9/4/17. Acetaminophen Overdose Modelling

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1 cetaminophen Overdose Modelling cetaminophen (generic name for tylenol) is a pain reliever and fever reducer, used to treat many condi@ons such as headache, muscle aches, arthri@s, backache, toothaches, colds, and fevers Unfortunately on its own, or in combina@on with other substances (hydrocodone, morphine, etc.), acetaminophen is widely used as a recrea@onal drug Overdose, in many cases, causes acute liver failure; ohen, within days, it progresses to cause death unless an immediate liver transplanta@on is performed We use mathema@cs to bejer understand the consequences of acetaminophen overdose to the func@oning of the liver cetaminophen Overdose math model Research problem: How, and in which cases, can a pa@ent who overdosed on acetaminophen be saved? [here we focus on understanding mathema@cs; we do not worry about biology] 1

2 The authors introduce a mathema&cal model to track the amount of acetaminophen (denoted by PP) whose toxicity is caused by the forma@on of N- acetyl- p- benzoquinoneimine (denoted by NPQI). NPQI causes damage to hepatocytes (liver cells) which form about 75-80% of the mass of the liver. The authors name the model MLD = Model for cetaminophen- induced Liver Damage They use a system of differen@al equa@ons, and known clinical data to calibrate their model. That damage, if not treated immediately, causes serious problems, including death. to calibrate means to assign numeric values to parameters (details follow) Building the MLD model nalyze the interac@on of PP with processes and chemical substances involved and then develop mathema@cal rela@onships Next slide: flow chart of interac@ons Slide aher: transla@ng interac@ons into a system of differen@al equa@ons 2

3 Look at the first dt = α H δ a We will learn math which will help us understand all this! Deriva@ve = rate of change, so this equa@on describes how = amount of PP changes (t in days we figured this out by reading the paper) What do the remaining symbols mean? dt = α H δ a H = number of func@onal hepatocytes context implies that this is a func&on of &me; the remaining symbols denote posi&ve constants (called parameters) Determining the values of these (and other) parameters in a model is called calibra&on. 3

4 Thus, in dt = α H δ a and H are func@ons of t, and the remaining three symbols on the right side denote posi@ve constants (parameters) Look at the equa@on closer what can we say? dt = α H δ a The deriva&ve of is nega&ve, thus the model predicts that will decrease Context implies that and H are posi&ve func@ons s mathema&cians, that s all we need to know! When H is very small in math we use H = 0 Interac@ons dt = δ a dt = α H δ a The rate of change of is a mul@ple of, that is, it is propor&onal to This is a model of exponen&al decay (more about it soon) Later, we will learn that the product H describes interac@ons between and H. Thus (look at the first term on the right side) we say that as H interacts with, it contributes to a decrease in (the paper says is cleared by H ) 4

5 Results are ohen given in the form of diagrams dt = α H δ a this is called a differen@al equa@on to solve, we have to un- apply the deriva@ve, i.e., we have to use integra@on [we do not worry about what they represent] Conclusion of the paper: MLD is useful! MLD contributes to the original research ques@on by providing a method to es@mate overdose elapsed since overdose, and outcome from pa@ent laboratory values commonly available on admission in cases of acute liver failure due to PP overdose Important message Drug overdose can (and does) seriously impact our health and well- being Important message: We need to learn math in order to understand a vastly increasing number of publica@ons in biology and health sciences which use mathema@cs and sta@s@cs Even if you do not plan to become a researcher, you will need to read and understand all kinds of documents, manuals, and reports which use quan@ta@ve informa@on 5

6 Is this going to be on the test? Yes, but only aher we cover all math that s necessary We will be explicit about what each test covers (as well, there are sample tests in your coursepack) 6