Modelling the Amount of a Drug in the Bloodstream - Lab Report Example

Math Portfolio- Modelling the Amount of a Drug in the Bloodstream Math Portfolio- Modelling the Amount of a Drug in the Bloodstream Introduction Thismathematics portfolio will model the amount of drug in the bloodstream based on the data given for a drug for treating malaria in the bloodstream over the 10-hour period following an initial dose of 10 μg. After modelling the amount of drug in bloodstream, a comparison will be made in-between graph of model function and graph of data given so that suitability of the model function can be determined (part A). After that in part B, various cases for malaria patient who is instructed to take 10 μg drug doses after every six hours will be taken. In addition, maximum and minimum drug dose for 24-hour period will be calculated based on model function. At last, cases will be taken for the patient, for no further doses taken and doses continued to be taken for every six-hours for period of one week. PART A Finding a Suitable Function to Model the Given Data Table 1 shows the data (given) for the amount of drug (in µg) after equal interval of half an hour for 10-hour period. Figure 1 is drawn based on this given data i.e. table 1. Table 1: Amount of drug (y) after time (t) Time in hours (t) Amount of drug in µg (y) 0 10 0.5 9 1 8.3 1.5 7.8 2 7.2 2.5 6.7 3 6 3.5 5.3 4 5 4.5 4.6 5 4.4 5.5 4 6 3.7 6.5 3 7 2.8 7.5 2.5 8 2.5 8.5 2.1 9 1.9 9.5 1.7 10 1.5 Figure 1: The amount of a drug in the bloodstream (given) As given, from figure 1, it is clear that the rate of the decrease of the drug is approximately proportional to the amount remaining. In addition, the graph is similar to an exponential decay graph because the amount is subject to exponential decay as it decreases at a rate proportional to its value. Therefore, for modelling exponential decay function will be used. The general equation of an exponential decay function is written as below: Where, is initial value at t = 0, is decay constant, is the time and is the amount remaining after time. Like (pie), is also an irrational number, which is used in natural logarithmic (ln) and the value of is approximately 2.7182818284590452353602874713527. Let us use the natural exponential decay function, for modelling the amount of drug in bloodstream. At the start, the drug given to malaria patient is 10 µg. Therefore, the value of will be 10. For simplicity let us use instead of. Now, the equation can be written as: The data given is not precise, therefore, for each interval (hours) the value of (decay constant) will be different. For calculating approximate value of, the value of from each data will be calculated and their average will be taken. Simplifying function, for calculation of decay constant. There are 20 records for the amount of drug remained, therefore, the value of decay constantcalculated for each record will be different. The above derived function for can be used for calculating different values of for the given data. For example, for first records the value of will be: Taking only five decimal places, the value of will be 0.21072. Similarly, the values of for other records can be calculated that is shown in table 2. Table 2: The values of for different records Time in hours (t) Amount of drug in µg (y) 0.5 9 0.21072 1 8.3 0.18633 1.5 7.8 0.16564 2 7.2 0.16425 2.5 6.7 0.16019 3 6 0.17028 3.5 5.3 0.18139 4 5 0.17329 4.5 4.6 0.17256 5 4.4 0.16420 5.5 4 0.16660 6 3.7 0.16571 6.5 3 0.18523 7 2.8 0.18185 7.5 2.5 0.18484 8 2.5 0.17329 8.5 2.1 0.18361 9 1.9 0.18453 9.5 1.7 0.18652 10 1.5 0.18971 From table 2 values of, the average value of will be: = = 0.17754 (Taking only five decimal places.) Therefore, the equation of model function will be: Comparison of Graph of Model Function and Graph of Data Given Figure 2: Graph of model function and the data given Figure 2 shows the Graph of model function and the data given for the amount of drug in the bloodstream over a period of 10 hours. Now, from figure 2, it can be seen that both graph of model function and graph of data give follows approximately same path. There is minor difference in-between period 2-3 and 5-6 hours. However, for rest of the points (period), model graph function follows the same path as graph of data given. Suitability of the Model Function From figure 2, graph of model function and the data given for amount of drug in bloodstream over a period of 10 hours, it can be said that the model function follows the same path (trend) as amount of drug in bloodstream over a period of 10 hours. Therefore, the model function is suitable for modelling the amount of drug in bloodstream. PART B (A Patient Is Instructed To Take 10 µg Of This Drug Every Six Hours) The Amount of the Drug