Linear pharmacokinetic model of first order metabolism in the liver

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In this study we demonstrate the added value of mathematical model of metabolism for drug modi cation into metabolites. We show that for speci ed parameter val-ues, the model proposed by Polkings et al. (2005) can be substituted into a proposed metabolism model, which can describe the dynamics of drug change in the liver. When we ingest a drug into our body, the body absorbs the drug into the blood stream. Then the body breaks that drug into smaller pieces called metabolites. This conversion of drugs into metabolite is the third phase of the Pharmacokinetic phenomenon (ADME). Mathematical modeling of Pharmacoki-netics (PK) is the rate of change in concentration of a medical drug as it goes through di erent compartments in the human body. In the third phase of drug processing in the body, it is expected that the absorbed drug has to undergo bio-transformation in the liver before nal excretion of the metabolites through the kidney or any other excretory system. This third phase of PK (drug metabolism) is an inevitable processing stage of a drug in order to prevent toxicity build-up due to re-absorption of un-metabolised active substrate (drug concentration). This work is an extension of the study done by Polking, Boggess and Arnold (2005). Their research ndings shows that drug substrate in the human organ or tissue can be analytically determined using rst-order di erential equation. However their model encapsulates only the rst two phases (AD) of the whole (ADME) process and moreover their model hardly tells us the fate of the active substrate after pharmacological action (healing e ect) in the human organ (tissue). We develop a rst-order di erential equation model characterizing the metabolism iii reaction in the liver after direct transportation of active substrate from the tis-sue compartment. Primarily, this goal is achieved by using principles in mixing problems, methods of integration factor and integration by parts. This study combines the rst three phases (ADM) of (ADME) and our analysis reveals and also demonstrate the critical conditions under which liver metabolism transpires. The result of our analysis will help improve medical dosing treatment strategies for non-linear drugs.
A thesis submitted to the Department of Mathematics, Kwame Nkrumah University of Science and Technology in partial fulfillment of the requirements for the degree of Master of Philosophy,