Closed form solutions and dominant elimination ... - Springer Link

J Pharmacokinet Pharmacodyn (2015) 42:151–161 DOI 10.1007/s10928-015-9407-3

ORIGINAL PAPER

Closed form solutions and dominant elimination pathways of simultaneous first-order and Michaelis–Menten kinetics Xiaotian Wu • Jun Li • Fahima Nekka

Received: 12 September 2014 / Accepted: 4 February 2015 / Published online: 13 February 2015 Ó Springer Science+Business Media New York 2015

Abstract The current study aims to provide the closed form solutions of one-compartment open models exhibiting simultaneous linear and nonlinear Michaelis–Menten elimination kinetics for single- and multiple-dose intravenous bolus administrations. It can be shown that the elimination half-time (t1=2 ) has a dose-dependent property and is upper-bounded by t1=2 of the first-order elimination model. We further analytically distinguish the dominant role of different elimination pathways in terms of model parameters. Moreover, for the case of multiple-dose intravenous bolus administration, the existence and local stability of the periodic solution at steady state are established. The closed form solutions of the models are obtained through a newly introduced function motivated by the Lambert W function.

Keywords Closed form solutions Simultaneous elimination Elimination half-time Dominant role Lambert W function

X. Wu Department of Mathematics, Shanghai Maritime University, Shanghai 201306, China X. Wu J. Li F. Nekka (&) Faculte´ de Pharmacie, Universite´ de Montreál, Montreál, QC H3T1J4, Canada e-mail: [email protected] J. Li F. Nekka Centre de Recherches Mathe´matiques, Universite´ de Montreál, Montreál, QC H3C3J7, Canada

Introduction Substances that manifest parallel elimination mechanisms are not rare, including growth factors and ethanol [1–5]. Their elimination generally involves a linear elimination through organs (kidney, lung, skin, etc.), combined simultaneously with a non-linear elimination that metabolizes and clears the parent products. This kind of elimination kinetics are typically exhibited by hormone drugs, such as granulocyte colony-stimulating factor (G-CSF), which are used in combination with chemotherapeutic agents to stimulate the production of hematopietic cells when their number is reduced [1–4]. While several models of these kinetics have been reported, their systematic analysis allowing for a quantitative characterization of the interaction between these two elimination pathways is still lacking. To model the pharmacokinetics (PK) of such drugs, compartment models, considering a first-order and a Michaelis–Menten elimination at the same time, are widely used [2, 4, 6]. While the major focus of the literature is concerned with the estimation of model parameters from data, several papers have reported analytic solutions of compartmental models involving a single elimination pathway, Michaelis–Menten in particular [7]. However, though the involvement of more than one pathway of different natures can potentially influence the drugs performance, analytic solutions of models with simultaneous elimination are still to be established. It is well known that analytic solutions are preferable since they provide a clear and direct way to explore the underlying mechanisms, reveal intrinsic relationships between different model components, and easily derive expressions of related pharmacological indices. Whereas the area under the concentration time curve and mean residence time have been studied within this context [8, 9], the expressions of other pharmacological indices,

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such as the elimination half-time and multi-dose characteristics are still to be established. Moreover, since the two pathways can have different contributions to drug disposition, distinguishing their relative roles in terms of model parameters can greatly inform the process of drug development and design [10]. When drug disposition is described by linear mechanisms using compartment models, closed forms of drug concentrations are generally obtained using Laplace transforms. In the nonlinear case involving Michaelis– Menten elimination, there is no standard mathematical tool that can be easily applied. Many efforts have been put so far to numerically estimate the model parameters and predict drug concentrations using, for example, Runge– Kutta method for differential equations, or root-solving methods, such as Levenberg-Marquardt algorithm, for their associated integrated forms [11–13]. Meanwhile, research interests to find closed form solutions of these nonlinear elimination models have never slowed down. Schnell and Mendoz [14] were the first to use Lambert W function for a model of enzyme reaction processes with Michaelis– Menten kinetics [14, 15]. More recently, Tang and Xiao [7] provided a detailed closed form solution for the one compartment model with a Michaelis–Menten elimination pathway alone, under different administrations, precisely for zero-order infusion, single and multiple dose intravenous bolus [7]. In this paper, we will provide the closed form solutions of simultaneous first-order and Michaelis– Menten elimination for both single- and multiple-dose intravenous bolus administrations. Inspired by the Lambert W function, we here propose a new function to explicitly express the drug concentration in a closed form for models with the two parallel pathways, in the case of single and multiple intravenous bolus administrations. This well defined function can be implemented into software packages such as Matlab for symbolic and numerical purposes. As a fallout, owing to this function, we are able to provide the explicit expressions of those PK indices mentioned above. More importantly, for the first time, we analytically distinguish the dominant contribution of each elimination pathway along the time course of drug disposition. This characterization, driven by model parameters, offers a valuable means to explore drug disposition in different therapeutic situations.

