Pharmacokinetics: Repeated Intermittent Dosing and Drug Accumulation

Drug Accumulation and Steady-State Principles

• Prevalence of Repeated Dosing: In clinical practice, most drugs are administered as multiple doses rather than single doses to maintain therapeutic effects. Common administration methods include:

  • Intravenous (IV) infusion.

  • IV bolus injections.

  • Extravascular administration (e.g., oral, intramuscular).

• Mechanism of Accumulation: Accumulation occurs when a drug is administered repeatedly. It is fundamentally determined by the balance between two competing rates:

  • The rate of administration (drug entry).

  • The rate of elimination (drug removal).

• Definition of Steady-State ($C_{ss}$): Steady-state is the equilibrium achieved when the rate of drug entry into the body equals the rate of drug elimination.

  • At steady-state: $\text{Rate of entry} = \text{Rate of elimination}$.

  • Rate of Entry: Calculated as $F \times \text{Dose Rate}$ (e.g., $mg/hr$).

  • $F$ represents bioavailability, ranging from $0$ to $100\%$ ($0$ to $1.0$).

  • Rate of Elimination: Calculated as $C_{ss} \times CL$.

• The Role of Clearance ($CL$): For a given dose rate, the clearance of the drug determines the steady-state concentration according to the formula:

  • $C_{ss} = \frac{F \times \text{Dose Rate}}{CL}$

Dynamics of Repeated Intermittent Dosing

• Fluctuations: Unlike continuous infusions which maintain a constant level, repeated intermittent doses result in concentrations that fluctuate between peaks (maximums) and troughs (minimums).

• Hypothetical Average: These fluctuations occur around a hypothetical average concentration ($C_{ss,ave}$). This average is the same concentration that would be achieved if the total dose over a specific time period were administered via a continuous infusion.

• Influence of Dosing Interval ($\tau$):

  • If the same dose is given less frequently (increasing the dosing interval), there is more time for the drug to be eliminated before the next dose is administered.

  • A longer time between doses leads to less accumulation because the overall dose rate ($\text{Dose} / \tau$) has decreased.

Impact of Pharmacokinetic Constants on Accumulation

• Half-Life ($t_{1/2}$) and the Elimination Rate Constant ($k$):

  • Shorter Half-Life: If a drug has a shorter half-life for a given dosing interval, less accumulation will be seen. This is because the drug is eliminated more effectively between doses.

  • Larger $k$: Since $k = \frac{\ln(2)}{t_{1/2}}$, a larger elimination rate constant implies a shorter half-life, leading to less accumulation.

  • Causes of Short Half-Life: This can be caused by higher clearance ($CL$) or a smaller volume of distribution ($V_d$).

• Longer Half-Life: Conversely, a decrease in the elimination rate constant ($k$)—often brought about by a larger volume of distribution ($V_d$)—results in a longer half-life. For a given dose and interval, the patient will accumulate more drug because a larger amount remains from the previous dose when the next one is administered.

Quantitative Measures of Drug Accumulation and Concentration

• Amount of Drug in the Body ($A$):

  • After the first dose, the maximum amount is: $A_{max, 1} = D$ (where $D$ is the dose).

  • At steady-state, the maximum amount is: $A_{max, ss} = \frac{D}{(1 - e^{-k\tau})}$.

  • The ratio of the maximum amount at steady-state to the first dose is the Accumulation Ratio:

  • $\text{Accumulation Ratio} = \frac{A_{max, ss}}{A_{max, 1}} = \frac{1}{(1 - e^{-k\tau})}$.

• Extravascular Dosing Adjustments: For non-IV doses, the dose ($D$) in formulas must be replaced with the bioavailable dose:

  • $F \times D$.

• Steady-State Minimum and Maximum Amounts:

  • Maximum Amount at steady-state: $A_{max, ss} = \frac{F \times D}{(1 - e^{-k\tau})}$.

  • Minimum Amount at steady-state: $A_{min, ss} = \frac{F \times D \times e^{-k\tau}}{(1 - e^{-k\tau})}$.

• Steady-State Concentrations ($C$): To convert amount to concentration, divide by the volume of distribution ($V_d$):

  • $C_{max, ss} = \frac{F \times D}{V_d \times (1 - e^{-k\tau})}$.

  • $C_{min, ss} = \frac{F \times D \times e^{-k\tau}}{V_d \times (1 - e^{-k\tau})}$.

Fluctuations and Dosing Intervals

• Definition of Tau ($\tau$): $\tau$ represents the dosing interval, or the exact time elapsed between consecutive doses.

• Quantifying Fluctuation: The fluctuation can be expressed as a ratio between the peak and the trough.

  • $\text{Fluctuation} = e^{k\tau}$.

  • For example, if the ratio is $2$, the peak concentration ($C_{max}$) is double the trough concentration ($C_{min}$).

• Determinants of Fluctuation Severity:

  • Rate Constant ($k$): A greater elimination rate constant lead to more significant fluctuations. More drug is eliminated in the time between doses, creating a larger gap between the peak and the trough.

  • Half-Life ($t_{1/2}$): A longer half-life results in less elimination during a given dosing interval, leading to smaller fluctuations.

  • Volume of Distribution ($V_d$): A larger volume of distribution can lead to a longer half-life, which in turn reduces the extent of fluctuation.

Summary of Intermittent Dosing Equations

• Average Steady-State Concentration ($C_{ss,ave}$):

  • $C_{ss,ave} = \frac{R_{0}}{CL}$

  • $C_{ss,ave} = \frac{F \times \text{Dose Rate}}{CL}$

  • $C_{ss,ave} = \frac{F \times \frac{D}{\tau}}{CL} = \frac{F \times D}{CL \times \tau}$

• Peak Steady-State Concentration ($C_{max,ss}$):

  • $C_{max,ss} = \frac{F \times D}{V_d} \times \frac{1}{(1 - e^{-k\tau})}$

• Trough Steady-State Concentration ($C_{min,ss}$):

  • $C_{min,ss} = \frac{F \times D}{V_d} \times \frac{e^{-k\tau}}{(1 - e^{-k\tau})}$

• Accumulation Ratio:

  • $\frac{A_{max,ss}}{A_{max,1}} = \frac{1}{(1 - e^{-k\tau})}$