exist a non-singular matrix T such that (Sontag 1990): ~ AT = TA ~ B = TB ~ CT = C

(5)

(3)

The subscript p designates that the behaviour of the system is implicitly dependent upon the parameter vector p . The observed output defines an equivalence relation1 on the parameter space with classes of indistinguishable parameter vectors [ p ] . Thus: p ~ p' ⇔ y p ≡ y p' .

Only parameter in the controllable and observable component of a component model can possibly be identifiable (Cobelli and Romanin-Jacur 1976). Therefore controllability and observability are verified in order to avoid trivial cases of nonidentifiability. For linear systems of low complexity, as will be considered in this paper, these criteria are easily checked using a set of ‘geometric’ criteria (Godfrey and Chapman 1990) that are reproduced in a simplified form for the models considered below:

(4) 1.

Thus we have that (Evans et al 2002):

2

1

The two systems are structurally observable (Cobelli and Romanin-Jacur 1976). The two systems are structurally controllable. There exists a non-singular matrix T such that the systems are similar (see below).

Partitions the set of all possible parameter values into classes that give the same observed behaviour.

A compartmental model is structurally controllable if and only if there is a path to every compartment from an input.

Possibly infinite number but no ‘close’ neighbouring parameter sets give the same observed behaviour. For more information see (Evans et al 2002).

2.

A compartmental model is structurally observable if and only if there is a path from each compartment to at least one observed compartment.

F. Dose

2

3.

1

1/V

k10

By path is meant a route along the arrows of flux in the compartmental diagram. This method (5) can therefore be used to find different parameter values that give the same output. In the sequel, a tilde denotes an alternative parameter value.

ka

= observation

Fig. 1. One-compartment model with first order absorption. This model represents the one compartment distribution of a compound after absorption from the gut.

CONSTRAINED STRUCTURES

It is sometimes possible to carry out the ‘same’ experiment several times on a system in which it can be assumed a priori that some, but not all, of its rate constants change between experiments. The models representing each experimental observation thus share some common rate constant values. A parallel experiment structure is constructed as follows: the basic pharmacokinetic model is reproduced a number of times corresponding to the number of experiments to be performed. The new model structure is reparameterised with a constrained parameterisation representing prior knowledge of the changes in parameter values between experiments. This forms a much more constrained structure than the individual model. In order to perform structural identifiability analysis, it is necessary to formulate the concept of a parallel experiment in the form of a linear system. Consider a single pharmacokinetic model of the form (1) represented by the triple ( A( p), B ( p), C ( p) ) (in pharmacokinetic experiments it is reasonable to assume that the initial condition is the zero state). Let superscripts represent the experiment number. Then the parallel experiment structure representing n experiments and parameterised by P' may be represented by the triple ( A' ( p'), B' ( p' ), C ' ( p' )) where  A( E 1 ( p' )) 0  0   A' ( p' ) =  0 … 0  n   0 0 A ( E ( p ' ))  

(6)

 B( E 1 ( p ' )) 0  0   B' ( p ' ) =  0 … 0  n   0 ( ( ' )) 0 B E p  

(7)

C ( E 1 ( p ' )) 0  0   C ' ( p' ) =  0  0  n   0 C E p 0 ( ( ' ))  

(8)

Here E i : P' → P

i = 1...n

(9)

is a map between the constrained parallel experiment parameters and the individual model parameters. Notice that dimension (P’) < n·dimension(P) ,

(10)

which is as a result of the constraints. Thus the systems

( A( p), B( p), C ( p) ) are

unconnected

but the functions E i represent the a priori assumptions of common and changing parameter values. Notice that if ( A( p ), B ( p), C ( p) ) is controllable is and observable then ( A' ( p'), B ' ( p ' ), C ' ( p' )) controllable and observable. This is because of the block structures defined in (6)–(8) result in the individual unconnected structures being controllable and observable.

The

parallel

experiment

structure

( A' ( p '), B' ( p' ), C ' ( p' )) is now of the form (1) and may

be analysed using the similarity criteria (5).

4.

THREE CASE STUDIES

Three different models are now examined with respect to parallel experiments in order to demonstrate the mathematical ideas discussed above. It is shown how individually unidentifiable or locally identifiable models may be rendered globally structurally identifiable by considering them in the context of some parallel experiment. Structural observability, controllability and identifiability analysis were performed using MATHEMATICA (Wolfram Research Inc., Illinois, U.S.A.). Solutions of the simultaneous equations in (5) could not be found automatically, a large amount of operator ‘supervision’ was required to guide the software to the solution set.

