Metabolite Pharmacokinetic Model for Ivabradine - Springer Link

Journal of Pharmacokinetics and Pharmacodynamics, Vol. 28, No. 1, 2001

PHARMACOMETRICS An Identifiability Analysis of a Parent–Metabolite Pharmacokinetic Model for Ivabradine Neil D. Evans,1 Keith R. Godfrey,1,4 Michael J. Chapman,2 Michael J. Chappell,1 Leon Aarons,3 and Stephen B. Duffull3 Received November 9, 1999—Final August 30, 2000 The paper considers the structural identifiability of a parent–metabolite pharmacokinetic model for iûabradine and one of its metabolites. The model, which is linear, is considered initially for intraûenous administration of iûabradine, and then for a combined intraûenous and oral administration. In both cases, the model is shown to be unidentifiable. Simplification of the model (for both forms of administration) to that proposed by Duffull et al. (1) results in a globally structurally identifiable model. The analysis could be applied to the modeling of any drug undergoing first-pass metabolism, with plasma concentrations aûailable from drug and metabolite. KEY WORDS: compartmental models; identification; ivabradine; linear systems; pharmacokinetics.

INTRODUCTION Structural identifiability is concerned with whether the parameters of a postulated model can be identified from a specified experiment or experiments with perfect input–output data. As such, it is a very important concept in mathematical modeling in pharmacokinetics and pharmacodynamics, for which there are usually a very limited number of input and output locations. For linear models, there are several possible approaches for analyzing structural identifiability, the most familiar being those based on the Laplace The authors gratefully acknowledge the support of the Engineering and Physical Sciences Research Council of the U.K. under grant GR兾M119143. S.B.D. was supported by grant BIOMED PL 962640 from the European Community. 1 School of Engineering, University of Warwick, Coventry CV4 7AL, England. 2 School of MIS-Mathematics, Coventry University, Coventry CV1 5FB, England. 3 School of Pharmacy and Pharmaceutical Sciences, University of Manchester, Manchester M13 9PL, England. 4 To whom correspondence should be addressed. 93 1567-567X兾01兾0200-0093$19.50兾0  2001 Plenum Publishing Corporation

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transformation of the output (or outputs) (2,3), or on the use of a similarity transformation approach (3,4). A comparison between the application of the two approaches to a particular compartmental model is given in Godfrey and Chapman (5). It is difficult to know in advance which of the two approaches will prove the easier for a particular model with a single input and single output (i.e., measurement location), but the similarity transformation approach often proves the easier as the number of inputs and兾or outputs is increased. Set against this is the need to perform controllability and observability checks when using the similarity transformation approach. Such checks are not needed for the Laplace transformation approach (provided any possible simplification between the numerator and denominator of any transfer function involved in the model has been performed), which works directly on the input–output relationship(s). For linear models, the identifiability analysis will not depend on the input waveform which is assumed to be parameter independent and persistently exciting (i.e., that its Laplace transform does not vanish over any real interval). In this paper, the structural identifiability of the model of Fig. 1 for the pharmacokinetics of ivabradine (a bradycardic agent developed for the prevention of myocardial ischemia) and one of its metabolites S-18982, for both intravenous and oral administration of ivabradine, is considered. The pharmacokinetic (PK) schematic as shown in Fig. 1 describes a combined two-compartment parent (ivabradine) and metabolite (S-18982) model. Ivabradine is administered either by the oral or intravenous route.

Fig. 1. The full model. There are two measurement locations, namely, Compartments I1 and S1 .

