Functions and Graphs Exponential function
A patient recovering from a heart condition, and one fully fit volunteer are to take a regular treadmill stress test (www.heartsite.com/html/regular_stress.html) to assess their fitness for a marathon run.
1) It is observed that the heart rate (HR) of the volunteer increases linearly in time (t), ie. according to the function HR(t) = at+b. His initial heart rate is 70 beats per minute, but reaches 120 beats per minute at the end of stage 1.
a. Make a guess at the values of the parameters a and b, and plot the function HR(t) for the entire duration of the first stage (report your initial guesses and the plot).
b. Adjust the values of a and b until the function fits the observations (report your final result for a and b, and the corresponding plot).
2) In the patient, the initial heart rate is 60 beats per minute, and the final heart rate at the end of stage 1 is 210 beats per minute. An intermediate measurement at 1 minute shows a heart rate of 90 beats per minute.
a. Assume that the function HR(t) is linear, and guess the values of a and b using the same method as in the volunteer. Which values of a and b give you the best description of these data? (Report the values and the plot, and explain why these values are the best).
b. Can you find a function that describes the observed data even better? (Report the formula of the function and the plot, and explain why it is better).
c. If the patients’ target heart rate is 150 beats per minute, at which time should the test have been stopped?
Exponential function
After intravenous injection of a certain dose D (mmol) of indicator, the concentration C (mmol/ml) in the blood plasma decays due to renal clearance. This decay is exponential, and it depends on the body’s extracellular fluid volume (V, in litre) and the glomerular filtration rate (GFR, in ml/min) of the kidneys. The formula is C(t) = C0 * exp(-t*GFR/V).
1) Search on-line or in the literature for a typical value of GFR and V, and plot the function C(t) for the first 3 hours. Assume the concentration immediately after injection is 0.1 mmol/ml (report the values you found, your sources, and the plot).
2) A measurement is now performed in an adult patient with suspicion of reduced kidney function. Concentrations are measured in blood samples taken at 1hr, 2hrs, and 3hrs after injection, giving values of 85.4, 72.8, and 61.9 µmol/ml. Assuming the same value for V as before, what is the GFR of this patient? (report the result, and how you found it).
PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET A GOOD DISCOUNT