in the Bloodstream over a 24-Hour Period From figure 1, it can be seen that the amount of the drug remained in the bloodstream after six-hour period for the given data is 3.7 µg. Therefore, after six hour, the decrease in amount of drug is 6.3 µg (10 - 3.7 = 6.3). Assuming this decrease is constant for further doses taken, the amount of the drug in the bloodstream over a period of 24-hour at start and end of every period is given in table 3 and the graph of this is presented in figure 3. Table 3: Amount of drug over a period of 24-hour based on the data given Periods Amount of drug before addition of new dose (µg) Amount of drug after addition of new dose (µg) 0-6 0 10 6-12 3.7(min) 13.7 12-18 7.4 17.4 18-24 11.1 21.1(max) 24-30 14.8 --- Figure 3: Amount of drug over a period of 24-hour based on the data given The Amount of the Drug in the Bloodstream over a 24-Hour Period based on model function The amount of drug remained after an initial dose of 10 µg based on the model function will be: Now this amount will be added to new dose, therefore, at start of seventh hour the amount of drug in the bloodstream will be equal to 13.5 µg. Similarly, for rest of periods the amount of the drug in the bloodstream before addition of new dose and after addition of new dose is calculated and presented in table 4. Table 4: Amount of drug over a period of 24-hour based on model function Periods Amount of drug before addition of new dose (µg) Amount of drug after addition of new dose (µg) 0-6 0 10 6-12 3.5(min) 13.5 12-18 4.7 14.7 18-24 5.1 15.1(max) 24-30 5.2 --- For drawing graph of this situation, the initial amount of drug will be taken different for each period in model function derived earlier. Now, the general function for this situation will be: Where is the value of drug dose at start of period, and p is the period. Therefore, function for period 1, 2, 3 and four will be: Based on above function for every period, the graph of the amount of the drug in the bloodstream for a period of 24-hour is shown in figure 4. Figure 4: The amount of the drug in the bloodstream for a period of 24-hour The maximum and minimum amounts during 24-hour periods Based on the model functionthe maximum and minimum amount of the drug in the bloodstream are: Maximum amount = 15.1 μg (at the start of fourth period). Minimum amount = 3.5 μg (at the end of first period). The Amount of the Drug in the Bloodstream for the Next Week The model function is an exponential function. The value of the part will be never zero for any value of t. As the values of t increases, the value of the function will decreases; however, theoretically it will never become zero. This means that there will be always some amount of the drug left in the bloodstream of the patient. Figure 5 shows this situation. Figure 5: The amount of drug in the bloodstream for next week with no further doses If patient continues to take drug doses after every six-hour period for a week than the data and will be similar to table 4 and graph will be similar to graph of figure 4. The calculation will be done using model functionas done earlier in table 4. Table 5 (similar to table 4) represents this situation for a period of one week. From table 5, it can be seen that after certain period, the amount of drug before addition of new dose and after addition of new dose becomes constant and there is no further increase in it. Figure 6 shows the graph of this situation when drug doses continue to be taken every six hours. Table 5: Amount of drug for a week based on model function Periods Amount of drug before addition of new dose (µg) Amount of drug after addition of new dose (µg) 0-6 0 10 6-12 3.5(min) 13.5 12-18 4.7 14.7 18-24 5.1 15.1 24-30 5.2 15.2(max) 30-36 5.2 15.2(max) 36-42 5.2 15.2(max) 42-48 5.2 15.2(max) 48-54 5.2 15.2(max) 54-60 5.2 15.2(max) ---- --- --- Figure 6: The amount of the drug in the bloodstream for a week based on model Conclusion This portfolio investigated the amount of the drug in the bloodstream for malaria patient. From investigation, it was found that decrease of amount of drug is proportional to amount remaining and this can be modelled using exponential decay function. In addition, it was also found that amount of the drug in the bloodstream of the malaria patient never becomes zero after taking an initial dose once a dose is taken by malaria patient. Reference: Exponential decay, accessed on January 31, 2008 from Ian Garbett 2001, Radioactive decay and exponential laws, accessed on January 31, 2008 from Exponential growth and decay, accessed on January 31, 2008 from Exponential Functions: The "Natural" Exponential "e", accessed on January 31, 2008 from Applications of exponential and logarithmic functions, accessed on January 31, 2008 from Read More