Modelling the Amount of a Drug in the Bloodstream - Term Paper Example

Math Portfolio: Modelling the Amount of a Drug in the Bloodstream This math portfolio investigates and models the amount of a drug for treating malaria in the bloodstream over a period following an initial dose. In the first part A, initially an investigation will be made for finding a suitable function to model data given over a period of 10 hours following an initial dose of 10 μg. After that, graph of this model function will be compared with the data given so that suitability of the model function can be derived (known). In the second part B, the affect of doses given every six hour will be checked with and without using model function. In addition, it will be also checked that what happen if no further doses is given to malaria patient over a week period after an initial dose of 10 μg. Table 1: Amount of a Drug in the Bloodstream (data given, from graph) No. Time in Hours (t) Amount of drug in µg (y) 0 0 10 1 0.5 9 2 1 8.3 3 1.5 7.8 4 2 7.2 5 2.5 6.7 6 3 6 7 3.5 5.3 8 4 5 9 4.5 4.6 10 5 4.4 11 5.5 4 12 6 3.7 13 6.5 3 14 7 2.8 15 7.5 2.5 16 8 2.5 17 8.5 2.1 18 9 1.9 19 9.5 1.7 20 10 1.5 Table 1 contains the data given for amount of a drug in the bloodstream over a period of 10 hour following an initial dose of 10 μg and figure 1 is reproduced for the given graph based on table 1. Figure 1: Amount of a Drug in the Bloodstream (data given) PART A 1. Use this information to help you find a suitable function to model this data. From The graph of the given data, it is obvious that the amount of drug in the bloodstream decreases as the time passes. From investigation, it is found that it is similar to the graph of an exponential decay function (for example radioactive decay graph). The general equation of such an exponential decay function is: Where, is the initial amount , is the amount after time t, k is the decay constant, and b is the constant and can have value 2, 3, 4, … or e (mathematical constant, approximately equal to 2.718281828459). For the calculation purpose, it useful to take the value of b as e, therefore, the equation becomes: For our model function, let and, therefore equation becomes Where, is the initial amount of the drug , is the amount of drug after time t, is the decay constant, From data given, at the start, the amount is 10 µg, therefore the value of initial amount will be 10 µg i.e. . Putting this value of in the model function, the model function becomes: Now for determining the value of decay constant, solving above model function for. From the above function, the different values of can be determined for the given data that is summarized in table 2. Since, there are different values of for every data points, therefore, for our model function taking average value (mean value) of from the calculated values of ,. Average value of k = (Taking only four decimal places) Therefore model function will be: Table 2: Different values of decay constant (k) for the given data No. Time in Hours (t) Amount of drug in µg (y) Value of decay constant 0 0 10 1 0.5 9 0.2107 2 1 8.3 0.1863 3 1.5 7.8 0.1656 4 2 7.2 0.1643 5 2.5 6.7 0.1602 6 3 6 0.1703 7 3.5 5.3 0.1814 8 4 5 0.1733 9 4.5 4.6 0.1726 10 5 4.4 0.1642 11 5.5 4 0.1666 12 6 3.7 0.1657 13 6.5 3 0.1852 14 7 2.8 0.1819 15 7.5 2.5 0.1848 16 8 2.5 0.1733 17 8.5 2.1 0.1836 18 9 1.9 0.1845 19 9.5 1.7 0.1865 20 10 1.5 0.1897 2. Draw a graph of your function and compare your graph to the one above. Figure 2 shows the Graph of model function and data given. From figure 2 , it can be seen that the graph of model function and data given are similar and approximately follows the same path. Some minor deviations may be because of the error in colleting the data for the amount of drug in the bloodstream over a period. Figure 2: Graph of model function and data given 3. Comment on the suitability of the model. The model functionis suitable for the modeling of amount of the drug in the bloodstream. The suitability of the model function is also derived from the comparison of the graph of model function and data given and both are similar. PART B A patient is instructed to take 10 μg of this drug every six hours. 