Closed form solutions for the simultaneous elimination models Single intravenous bolus administration The drug disposition model that we discuss in this paper is described as follows

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J Pharmacokinet Pharmacodyn (2015) 42:151–161

Vd

dCðtÞ Vmax CðtÞ ¼ CLl CðtÞ ; dt Km þ CðtÞ

t [ 0;

Cð0þ Þ ¼ D=Vd ¼ C0 :

ð1Þ ð2Þ

where CðtÞ is the drug plasma concentration at time t, Vd is the apparent volume of distribution, D is the dose amount, CLl is the clearance for the linear elimination process, Vmax is the theoretical maximum rate of the process, and Km , known as the Michaelis–Menten constant, is the concentration value when the rate change reaches Vmax =2, C0 is the drug concentration at time t ¼ 0. Eq. 1 can be transformed to Vd Km þ CðtÞ dCðtÞ ¼ dt: 2 max CLl C ðtÞ þ ðKm þ VCL ÞCðtÞ

ð3Þ

l

A partial fraction decomposition and a rearrangement of Eq. 3 lead to p1 p2 þ dCðtÞ ¼ dt ð4Þ CðtÞ CðtÞ þ b where Vd CLint Vd ; p2 ¼ ; CLl þ CLint CLl CLl þ CLint CLl þ CLint b ¼ Km CLl

p1 ¼

ð5Þ

in which CLint ¼ Vmax =Km represents the intrinsic clearance of Michaelis–Menten kinetics [16]. CLl þ CLint is the total body clearance in the absence of drug. Hence p1 represents the corresponding time for the drug to be eliminated, and p2 represents this time (p1 ) modulated by the ratio of contribution of the nonlinear elimination to the linear pathway. b can be seen as the concentration of a model having a linear elimination coefficient CLl =Vd , that gives the same rate of change at Km , for the same model, but the latter with a total elimination rate ðCLl þ CLint Þ=Vd . More explicitly, if we consider the two models dC1 CLl ¼ C1 ðtÞ dt Vd

ð6Þ

and dC2 CLl þ CLint ¼ C2 ðtÞ; dt Vd

ð7Þ

then b is the concentration of Eq. 6 such that dC1 dC2 ¼ : dt C1 ¼b dt C2 ¼Km Integrating Eq. 4 from time 0þ to time t, we obtain p1 ln CðtÞ þ p2 lnðCðtÞ þ bÞ ¼ p1 ln Cð0þ Þ þ p2 lnðCð0þ Þ þ bÞ t; ð8Þ


153

which is equivalent to p2

p1

CðtÞ ðCðtÞ þ bÞ ¼

C0p1 ðC0

p2

þ bÞ expðtÞ:

ð9Þ

Equation 9 is a transcendental equation, hence its solution cannot be expressed in a conventional way. In order to obtain the explicit expression of Eq. 9, we introduce below a new function, X, motivated by the well known Lambert W function [15] that has recently been applied to the case of Michaelis–Menten elimination pathway alone [7, 14].

we only have a linear elimination at play, thus Vmax can be considered as zero. Then Eqs. 5 reduce to p1 ¼ Vd =CLl , p2 ¼ 0 and b ¼ Km . By the definition of X function (Eq. Vd h iCL Vd 10), it is easy to see that Xðt; CLl ; 0Þ l ¼ t, which is CLl

Vd equivalent to Xðt; CL ; 0Þ ¼ t Vd . Hence Eq. 14 reduces to l ! Vd D CLl Vd expðtÞ; ;0 CðtÞ ¼ b X bVd CLl

"

Definition 1 We define Xðt; p; qÞ as the solution of the following equation p q ð10Þ Xðt; p; qÞ Xðt; p; qÞ þ 1 ¼ t

¼b ¼

where p [ 0 and q 0 are given constants, t is a variable. In fact, the left side of Eq. 10 has the form of f ðx; p; qÞ ¼ xp ð1 þ xÞq , which is positive and monotonically increasing for x [ 0, ensuring that X is a welldefined function. Moreover, we have 1 d 1 p q X ðt; p; qÞ ¼ þ [ 0; dt t Xðt; p; qÞ 1 þ Xðt; p; qÞ ð11Þ which indicates that X function is smooth and strictly increasing for t [ 0. In the remaining of the article, we make use of X function to find the closed form of one compartment models (Eq. 1), for single- and multiple-dose intravenous bolus administrations. Dividing both sides of Eq. 9 by bp1 þp2 leads to p2 p1 p2 CðtÞ p1 CðtÞ C0 C0 þ1 ¼ þ 1 expðtÞ: b b b b ð12Þ Using the newly introduced X function, the solution of Eqs. 1–2 can be obtained in terms of the initial drug concentration and model parameters as follows p2 p1 C0 C0 CðtÞ ¼ b X þ 1 expðtÞ; p1 ; p2 ; t 0: b b ð13Þ Reparametrization with D=Vd instead of C0 , we finally obtain the closed form solution of the model defined by Eqs. 1–2 CðtÞ ¼ b X

