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Variability and singularity arising from poor compliance in a pharmacokinetic model II: the multi-oral case

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Abstract

We propose a stochastic model for the drug concentration in the case of multiple oral doses and in a situation of poor patient adherence. Our model is able to take into account an irregular drug intake schedule. This article is the second in a series of three. It presents a multi-oral version of the results given in Lévy-Véhel and Lévy-Véhel (J Pharmacokinet Pharmacodyn 40(1):15–39, 2013), that dealt with the multi-IV bolus case. Under the assumption that the irregular dosing schedule follows a Poisson law, we study features of the drug concentration that have practical implications, such as its variability and the regularity of its cumulative probability distribution, which describes its predictive power with respect to the mean behaviour. We consider four variants: continuous-time, with either deterministic or random doses, and discrete-time, also with either deterministic or random doses. Our computations allow one to assess in a precise way the effect of various significant parameters such as the mean rate of intake, the elimination rate, the absorption rate and the mean dose. They quantify how much poor adherence will affect the efficacy of therapy. To appreciate this impact, we provide detailed comparisons with the variability of concentration in two reference situations: a fully adherent patient and a population of fully adherent patients with log-normally distributed pharmacokinetic parameters. Besides, the discrete-time versions of our models reveal unexpected links with objects which have been studied in the mathematical literature under the name of infinite Bernoulli convolutions (Erdós, Am J Math 61:974-975, 1939). This allows us to quantify the fact that, when the random dosing schedule is too sparse, the concentration behaves in a very erratic way. Our results complement the ones in Lévy-Véhel and Lévy-Véhel (J Pharmacokinet Pharmacodyn 40(1):15–39, 2013) and help understanding the consequences of poor adherence. They may have practical outcomes in terms of drug dosing and scheduling.

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Notes

  1. This means that the integral of the square of the probability density function is finite.

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Acknowledgments

The research of L.J. Fermín was supported by project DIGITEO DIM, ANIFRAC: Uncertainties in processes with fractal characteristics, by a research grant from the project DIUV 2/2011 of the Universidad of Valparaíso, and partially supported by projects Anillo ACT1112 CONICYT-CHILE, and MathAmSud 16MATH03.

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Correspondence to Lisandro J. Fermín.

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L. J. Fermín was partially supported by projects: DIGITEO DIM ANIFRAC, DIUV REG.N2/2011 U. Valparaíso, Anillo ACT1112 CONICYT-CHILE, and MathAmSud 16MATH03.

Appendix: Proof of technical results

Appendix: Proof of technical results

Proof of Proposition 1

The characteristic function \(\varphi _t\) is easily obtained by applying Campbell’s theorem (Kingman 1993). In our case, this leads to

$$\begin{aligned}\begin{array}{lll} \varphi _t(\theta )&{} = &{} \displaystyle {\exp \left\{ \lambda \int _0^t \left( \exp \left\{ i \theta \alpha \left( e^{-k_e(t-x)} - e^{-k_a(t-x)}\right) \right\} -1\right) dx \right\} }\\ &{} &{} \times \displaystyle { \exp \left\{ i \theta \alpha \left( e^{-k_e t} - e^{-k_a t}\right) \right\} .} \end{array} \end{aligned}$$

where we recall that \(\alpha := \frac{F D}{V_d}\frac{k_a}{k_a-k_e}\) and the chance of variable \(u=e^{-(t-x)}\).

The change of variable \(u=e^{-(t-x)}\) leads to the formula 14. \(\square \)

Proof of Proposition 2

This is a direct application of the classical bound; see e.g. Loeve (1977, p. 209):

$$\begin{aligned} \mathbb {P}(C(t) \ge \gamma ) \le 7\gamma \int _0^{\frac{1}{\gamma }} \left( 1 - Re(\varphi _t(\theta )) \right) d\theta , \end{aligned}$$

valid for \(\gamma > 0\), and where Re denotes the real part. Indeed,

$$\begin{aligned} \begin{array}{lll} Re(\varphi _t(\theta ))&{} =&{} \cos \left( \theta \alpha (e^{-k_e t} - e^{-k_a t}) + \lambda \int _{e^{-t}}^1 \frac{ \sin \left( i \theta \alpha \left( u^{k_e} - u^{k_a}\right) \right) }{u} du \right) \\ &{}&{} \times \exp \left\{ \lambda \int _{e^{-t}}^1 \frac{cos\left( \theta \alpha \left( u^{k_e} - u^{k_a}\right) \right) -1}{u} du\right\} , \end{array} \end{aligned}$$

and routine estimates yield (17). Inequality (18) follows in a similar way. Finally, (19) is simply Chebychev inequality. \(\square \)

Proof of Proposition 3

When t tends to infinity, \(\varphi _t\) tends pointwise to \(\varphi \), which is continuous at \(\theta = 0\). By Lévy’s theorem, this implies the limit result.