4.1 One-compartment absorption

model

with

first

order

As a simple example to illustrate the process of analysis, the model in Fig. 1 is considered first. This model represents the one compartment distribution of a compound (compartment 1) after absorption from the gut (compartment 2). This model is commonly referred to as a ‘one-compartment model with first order absorption’. It can be seen that from the rules presented above that the model is both controllable and observable, thus the similarity transformation method may be applied. The resulting structural identifiability status of this model is well known, however the analysis is presented here so that the connection between a model and potential parallel experiment structures may be appreciated. The system in Fig 1 may be written in the form (1) thus − k A =  10  0

ka  , − ka 

(11)

0 B = , F  C = [1 / V

0] .

 ka F  k10 T= ~ F  k10 − k a  k 10 

(13)

 0 . 1  

p new = (V / F , k a , k10 ) .

(15)

This can be seen to be locally identifiable because (5) implies that (16)

k10

1/V1

Fig. 2. ‘Classical’ two compartment Pharmacokinetic model with a third compartment representing the absorption of an orally administered dose. Consider the case now that the same drug is dosed orally using two different formulations. It can therefore be assumed that the body pharmacokinetics parameters V and k10 are constant between the two experiments, but that the bioavailability F and absorption rate k a will vary. The parallel experiment structure can be written in the form (8)(10): − k10  0 A( p' ) =  0   0 0  1 F B( p' ) =  0   0

~ ka t V F = 11 = ~. k10 t 22 F V

(17) V  , k a , k10  F  

This means that the two solutions are   , k10 , k a  . 

k a1

0

− k a1

0

0

− k10

0

0

0   0  k a2   − k a2 

(18)

0   0  0   F 2 

(19)

1 / V C ( p' ) =   0

0

0

0 1/ V

0  0

(20)

)

(21)

where

(

p ' = V , k10 , k 1a , k a2 , F 1 , F 2 .

and

( E (V , k

) ( ) = (F

E 1 V , k10 , k 1a , k a2 , F 1 , F 2 = F 1 , V , k a1 , k10 1 2 1 2 10 , k a , k a , F , F

2

)

, V , k a2 , k10

)

(22)

An analysis of this structure yields that T in (5) must be

and so that

 Fk a

2

1

k21

2

~ = V /V

 Vk10

k12

ka

3

(14)

It can be seen that a new parameter vector may be ~ generated from p by varying F . The model becomes locally identifiable by considering the parameterisation

and 

Dose

(12)

The resulting simultaneous equations (5) may be solved to show that the model is unidentifiable. There are, for a given p = (F , V , k a , k10 ) an infinite number of possible matrices T of the form

t11

F. Dose

1 0  F 1 0 1 T = ~1 F 0 0  0 0

0 0  0 0 . 1 0  0 1

(23)

This means that the parameterisation p’ is unidentifiable. However the uniquely identifiable parameter combinations are:

Table 1. Parameters used in the parent-metabolite model shown in fig. 3.

(1- fm) · Cl / vp Dose

PG

k12

F · ka

PP

PC

Parameter Vp

k21

(1- F) · ka

fm · Cl / vp

Vm MC Clm/vm

Fig. 3. Compartmental diagram of the parentmetabolite model used to model the pharmacokinetics of dextromethorphan and dextrophan.

(

p ' new = V / F 1 ,V / F 2 , k 1a , k a2 , k10

).

(24)

Thus the parallel experiment has eliminated the local identifiable indeterminacy between the absorption rate constant and the rate of elimination. 4.2 Classical Two Compartment Model with Oral Dosing Another common structural identifiability issue is with two compartment models following absorption of an oral dose (Fig 2). Such models have a triexponential impulse (bolus dose) response. A structural identifiability analysis (Godfrey et al 1980) demonstrates that there are 3 solution sets of parameters and thus the model is locally identifiable. This is by again considering V/F as a parameter. Consider the case where the same compound is dosed orally on two separate occasions where the formulation is different (Nelson and Schaldemose 1959). This might be as a result of different crystal states, vehicle or tablet form. In this case it can be assumed that post absorption the pharmacokinetics, represented by V1 , k12 , k 21 and k10 will be the same in both experiments: the same compound is dosed. However the absorption kinetics, as represented by k a and F, will vary. An analysis of this proposed parallel experiment shows that the disposition parameters k12 , k 21 and k10 are globally identifiable, as are the two absorption rates k 1a and k a2 . The two combination parameters V1 / F 1 and V1 / F 2 are also shown to be globally identifiable.