D bVd

p1

p2 D þ 1 expðtÞ; p1 ; p2 ; bVd

D bVd

l #CL Vd

Vd CL

l

ð15Þ

expðtÞ

D CLl D expð tÞ ¼ expðkel tÞ: Vd Vd Vd

which is exactly the known closed form solution of the first-order elimination model. Here kel is the associated first-order elimination rate constant. Another extreme case is when there is only a capacitylimited Michaelis–Menten elimination pathway involved, which implies that CLl ¼ 0. However, we cannot directly deduce the closed form solution of the Michaelis–Menten case from Eq. 14, since b and p2 are not defined in this case. However, we can show that the Michaelis–Menten closed form can be obtained as a limiting case of the general form of Eq. 14 when CLl ! 0. In fact, the solution given by Eq. 14 is identical to p2 p2 CðtÞ C0 þ 1 ¼ ðC0 Þp1 þ 1 expðtÞ ð16Þ ðCðtÞÞp1 b b by multiplying both sides of Eq. 12 with bp1 . Letting CLl ! 0, we have ( p2 CðtÞb )CLlim!0fp2 CðtÞ b g l CðtÞ CðtÞ þ1 þ1 lim ¼ lim : CLl !0 CLl !0 b b As the parameter p1 tends to CLVdint and both p2 and b tend to infinity when CLl ! 0, we have p2 CðtÞ Vd CLint CðtÞ ¼ lim lim CLl !0 CLl !0 CLl þ CLint Km ðCLl þ CLint Þ b Vd CðtÞ : ¼ Vmax 1

Combining this with the property limð1 þ xÞx ¼ e, we have x!0

t 0;

ð14Þ where p1 , p2 , b are the same as defined in Eqs. 5. There are two extreme cases for this simultaneous firstorder and Michaelis–Menten elimination model. Suppose

p2 CðtÞ Vd CðtÞ þ1 ¼ exp : CLl !0 b Vmax p2 C0 Vd C0 þ 1 ¼ exp Similarly, lim . Under the CLl !0 b Vmax previous deduction, when CLl ! 0, Eq. 16 becomes lim

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J Pharmacokinet Pharmacodyn (2015) 42:151–161 Vd

ðCðtÞÞCLint expð

Vd CðtÞ Þ Vmax

Vd C0 Vmax t Þ: Vmax

Vd

¼ ðC0 ÞCLint expð

ð17Þ

1 int Taking Eq. 17 to the power CL Vd and then multiplying by Km , this reduces to C CðtÞ C V t CðtÞ 0 ð18Þ exp exp 0 V : ¼ Km Km Km Km

p2 CðtÞ p1 CðtÞ þ1 b b p2 p Cððn 1Þsþ Þ 1 Cððn 1Þsþ Þ þ1 ¼ b b

ð23Þ

expððt ðn 1ÞsÞÞ

max d

The closed form solution of Eq. 18 can be given using Lambert W function [15] as D D Vmax t CðtÞ ¼ Km W exp ; ð19Þ V d Km Km Vd which is consistent with the solution given in [7, 14]. More explicitly, we have p2 D p1 D þ 1 expðtÞ; p1 ; p2 CðtÞ ¼ b X bVd bVd D D Vmax t ! Km W exp ; as CLl ! 0: Vd Km K m Vd Remark Note that Lambert W function can be considered as a limit case of X functions, however it does not belong to the family of X functions. Moreover, our proposed X function can be used to express closed form solutions of all cases of elimination: single first-order elimination, single Michaelis–Menten elimination and their combined version as the one considered here. Multiple-dose intravenous bolus administration X function can also be used to express the closed form of the simultaneous elimination model in the case of multiple-dose intravenous bolus administration. Indeed, considering a dosing interval s, the drug disposition can be described as Vd

dCðtÞ Vmax CðtÞ ¼ CLl CðtÞ ; t 6¼ ðn 1Þsþ ; dt Km þ CðtÞ ð20Þ þ

Cððn 1Þs Þ ¼ Cððn 1ÞsÞ þ D=Vd ; t ¼ ðn 1Þs

þ

ð21Þ for n ¼ 1; 2; , Cð0Þ ¼ 0 and Cððn 1Þsþ Þ refers to the concentration at time ðn 1Þs immediately following an administered dose D. Integrating from ðn 1Þsþ to t in Eq. 4, the integration of Eqs. 20–21 leads to CðtÞp1 ðCðtÞ þ bÞp2 ¼ Cððn 1Þsþ Þp1 ðCððn 1Þsþ Þ þ bÞp2 expððt ðn 1ÞsÞÞ: Dividing both sides of Eq. 22 by bp1 þp2 yields

123

ð22Þ

def

¼ gðCððn 1Þsþ Þ; tÞ:

Based on the definition of X function (Eq. 10), the solution is obtained as

CðtÞ ¼ b X g Cððn 1Þsþ Þ; t ; p1 ; p2 ; t 2 ððn 1Þs; ns: ð24Þ In particular, the trough drug concentration is CðnsÞ ¼ b X ðgðCððn 1Þsþ Þ; nsÞ; p1 ; p2 Þ;

ð25Þ

and the concentration immediately after a dose D is D Cðnsþ Þ ¼ b X ðgðCððn 1Þsþ Þ; nsÞ; p1 ; p2 Þ þ Vd ð26Þ D ¼ CðnsÞ þ : Vd To find the steady-state concentration at any time t, we write Cn ¼ CðnsÞ. Then Eq. 25 reduces to the following difference equation Cn ¼ b X ðgðCn1 þ C0 ; nsÞ; p1 ; p2 Þ: ð27Þ Using the definition of X function, Eq. 27 becomes 1s s Cn Cn þ b ¼ expðkel sÞ Cn1 þ C0 þ b Cn1 þ C0

ð28Þ

CLint , C0 , b, kel and s are defined as beCLl þ CLint fore. Let n tend to infinity, the steady-state solution C verifies the following equation 1s s C C þ b ð29Þ ¼ expðkel sÞ: C þ C0 þ b C þ C0 where s ¼

