{"title":"First-order compartment model solutions – Exponential sums and beyond","authors":"Cyprian Świętaszczyk , Lars Jødal","doi":"10.1016/j.vascn.2024.107534","DOIUrl":null,"url":null,"abstract":"<div><p>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.</p></div>","PeriodicalId":16767,"journal":{"name":"Journal of pharmacological and toxicological methods","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1056871924000443/pdfft?md5=f175badb789b2a97176662cc39201e98&pid=1-s2.0-S1056871924000443-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of pharmacological and toxicological methods","FirstCategoryId":"3","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1056871924000443","RegionNum":4,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"PHARMACOLOGY & PHARMACY","Score":null,"Total":0}
引用次数: 0
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.
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.