On the other hand, note that the distribution of C invariant by time reversal: looking “backwards” in time, one see that C(t) has the same law as

$$\begin{aligned} \tilde{C}(t) = \frac{F D}{V_d} \frac{k_a}{k_a - k_e} \sum _i \left( e^{-k_e T_i} - e^{-k_a T_i}\right) \mathbbm {1}_{(t\ge T_i)}. \end{aligned}$$

When t tends to infinity, the random variables \(\tilde{C}(t)\) converge almost surely to

$$\begin{aligned} \tilde{C} = \frac{F D}{V_d} \frac{k_a}{k_a - k_e} \sum _i \left( e^{-k_e T_i} - e^{-k_a T_i}\right) , \end{aligned}$$

which therefore has the same distribution as C. Thus, we can write \(C=\tilde{C}\), since we are only interested in distributional properties. \(\square \)

Proof of Proposition 4

One computes:

$$\begin{aligned} |\varphi (\theta )| = \exp \left\{ - \lambda \int _0^1 \frac{ 1 - cos\left( \theta \alpha \left( u^{k_e} - u^{k_a}\right) \right) }{u} du \right\} =: \exp \left\{ -\lambda I(\theta ) \right\} . \end{aligned}$$

Set \(0< \delta < 1\), and decompose the integral \(I(\theta )\) as follows:

$$\begin{aligned} I(\theta ) = I_1(\theta ) - I_2(\theta ) - log(\delta ), \end{aligned}$$
(31)

where

$$\begin{aligned} \begin{array}{lll} I_1(\theta ) &{} = &{} \displaystyle { \int _0^{\delta } \frac{ 1 - cos\left( \theta \alpha \left( u^{k_e} - u^{k_a}\right) \right) }{u} du,}\\ &{}&{}\\ I_2(\theta ) &{} = &{} \displaystyle {\int _{\delta }^1 \frac{ cos\left( \theta \alpha \left( u^{k_e} - u^{k_a}\right) \right) }{u} du.} \end{array} \end{aligned}$$
(32)

When \(0 \le u \le \delta \), we have that \(\alpha \left( u^{k_e} - u^{k_a}\right) \sim |\alpha | u^{k^*}\) with \(k^*= \min \{k_e, k_a\}\), and thus

$$\begin{aligned} \begin{array}{lll} I_1(\theta ) &{} \sim &{} \displaystyle { \int _0^{\delta } \frac{ 1 - cos\left( \theta \alpha u^{k^*} \right) }{u} du }\\ &{}&{}\\ &{} \sim &{} \displaystyle { \frac{1}{k^*} log(\theta |\alpha |) + \frac{\gamma _e}{k^*} + log(\delta ) + O\left( \frac{1}{\theta }\right) }, \end{array} \end{aligned}$$
(33)

where \(\gamma _e\) is the Euler constant.

Denote \(h(u)= \alpha \left( u^{k_e} - u^{k_a}\right) \). The function h has a global maximum at \(u_0=\left( \frac{k_e}{k_a}\right) ^{\frac{1}{k_a - k_e}}\) with \(h^{''}(u_0)<0\). Stationary phase arguments imply that, when \(\theta \) tends to infinity,

$$\begin{aligned} \begin{array}{lll} I_2(\theta ) &{} \sim &{} \displaystyle { Re\left( \int _{\delta }^1 \frac{ e^{i \theta \alpha ( u^{k_e} - u^{k_a} )} }{u} du \right) } \\ &{}&{}\\ &{} \sim &{} \displaystyle { Re\left( \frac{ e^{i ( \theta h(u_0) - \pi /4)} \sqrt{2\pi }}{u_0 \sqrt{\theta |h^{''}(u_0)|}} \right) }\\ &{}&{}\\ &{}=&{} \displaystyle { \frac{cos(\theta h(u_0) - \pi /4)}{u_0} \sqrt{ \frac{2 \pi }{ \theta |h^{''}(u_0)| }} }\\ &{}&{}\\ &{} = &{} \displaystyle { O\left( \frac{1}{\sqrt{\theta }}\right) }. \end{array} \end{aligned}$$
(34)

Formulas (31), (32), (33) and (34) entail that

$$\begin{aligned} I(\theta ) = \frac{1}{k^*} log(\theta |\alpha |) + \frac{\gamma _e}{k^*} + O\left( \frac{1}{\sqrt{\theta }}\right) . \end{aligned}$$

This implies that there exists \(K>0\) such that \(|\varphi (\theta )|\sim K|\theta |^{-\mu ^*}\) when \(\theta \) tends to infinity. \(\square \)