By introducing another well-defined function that we denote by Y, we can show that Eq. 29 has a unique positive solution C which is given by D C ¼ Y expðkel sÞ; s; ; b : ð30Þ Vd The definition of Y and proof of the solution are detailed in Appendix A. Finally, the unique periodic solution of Eqs. 20–21 can be given by p2 D p1 D C þ Vd C þ Vd CðtÞ ¼ b X þ1 b b expððt ðn 1ÞsÞÞ; p1 ; p2 Þ; t 2 ððn 1Þs; ns: ð31Þ


155

It is important to emphasize here that, contrary to the case of the single Michaelis–Menten elimination, where the solution exists only under the condition D=s\Vmax [7], we have shown that the steady-state solution of the simultaneous elimination model exists without restriction. Moreover, we have shown the local stability of the periodic solution (31) (Appendix B). These periodic solutions at steady-state which abided by Eqs. 20–21 generated by X function, for two dose regimens are shown in Fig. 1. It is natural to investigate the impact of dose D and dosing interval s on the drug kinetics at steady-state. In particular, one may be interested in the steady-state minimum and maximum concentrations and their ranges, i.e., ss ss ss ss Cmin , Cmax and Cmax Cmin . At the steady-state, these concentrations can be given by p2 D p1 D C þ Vd C þ Vd ss þ1 Cmin ¼ b X b b def

expðsÞ; p1 ; p2 Þ ¼ C ; and p2 D p1 D C þ Vd C þ Vd þ1 b b D expðsÞ; p1 ; p2 Þ þ Vd D def ¼C þ ¼C ; Vd

ss Cmax ¼bX

D : Vd

ss ss and Cmax We can notice as displayed in Fig. 2(a) that Cmin increase in a non-linear way with respect to dose. However, for a fixed dose, these concentrations are decreasing with respect to dosing interval. Interestingly, the concentration

7

Concentration (ug/ml)

4 3 2 1

0

The half-time of elimination t1=2 is the time required for the drug concentration at any time point to decrease by onehalf [17]. When only a linear elimination is involved, this parameter is known to be independent of the concentration and given by ln 2=kel . We now consider the time required for a concentration to reduce by half. Suppose Cðt0 Þ to be the initial concentration, then replacing Cð0þ Þ by Cðt0 Þ and CðtÞ by Cðt0 Þ=2 in Eq. 8 and solving for time t to find t1=2 , we have t1=2 ¼ ðp1 þ p2 Þ ln 2 þ p2 ln

Cðt0 Þ þ b Cðt0 Þ þ 2b

1 Vmax ln 2 þ kel kel ðKm kel Vd þ Vmax Þ Vd kel Cðt0 Þ þ Vd Km kel þ Vmax ln Vd kel Cðt0 Þ þ 2Vd Km kel þ 2Vmax Vd Vd Vmax ¼ ln 2 þ CLl CLl ðKm CLl þ Vmax Þ CLl Cðt0 Þ þ CLl Km þ Vmax : ln CLl Cðt0 Þ þ 2CLl Km þ 2Vmax

5

0

Since two elimination pathways are present simultaneously within the same model, their relative contributions to the drug disposition may vary for different concentration ranges. In this section, we will characterize this elimination by explicitly expressing their half-times and determining their roles in shaping the drug kinetics.

¼

τ = 1 hr τ = 2 hr

6

Characterization of elimination pathways

Half-time of elimination

and their range of concentrations ss ss Cmax Cmin ¼

range is proportional to dose regardless of dose interval. Mathematical details can be found in Appendix A. It is also interesting to consider how different levels of linear elimination can influence the steady-state minimum and maximum concentrations as well as the range of concentrations. In the case of a large intrinsic clearance CLint , the concentration curves are barely sensitive to the linear elimination kel , as shown in Fig. 2(b). However, when the intrinsic clearance is small, these concentrations are relatively stable for a large kel but become very sensitive when kel is reduced. Since the range of concentrations is D=Vd , it is independent of the elimination.

10

20

30

40

Time (hr)

Fig. 1 Time courses of drug concentration for Eqs. 20–21 under two dose regimens. Red curve: s ¼ 1 h; blue curve: s ¼ 2 h. D ¼ 2 lg, CLl ¼ 0:2 ml/h, Km ¼ 0:8 lg/ml, Vmax ¼ 1 lg/h, Vd ¼ 1 ml

ð32Þ

This clearly indicates that t1=2 depends on the original drug concentration Cðt0 Þ (thus dose amount in the body). In fact, t1=2 is monotonically increasing with respect to drug concentration in the sense that a higher concentration requires more time to be eliminated. Interestingly, we observe that the half-time of elimination can have any value from the interval

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156


ss ss Fig. 2 Left panel Cmin , Cmax and the range versus dose for different dose intervals s: s ¼ 1 h for the brown riband and s ¼ 3 h for the green riband, CLl ¼ 0:2 ml/h, Vmax ¼ 1 lg/h. Right ss ss , Cmax and the range panel Cmin versus CLl for different CLint : Vmax ¼ 2 lg/h for the brown riband and Vmax ¼ 0:1 lg/h for the green riband, D ¼ 4 lg, s ¼ 2 h. For both cases, Km ¼ 0:8 lg/ml and Vd ¼ 1 ml. Solid ss and dashed lines refer to Cmin ss lines refer to Cmax

