3.3 Pharmacokinetic Modeling, Parameters, and Dosing Calculations

3.3 Pharmacokinetic Modeling, Parameters, and Dosing Calculations

Clinical pharmacokinetics applies mathematical models to describe and predict the time course of drug concentrations in biological fluids. Mastery of kinetic modeling and dosing equations enables the pharmacist to design individualized dosage regimens that optimize therapeutic efficacy while preventing toxicity.

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1. Compartmental Pharmacokinetic Models

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|                        ONE-COMPARTMENT vs. TWO-COMPARTMENT MODELS                   |
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|  ONE-COMPARTMENT OPEN MODEL:                                                        |
|                                                                                     |
|    [ Dose (IV / Oral) ] ===> [ Central Compartment (Vd) ] ===> [ Elimination (ke) ] |
|                                                                                     |
|  TWO-COMPARTMENT OPEN MODEL:                                                        |
|                                                                                     |
|                                  k12                                                |
|    [ Central Compartment (V1) ] <===> [ Peripheral Compartment (V2) ]               |
|             |                    k21    (Muscle, Adipose, Bone)                     |
|             v ke                                                                    |
|     [ Elimination ]                                                                 |
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One-Compartment Open Model

Assumes the human body behaves as a single, instantaneously well-mixed kinetic compartment. Following an intravenous bolus injection, drug distribution between vascular fluid and tissues occurs instantaneously:

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

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

Where $C_0 = \text{Dose} / V_d$, and $k_e$ is the first-order elimination rate constant.

Two-Compartment Open Model

Drugs do not distribute instantaneously into all tissues. Tissues with high blood flow (heart, liver, lungs, kidneys, brain) comprise the Central Compartment ($V_1$), while poorly perfused tissues (muscle, skin, adipose) comprise the Peripheral Compartment ($V_2$).

Following an IV dose, the plasma concentration decline is biexponential:

$C(t) = A \cdot e^{-\alpha \cdot t} + B \cdot e^{-\beta \cdot t}$

• Distribution Phase ($\alpha$-phase): Rapid initial decline in plasma concentration driven by drug moving from the central compartment into peripheral tissues.
• Elimination Phase ($\beta$-phase): Slower terminal decline reflecting irreversible drug elimination from the body once equilibrium between central and peripheral compartments is established.
• Clinical Relevance: Therapeutic drug concentrations must be sampled during the post-distribution $\beta$-phase (e.g., waiting $6-8\text{ hours}$ post-dose for oral digoxin, or $1-2\text{ hours}$ post-infusion for vancomycin).

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2. Elimination Kinetics: First-Order, Zero-Order, and Michaelis-Menten

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|                         ELIMINATION KINETICS PROFILES                               |
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|  First-Order Kinetics (Linear)       : dC/dt = -ke * C                              |
|  - Constant Fraction eliminated per unit time (e.g., 20%/hr)                        |
|  - Constant t(1/2) and Constant Clearance (CL)                                      |
|  - Css is directly proportional to dose (Double dose -> Double Css)                 |
|                                                                                     |
|  Zero-Order Kinetics (Saturable)     : dC/dt = -k0                                  |
|  - Constant Amount eliminated per unit time (e.g., 10 mg/hr)                        |
|  - t(1/2) increases as concentration increases; Clearance decreases                 |
|  - E.g., High-dose Ethanol, Aspirin overdose                                        |
|                                                                                     |
|  Michaelis-Menten Kinetics (Mixed)   : dC/dt = -(Vmax * C) / (Km + C)               |
|  - At C << Km: Behaves as First-Order                                               |
|  - At C >> Km: Behaves as Zero-Order (Saturated enzymes)                            |
|  - Small dose increase -> Disproportionate surge in Css                             |
|  - Classic Example: Phenytoin                                                       |
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Phenytoin Michaelis-Menten Dynamics

Phenytoin biotransformation by hepatic CYP2C9 and CYP2C19 becomes saturated within the standard therapeutic range ($10 - 20\text{ mg/L}$ or $40 - 80\text{ \mu mol/L}$):

$\text{Daily Dose Rate } (R) = \frac{V_{\text{max}} \cdot C_{ss}}{K_m + C_{ss}}$

$\text{Steady-State Concentration } (C_{ss}) = \frac{K_m \cdot R}{V_{\text{max}} - R}$

Where typical population parameters are $V_{\text{max}} \approx 7\text{ mg/kg/day}$ ($400 - 600\text{ mg/day}$) and $K_m \approx 4\text{ mg/L}$ ($16\text{ \mu mol/L}$).

Clinical Warning: As $R$ approaches $V_{\text{max}}$, the denominator ($V_{\text{max}} - R$) approaches zero, precipitating an exponential surge in $C_{ss}$. When phenytoin levels are $12\text{ mg/L}$, increasing the daily dose by as little as $25 - 50\text{ mg/day}$ can double the plasma concentration, causing neurotoxicity.