Proof of Proposition 5

From (23), one deduces easily the mean and variance of \(C^{rd}\):

$$\begin{aligned} (\varphi ^{rd}_t)'(\theta ) = \varphi ^{rd}_t(\theta )\left[ \lambda \int _0^t \int _A i\alpha u h(t-x)e^{i\theta \alpha u h(t-x)}\nu (x,du) dx + i \alpha h(t) \right] , \end{aligned}$$

and thus

$$\begin{aligned} (\varphi ^{rd}_t)'(0) = i\alpha \left( \mu _e (1- e^{-k_e t}) - \mu _a (1-e^{-k_a t}) \right) \mathbb {E}_x(D_1) + i\alpha \left( e^{-k_e t} - e^{-k_a t}\right) . \end{aligned}$$

where \(\mathbb {E}_x(D_1)=\int _Au \nu (x,du)\) is the expectation of \(D_i\) knowing that \(T_i=x\), which is equal to one for any x by assumption. Thus we find that:

$$\begin{aligned} \mathbb {E}(C^{rd}(t)) = \mathbb {E}(C(t)). \end{aligned}$$

Likewise,

$$\begin{aligned} (\varphi ^{rd}_t)''(\theta ) \!= & {} \!\varphi ^{rd}_t(\theta ) \left[ \left( \lambda \int _0^t \int _A i\alpha u h(t-x) e^{i\theta \alpha u h(t-x)} \nu (x,du) dx - \alpha ^2 h^2(t-x) \right) ^2 \right. \\&\left. - \lambda \int _0^t \int _A \alpha ^2 u^2 h^2(t-x) e^{i\theta \alpha u h(t-x)} \nu (x,du) dx \right] . \end{aligned}$$

Using that, by definition, \(\int _A u^2\nu (x,du) = \mathbb {E}_x(D_1^2)\), this entails:

$$\begin{aligned} (\varphi ^{rd}_t)''(0)= & {} - \mathbb {E}(C(t))^2 - \lambda \alpha ^2 \int _0^t h^2(t-x) \int _A u^2 \nu (x,du) dx \\= & {} - \mathbb {E}(C(t))^2 - \lambda \alpha ^2 \int _0^t h^2(t-x) \mathbb {E}_x(D_1^2) dx \end{aligned}$$

or, since \((\varphi ^{rd}_t)''(0) = - \mathbb {E}(C(t))^2 - \mathrm {Var}(C^{rd}(t))\),

$$\begin{aligned} \mathrm {Var}(C^{rd}(t)) = \lambda \alpha ^2 \int _0^t h^2(t-x) \mathbb {E}_x(D_1^2) dx. \end{aligned}$$

However,

$$\begin{aligned} \mathbb {E}_x(D_1^2) = \mathrm {Var}_x(D_1) + \mathbb {E}_x(D_1)^2 = \mathrm {Var}_x(D_1) + 1 \end{aligned}$$

(\(\mathrm {Var}_x(D_1)\) denotes the variance of \(D_i\) knowing that \(T_i=x\)) and

$$\begin{aligned} \lambda \alpha ^2 \int _0^t h^2(t-x) dx = \mathrm {Var}(C(t)), \end{aligned}$$

thus

$$\begin{aligned} \mathrm {Var}(C^{rd}(t)) = \mathrm {Var}(C(t)) + \lambda \alpha ^2 \int _0^t h^2(t_x) \mathrm {Var}_x(D_1) dx. \end{aligned}$$

Since \(\mathrm {Var}_x(D_1) = \mathrm {Var}(D_1)\), one gets:

$$\begin{aligned} \mathrm {Var}(C^{rd}(t)) = \mathbb {E}(D_1^2)\mathrm {Var}(C(t)) = (1 + \mathrm {Var}(D_1))\mathrm {Var}(C(t)). \end{aligned}$$

\(\square \)

Proof of Proposition 6

Thanks to the assumption on \(\nu \), one computes

$$\begin{aligned} |\varphi ^{rd,u}(\theta )|= & {} \exp \left\{ -\lambda \int _A \int _{0}^1\frac{ 1- \cos ( \theta y \alpha (u^{k_e}- u^{k_a})) }{u} du \nu (dy) \right\} \\= & {} \exp \left\{ -\lambda \int _A I(\theta y) \nu (dy) \right\} . \end{aligned}$$

The proof of Proposition 4 shows that, when \(\theta \) tends to infinity,

$$\begin{aligned} -\lambda I(\theta y)\sim K -\mu ^*(\log (\theta )+\log (y)) \end{aligned}$$

for a certain constant K, with in addition \(-\lambda I(\theta y) - K +\mu ^*(\log (\theta )+\log (u))\) tending to 0 with a rate of convergence \(\frac{1}{\sqrt{\theta }}\). Thus, using the assumption on the logarithmic moment of \(\nu \) and taking \(q=\nu (A)\),

$$\begin{aligned} -\lambda \int _A I(\theta y) \nu (dy) \sim K -q\mu ^*\log (\theta )-\mu ^*\int _A \log (y) \nu (dy) \end{aligned}$$

and one finishes up the proof with the help of a dominated convergence argument to show that the difference between the right-hand side and the left-hand side in the equivalent above indeed tends to 0 when \(\theta \) tends to infinity. \(\square \)

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Fermín, L.J., Lévy-Véhel, J. Variability and singularity arising from poor compliance in a pharmacokinetic model II: the multi-oral case. J. Math. Biol. 74, 809–841 (2017). https://doi.org/10.1007/s00285-016-1041-1

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