Answers to Basic Principles Problems

1. Phenobarbital (a weak organic acid, pKa=7.2) given orally will be well absorbed from the stomach (pH=1.4) into the plasma (pH=7.4), but if given systemically (i.v.) will be poorly excreted into the stomach. Why is this the case? Defend your argument mathematically.

2. Would you expect morphine (weak base, pKa=8.8) to behave in a similar manner to phenobarbital when given orally or when given systemically? If not, why? Defend you answer mathematically.

3. What dose of lidocaine would a patient receive if given a 0.5 mL injection of a 2% lidocaine solution?

2% solution equals 2000mg in 100 mL or 20 mg/mL. A 0.5 mL volume of 20 mg/mL equals 10 mg.

4. What dose of epinephrine would a patient receive if given a 1.0 mL injection of a 1:100,000 epinephrine solution?

1: 100,000 equal 1 g in 100,000 mL of water (1mL weighs 1 g) which equals 0.01 mg/mL or 10µg/mL. Injection of 1.0 mL is 10 µg.

5. A single i.v. dose of thiopental is associated with a duration of action of 20-30 minutes. A second i.v. dose given immediately upon awakening is associated with a duration of 2-4 hours. Explain why the duration of action is prolonged with the second dose. (Include a description of how thiopental's pharmacologic action is normally terminated.)

Thiopental's duration of action following dose one is terminated by redistribution of thiopental from rapidly perfused tissues (brain) to other areas of the body that are more slowly perfused by blood (muscle, fat). The second dose has a longer duration of action since the rate of redistribution of dose two is slower because concentration gradients between compartments are not as steep as with the first dose. (Refer to Fick's principle and note that the driving force for passive diffusion is C1-C2.)

6. If a steady state plasma level of sulfisoxazole of 80 ng/mL is reached during a constant intravenous infusion, how long will it take for the plasma level of drug to fall to 2 ng/mL when the infusion is stopped? (sulfisoxazole's t1/2 = 69.3 min.)

368.9 min
There are several ways to solve this problem:

lnC_t = lnC₀- ket; ln2 = ln80 - 0.01(t); 0.693 = 4.3820 - 0.01(t); -3.689 = -0.01t
t = 368.9 minutes

f = 1-e^-ket; fractional shift is the percentage change in the concentration; a shift from 80 to 2 ng/mL represents a 97.5 % change or f = 0.975
0.975 = 1-e^-ket; -0.025 = -e^-ket; 0.025 = e^-ket; ln0.025 = -0.01t; -3.689 = -0.01t
t = 368.9 minutes

Patient R.G. is to be infused with tetracycline for rickettsial infection. The half-life of the drug in this patient is 11 hours and its volume of distribution is 100 liters. (Use this information for questions 7-16)

7. If you decide to infuse the drug at a rate of 500 mg/min, what would be the steady state plasma concentration of tetracycline?

4.76 µg/ml The basic equation is Css = Input/ (ke (Vd)
ke = -0.693/ half-life = -0.693/11 hours = 0.063 hr^-1

Css = 500 mg/min ÷ (0.063 hr^-1 x 100 L) = 30,000 ÷ (0.063 hr^-1 x 100,000 mL)
30,000 mg/hr ÷ 6,300 mL hr^-1
Css = 4.76 mg/mL or some consistent multiple of this

8. What would the steady state plasma level be if you double the infusion rate? Triple the infusion rate?

9.52 µg/ml ; 14.29 µg/ml
Since none of the parameters have changed except the input we can simply multiply the steady state concentration in #7 by the factor by which the infusion rate is increased.

9. For an infusion rate of 500 mg/min., how long would it take to reach 50% of the final plateau level?

11 hours
There are several ways to answer this question. The equations given in #6 could be used or the rule that drugs reach their steady levels at the same rate as they disappear. So it would take one half-life or 11 hours for tretracycline to reach 50% of the steady state.

10. What infusion rate would be required if you wanted to establish a tetracycline concentration of 10 mg/mL in the plasma at steady state?

1.05 mg/min
Answering this question requires application of the steady state equation
10 mg/mL = Input ÷ 6,300 mL hr^-1
Input = 6,300 mL hr^-1 x 10 mg/mL = 63,000 mg/hr or 1050 mg/min or 1.05 mg/min

11. How long would it take to reach 90% of the plateau concentration?

36.5 hours
Use of the equation which describes the fractional shift in steady state levels is useful here.
f = 1-e^-ket;
0.90 = 1-e^-.063t; -0.1 = -e^-.063t; 0.1 = e^-.063t; ln0.1 = -0.063t; -2.302 = -0.063t;
36.5 hours

12. If you stop infusing tetracycline when the plasma concentration is 6 mg/mL how long would it take for the plasma level of tetracycline to fall to 0.75 microg/mL?

33 hours
same logic is applied to problem as is used in problem # 6

13. How long would it take to fall from 6 mg/mL to 0.45 microg/mL?

41.1 hours

14. What would be the plasma concentration of tetracycline at steady state?

3.82 µg/ml
To answer this question requires the application of the equation which describes the relationship between steady state and oral repeated dosing taking into account the degree of bioavailability.

15. What would be the plasma concentration of tetracycline at steady state if the same dose was given every 6 hrs? Every 12 hrs? Every 48 hrs?

15.28 µg/ml ; 7.64 µg/ml ; 1.91 µg/ml
Since the dosing interval is being changed from 24 hours we can easily calculate what the new steady states should be. Decreasing the interval from 24 to 6 hours is the same as increasing the dose by 4. Multiplying 3.82 by 4 gives 15.28. Likewise for the other changes in dosage interval.

16. What would be steady state level if the dose was changed from 750 mg/day to 375 mg/day? 1500 mg/day?

1.91 µg/ml ; 7.64 µg/ml
This question is asking what happens to the steady state when we change the dose by a factor of one-half or by two.

40 hours

18. How long would it take for a drug that saturates its elimination system to fall from 200 microg/mL to 100 microg/mL? (ke = 10 microg/mL/hr)

10 hours

19. A patient is to receive the antibiotic, tobramycin. The clearance and Vd of tobramycin in this patient are 80 mL/min and 40 L, respectively. What maintenance dose must be administered intravenously, every 6 hours to eventually obtain a steady state plasma concentration of 4 mg/L?

115.2 mg given every 6 hours
This is an application of the steady state equation in which one calculates the dose administered per minute by infusion and then multiplies it by 360 to calculate the amount every 6 hours.

20. Drug X has a narrow therapeutic index: the minimum toxic plasma concentration is 150 mg/L while the minimum therapeutic plasma concentration is 100 mg/L. The half-life is 6 hours. It is essential to maintain the plasma concentration above the minimum therapeutic level. The most appropriate dosing regimen would be how many times a day?

Should be given every 3.5 hours (or 7 times a day) to maintain concentration between 150 mg/L and 100 mg/L.
The solution to this problem is realizing just what is being asked. We do not need to know what dose is being administered. The question is asking how frequently the dose must be given in order to keep the trough concentration at or above 100 mg/L. The upper limit is 150 mg/L because of the toxicity of the drug. The solution is to determine the length of time it takes for the drug level to fall from 150 mg/L to 100 mg/L and the equation:

lnC_t = lnC₀- ket