CL CLm k12 k 21 ka fm F

An important example of parallel experimental design is the inhibition of metabolism by co-administering a suitable enzyme inhibitor (Moghadamnia et al 2003). The dextrorotatory morphinan codeine synthetic analogue, dextromethorphan (DEX) is an anti-tussive

central central

Transfer rate constant Oral dose absorption rate constant Fraction metabolised Bioavailability of parent

therapeutic agent. DEX metabolise to a major active metabolite dextrorphan (DOR). Blood concentration levels for DEX and DOR were recorded separately after an oral dose of DEX. The proposed model structure for a single experiment is detailed in Fig. 3. The first pass effect is incorporated into the model by connecting the parent and metabolite central compartments, PC and M C , to the oral dose compartment PG . The parameterisation of this model is

(

p = V p ,Vm , CLm , k12 , k 21 , k a , f m , CL, F

)

(25)

See table 1 for an explanation of these parameters. Structural identifiability analysis of this model shows it to be unidentifiable with respect to the proposed inputs and outputs. The identifiable parameter combinations are:  V p Vm CL CLm Ff m  , , , , p new =  F 1 − F V p Vm 1 − F   k12 , k 21 , k a

 , .  

(26)

As it will be shown, this situation may be remedied by creating a parallel experiment design. The orally administrated regimen was divided into two parts, DEX was dosed with either 50 mg of quinidine sulphate or placebo. Quinidine is an inhibitor of DEX conversion to DOR (Moghadamnia et al 2003). Thus the parallel experiment using the same model structure was: 1.

4.3 Parent-Metabolite Model with Oral Dose

Meaning Volume of parent drug compartment Volume of metabolite compartment Total clearance of parent drug Total clearance of metabolite Transfer rate constant

2.

DEX (30mg), quinidine placebo anteceded at 1 hour. DEX (30mg), quinidine sulphate 50mg anteceded at 1 hour.

The constraints that are placed on the parallel experiment model parameterisation are that the parameters will remain constant for the two experiments except for those influenced by the rate of metabolism. These are the bioavailability F, the

clearance of the parent-drug CL and the fraction metabolised f m . There is some evidence that quinidine can affect active transport of drugs across the gut wall, however for the analysis here it is assumed that the absorption rate is constant. Thus the parallel structure model has the parameterisation V p , Vm , CLm , k12 , k 21 , k a ,   p' =   f 1 , f 2 , CL1 , CL2 , F 1 , F 2   m m 

(27)

This parallel structure with parameterisation (27) is then globally structurally identifiable.

5. CONCLUSIONS Parallel experiments to enhance the structural identifiability of pharmacokinetic models have been implicitly discussed previously (Nelson and Schaldemose 1959; Moghadamnia et al 2003). However, there appears to have never been a formal formulation and analysis of this experimental design problem. A preliminary formulation has been presented here that places the concept of a parallel experiment in the context of a single constrained model structure. Three case studies have been examined in order to illustrate the constrained model concept. The parallel experimental design has been shown to be beneficial with regards to structural identifiability. It is apparent as well that multiple experiments will be beneficial from a system identification point of view. There are other potential examples of parallel experiments. In (Yuasa et al 1995) it is discussed how anaesthesia can influence the rate of absorption and elimination of certain pharmaceuticals. This has a physiological justification due to the lowered blood pressure that anaesthesia induces. In (Brown et al 2004) it is noted that the pharmacokinetics of a drug, especially metabolism and elimination, alters dependent on whether the individual is at rest or exercising. Again this is mainly influenced by blood pressure and flow rate. Incorporation of prior knowledge into parallel experiment model structures with constrained parameterisation allows sufficient information to be present in the input-output behaviour to give unique parameter estimates. It is apparent from the results presented here that parallel experiment strategies can be very powerful in providing globally structurally identifiable pharmacokinetic models.

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