Analysis of a Parent–Metabolite PK Model for Ivabradine

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After oral administration some ivabradine may be eliminated from the gut compartment without reaching the systemic circulation as either intact parent or S-18982. This pathway represents either the production of other metabolites that were not assayed in the original study or fecal elimination of ivabradine. In the original PK study of Duffull et al. (1) it was believed that the fraction of drug that met this fate was not identifiable and it was assumed that all drug administered orally reached the systemic circulation as either intact ivabradine or was metabolized on first-pass to S-18982. A first-order process was used to describe the presentation of both ivabradine and S-18982 to the systemic circulation and the rate constants for both compounds were considered to be the same. In this setting the rate-limiting step of appearance of S-18982 was believed to be the presentation of ivabradine to the liver and not the conversion of ivabradine to S-18982. The schematic also shows biotransformation of ivabradine to S-18982 and elimination to another, undetermined, fate. In the latter case this may be characterized by renal elimination or metabolism to other compounds. However, since no information was available to describe any of these processes, it was again believed that this process was unidentifiable. It is the purpose of the present work to formally determine the identifiability of the ivabradine model. The model is considered initially with elimination directly to the environment from Compartment I1 and from the Gut compartment included. Since there are two measurement locations (the responses in the plasma of both ivabradine and S-18982), the similarity transformation approach is used. After the initial check for controllability and observability, the structural identifiability of the model is analyzed first for the case where the responses (concentration of ivabradine in Compartment I1 and of S-18982 in Compartment S1) to an iv dose only are available, and then for the case where responses following both iv and oral doses are available. IDENTIFIABILITY ANALYSIS USING SIMILARITY TRANSFORMATION APPROACH If a given parameter in a parameterized model can be uniquely determined, given perfect, noise-free, input–output data, then it is said to be globally identifiable. If the parameter can take any of a countable number of values it is said to be locally identifiable. Otherwise the parameter is said to be unidentifiable. A globally identifiable model is one for which all of the parameters are globally identifiable. The model is said to be unidentifiable if any parameter is unidentifiable, or locally identifiable if no parameter is unidentifiable but at least one is locally identifiable. A model structure that is shown to be

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globally, or locally, identifiable will typically have degenerate parameter values, i.e., values that are solutions of polynomial equation(s), where the analysis proves invalid. Since such atypical cases have zero measure, they are not considered. These atypical cases include the situation where relations exist between parameter values, for example, in the context of compartmental models, if one rate constant is a known multiple of another. In this paper, it is assumed that all rate constants are independent. To apply the similarity transformation approach to a linear system, the system is described by dx

GAxCBu

dt yGCx

(1a) (1b)

where B is the input matrix, C is the output matrix and A is the (nBn) system matrix. As a technical point, it is assumed that two different sets of parameter values cannot give rise to the same matrices A, B, and C. Because this approach does not directly use the input–output relationships, it is necessary before proceeding to the identifiability analysis to check that the system is controllable and observable. For a linear PK model with a single input and with no traps, i.e., both drug and metabolite concentrations decay asymptotically to zero once the input has finished, controllability is assessed by checking that an input will affect all compartments (6,7,8). The addition of a second input to a controllable system will still result in a controllable system. Similarly, observability is assessed by checking that substance (drug or metabolite) can move from any compartment to a measured compartment. (Note that in the model considered in this paper, the drug to metabolite conversion is considered as a substance flow from Compartment I1 to Compartment S1.) Provided a system is controllable and observable, it is then possible to proceed with the identifiability analysis using the similarity transformation approach. This utilizes the fact (3,4) that, for another model, characterized ¯ ), to have identical input–output behavior, it is necessary and ¯ , B¯ , C by (A sufficient that there exists a nonsingular matrix T such that TB¯ GB

(2a)

¯ GCT C

(2b)

¯ GAT TA

(2c)

¯ ) is assumed to have the same structure ¯ , B¯ , C The model characterized by (A as the original, but possibly different parameter values. Hence if any entry in A (respectively, B or C) is zero then the corresponding entry in the matrix


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Fig. 2. The model for the iv dose experiment. The two compartments corresponding to the drug are 1 and 3. Compartments 2 and 4 correspond to the metabolite S-18982.