70

40

(a) Concentration (ug/ml)


(b)

35

60 50

τ= 1 hr 40 30 20

τ = 3 hr

10

30 25 20 15 10 CLint = 0.125 ml/hr 5 CLint = 2.5 ml/hr

0

2

4

6

8

10

12

0 0.05

0.1

0.15

0.25

0.3

0.35

It would be important to recover from Eq. 32, the halftimes for each elimination pathway alone. If we take Vmax as zero, then the model (Eqs. 1–2) reduces to a linear elimination model, and Eq. 32 reduces to ln 2=kel . However, to recover the case of Michaelis–Menten elimination alone, we cannot directly take the value of kel as zero in Eq. 32. Instead, we can consider CLl approaching 0, then t1=2 decays to that known for the Michaelis–Menten elimination alone. Indeed, Eq. 32 can be rearranged as Cðt0 Þ t1=2 ¼ p1 ln 2 þ p2 ln þ1 : ð33Þ Cðt0 Þ þ 2b

FO MM FO&MM 10

Half−time (hr)

0.2

CL l (ml/hr)

Dose (ug)

5

Letting CLl ! 0, we have 0

10

20

30

40

Dose (ug)

Fig. 3 Half-time versus dose: first order elimination (FO), Michaelis– Menten elimination (MM) and mixed elimination (FO&MM). The parameters are: Vd ¼ 1 ml, Vmax ¼ 1 lg/h, Km ¼ 1:2 lg/ml and CLl ¼ 0:5 ml/h. The interval of half-time for the simultaneous elimination model is ð0:5199; 1:3863Þ

Vd Vd ln 2; ln 2 : CLd þ CLint CLl

Indeed, the extreme values of this interval are the limit cases of t1=2 as Cðt0 Þ goes to zero or infinity in Eq. 32, that and is lim t1=2 ¼ p1 ln 2 ¼ Vd =ðCLl þ CLint Þ ln 2 Cðt0 Þ!0

lim t1=2 ¼ ln 2=kel . In order to have a clear idea of the

Cðt0 Þ!1

meaning of these two extreme values, consider the case of low concentrations, where the model (Eqs. 1–2) can be approximated by a linear pathway with elimination constant ðCLl þ CLint Þ=Vd and the corresponding half-time as the lower bound of the interval. On the other side, for a high concentration, the Michaelis–Menten pathway is saturated and the elimination of model (Eqs. 1–2) can be approximated by a linear pathway with a rate constant kel and a half-time as the upper bound of the interval.

123

p1 ln 2 !

Vd ln 2 CLint

and

Cðt0 Þ þ1 p2 ln Cðt0 Þ þ 2b

!

Vd Cðt0 Þ 2Vmax 1

0Þ using the property limð1 þ xÞx ¼ e with x ¼ CðtCðt . Then 0 Þþ2b

x!0

we have t1=2 ¼

Vd Vd Cðt0 Þ ln 2 þ 2Vmax CLint

which is consistent with the conclusion in [7]. Figure 3 illustrates the change of elimination half-time t1=2 with respect to dose considering simultaneous elimination model along with the two corresponding extreme cases. As shown in Fig. 3, t1=2 of model (Eqs. 1–2) increases with dose but not proportionally. At low concentrations, this increase is more rapid and reaches a plateau to become indistinguishable from the linear elimination case. However, the saturating process involved in the Michaelis–Menten elimination can make its t1=2 rise to infinity, which contrasts with the linear or simultaneous cases. In fact, it is the linear elimination which keeps the half-time of the simultaneous

J Pharmacokinet Pharmacodyn (2015) 42:151–161 4 D = 100 ug

D = 1000 ug

800

Dominant Contribution t max 1/2

3

600

2

400

Half−time (hr)


D = 10 ug

Concentration

1000

157

First order elimination First order elimination

Ctr

1

Michaelis−Menten elimination

t min

200

1/2

0 0 0

10

20

30

0

1

Time (hr)

l

Fig. 4 Time course of drug concentrations and the associated halftime curves for three different single doses of Eq. 1. Solid curves correspond to the left y-axis representing drug concentrations, and dash curves correspond to the right y-axis representing the corresponding half-times during drug disposition. The other parameters are given by Vd ¼ 1 ml, Vmax ¼ 1 lg/h, Km ¼ 0:8 lg/ml and CLl ¼ 0:2 ml/h

model under control. These mathematical driven observations are in line with previous experimental results [18, 19]. For example, authors in [19] observed that an increase of 10 fold in dose of filgrastim (from 100 to 1000 ug/kg) results in 1.5-fold increase in t1=2 (from 1.7 to 3.1 h) for the wild-type mice. Figure 4 illustrates the decay of drug concentration for three different doses along with the corresponding curves of half-time. Moreover, lower-bound and upper-bound of half min max times, t1=2 and t1=2 , are shown. Dominant role of elimination pathways

through linear and Michaelis–Menten pathways,

respectively. In the situation where CLl CLint , we have CLl CðtÞ CLint CðtÞ [

Vmax CðtÞ : Km þ CðtÞ

ð34Þ

Equation 34 indicates that the rate of drug amount linearly removed is larger than that through Michaelis–Menten; in this case we say that the pathway of first order elimination

int

Fig. 5 Dominant contribution of the first-order and Michaelis– Menten elimination pathways in model (Eq. 1)

is dominant. On the opposite, when CLl \CLint , let us first consider an equal rate of drug amount eliminated by each pathway, which means CLl CðtÞ ¼

