Chapter 3: Pharmacokinetics

The effective drug concentration is the concentration of a drug at the receptor site. In patients, drug concentrations are more readily measured in the blood. Except for topically applied agents, the concentration at the receptor site is usually proportional to the drug’s concentration in the plasma or whole blood at equilibrium. The plasma concentration is a function of the rate of input of the drug (by absorption) into the plasma, the rate of distribution, and the rate of elimination. If the rate of input is known, the remaining processes are well described by 2 primary parameters: apparent volume of distribution (V_d) and clearance (CL). These parameters are unique for a particular drug and a particular patient but have average values in large populations that can be used to predict drug concentrations.

The volume of distribution (V_d) relates the amount of drug in the body to the plasma concentration according to the following equation:

(1)

The calculated parameter for the V_d has no direct physical equivalent; therefore, it is usually denoted as the apparent V_d. Because the size of the compartments to which the drug may be distributed can vary with body size, V_d is sometimes expressed as V_d per kilogram of body weight (V_d/kg). A drug that is completely retained in the plasma compartment (Figure 3–1) will have a V_d equal to the plasma volume (about 4% of body weight). The V_d of drugs that are normally bound to plasma proteins such as albumin can be altered by liver disease (through reduced protein synthesis) and kidney disease (through urinary protein loss). On the other hand, if a drug is avidly bound in peripheral tissues, the drug’s concentration in plasma may drop to very low values even though the total amount in the body is large. As a result, the V_d may greatly exceed the total physical volume of the body. For example, 50,000 liters is the average V_d for the drug quinacrine in persons whose average physical body volume is 70 liters.

Clearance (CL) relates the rate of elimination to the plasma concentration:

(2)

For a drug eliminated with first-order kinetics, clearance is a constant; that is, the ratio of rate of elimination to plasma concentration is the same over a broad range of plasma concentration (Figure 3–2). As in the case of V_d, clearance is sometimes expressed as CL per kg of body weight. The magnitudes of clearance for different drugs range from a small percentage of the blood flow to a maximum of the total blood flow to the organs of elimination. Clearance depends on the drug, blood flow, and the condition of the organs of elimination in the patient. The clearance of a particular drug by an individual organ is equivalent to the extraction capability of that organ for that drug times the rate of delivery of drug to the organ. Thus, the clearance of a drug that is very effectively extracted by an organ (ie, the blood is completely cleared of the drug as it passes through the organ) is often flow-limited. For such a drug, the total clearance from the body is a function of blood flow through the eliminating organ and is limited by the blood flow to that organ. In this situation, other conditions—cardiac disease, or other drugs that change blood flow—may have more dramatic effects on clearance than disease of the organ of elimination. Note that for drugs eliminated with zero-order kinetics (see Figure 1–3, right), elimination rate is constant and clearance is not constant.

Most drugs in clinical use obey the first-order kinetics rule described in the text (See Chapter 1). Can you name 3 important drugs that do not? The Skill Keeper Answer appears at the end of the chapter.

Half-life (t_1/2) is a derived parameter, completely determined by V_d and CL. Like clearance, half-life is a constant for drugs that follow first-order kinetics. Half-life can be determined graphically from a plot of the blood level versus time (eg, Figure 1–4) or from the following relationship:

One must know both primary variables (V_d and CL) to predict changes in half-life. Disease, age, and other variables usually alter the clearance of a drug much more than they alter its V_d. The half-life determines the rate at which blood concentration rises during a constant infusion and falls after administration is stopped (Figure 3–3). The effect of a drug at 87–90% of its steady-state concentration is clinically indistinguishable from the steady-state effect; thus, 3–4 half-lives of dosing at a constant rate are considered adequate to produce the effect to be expected at steady state.