¯ ) is also zero. If these three equations taken together ¯ (respectively, B¯ or C A imply that TGIn (the nBn identity matrix), then the model is globally identifiable. The use of this approach is illustrated below. THE INTRAVENOUS DOSE MODEL For the intravenous dose experiment, the model of Fig. 1 reduces to the compartmental model of Fig. 2, in which Compartments 1 and 3 contain ivabradine, Compartments 2 and 4 contain the metabolite S-18982, and the link from Compartment 1 to Compartment 2 represents biotransformation (at a first-order rate) from ivabradine to S-18982. As noted above, the similarity transformation approach considers the triple (A, B, C), where B is the input matrix, C is the output matrix and A is the system matrix. Conventionally, the order of subscripts in the pharmacokinetics literature is the reverse of that used in the system matrix, so that aij Gkj i for all i ≠ j. For the model of Fig. 2, the system matrices are (2)

冢

a11 0 a13 0 a21 a22 0 a24 AG a31 0 a33 0 0 a42 0 a44

冣

(3a)

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where a11 G−(a01Ca21Ca31)G−(k1eCk12Ck13)

(3b)

a22 G−(a02Ca42)G−(k2eCk24)

(3c)

a33 G−a13 G−k31

(3d)

a44 G−a24 G−k42

(3e)

and

冢冣

1 0 BG , 0 0

CG

冢0

c1

冣

0 0 0 c2 0 0

(4)

where c1 G1兾V1 and c2 G1兾V2 ; V1 and V2 being the apparent volumes of distribution of ivabradine and S-18982, respectively. Let p, given by pG(k1e k12 k13 k2e k24 k31 k42 V1 V2 )T denote the vector of unknown positive parameters. Note that, in this particular case, B does not depend on p and so B¯ GB. By inspection of Fig. 2, the input affects all four compartments, so that the system is controllable, while all four compartments affect the metabolite measurement, so that the system is also observable. Let T be a nonsingular matrix of the form

TG

冢

t11 t12 t13 t14 t21 t22 t23 t24 t31 t32 t33 t34 t41 t42 t43 t44

冣

then Eq. (2a) implies that t11 G1, and t21 Gt31 Gt41 G0. In an identifiability ¯ is assumed to have the same structure as C so that, analysis the matrix C ¯ denoted by c¯ i , c¯ i G0 if ci G0 and c¯ i ≠ 0 if ci ≠ 0. Therewith the entries in C fore Eq. (2b) implies that c¯ 1 Gc1 ,

t12 Gt13 Gt14 Gt23 Gt24 G0,

and so

冢

1 0 0 c¯ 2 兾c2 TG 0 t32 0 t42

冣

0 0 0 0 . t33 t34 t43 t44

t22 G

c¯ 2 c2


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¯ is assumed to have the same strucSimilarly, in an identifiability analysis, A ture as A (i.e., a¯ i j G0 if and only if ai j G0) so that Eq. (2c) implies that

冢

a¯ 11 0 a¯ 13 0 (c¯ 2 兾c2 )a¯ 22 0 (c¯ 2 兾c2 )a¯ 24 (c¯ 2 兾c2 )a¯ 21 a¯ 21 t32Ca¯ 31 t33 a¯ 22 t32Ca¯ 42 t34 a¯ 33 t33 a¯ 24 t32Ca¯ 44 t34 a¯ 21t42Ca¯ 31 t43 a¯ 22 t42Ca¯ 42 t44 a¯ 33 t43 a¯ 24 t42Ca¯ 44 t44

冢

冣

冣

a11 a13 t32 a13 t33 a13 t34 a21 (c¯ 2 兾c2 )a22Ca24 t42 a24 t43 a24 t44 G a31 a33 t32 a33 t33 a33 t34 0 (c¯ 2 兾c2 )a42Ca44 t42 a44 t43 a44 t44 where a¯ i j Gkr ji , i ≠ j, and a¯ 0i Gkr ie . Comparing the components in the (1,2), (1,4), and (2,3) positions implies that t32 Gt34 Gt43 G0; since t43 G0, comparing components in the (4,1) positions implies that t42 G0. Note that for T to be nonsingular there must be no identically zero rows or columns so that t33 and t44 must be nonzero. Substituting these values into the above matrix equation yields the following: a¯ 11Ga11

(5a)

a¯ 13Ga13 t33

(5b)

a¯ 21c¯ 2Ga21c2

(5c)

a¯ 22Ga22

(5d)

a¯ 24 c¯ 2G(a24 c2 )t44

(5e)

a¯ 31 t33Ga31

(5f)

a¯ 33Ga33

(5g)

冢 ¯c¯ 冣 t a42 2

a42 c2

(5h)

a¯ 44Ga44

(5i)

G

44

Note that, if a¯ i j Gai j for some i, j, then any two models with the same input– output map must have equal ai j terms. This means that ai j is unique for a given input–output map, that is, the parameter kj i (kj i Gai j , i ≠ j ) is globally identifiable. A locally identifiable parameter is one for which there exist a countable number of solutions for the corresponding a¯ i j term. However, if there exist an uncountable number of solutions then the parameter is unidentifiable.