Vmax CðtÞ : Km þ CðtÞ

ð35Þ

Equation 35 yields a solution Ctr as follows Ctr ¼

Vmax CLint CLl Km ¼ Km : CLl CLl

ð36Þ

When the drug concentration is lower than the threshold value Ctr , the rates of drug amounts eliminated from both pathways satisfy the following relation CLl CðtÞ ¼

The two elimination pathways being simultaneously involved does not necessarily mean that their contributions to the drug disposition are equal nor constant through the whole PK process. In fact, one pathway can dominate over the other. There could also be situations where their roles are mutually switched. In this section, using mathematical analysis, we will reveal the dominant role of each elimination pathway in terms of model parameters. This will set up a solid foundation for a better understanding of the mechanisms of drugs exhibiting such kinetics, hormone drugs for example [2, 18]. From the model (Eq. 1), it is obvious that, at any time t, the amount rates of eliminated drugs are CLl CðtÞ and Vmax CðtÞ , Km þ CðtÞ

Relative ratio r = CL /CL

Vmax Vmax CðtÞ; CðtÞ\ Km þ Ctr Km þ CðtÞ

ð37Þ

which indicates the dominance of the nonlinear Michaelis– Menten elimination over the linear one. When the drug concentration becomes larger than the threshold value Ctr , we obtain the following inverse relationship CLl CðtÞ ¼

Vmax Vmax CðtÞ; CðtÞ [ Km þ Ctr Km þ CðtÞ

ð38Þ

which shows a mutual switch in terms of dominance between the two pathways. Figure 5 illustrates the dominant role of each elimination pathway in terms of the threshold concentration and model parameters. Remark Denote r ¼ CLl =CLint , then this dominance behavior can be determined into two steps: if r [ 1, then the linear elimination dominates regardless of the concentration level; otherwise, Ctr plays the decisive role with dominance of Michaelis–Menten for concentrations below Ctr , and linear dominance otherwise. In order to have a clear idea of the dominant roles discussed above, we consider different values of the model

123

158


parameters CLl , Vmax and Km . Figure 6(a) indicates a case of dominance of linear elimination where CLl [ CLint , in which we can see that the concentration time curve with linear elimination alone is close to that of simultaneous model (Eqs. 1–2). This behavior is in fact independent of the initial dose. Figure 6(b) indicates a case of dominance of linear elimination where CLl ¼ CLint . In the case of CLl \CLint and C0 \Ctr ¼ 4:2 lg/ml, Michaelis–Menten elimination dominates as depicted in Fig. 6(c). However, if a large dose is initially administrated such that C0 [ Ctr , we observe that, at the beginning, the concentration time curve of the first-order elimination alone is close to that of the simultaneous model, indicating thus a linear dominance. Once the drug concentration decreases below the threshold Ctr , the concentration time curve of the Michaelis–Menten elimination alone is close to that of the simultaneous model, which indicates the dominance of Michaelis–Menten elimination, as depicted in Fig. 6(d). Our theoretical findings give a rational to previous experimental results reported in [2, 18], where the renal elimination has been found to play a major role when the drug concentration is high, and conversely, the Michaelis– Menten elimination was dominant for lower concentrations. Weak dominance As shown in Fig. 5, if CLl \CLint , the elimination dominance transfers from one pathway to the other, driven by

3

(a)

3

(b)

CLl(0.5) > CLint(0.2


Fig. 6 Dominant roles of elimination pathways of model (Eq. 1): FO, MM and FO&MM have the same meanings as in Fig. 3. a First-order elimination is dominant, kel ¼ 0:5 h. b Firstorder is dominant, CLl ¼ 0:2 ml/h. In a and b, D ¼ 2 lg, Vmax ¼ 1 lg/h, Km ¼ 5 lg/ml, thus CLint ¼ 0:2 ml/h. c Michaelis–Menten is dominant, where D ¼ 2 lg and C0 \Ctr . d First-order elimination is dominant first and then Michaelis–Menten elimination is dominant, D ¼ 8 lg and C0 [ Ctr . In c and d, CLl ¼ 0:2 ml/h, Vmax ¼ 1 lg/h, Vd ¼ 1 ml, Km ¼ 0:8 lg/ml

the concentration level. For drug concentration larger than a critical threshold Ctr , the linear pathway dominates and vice-versa. For concentrations far above or below Ctr , the dominance roles can be clearly set and the model consequently be simplified to keep only the dominant elimination pathway. However, for concentrations around Ctr , it is less obvious, and not necessary, to determine which pathway is clearly dominant. We will use weak dominance to refer to this situation. Since different dominance scenarios can exist and influence the PK parameters estimation, a clear classification into linear, Michaelis–Menten or weak dominance is important. The critical concentration Ctr is inversely increasing with CLl , and the associated weak dominance region increases as well. For example, we can define a weak dominance for concentrations within an interval of 20 % around Ctr . Figure 7 shows the evolution, for a constant value of CLint ¼ 1 ml/h, of the weak dominance in terms of CLl . When CLl take decreasing values as 0:7, 0:5 and 0:3 ml/h, the associated Ctr are 0:43, 1 and 2:33 l/ml, with the corresponding weak dominance intervals evolving from ½0:34; 0:51, ½0:8; 1:2 to ½1:86; 2:8 (unit: ug/ml), respectively. It is clear that for patients of different linear elimination, their Ctr values and their associated weak dominance intervals could be different, implying a the necessity for a careful consideration when estimating PK parameters, even when these patients share similar observations.