The bioavailability of a drug is the fraction (F) of the administered dose that reaches the systemic circulation. Bioavailability is defined as unity (or 100%) in the case of intravenous administration. After administration by other routes, bioavailability is generally reduced by incomplete absorption (and in the intestine, expulsion of drug by intestinal transporters), first-pass metabolism, and any distribution into other tissues that occurs before the drug enters the systemic circulation. Even for drugs with equal bioavailabilities, entry into the systemic circulation occurs over varying periods of time, depending on the drug formulation and other factors. To account for such factors, the concentration appearing in the plasma is integrated over time to obtain an integrated total area under the plasma concentration curve (AUC, Figure 3–4).

Removal of a drug by an organ can be specified as the extraction ratio, that is, the fraction or percentage of the drug removed from the perfusing blood during its passage through the organ (Figure 3–5). After steady-state concentration in plasma has been achieved, the extraction ratio is one measure of the elimination of the drug by that organ.

Drugs that have a high hepatic extraction ratio have a large first-pass effect and the bioavailability of these drugs after oral administration is low.

The oral route of administration is the most likely to have a large first-pass effect and therefore low bioavailability (See Chapter 1). What tissues contribute to this effect? The Skill Keeper Answer appears at the end of the chapter.

A dosage regimen is a plan for drug administration over a period of time. An optimal dosage regimen results in the achievement of therapeutic levels of the drug in the blood without exceeding the minimum toxic concentration. To maintain the plasma concentration within a specified range over long periods of therapy, a schedule of maintenance doses is used. If it is necessary to achieve the target plasma level rapidly, a loading dose may be used to “load” the V_d with the drug. Ideally, the dosing plan is based on knowledge of both the minimum therapeutic and minimum toxic concentrations for the drug, as well as its clearance and V_d.

A. Maintenance Dosage

Because the maintenance rate of drug administration is equal to the rate of elimination at steady state (this is the definition of steady state), the maintenance dosage is a function of clearance (from Equation 2).

Note that V_d is not involved in the calculation of maintenance dosing rate. The dosing rate computed for maintenance dosage is the average dose per unit time. When performing such calculations, make certain that the units are in agreement throughout. For example, if clearance is given in mL/min, the resulting dosing rate is a per minute rate. Because convenience of administration is desirable for chronic therapy, doses should be given orally if possible and only once or a few times per day. The size of the daily dose (dose per minute × 60 min/h × 24 h/d) is a simple extension of the preceding information. The number of doses to be given per day is usually determined by the half-life of the drug and the difference between the minimum therapeutic and toxic concentrations (see Therapeutic Window, below).

If it is important to maintain a concentration above the minimum therapeutic level at all times, either a larger dose is given at long intervals or smaller doses at more frequent intervals. If the difference between the toxic and therapeutic concentrations is small, then smaller and more frequent doses must be administered to prevent toxicity.

B. Loading Dosage

If the therapeutic concentration must be achieved rapidly and the V_d is large, a large loading dose may be needed at the onset of therapy. This can be calculated from the following equation:

Note that clearance does not enter into this computation. If the loading dose is large (V_d much larger than blood volume), the dose should be given slowly to prevent toxicity due to excessively high plasma levels during the distribution phase.

The therapeutic window is the safe range between the minimum therapeutic concentration and the minimum toxic concentration of a drug. These data are used to determine the acceptable range of plasma levels when designing a dosing regimen. Thus, the minimum effective concentration usually determines the desired trough levels of a drug given intermittently, whereas the minimum toxic concentration determines the permissible peak plasma concentration. For example, the drug theophylline has a therapeutic concentration range of 8–20 mg/L but may be toxic at concentrations of more than 15–20 mg/L. The therapeutic window for a particular patient might thus be 8–16 mg/L (Figure 3–6). Unfortunately, for some drugs the therapeutic and toxic concentrations vary so greatly among patients that it is impossible to predict the therapeutic window in a given patient. Such drugs must be titrated individually in each patient.