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Since c¯ 1 Gc1 , the parameter V1 (=1兾c1 ) is unique, that is, V1 is globally identifiable. Equations (5g) and (3d) imply that a¯ 13 Ga13 (i.e., k31 is globally identifiable), so that, from Eq. (5b), it is seen that t33 G1. Therefore, since t33 G1, Eq. (5f) implies that a¯ 31 Ga31 . Equations (5i) and (3e) give a¯ 24 Ga24 and so Eq. (5e) implies that t44 Gc¯ 2 兾c2 . Therefore Eq. (5h) implies that a¯ 42 G a42 and so, since from Eqs. (5d) and (3c) (a¯ 02Ca¯ 42 )G(a02Ca42 ), it is seen that a¯ 02 Ga02 . Hence the parameter k2e is globally identifiable. Since a¯ 31 G a31 , Eqs. (5a) and (3b) imply that (a¯ 01Ca¯ 21)G(a01Ca21). Summarizing, the following individual parameters are globally identifiable from this experiment: k2e , k31 , k42 , k13 , k24 , and V1 . The remaining individual parameters, namely, k1e , k12 , and V2 , are unidentifiable, but the following combinations of these parameters are globally identifiable: k12 兾V2 and (k1eCk12 ). (Note that if V2 were known a priori, then all the rate constants in the model would be globally identifiable.) In the model postulated in Duffull et al. (1), elimination of ivabradine directly to the environment was assumed not to take place (i.e., k1e G0), in which case, the remaining parameters k12 and V2 are globally identifiable. THE COMBINED INTRAVENOUS/ORAL DOSE MODEL For the combined iv兾oral dose model, the model of Fig. 1 corresponds to the compartmental model of Fig. 3, in which Compartments 1 to 4 are as in Fig. 2.

Fig. 3. The model for the iv and oral dose experiment. Compartment 5 corresponds to the gut. The input for the iv dose is u1 and that for the oral dose is u2 .


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From Compartment 5, representing the Gut, ivabradine is assumed to be transferred at a first-order rate to Compartment 1, while the link from Compartment 5 to Compartment 2 represents biotransformation, at a firstorder rate, from ivabradine to S-18982. The elimination from Compartment 5 directly to the environment is used to account for ivabradine which is neither absorbed into the bloodstream, nor metabolized in the Gut into S18982, which is itself subsequently absorbed into the blood stream; this elimination is omitted in the model postulated in Duffull et al. (1). For the model of Fig. 3, therefore,

冢

a11 0 a13 a21 a22 0

0 a15 a24 a25

冣

AG a31 0 a33 0 0 0 a42 0 a44 0 0 0 0 0 a55

(6a)

where a11 , a22 , a33 , and a44 are given by Eqs. (3b)–(3e), as before, and a55 GA(a05Ca15Ca25 )G−(k5eCk51Ck52 )

(6b)

With both iv input and oral input, and (as before) measurement of concentrations of ivabradine and S-18982, the matrices B and C are given by

冢冣

1 0 BG 0 0 0

0 0 0 , 0 1

CG

冢0

c1

冣

0 0 0 0 c2 0 0 0

(7)

where it is assumed that the iv and oral doses are not applied simultaneously. The vector of unknown parameters, again denoted by p, is equal to pG(k1e k12 k13 k2e k24 k31 k42 k5e k51 k52 V1 V2 )T By inspection of Fig. 3, the oral input affects all five compartments so that the system is controllable, while all five compartments affect the metabolite measurement so that the system is also observable. Therefore the similarity transformation approach can be applied to test the structural identifiability of this model.