Ll(0.2) = CLint(0.2)

2

2

1

1

0

0

3

(c) CLl(0.2) < CLint(1.25)

8

(d) CLl(0.2) < CLint(1.25) Ctr=4.2 < C(0)

6

2 FO

4

MM FO&MM

1

2

0

0

2

4

6

Time (hr)

123

8

10

12

0

0

2

4

6

Time (hr)

8

10

12


159

becomes a special case of our derived expression of halftime. The closed form solutions have the advantage to make it possible to analytically derive other multiple-dose PK characteristic parameters such as the persistence factor (PF), the loss factor (LF) and the accumulation factor (AF) [10]. For the classical first-order elimination, they are expressed as

5


4 CL int=1.0 ml/hr

3

2

PF ¼ ekel s ;

1

0 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CL l (ml/hr)

Fig. 7 Region of weak dominance within an interval of 20 % of Ctr . The banded region represents the weak dominance region by varying CLl , and three error bars represents the interval of weak dominance at CLl ¼ 0:3; 0:5; 0:7 ml/h, Vd ¼ 1 ml. Km ¼ 1 lg/ml, Vmax ¼ 1 lg/h such that CLint ¼ 1 ml/h

Discussion

LF ¼ 1 ekel s

AF ¼

1 : 1 ekel s

It is obvious that all these factors depend on the dosing time interval, so that a frequent dosing leads to a rapid drug accumulation in the body and quickly achieves the steadystate. Using the closed form solution given in [7], we are able to express the respective factors for the Michaelis–Menten elimination alone as PF mm ¼ eu ;

LF mm ¼ 1 eu

V

In this paper, we proposed a new mathematical approach to obtain the closed form solutions of the one compartment model with simultaneous first-order and Michaelis–Menten elimination. In order to reach this goal, we introduced a well-defined function that we named X, inspired by the Lambert W function, to express the algebraic solution of this model. Using the X function, we were able to explicitly express the concentration-time curves for the cases of single- and multiple-dose intravenous bolus administration. Compared to the existing results for models having only one type of elimination, we have been able to provide a non-trivial generalization by extending their expressions to the simultaneous elimination case. More specifically, for the linear elimination alone, the well-known solution is a special case of X function. Moreover, the solution of the Michaelis–Menten elimination alone, which is expressed in terms of Lambert W function, is a limiting case of our general formula (Eq. 14) when CLl ! 0. However, we have to mention that the family of X functions does not include the Lambert W function, the latter being a limit case. We have also provided an analytical expression and established the stability of the periodic solution for the case of steady-state multiple-dose. We have also obtained the elimination half-time of the simultaneous model which is expressed as dose dependency and is upper and lower bounded. Contrary to the linear elimination case where the half-time is constant, higher concentration herein will need longer time to decrease by half. Moreover, we confirmed that the half-time for the Michaelis–Menten elimination alone given in [7]

and

and

AF mm ¼

1 ; 1 eu

D

is a positive constant. Additional where u ¼ max s Km Vd V d Km to the dosing time interval, these factors depend on dose too. Of particular interest, we notice that the accumulation factor is monotonically increasing with respect to dose D, thus giving rise to potential risk of toxicity. Now, for the simultaneous elimination model that we are discussing in this paper, these factors can be derived using the closed form solutions (Eq. 31) as ss Cmin ; þ D=Vd 1 ¼ ; 1 PF l&mm

PF l&mm ¼ and

AFl&mm

ss Cmin

LF l&mm ¼ 1 PF l&mm

where l&mm refers to simultaneous linear and Michaelis– Menten elimination pathways and the expression of the ss Cmin is given by C as shown in Eq. 30. It is clear that these factors are nonlinear to dose and dosing time interval. The higher the given dose, the larger is PF l&mm and the smaller is LF l&mm . Moreover, a shorter dosing time interval leads to a more rapid accumulation of the drug in the body, requiring thus less time to approach the steady-state (Fig. 8). It was also important to investigate the role of each elimination pathway in order to understand their mutual contributions. We found that two parameters, namely the ratio r and a threshold concentration Ctr play a major role in delineating the predominant roles of linear and Michaelis–Menten elimination. This can in fact be crucial in the process of PK modeling where the quality of parameters estimation can heavily depend on the levels of concentration.

123

160


Acknowledgments The authors thank the referees for their careful reading of the manuscript and valuable comments they made that helped us to improve the paper. The authors acknowledge the financial support of NSERC-Industrial Chair in Pharmacometrics, FRQNT, NSERC, Mprime, Novartis, Pfizer and InVentiv Health.

interval s is fixed, C would be monotonically increasing with the dose D (Fig. 9).

Appendix B In the following, we show that the periodic solution given by Eqs. 31 is locally asymptotically stable.