Renal disease or reduced cardiac output often reduces the clearance of drugs that depend on renal elimination. Alteration of clearance by liver disease is less common but may also occur. Impairment of hepatic clearance occurs (for high extraction drugs) when liver blood flow is reduced, as in heart failure, and in severe cirrhosis and other forms of liver failure. Because it is important in the elimination of drugs, assessing renal function is important in estimating dosage in patients. The most important renal variable in drug elimination is glomerular filtration rate (GFR), and creatinine clearance (CL_cr) is a convenient approximation of GFR. The dosage in a patient with renal impairment may be corrected by multiplying the average dosage for a normal person times the ratio of the patient’s altered creatinine clearance (CL_cr) to normal creatinine clearance (approximately 100 mL/min, or 6 L/h in a young adult).

(6)

This simplified approach ignores nonrenal routes of clearance that may be significant. If a drug is cleared partly by the kidney and partly by other routes, Equation 6 should be applied to the part of the dose that is eliminated by the kidney. For example, if a drug is 50% cleared by the kidney and 50% by the liver and the normal dosage is 200 mg/d, the hepatic and renal elimination rates are each 100 mg/d. Therefore, the corrected dosage in a patient with a creatinine clearance of 20 mL/min will be:

(7)

Renal function is altered by many diseases and is often decreased in older patients. CL_cr can be measured directly, but this requires careful measurement of both serum creatinine concentration and a timed total urine creatinine. A common shortcut that requires only the serum (or plasma) creatinine measurement (S_cr) is the use of an equation. One such equation in common use is the Cockcroft-Gault equation:

The result is multiplied by 0.85 for females. A similar equation for GFR is the MDRD equation:

1. Mr Jones has zero kidney function and is undergoing hemodialysis while awaiting a kidney transplant. He takes metformin for type 2 diabetes mellitus and was previously stabilized (while his kidney function was adequate) at a dosage of 500 mg twice daily, given orally. The plasma concentration at this dosage with normal kidney function was found to be 1.4 mg/L. He has been on dialysis for 10 days and metformin toxicity is suspected. A blood sample now shows a metformin concentration of 4.2 mg/L. What was Mr. Jones’ clearance of metformin while his kidney function was normal?

   • (A) 238 L/d

   • (B) 29.8 L/h

   • (C) 3 L/d

   • (D) 238 L/h

   • (E) 30 L/min

2. Ms Smith, a 65-year-old woman with pneumonia, was given tobramycin, 150 mg, intravenously. After 20 minutes, the plasma concentration was measured and was found to be 3 mg/L. Assuming no elimination of the drug in 20 minutes, what is the apparent volume of distribution of tobramycin in Ms Smith?

   • (A) 3 L/min

   • (B) 3 L

   • (C) 50 L

   • (D) 7 L

   • (E) 0.1 mg/min

3. St John’s Wort, a popular botanical remedy, is a potent inducer of hepatic phase I CYP3A4 enzymes. Verapamil and phenytoin are both eliminated from the body by metabolism in the liver. Verapamil has a clearance of 1.5 L/min, approximately equal to liver blood flow, whereas phenytoin has a clearance of 0.1 L/min. Based on this fact, which of the following is most correct?

   • (A) St John’s Wort will increase the half-life of phenytoin and verapamil

   • (B) St John’s Wort will decrease the volume of distribution of CYP3A4 substrates

   • (C) St John’s Wort will decrease the hepatic extraction of phenytoin

   • (D) St John’s Wort will decrease the first-pass effect for verapamil

   • (E) St John’s Wort will increase the clearance of phenytoin

4. A 55-year-old man with severe rheumatoid arthritis has elected to participate in the trial of a new immunosuppressive agent. It is given by constant intravenous infusion of 8 mg/h. Plasma concentrations (Cp) are measured with the results shown in the following table.

   Table Graphic Jump Location

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   Time After Start of Infusion (h)	Plasma Concentration (mg/L)
   1	0.8
   2	1.2
   8	3.0
   10	3.6
   20	3.84
   40	4.0

   What conclusion can be drawn from these data?

