Uncertain pharmacokinetic model based on uncertain differential equation
Introduction
As a branch of pharmacology, pharmacokinetics [6] investigates the drug and its metabolite kinetics following its introduction into the body. Pharmacokinetics plays an important role in developing new drugs and selecting the dose regimen in order to get the valid, secure and economic measure to prevent and charm away disease. Traditionally, these complex physiologic spaces or processes were expressed by a system of ordinary differential equations without uncertainty incorporated into models [20]. However real biological systems always be exposed to noises that are not completely understood or not feasible to model explicitly [1]. Considering such noises as random variables, stochastic pharmacokinetic models [3], [17], [18] driven by the Brownian motion became alternatives to above deterministic models. Unfortunately, stochastic differential equations may fail to model many time varying systems [12] due to properties of the Brownian motion. For example, a counterintuitive result for a stochastic pharmacokinetic model, i.e., the drug concentration increases or decreases with almost equal probability at any given time, is presented in Section 8. These facts motivate us to find more reasonable approaches to better describe the pharmacokinetics with noises.
Uncertainty theory [8], [10] based on the normality, duality, subadditivity and product axioms deals with personal belief degrees rationally. Under the framework of uncertainty theory, Liu process [10] was designed as a stationary independent increment process whose increments are normal uncertain variables rather than random variables. As counterparts of stochastic differential equations, uncertain differential equations [9] driven by Liu process reasonably model dynamic systems. Many researchers have made a significant number of contributions in this area. Chen and Liu [2] proved the existence and uniqueness theorem for uncertain differential equations under linear growth condition and Lipschitz condition. Some scholars were devoted to various kinds of stability properties [19], [25], [28]. Besides, Chen and Liu [2], Liu [14], and Yao [27] derived analytic solutions for some forms of uncertain differential equations. Yao and Chen [26] connected an uncertain differential equation with a family of ordinary differential equations using Yao-Chen formula, which opens the door of numerical methods [22], [23], [32] for uncertain differential equations. As an extension, Yao [29] combined uncertain differential equations with uncertain renewal processes to model discontinuous uncertain dynamic systems. Other extensions include high-order uncertain differential equations [30], uncertain delay differential equations [7], uncertain partial differential equations [24] and so on. A comprehensive and detailed exposition of uncertain differential equation can be found in Yao’s book [30]. For this theory to achieve its full potential, Yao and Liu [31] gave the moment estimations for unknown parameters in uncertain differential equations. Later, parameter estimations in uncertain delay differential equations [16] and generalized moment estimation [15] were proposed. Uncertain differential equation achieves fruitful results not only in theory but also in applications. Interested readers can refer to [13].
This paper joins the research stream by applying the tool of uncertain differential equations to pharmacokinetics to depict spontaneous erratic variations in the metabolic processes. The rest of this paper is arranged as follows. In Section 2, we will propose an uncertain pharmacokinetic model for mono-compartmental drugs administered with intravenous administration and derive the uncertain measure that the drug concentration is larger than the minimum effective concentration. Sections 3 and 4 are going to present uncertainty distributions, expected values and confidence intervals of the half-life and the area under the curve, respectively. Following that, moment estimations for unknown parameters in this uncertain pharmacokinetic model will be suggested in Section 5. A numerical example will be shown in Section 6. Section 7 is going to illustrate our methods by a real data analysis. Then a paradox of stochastic pharmacokinetic models will be presented in Section 8 to show that stochastic differential equations are not suitable for pharmacokinetics. Finally, Section 9 is going to give some conclusions and discussions.
Section snippets
Uncertain pharmacokinetic models
All drugs initially distribute into a central compartment before distributing into the peripheral compartment. If a drug rapidly equilibrates with the tissue compartment, for example aminoglycosides, then for practical purposes, we can use the mono-compartment model. Intravenous administration is used under situations such as drugs absorbed poorly from other routes, an immediate effect is desired or a high degree of accuracy for dose is needed. Many drugs are administered intravenously,
Properties of half-life
Half-life of a drug is the time required for the drug concentration to reduce to half of its initial value. In general, the effect of the drug is considered to have a negligible therapeutic effect after 4 half-lives, that is, when only of the original dose remains in the body. Theorem 3.1 The half-life for the uncertain pharmacokinetic model (1), i.e.,has an uncertainty distribution Proof Because is the time required for the drug
Area under the curve
In the field of pharmacokinetics, the area under the curve () [21] is the integral of drug concentration with respect to time, i.e., where is the drug concentration at time . The represents the total drug exposure across time, and becomes useful for knowing the average concentration over a time interval. Next, we give the inverse uncertainty distribution, the expected value and the confidence interval for at any given time . Theorem 4.1 The for the uncertain
Parameter estimation
As we can see, there are two unknown parameters and in the uncertain pharmacokinetic model (1). In this section, we employ the method of moments [31] to estimate these unknown parameters.
Give positive observations of the solution of at times with respectively before the drug is completely eliminated from the body. Note that Eq. (1) has a difference formwhich can be rewritten as
Numerical example
In this section, we document a numerical example to show our methods in detail. Example 6.1 Suppose that and in the uncertain pharmacokinetic model (1). Then we obtainAccording to Theorem 2.1, we get the drug concentration as It follows from Corollary 2.1 that at any given time the uncertainty distribution of the drug concentration iswhich is shown in Fig. 1. Figs. 2 and 3 show the
Real data analysis
In this section, we use one concentration-time profile of a single intravenous 18mg Nicorandil injection obtained from one acute heart failure patient, which can be found in [5]. Since the profile is almost a straight line on the logarithmic scale, we adopt the uncertain pharmacokinetic model (1), i.e.,It follows from Theorem 5.1 that unknown parameters and have moment estimations and respectively, and we obtain the estimated uncertain
Paradox of stochastic pharmacokinetic models
In this section, we present a paradox for the stochastic pharmacokinetic model for mono-compartmental drugs administered with intravenous administration. Based on stochastic differential equations, the pharmacokinetic model for mono-compartmental drugs administered with intravenous administration is deduced aswhere is the drug concentration at time are given nonnegative constants, and is the Brownian motion. Note that the solution for model (3) is
Conclusion
In order to rationally describe noises in pharmacokinetics, we proposed an uncertain pharmacokinetic model using uncertain differential equations. For mono-compartmental drugs administered with intravenous administration, some properties of the drug concentration, the half-life and the area under the curve were considered. What’s more, moment estimations for unknown parameters in this uncertain pharmacokinetic model were presented. Following that, a numerical example and a real data analysis
Acknowledgments
This work was supported by National Natural Science Foundation of China (No. 11771241) and National Natural Science Foundation of China (No. 62073009).
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