1. Sketch a diagram to show the amount of the drug in the bloodstream over a 24-hour period and state any assumption made. From given graph of amount of drug in the bloodstream for 10-hour period following an initial dose of 10 μg, it can be seen that amount of drug remained in the bloodstream after six-hour period is equal to 3.7 μg. Therefore, in six-hour period the amount of drug decay is 6.3 μg. Assuming this decay-rate is constant for further period, when a patient is instructed to take 10 μg of this drug every six hours. Then, the amount of drug at start and at end of each period will be as given in table 3. The maximum amount of drug 21.1 μg in the bloodstream will be at start of fourth period (i.e. 18-24 hour period) and the minimum amount of drug 3.7 μg in the bloodstream will be at end of first period (i.e. 0-6 hour period). Table 3: Amount of drug (based on data given) for 24-hour period Period Initial amount at start of period (µg) Amount at end of period (µg) 0-6 (1st ) 10 3.7 (min) 6-12 (2nd ) 13.7 7.4 12-18 (3rd ) 17.4 11.1 18-24 (4th ) 21.1 (max) 14.8 Figure 3 shows the sketch (graph) of the amount of the drug in the bloodstream over a 24-hour period assuming constant decay-rate of 6.3 μg over a period of six hours. Figure 3: Amount of drug (based on data given) for 24-hour period 2. Use your GDC or graphing software and your model from part A to draw an accurate graph to represent this situation. Table 4 summarizes the amount of drug in the bloodstream at start and at end of each period using the model function derived in part A. Table 4: Amount of drug based on model function for 24-hour period Period Initial amount at start of period (µg) Amount at end of period (µg) 0-6 (1st ) 10 3.45 (min) 6-12 (2nd ) 13.45 4.64 12-18 (3rd ) 14.64 5.05 18-24 (4th ) 15.05 (max) 5.19 Using model function, the maximum amount of drug 15.05 μg in the bloodstream will be at start of fourth period (i.e. 18-24 hour period) and the minimum amount of drug 3.45 μg in the bloodstream will be at end of first period (i.e. 0-6 hour period). Figure 4 shows the graph of the amount of the drug in the bloodstream over a 24-hour period Using model function. Figure 4: Amount of drug based on model function for 24-hour period 3. State the maximum and minimum amounts during this period. Maximum amounts of Drug = 15.05 μg Minimum amounts of Drug = 3.45 μg 4. Describe what would happen to these values over the next week if: (a) No further doses are taken. Figure 5: Amount of drug for week period (no further dose) If, after first dose no further dose is given to malaria patient, than the graph of amount of drug in the bloodstream for a week period will be given by figure 5. From figure 5, it can be seen that the amount of drug in the bloodstream never becomes zero and some amount of drug always remains in the bloodstream. The reason for this is that the value of exponential function never becomes zero. (b) Doses continue to be taken every six hours. The graph of the amount of drug in the bloodstream at start and at end of each period using the model function for a week period will be similar to figure 4. In this case, it will be plotted for a week period instead of 24-hour period. The table 5 summarizes the amount of drug in the bloodstream at start and at end of each period using the model function for a week period. Here, it can be seen that after certain period the amount of drug at start and at end of period becomes constant (i.e. 15.26 μg and 5.26 μg). Table 5: Amount of drug based on model function for a week period Period Initial amount at start of period (µg) Amount at end of period (µg) 0-6 (1st ) 10 3.45 (min) 6-12 (2nd ) 13.45 4.64 12-18 (3rd ) 14.64 5.05 18-24 (4th ) 15.05 5.19 24-30 (5th ) 15.19 5.24 30-36 (6th ) 15.24 5.25 36-42 (7th ) 15.25 5.26 42-48 (8th ) 15.26 (max) 5.26 48-54 (9th ) 15.26 (max) 5.26 54-60 (10th ) 15.26 (max) 5.26 ---- ---- ----- Figure 6 shows the graph of the amount of the drug in the bloodstream over a week period using model function. Figure 6: Amount of drug based on model function for week period References: Radioactive Curves and Calculations, retrieved on February 28, 2008 from http://curvebank.calstatela.edu/radiodecay/radiodecay.htm How models are derived (Example model 2. Exponential decay), retrieved on February 28, 2008 from http://www.graphpad.com/curvefit/how_to_derive.htm Graphs of Exponential Growth/Decay, retrieved on February 28, 2008 from http://serc.carleton.edu/introgeo/teachingwdata/Graphsexponential.html Exponential Decay, retrieved on February 28, 2008 from http://www.intmath.com/Exponential-logarithmic-functions/2_Graphs-exp-log-fns.php Read More