   • (A) Clearance is 2 L/h

   • (B) Doubling the rate of infusion would result in a plasma concentration of 16 mg/L at 40 h

   • (C) Elimination follows zero-order kinetics

   • (D) Half-life is 8 h

   • (E) Volume of distribution is 30 L

5. You are the only physician in a clinic that is cut off from the outside world by violent storms, flooding, and landslides. A 15-year-old girl is brought to the clinic with severe asthmatic wheezing. Because of the lack of other drugs, you decide to use intravenous theophylline for treatment. The pharmacokinetics of theophylline include the following average parameters: V_d 35 L; CL 48 mL/min; half-life 8 h. If an intravenous infusion of theophylline is started at a rate of 0.48 mg/min, how long would it take to reach 93.75% of the final steady-state concentration?

   • (A) Approximately 48 min

   • (B) Approximately 7.4 h

   • (C) Approximately 8 h

   • (D) Approximately 24 h

   • (E) Approximately 32 h

6. A 74-year-old retired mechanic is admitted with a myocardial infarction and a severe acute cardiac arrhythmia. You decide to give lidocaine to correct the arrhythmia. A continuous intravenous infusion of lidocaine, 1.92 mg/min, is started at 8 am. The average pharmacokinetic parameters of lidocaine are: V_d 77 L; clearance 640 mL/min; half-life 1.4 h. What is the expected steady-state plasma concentration?

   • (A) 40 mg/L

   • (B) 3.0 mg/L

   • (C) 0.025 mg/L

   • (D) 7.2 mg/L

   • (E) 3.46 mg/L

7. A new drug is under study in phase 1 trials. It is found that this molecule is avidly taken up by extravascular tissues so that the final total amount in the extravascular compartment at steady state is 100 times the amount remaining in the blood plasma. What is the probable volume of distribution in a hypothetical person with 8 L of blood and 4 L of plasma?

   • (A) Insufficient data to calculate

   • (B) 8 L

   • (C) 14.14 L

   • (D) 100 L

   • (E) 404 L

8. A 63-year-old woman in the intensive care unit requires an infusion of procainamide. Its half-life is 2 h. The infusion is begun at 9 am. At 1 pm on the same day, a blood sample is taken; the drug concentration is found to be 3 mg/L. What is the probable steady-state drug concentration after 16 or more hours of infusion?

   • (A) 3 mg/L

   • (B) 4 mg/L

   • (C) 6 mg/L

   • (D) 9.9 mg/L

   • (E) 15 mg/L

9. A 30-year-old man is brought to the emergency department in a deep coma. Respiration is severely depressed and he has pinpoint pupils. His friends state that he self-administered a large dose of morphine 6 h earlier. An immediate blood analysis shows a morphine blood level of 0.25 mg/L. Assuming that the V_d of morphine in this patient is 200 L and the half-life is 3 h, how much morphine did the patient inject 6 h earlier?

   • (A) 25 mg

   • (B) 50 mg

   • (C) 100 mg

   • (D) 200 mg

   • (E) Not enough data to predict

10. Gentamicin, an aminoglycoside antibiotic, is sometimes given in intermittent intravenous bolus doses of 100 mg 3 times a day to achieve target peak plasma concentrations of about 5 mg/L. Gentamicin’s clearance (normally 5.4 L/h/70 kg) is almost entirely by glomerular filtration. Your patient, however, is found to have a creatinine clearance one third of normal. What should your modified dosage regimen for this patient be?

    • (A) 20 mg 3 times a day

    • (B) 33 mg 3 times a day

    • (C) 72 mg 3 times a day

    • (D) 100 mg 2 times a day

    • (E) 150 mg 2 times a day

1. Examination questions often provide more information than is needed—to test the student’s ability to classify and organize data. In question 1, the data provided for Mr Jones on dialysis is irrelevant, even though choice A, 238 L/d, is the correct clearance while on dialysis. By definition, clearance is calculated by dividing the rate of elimination by the plasma concentration:

   

   The answer is B.

2. The volume of distribution (V_d) is the apparent volume into which the loading dose is distributed. It is calculated by dividing the dose by the resulting plasma concentration, C_p:

   

   The answer is C.