Appendix A Definition 2 For given parameters v 2 ð0; 1Þ, a 2 ð0; 1Þ and b 2 ð0; 1Þ, let us define Yðt; v; a; bÞ as the inverse 1v v s sþb function of hðsÞ ¼ , for s [ 0, sþa sþaþb which means hðYðt; v; a; bÞÞ ¼ t:

ð39Þ

For any given v 2 ð0; 1Þ, a 2 ð0; 1Þ and b 2 ð0; 1Þ, the derivative of hðsÞ is að1 vÞ av h0 ðsÞ ¼ hðsÞ þ [ 0; ð40Þ sðs þ aÞ ðs þ a þ bÞðs þ aÞ Thus hðsÞ is a continuous, differentiable and one-to-one map with respect to the variable s [ 0 such that the inverse function of hðsÞ exists and is globally unique. Moreover, we can verify that hð0Þ ¼ 0 and lim hðsÞ ¼ 1. s!1

In the following, we show how we use hðsÞ and ekel s to ensure the existence and uniqueness of C . Here for each dosing interval s we have ekel s 2 ð0; 1Þ such that there exists a unique C satisfying Eq. 29. Moreover, C is a monotonically decreasing function with respect to dosing interval s. In other words, the longer is the dosing interval, the smaller is the value of ekel s and then the smaller is the solution C to be obtained (Fig. 9). However, if the dosing

Theorem 1 The periodic solution given by Eqs. 31 is locally stable. Proof The model solutions described by Eqs. 20–21 will asymptotically approach the periodic solution given by Eqs. 31, for any given initial dose. In terms of Eq. 25, we define a stroboscopic map M : Cððn 1Þsþ Þ 2 ð0; 1Þ ! Cðnsþ Þ 2 ð0; 1Þ in the sense that Cðnsþ Þ ¼ b XðgðCððn 1Þsþ Þ; nsÞ; p1 ; p2 Þ þ D=Vd def

¼ MðCððn 1Þsþ ÞÞ: ð41Þ

We have shown that the fixed point of the map MðCððn 1Þsþ ÞÞ, denoted C ¼ C þ D=Vd and C expressed by Eq. 30, uniquely exists. Then solution C at equilibrium is locally stable provided that the following condition oMðCððn 1Þsþ ÞÞ ð42Þ \1: oCððn 1Þsþ Þ Cððn1Þsþ Þ¼C is satisfied. To show this, let z ¼ gðCððn 1Þsþ Þ; nsÞ, and according to the chain rule, we have

1 h(C*(Dsmall))

0.7

Persistence factor

0.8

τ = 2 hr

0.6

h(C*(Dlarge)) e−kelτ

0.6 0.4 τ = 4 hr

0.4

0.2

0.2 τ = 6 hr

0 0 1

5

10

20

30

0

C*(Dsmall)

4

C*(Dlarge)

8

10

Dose (ug)

Fig. 8 Persistence factor PF l&mm of model (Eqs. 20–21) versus dose size D for different dosing time intervals s. Vd ¼ 1 ml, Vmax ¼ 1 lg/h, Km ¼ 0:8 lg/ml, CLl ¼ 0:2 ml/h

123

Fig. 9 Unique existence of the solution C of Eq. 29: C is an increasing function with respect to dose D, here we take CLl ¼ 0:4 ml/h, Km ¼ 1 lg/ml, Vmax ¼ 0:4 lg/h, Vd ¼ 1 ml; s ¼ 1 h; Dsmall ¼ 1 lg and Dlarge ¼ 3 ug


161

oMðCððn 1Þsþ ÞÞ oCððn 1Þsþ Þ Cððn1Þsþ Þ¼C oz ¼ b Xz0 ðz; p1 ; p2 Þ : þ þ oCððn 1Þs Þ Cððn1Þsþ Þ¼C Cððn1Þs Þ¼C

2. 3.

ð43Þ On the one hand, replacing t by ns in the definition of g function, we obtain oz oCððn 1ÞsÞ Cððn1Þsþ Þ¼C p2 p1 C þ C0 s C þ C0 þ1 ¼e ð44Þ b b p1 p2 þ : C þ C0 C þ b þ C0 On the other hand, it follows from Eq. 25 that we have C Xðz; p1 ; p2 Þ : ¼ ð45Þ Cððn1Þsþ Þ¼C b Moreover, the definition of X function yields z Cððn1Þsþ Þ¼C p2 C þ C0 p1 C þ C0 þ 1 es : ¼ b b Substituting Eqs. 45 and 46 into Eq. 11 yields Xz0 ðz; p1 ; p2 Þ Cððn1Þsþ Þ¼C 1 i ¼ h p2 Cððn1Þsþ Þ¼C z Xðz;pp11 ;p2 Þ þ 1þXðz;p 1 ;p2 Þ ¼

C þC0 b

p1

C þC0 b

es þ1

4.

5.

6.

7.

8.

9.

ð46Þ 10.

11. 12. 13.

ð47Þ 14.

p2 h

bp1 C

2 þ Cbp þb

i:

15.

Substituting Eqs. 44, 47 into Eq. 43, we have p1 p2 oMðCððn 1Þsþ ÞÞ C þC0 þ C þbþC0 ¼ \1; p1 p2 oCððn 1Þsþ Þ Cððn1Þsþ Þ¼C C þ C þb

16.

ð48Þ

17.

which implies the local stability of the steady-state C . h

18.

19.

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