First-order compartment model solutions – Exponential sums and beyond

IF 1.3 4区 医学 Q4 PHARMACOLOGY & PHARMACY Journal of pharmacological and toxicological methods Pub Date : 2024-07-01 DOI:10.1016/j.vascn.2024.107534
Cyprian Świętaszczyk , Lars Jødal
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Abstract

First-order compartment models are common tools for modelling many biological processes, including pharmacokinetics. Given the compartments and the transfer rates, solutions for the time-dependent quantity (or concentration) curves can normally be described by a sum of exponentials. This paper investigates cases that go beyond simple sums of exponentials. With specific relations between the transfer rate constants, two exponential rate constants can be equal, in which case the normal solution cannot be used. The conditions for this to occur are discussed, and advice is provided on how to circumvent these cases. An example of an analytic solution is given for the rare case where an exact equality is the expected result. Furthermore, for models with at least three compartments, cases exist where the solution to a real-valued model involves complex-valued exponential rate constants. This leads to solutions with an oscillatory element in the solution for the tracer concentration, i.e., there are cases where the solution is not a simple sum of (real-valued) exponentials but also includes sine and cosine functions. Detailed solutions for three-compartment cases are given. As a tentative conclusion of the analysis, oscillatory solutions appear to be tied to cases with a cyclic element in the model itself.

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一阶区间模型解决方案--指数和及其他。
一阶分区模型是模拟许多生物过程(包括药代动力学)的常用工具。给定隔室和转移率后,随时间变化的数量(或浓度)曲线的解通常可以用指数和来描述。本文研究的情况超出了简单的指数之和。在传递速率常数之间存在特定关系的情况下,两个指数速率常数可能相等,在这种情况下就不能使用正常的解法。讨论了出现这种情况的条件,并就如何规避这些情况提供了建议。对于预期结果完全相等的罕见情况,给出了一个解析解的例子。此外,对于至少有三个隔室的模型,存在实值模型解涉及复值指数速率常数的情况。这导致示踪剂浓度的解中含有振荡元素,即解不是简单的(实值)指数和,还包括正弦和余弦函数。本文给出了三室情况的详细解法。作为分析的初步结论,振荡解似乎与模型本身具有循环元素的情况有关。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of pharmacological and toxicological methods
Journal of pharmacological and toxicological methods PHARMACOLOGY & PHARMACY-TOXICOLOGY
CiteScore
3.60
自引率
10.50%
发文量
56
审稿时长
26 days
期刊介绍: Journal of Pharmacological and Toxicological Methods publishes original articles on current methods of investigation used in pharmacology and toxicology. Pharmacology and toxicology are defined in the broadest sense, referring to actions of drugs and chemicals on all living systems. With its international editorial board and noted contributors, Journal of Pharmacological and Toxicological Methods is the leading journal devoted exclusively to experimental procedures used by pharmacologists and toxicologists.
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