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3. Fundamental Pharmacokinetic Parameters and Formulas

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|                        KEY PHARMACOKINETIC FORMULAS                                 |
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|  1. Elimination Rate Constant (ke)    : ke = ln(C1 / C2) / (t2 - t1) = CL / Vd      |
|  2. Elimination Half-Life (t1/2)      : t1/2 = 0.693 / ke = (0.693 * Vd) / CL       |
|  3. Systemic Clearance (CL)           : CL = ke * Vd = Dose(IV) / AUC0-inf          |
|  4. Loading Dose (LD)                 : LD = (Vd * Ctarget) / F                     |
|  5. Maintenance Dose (MD)             : MD = (CL * Ctarget * tau) / F               |
|  6. Average Steady State (Css,avg)    : Css,avg = (F * Dose) / (CL * tau)           |
|  7. Accumulation Factor (Racc)        : Racc = 1 / (1 - e^(-ke * tau))              |
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Mathematical Interrelationships

1. Elimination Rate Constant ($k_e$):

$k_e = \frac{\ln(C_1) - \ln(C_2)}{t_2 - t_1} = \frac{\ln(C_1 / C_2)}{\Delta t}$

1. Elimination Half-Life ($t_{1/2}$):

$t_{1/2} = \frac{\ln(2)}{k_e} = \frac{0.693}{k_e} = \frac{0.693 \cdot V_d}{CL}$

Steady-State Principle: The time required to achieve steady-state during repeated dosing depends only on the elimination half-life ($t_{1/2}$), not on the dose or dosing interval:

• $1 \times t_{1/2} = 50%$ of steady state
• $2 \times t_{1/2} = 75%$ of steady state
• $3.32 \times t_{1/2} = 90%$ of steady state
• $4 \text{ to } 5 \times t_{1/2} = 93.75% - 96.88%$ (clinically considered true steady state)

1. Systemic Clearance ($CL$): Clearance represents the intrinsic volume of biological fluid completely cleared of drug per unit time (expressed in $\text{L/h}$ or $\text{mL/min}$):

$CL = k_e \cdot V_d = \frac{F \cdot \text{Dose}}{\text{AUC}_{0-\infty}}$

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4. Dosing Calculations: Loading Doses, Maintenance Doses, and Accumulation

Loading Dose ($LD$)

A loading dose rapidly achieves the target therapeutic plasma concentration without waiting for $4-5\text{ half-lives}$ of steady-state accumulation. It depends strictly on the volume of distribution ($V_d$):

$LD = \frac{V_d \cdot C_{\text{target}}}{F}$

If a baseline drug concentration ($C_{\text{initial}}$) is already present:

$LD = \frac{V_d \cdot (C_{\text{target}} - C_{\text{initial}})}{F}$

Maintenance Dose ($MD$)

The maintenance dose replaces drug lost via systemic clearance during each dosing interval ($\tau$):

$MD = \frac{CL \cdot C_{\text{target}} \cdot \tau}{F} = \frac{k_e \cdot V_d \cdot C_{\text{target}} \cdot \tau}{F}$

Peak and Trough Concentrations at Steady State (Intermittent IV Infusion)

For a drug administered as a short intravenous infusion of duration $T_{\text{inf}}$ with dosing interval $\tau$:

$C_{\text{max},ss} = \frac{\text{Dose} \cdot (1 - e^{-k_e \cdot T_{\text{inf}}})}{T_{\text{inf}} \cdot k_e \cdot V_d \cdot (1 - e^{-k_e \cdot \tau})}$

$C_{\text{min},ss} = C_{\text{max},ss} \cdot e^{-k_e \cdot (\tau - T_{\text{inf}})}$

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5. Renal Function Assessment: The Cockcroft-Gault Equation

In Canadian pharmacy practice and drug monographs, dosage adjustments for renally eliminated drugs (e.g., direct oral anticoagulants, aminoglycosides, vancomycin, gabapentin, low-molecular-weight heparins) are based on the Cockcroft-Gault estimated creatinine clearance (CrCl).

Ideal Body Weight (IBW) Formulas

$\text{IBW}_{\text{male}} = 50\text{ kg} + 2.3 \cdot (\text{Height in inches} - 60)$

$\text{IBW}_{\text{female}} = 45.5\text{ kg} + 2.3 \cdot (\text{Height in inches} - 60)$

• Non-Obese Patients ($Actual Weight \le IBW$): Use Actual Body Weight (ABW).
• Normal Overweight ($ABW = 1.0 - 1.2 \times IBW$): Use IBW.
• Obese Patients ($ABW > 1.2 \times IBW$ or $120%$ of IBW): Use Adjusted Body Weight (AdjBW):

$\text{AdjBW} = \text{IBW} + 0.4 \cdot (\text{Actual Weight} - \text{IBW})$

Cockcroft-Gault Equations

$\text{Using Canadian SI Units (Serum Creatinine } S_{cr} \text{ in } \mu\text{mol/L}):$

$\text{CrCl (mL/min)} = \frac{(140 - \text{Age}) \cdot \text{Weight (kg)}}{S_{cr} (\mu\text{mol/L})} \cdot 1.2 \text{ (males; multiply by } 0.85 \text{ for females)}$

$\text{Using Conventional Units (Serum Creatinine } S_{cr} \text{ in mg/dL}):$

$\text{CrCl (mL/min)} = \frac{(140 - \text{Age}) \cdot \text{Weight (kg)}}{72 \cdot S_{cr} (\text{mg/dL})} \cdot (0.85 \text{ for females, } 1.00 \text{ for males})$

(Conversion Factor: $S_{cr} \text{ in } \mu\text{mol/L} = S_{cr} \text{ in mg/dL} \times 88.4$).

Which constant does the exam expect? The PEBC reference sheet supplied on screen during the Evaluating Examination gives the male form as (140 minus age) x actual body weight (kg) x 1.2 divided by serum creatinine in micromol/L, with the female value obtained by multiplying the male result by 0.85 (an effective factor of about 1.02). Many textbooks print 1.23 and 1.04 instead; the two forms differ by under 3% and rarely change a dosing band, but answer with the sheet you are given.