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Let T be a nonsingular matrix of the form

冢

t11 t12 t13 t14 t15 t21 t22 t23 t24 t25 TG t31 t32 t33 t34 t35 t41 t42 t43 t44 t45 t51 t52 t53 t54 t55

冣

Equation (2a) implies that t11 Gt55 G1 and t15 Gt21 Gt25 Gt31 Gt35 Gt41 Gt45 Gt51 G0 while Eq. (2b) implies that c¯ 1 Gc1 ,

t12 Gt13 Gt14 Gt23 Gt24 G0, and t22 Gc¯ 2 兾c2

Therefore,

冢

1 0 0 c¯ 2 兾c2 TG 0 t32 0 0

t42 t52

0 0

0 0

0 0

t33 t34 0 t43 t44 0 t53 t54 1

冣

The final condition, Eq. (2c), gives rise to the following 25 equations, which will be grouped by column order. By doing so, the property of the system matrix A (that minus the sum of each column is equal to the elimination rate constant) is utilized, while each of the (nonzero) unknowns a¯ i j appears in one and only one set of equations. Column 1 gives rise to a¯ 11Ga11

(8a)

c¯ 2 a¯ 21Ga21 c2

(8b)

t32 a¯ 21Ct33 a¯ 31Ga31

(8c)

t42 a¯ 21Ct43 a¯ 31G0

(8d)

t52 a¯ 21Ct53 a¯ 31G0

(8e)


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Column 2 gives rise to 0Gt32 a13Ct52 a15 c¯ 2 c¯ 2 a¯ 22G a22Ct42 a24Ct52 a25 c2 c2 t32 a¯ 22Ct34 a¯ 42Gt32 a33 t42 a¯ 22Ct44 a¯ 42G

c¯ 2 a42Ct42 a44 c2

t52 a¯ 22Ct54 a¯ 42Gt52 a55

(9a) (9b) (9c) (9d) (9e)

Column 3 gives rise to a¯ 13Gt33 a13Ct53 a15

(10a)

0Gt43 a24Ct53 a25

(10b)

t33 a¯ 33Gt33 a33

(10c)

t43 a¯ 33Gt43 a44

(10d)

t53 a¯ 33Gt53 a55

(10e)

Column 4 gives rise to 0Gt34 a13Ct54 a15

(11a)

c¯ 2 a¯ 24Gt44 a24Ct54 a25 c2

(11b)

t32 a¯ 24Ct34 a¯ 44Gt34 a33

(11c)

t42 a¯ 24Ct44 a¯ 44Gt44 a44

(11d)

t52 a¯ 24Ct54 a¯ 44Gt54 a55

(11e)

and Column 5 gives rise to a¯ 15Ga15

(12a)

c¯ 2 a¯ 25Ga25 c2

(12b)

t32 a¯ 25G0

(12c)

t42 a¯ 25G0

(12d)

t52 a¯ 25Ca¯ 55Ga55

(12e)

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From Eqs. (12c) and (12d) it is seen that t32 Gt42 G0. Substituting these into Eq. (8d), Eqs. (9a) and (9c) shows that t43 Gt52 Gt34 G0. Thus Eq. (8e) or Eq. (10b) implies that t53 G0, while Eq. (9e) or Eq. (11a) implies that t54 G 0. Since T is nonsingular both t33 and t44 are nonzero, for otherwise T would have a zero row (and column). Therefore the Eqs. (8a)–(12e) reduce to a¯ 11 Ga11 , a¯ 44 Ga44 ,

a¯ 21c¯ 2 Ga21c2 , a¯ 15 Ga15 ,

a¯ 22 Ga22 , c¯ 2 a¯ 25 Gc2 a25 ,

a¯ 33 Ga33 a¯ 55 Ga55

and t33 a¯ 31Ga31

(13a)

c¯ 2 a42 c2

(13b)

a¯ 13Gt33 a13

(13c)

c¯ 2 a¯ 24Gt44 a24 c2

(13d)

