Section 2.8: Applied Pharmacokinetics (PK): One-Compartment Models

Applied Pharmacokinetics (PK): One-Compartment Models

Pharmacokinetics describes the time course of drug absorption, distribution, metabolism, and excretion (ADME) in the body. The one-compartment open model is the simplest and most clinically utilized model. It views the body as a single, homogeneous unit in which the drug distributes instantly and uniformly upon administration. Although physiological systems are highly complex, this model provides excellent clinical utility for drugs that equilibrate rapidly between blood and tissue spaces, such as aminoglycosides and theophylline.

Assumptions of the One-Compartment Model

• Instantaneous Distribution: The drug distributes immediately throughout all tissues and body fluids. There is no delay in reaching equilibrium between plasma and tissues. This is a primary assumption of one-compartment models, simplifying clinical dosage adjustments.
• Homogeneity: Plasma concentration is in direct equilibrium with, and proportional to, tissue concentrations at all times.
• First-Order Elimination: The rate of drug elimination is directly proportional to the amount remaining in the body. A constant fraction of the drug is eliminated per unit of time.
• Open System: The drug is introduced, distributes, and is ultimately eliminated from the compartment via excretion (mainly renal) or metabolism (primarily hepatic), meaning it does not remain in the body indefinitely.

Core Pharmacokinetic Parameters

1. Elimination Rate Constant ($K_e$) and Half-Life ($t_{1/2}$)

The elimination rate constant ($K_e$, expressed in $\text{hr}^{-1}$) represents the fraction of drug removed from the body per unit of time. Under first-order kinetics, the decline in drug concentration over time is expressed as:

$C(t) = C_0 \cdot e^{-K_e \cdot t}$

Taking the natural logarithm of both sides yields the linear equation:

$\ln(C(t)) = \ln(C_0) - K_e \cdot t$

Graphing concentration versus time on a semi-logarithmic plot yields a straight line with a slope of $-K_e$.

Elimination Half-Life ($t_{1/2}$): The time required for the plasma concentration to decrease by exactly $50%$. Setting $C(t) = C_0 / 2$ in the decay equation yields:

$t_{1/2} = \frac{\ln(2)}{K_e} \approx \frac{0.693}{K_e}$

2. Volume of Distribution ($V_d$)

The apparent volume of distribution ($V_d$) relates the amount of drug in the body ($A_b$) to the measured plasma concentration ($C$):

$V_d = \frac{A_b}{C}$

Immediately following an IV bolus dose, before elimination begins:

$V_d = \frac{\text{Dose}}{C_0}$

$V_d is a proportionality constant reflecting the extent of drug distribution into tissues.

• Small $V_d$ (e.g., $0.1\text{ L/kg}$): Confined to vascular space (e.g., warfarin).
• Large $V_d$ (e.g., $>5\text{ L/kg}$): Distributes widely into tissues (e.g., digoxin).

3. Clearance ($Cl$)

Clearance is the volume of plasma cleared of drug per unit of time (expressed in $\text{mL/min}$ or $\text{L/hr}$). It is the most reliable index of drug elimination capacity and is calculated as:

$Cl = K_e \cdot V_d$

Clearance can also be determined from the Area Under the Curve ($AUC$), which represents total systemic drug exposure:

$Cl = \frac{S \cdot F \cdot \text{Dose}}{AUC}$

where $S$ is the salt fraction and $F$ is the bioavailability ($F = 1$ for IV).

4. Continuous IV Infusion & Steady-State ($C_{ss}$)

During continuous IV infusion at rate $R_0$ (in $\text{mg/hr}$), drug concentration accumulates until elimination rate equals infusion rate. This plateau is the steady-state concentration ($C_{ss}$):

$C_{ss} = \frac{R_0}{Cl}$

The time to reach $C_{ss}$ is governed solely by the drug's half-life ($t_{1/2}$), taking 4 to 5 half-lives of constant dosing.

5. Multiple IV Bolus Dosing

During multiple dosing at interval $\tau$, peak ($C_{max, ss}$) and trough ($C_{min, ss}$) steady-state concentrations are:

$C_{max, ss} = \frac{\text{Dose} / V_d}{1 - e^{-K_e \cdot \tau}}$

$C_{min, ss} = C_{max, ss} \cdot e^{-K_e \cdot \tau}$

Pharmacokinetic Parameter Reference Table

Step-by-Step Worked Clinical Example

A 65-year-old male patient weighing $80\text{ kg}$ is prescribed Gentamicin. His serum creatinine is $1.2\text{ mg/dL}$. Calculate the parameters and dose to achieve target peak ($C_{max}$) $8\text{ mcg/mL}$ and trough ($C_{min}$) $<1\text{ mcg/mL}$ at a $24$-hour dosing interval ($\tau = 24\text{ hr}$).

Step 1: Calculate Creatinine Clearance (CrCl)

$\text{CrCl} = \frac{(140 - 65) \cdot 80}{72 \cdot 1.2} \approx 69.4\text{ mL/min} \approx 4.16\text{ L/hr}$

Step 2: Estimate Volume of Distribution ($V_d$) and Elimination Constant ($K_e$)

Using aminoglycoside average $V_d$ of $0.25\text{ L/kg}$ of body weight:

$V_d = 0.25 \cdot 80 = 20\text{ L}$ $K_e = \frac{Cl}{V_d} = \frac{4.16\text{ L/hr}}{20\text{ L}} = 0.208\text{ hr}^{-1}$

Step 3: Determine Dose and Verify Trough

$\text{Dose} = C_{max} \cdot V_d = 8\text{ mg/L} \cdot 20\text{ L} = 160\text{ mg}$

$C(24) = 8\text{ mg/L} \cdot e^{-0.208 \cdot 24} \approx 0.05\text{ mcg/mL}$

This trough is safe ($<1\text{ mcg/mL}$), making $160\text{ mg}$ IV every $24\text{ hours}$ appropriate.