3. Induction of phase I metabolizing enzymes will decrease the half-life of substrates of these enzymes. P450 enzyme induction has no effect on volume of distribution. Hepatic extraction, the first-pass effect, and clearance for CYP3A4 substrates will be increased by inducers. However, the extraction of verapamil is already equal to the hepatic blood flow, so further increase in metabolism will not increase clearance of this drug. The answer is E.

4. By inspection of the data in the table, it is clear that the steady-state plasma concentration is approximately 4 mg/L. None of the measured concentrations is equal to one half of the steady state value, so the half-life is not immediately apparent. However, according to the constant infusion principle (Figure 3–3), 2 half-lives are required to reach 75% of the final concentration; 75% (3.0 mg/L) of the final steady-state concentration was reached at 8 h. If 8 h equals 2 half-lives, the half-life must be 4 h. Rearranging the equation for maintenance dosing (dosing rate = CL × Cp), it can be determined that the clearance (CL) = dosing rate/plasma concentration (Cp), or 2 L/h. The volume of distribution (V_d) can be calculated from the half-life equation (t_1/2 = 0.693 × V_d/CL) and is equal to 11.5 L. This drug follows first-order kinetics, as indicated by the progressive approach to the steady-state plasma concentration. The answer is A.

5. The approach of the drug plasma concentration to steady-state concentration during continuous infusion follows a stereotypical curve (Figure 3–3) that rises rapidly at first and gradually reaches a plateau. It reaches 50% of steady state at 1 half-life, 75% at 2 half-lives, 87.5% at 3, 93.75% at 4, and progressively halves the difference between its current level and 100% of steady state with each half-life. The answer is E, 32 h, or 4 half-lives.

6. The drug is being administered continuously and the steady-state concentration (Cp_ss) for a continuously administered drug is given by the equation in question 1. Thus,

   

   Rearranging:

   

   The answer is B.

7. Let Z be the amount in the blood plasma. If the amount in the rest of the body is 100 times greater, then the total amount in the body is 101Z. The concentration in the blood plasma (C_p) is Z/4 L. According to the definition:

   

   The answer is E.

8. According to the curve that relates plasma concentration to infusion time (Figure 3–3), a drug reaches 50% of its final steady-state concentration in 1 half-life, 75% in 2 half-lives, etc. From 9 am to 1 pm is 4 h, or 2 half-lives. Therefore, the measured concentration at 1 pm is 75% of the steady-state value (0.75 × Cp_ss). The steady-state concentration is 3 mg/L divided by 0.75, or 4 mg/L. The answer is B.

9. According to the curve that relates the decline of plasma concentration to time as the drug is eliminated (Figure 3–3), the plasma concentration of morphine was 4 times higher immediately after administration than at the time of the measurement, which occurred 6 h, or 2 half-lives, later. Therefore, the initial plasma concentration was 1 mg/L. Since the amount in the body at any time is equal to V_d × plasma concentration (text Equation 1), the amount injected was 200 L × 1 mg/L, or 200 mg. The answer is D.

10. If the drug is cleared almost entirely by the kidney and creatinine clearance is reduced to one third of normal, the total daily dose should also be reduced to one third. The answer is B.

The 3 important drugs that follow zero-order rather than first-order kinetics are ethanol, aspirin, and phenytoin (See Chapter 1).

The oral route of administration entails passage of the drug through the gastric and intestinal contents, the epithelium and other tissues of the intestinal wall, the portal blood, and the liver before it enters the systemic circulation for distribution to the body (See Chapter 1). Metabolism by enzymes in any of these tissues, expulsion by drug transporters, and excretion into the bile all may contribute to the first-pass effect of oral administration.

When you complete this chapter, you should be able to:

• ❑ Estimate the half-life of a drug based on its clearance and volume of distribution or from a graph of its plasma concentration over time.

• ❑ Calculate loading and maintenance dosage regimens for oral or intravenous administration of a drug when given the following information: minimum therapeutic concentration, minimum toxic concentration, oral bioavailability, clearance, and volume of distribution.

• ❑ Calculate the dosage adjustment required for a patient with impaired renal function.