t44 a¯ 42G

Since a¯ 33 Ga33 and a¯ 44 Ga44 , Eqs. (3d) and (13c) imply that t33 G1, while Eqs. (3e) and (13d) imply that t44 Gc¯ 2 兾c2 . Therefore Eq. (13a) implies that a¯ 31 Ga31 , and Eq. (13b) implies that a¯ 42 Ga42 . Finally a¯ 11 Ga11 , and a¯ 31 Ga31 imply that a¯ 01Ca¯ 21 Ga01Ca21 ; a¯ 22 Ga22 and a¯ 42 Ga42 imply that a¯ 02 Ga02 ; a¯ 55 Ga55 , and a¯ 15 Ga15 imply that a¯ 05Ca¯ 25 Ga05Ca25 . In summary, the following individual parameters and parameter combinations are globally identifiable from this experiment: k13 , k24 , k2e , k31 , k42 , k51 , V1 ,

k12 k52 , (k1eCk12), , (k5eCk52) V2 V2

(As before, if V2 were known a priori, then all of the rate constants in the model would be globally identifiable.) In the model proposed in Duffull et al. (1), elimination of ivabradine directly to the environment was assumed not to take place either from Compartment 1 or Compartment 5 (i.e., k1e G k5e G0), in which case the remaining parameters k12 , k52 , and V2 are globally identifiable. CONCLUSION The structural identifiability of a model for the pharmacokinetics of ivabradine and one of its metabolites (S-18982), for both intravenous and oral administration of the drug, has been considered. The model was based on that postulated in Duffull et al. (1) with, at least initially, elimination


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directly to the environment from Compartment I1 and from the Gut compartment included. By applying the similarity transformation approach for structural identifiability analysis it was found that the model was unidentifiable for both the intravenous only dose, and the combined intravenous and oral doses. In particular, it was shown that the model of Fig. 1 is globally identifiable if the volume of distribution V2 of S-18982 in the central compartment is known a priori. However, by simplifying the model in this paper to obtain that proposed in Duffull et al. (1), a globally structurally identifiable model is obtained. For the volume of distribution of S-18982 in the central compartment to be known a priori requires the administration of S-18982 directly to humans or the use of a scaled volume term from animal studies. This raises the interesting problem of carrying forward models into practice that are identifiable but mechanistically incorrect or inadvertently carrying forward an unidentifiable model that has greater mechanistic accuracy. In the former case as long as the assumptions are understood adequately and the model is not used for inference in situations that deviate from those of the original study there should be no problem. Indeed the simulation model developed by Duffull et al. (1) was shown to reflect adequately the concentration–time course of ivabradine and S-18982 when used to simulate a new data set. In the latter scenario it is important that all parameters (or, possibly, combinations of parameters) that have practical significance are globally identifiable, and so a formal identifiability analysis may be required. REFERENCES 1. S. B. Duffull, S. Chabaud, P. Nony, C. Laveille, P. Girard, and L. Aarons. A pharmacokinetic simulation model for ivabradine. Eur. J. Pharm. Sci. 10:285–294 (2000). 2. K. R. Godfrey. Compartmental Models and their Application, Academic Press, London and New York, 1983. 3. E. Walter. Identifiability of State Space Models, Springer-Verlag, Berlin, 1982. 4. E. Walter and Y. Lecourtier. Unidentifiable compartmental models: what to do? Math. Biosci. 56:1–25 (1981). 5. K. R. Godfrey and M. J. Chapman. Identifiability and indistinguishability of linear compartmental models. Math. Comput. Simul. 32:273–295 (1990). 6. M. J. Chapman and K. R. Godfrey. On structural equivalence and identifiability constraint ordering. In E. Walter (ed.), Identifiability of Parametric Models, Pergamon Press, Oxford, 1987, pp. 32–41. 7. Y. Hayakawa, S. Hosoe, M. Hayashi, and M. Ito. On the structural controllability of compartmental systems. IEEE Trans. Automat. Contr. AC–29:17–24 (1984). 8. R. M. Zazworsky and H. K. Knudsen. Controllability and observability of linear time invariant compartmental models. IEEE Trans. Automat. Contr. AC–23:872–877 (1978).