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

Modelling the Amount of a Drug in the Bloodstream This portfolio will investigate the rate of drop of a malaria-treating drug in a human bloodstream over a period. In part A, a graph showing the amount of a drug in the bloodstream over 10 hour period for initial dose of 10 µg will be used to find a suitable function to model the data given in graph. In part B, graphs will be made and discussed for a whole day considering the doses being continually administered during regular 6 hour periods based on the data given in the graph and using the model (function) derived in part A. Finally, graphs will be made and discussed for; what would happen over the next week if no further does were taken and what would happen if doses would continue to be taken every 6 hours. Part A Table 1, shows the approximate values for time and corresponding amount of drug from the graph (given). Table 1: Amount of a Drug in the Bloodstream 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 show the graph drawn based on table 1 data. Figure 1: Amount of a Drug in the Bloodstream after Time From the above figure 1, it is evident that the amount of drug decreases as time passes. By seeing the graph, it is also evident that that the rate at which the drug decreases in the bloodstream is proportionate to the amount remaining. Therefore, the data follows certain pattern that is of type of an exponential decay graph. [1] [2] [3] An exponential decay function can be represented by: Where A is the amount present at time t = 0, and k is a constant. From the given values, the initial amount at time = 0 is 10 µg. Therefore, the value of A will be 10. Since the data represents approximate values, therefore exact value of k cannot be determined for the pattern of amount remaining in the bloodstream over a period. However, an approximate value of k can be determined for each data points and than average value will be taken from these determined values of k. The value of k for a single data points can be determined by below mentioned example: For the values t = 0.5 and y = 9: Taking log on both sides The formula that can be used for determining different values of k is: Table 2: Different Values of k and Average value of k Time in Hours (t) Amount of drug in µg (y) Average value of k 1 0.5 9 0.21072 2 1 8.3 0.18633 3 1.5 7.8 0.16564 4 2 7.2 0.16425 5 2.5 6.7 0.16019 6 3 6 0.17028 7 3.5 5.3 0.18139 8 4 5 0.17329 9 4.5 4.6 0.17256 10 5 4.4 0.16420 11 5.5 4 0.16660 12 6 3.7 0.16571 13 6.5 3 0.18523 14 7 2.8 0.18185 15 7.5 2.5 0.18484 16 8 2.5 0.17329 17 8.5 2.1 0.18361 18 9 1.9 0.18453 19 9.5 1.7 0.18652 20 10 1.5 0.18971 Table 2 shows the calculation for different values of k for the data and an approximate average value from theses value. Now the average value of k is 0.177537 and the initial amount A is 10 µg, therefore the model function will be: Figure 2: Graph of function and given data points (table 1) From the above graph (figure2) of model function, it is evident that the model function follows the same pattern as shown in the given graph of ‘amount of a drug in a bloodstream’. At some points, given data points slightly differ from the graph function. However, it is evident that the model function represents the exact pattern for drug remained over a period in bloodstream. The model function can be used for determining the approximate value of drug remained at any given time. Therefore, it is evident that the model function is suitable for the given graph of ‘Amount of a Drug in the Bloodstream”. Part B If a patient is instructed to take 10 µg drug after every six hours, than initial amount of drug remained in bloodstream will be added to the new dose of 10 µg. From the given graph (figure 1), it is evident that after six hour the amount of drug remained is 3.7 µg. Therefore, the amount of drug used is 6.3 µg. Now let suppose that in every six-hour period the amount of drug used is 6.3 µg. Therefore, for the next time when new dose will be given than the total amount will be 13.7 µg. This pattern is summarized in the table 3 for amount of drug present in blood before addition of new dose and amount of drug present in blood after addition of new dose. Table 3: The Drug dosage for 24-hour period based on data given Hours Amount of drug present in blood before addition of new dose (µg) Amount of drug present in blood after addition of new dose (µg) 0 0 10 6 3.7 13.7 12 7.4 17.4 18 11.1 21.1 24 14.8 Figure 3: The Drug dosage for 24-hour period based on Table 3 Figure 3 shows the graph of the pattern described in table 3. From this graph it is evident that amount of drug after new dose (every six hour) increases for every preceding doses during 24-hour period. The maximum amount of drug during this period is 21.1 µg at the start of 19th hour (or after addition of fourth dose) and the minimum amount of drug during this period is 3.7 µg at the end of 6th hour (or before addition of second dose). An accurate graph for the drug usage considering a patient is instructed to take 10 µg of drug every six hours can be drawn using the model function. However, the value of A will be changes after addition of every new dose and that will be used by the model function. In general, for this case the model function will be represented by Table 4 shows, the different values of Amount of drug present in blood before addition of new dose and Amount of drug present in blood after addition of new dose for 24-hour period. Figure 4 is drawn based on table 4 data and using model function. Table 4: The Drug dosage for 24-hour period based on function Hours Amount of drug present in blood before addition of new dose (µg) Amount of drug present in blood after addition of new dose (µg) 0 0 10 6 3.45 13.45 12 4.64 14.64 18 5.05 15.05 24 5.19 Figure 4 shows the graph of the pattern described in table 4. From this graph it is evident that amount of drug after new dose (every six hour) increases for every preceding doses during 24-hour period. However, here increases are slightly less as compared to graph of figure 3. The maximum amount of drug during this period is 15.05 µg at the start of 19th hour or (after addition of fourth dose) and the minimum amount of drug during this period is 3.45 µg at the end of 6th hour (or before addition of second dose). Figure 4: The amount of the drug in the bloodstream over a 24-hour period The amount of the drug in the bloodstream over a week period when initially 10µg is given is drawn using the model function and is shown in figure 5. Here it is evident that amount of drug in bloodstream never reaches to zero as the value of in the model function never becomes zero. Figure 5: The amount of the drug in the bloodstream over a week period Table 5 shows, the different values of Amount of drug present in blood before addition of new dose and amount of drug present in blood after addition of new dose for a week period. Figure 6 is drawn based on table 5 data and using model function. Table 5: The Drug dosage for a week based on function Hours (t) Amount of drug present in blood before addition of new dose (µg) Amount of drug present in blood after addition of new dose (µg) 0 0 10 6 3.45 13.45 12 4.64 14.64 18 5.05 15.05 24 5.19 15.19 30 5.24 15.24 36 5.25 15.25 42 5.26 15.26 48 5.26 15.26 54 5.26 15.26 --- ---- ---- Figure 6: The amount of the drug in the bloodstream over a week From figure 6 (and table 5), it is evident that after certain period the amount of drug remained for every six period will not increase and become constant. This will be maximum amount of drug and it is equal to 15.26 µg. The minimum amount will be 3.45µg. References: [1] http://www.ugrad.math.ubc.ca/coursedoc/math102/keshet.notes/chapter9Notes.pdf accessed on November 23, 2007. [2] http://www.mathsyear2000.co.uk/alevel/pure/purtutloggro.htm accessed on November 23, 2007. [3] http://www.okc.cc.ok.us/maustin/Exp_growth/Exponential%20Growth.htm accessed on November 23, 2007. Read More