2.2 Drug Distribution

Distribution describes how a drug spreads from the bloodstream to tissues and organs after absorption. The volume of distribution (V_d) is a hypothetical volume that relates plasma concentration to total drug in the body. Plasma protein binding, tissue affinity, and barrier permeability determine how a drug distributes and where it accumulates.

Historical Context

Teorell (1937) first conceptualized distribution compartments in his seminal PK model. Goldstein (1949) formalized the volume of distribution as a mathematical construct. Ehrlich's concept of differential drug distribution led to Paul Ehrlich's "magic bullet" idea (1906) that drugs could be targeted to specific tissues, foreshadowing modern targeted drug delivery.

Derivation 1: Volume of Distribution

V_d is a proportionality constant relating the total amount of drug in the body to its plasma concentration. It is not a real physiological volume but a mathematical parameter.

Step 1: Definition from Mass Balance

After an IV bolus, total drug in body = Dose. The initial plasma concentration C_0 is measured before any elimination occurs:

\( V_d = \frac{\text{Total amount in body}}{\text{Plasma concentration}} = \frac{Dose}{C_0} \)

Step 2: Tissue Binding Relationship

In a two-compartment system (plasma volume V_P and tissue volume V_T), the drug distributes based on binding:

\( V_d = V_P + V_T \cdot \frac{f_{u,P}}{f_{u,T}} \)

where f_u,P = fraction unbound in plasma and f_u,T = fraction unbound in tissue. Drugs that bind extensively to tissue (small f_u,T) have very large V_d values.

Step 3: Interpretation

V_d approximately 3 L

Plasma volume only

Heparin, warfarin (highly protein-bound)

V_d approximately 14 L

Extracellular fluid

Aminoglycosides (polar, charged)

V_d approximately 42 L

Total body water

Ethanol (distributes evenly)

V_d > 42 L indicates extensive tissue binding. Chloroquine (V_d approximately 13,000 L) is not physically in 13,000 L of water; it is sequestered in tissue with very low plasma levels.

Derivation 2: V_d Estimation Methods

There are multiple ways to estimate V_d depending on the PK model and available data.

V_d by Area (V_d,ss)

For the steady-state volume of distribution, which accounts for distribution between compartments:

\( V_{d,ss} = \frac{Dose \cdot AUMC}{AUC^2} \)

where AUMC is the area under the first moment curve (integral of t * C(t)). This gives the true steady-state distribution volume, independent of elimination.

V_d from Terminal Phase

The apparent volume from the terminal elimination phase:

\( V_d = \frac{CL}{\beta} = \frac{Dose}{AUC \cdot \beta} \)

where beta is the terminal elimination rate constant. This overestimates the true V_d because it includes redistribution effects.

Relationship to Clearance and Half-Life

V_d connects clearance and half-life through the fundamental PK triangle:

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

Increasing V_d (more tissue distribution) prolongs half-life. Increasing CL (faster elimination) shortens it.

Derivation 3: Plasma Protein Binding

Only unbound (free) drug is pharmacologically active, can cross membranes, and is available for metabolism and excretion.

Step 1: Binding Equilibrium

Drug (D) binds plasma protein (P) with dissociation constant K_d:

\( D + P \rightleftharpoons DP \quad K_d = \frac{[D][P]}{[DP]} \)

Step 2: Fraction Unbound

The fraction unbound (f_u) at therapeutic concentrations (typically [D] is much less than K_d):

\( f_u = \frac{[D]_{free}}{[D]_{total}} = \frac{K_d}{K_d + [P]_{available}} \)

For most drugs: f_u = C_free / C_total. At therapeutic concentrations, f_u is approximately constant (linear binding).

Step 3: Clinical Significance

Albumin

Binds acidic drugs: warfarin (99% bound), phenytoin (90%), diclofenac (99.5%). Normal level: 3.5-5 g/dL.

Alpha-1 Acid Glycoprotein

Binds basic drugs: lidocaine (70%), propranolol (90%), imipramine (95%). Acute phase reactant: increases in inflammation.

Displacement interactions: When two highly bound drugs compete for the same binding site, displacement increases the free fraction transiently. However, this is rarely clinically significant for drugs cleared by the liver because the increased free drug is also more rapidly cleared, restoring the steady-state free concentration.

Derivation 4: Blood-Brain Barrier Permeability

The BBB is formed by tight junctions between endothelial cells of brain capillaries, excluding paracellular transport. Only transcellular (passive diffusion) and carrier-mediated transport allow CNS drug delivery.

Step 1: Permeability-Surface Area Product

BBB penetration is characterized by the permeability-surface area product (PS):

\( PS = P \cdot S_{BBB} \quad (\text{mL/min/g brain}) \)

where P is the transcellular permeability coefficient and S_BBB is the capillary surface area.

Step 2: Lipophilicity Correlation

For passive diffusion across the BBB, permeability correlates with lipophilicity up to a point:

\( \log P_{BBB} = a \cdot \log P_{oct} + b - c \cdot MW \)

where P_oct = octanol-water partition coefficient, MW = molecular weight, and a, b, c are empirical constants. Optimal logP for BBB penetration is approximately 1.5-2.7.

Step 3: P-glycoprotein Efflux

P-gp (MDR1) at the BBB actively pumps drugs back into the blood. Net brain uptake:

\( J_{net} = PS \cdot f_{u,P} \cdot C_P - \frac{V_{max,Pgp} \cdot C_{brain}}{K_{m,Pgp} + C_{brain}} \)

P-gp substrates (loperamide, many chemotherapy agents) achieve minimal brain concentrations despite adequate lipophilicity. This is why loperamide acts as a peripheral opioid (gut) but not centrally.

Derivation 5: Tissue Distribution Kinetics

Drug distribution to tissues follows perfusion-rate limited or permeability-rate limited kinetics depending on the drug's properties.

Perfusion-Limited Distribution

For small lipophilic drugs that freely cross capillary walls, distribution rate depends on blood flow to the tissue:

\( \frac{dC_T}{dt} = \frac{Q_T}{V_T}\left(C_{art} - \frac{C_T}{K_p}\right) \)

where Q_T = tissue blood flow, K_p = tissue:plasma partition coefficient. Highly perfused organs (brain, heart, liver) equilibrate in minutes; poorly perfused tissue (fat, bone) takes hours.

Tissue Partition Coefficient

K_p determines the equilibrium concentration ratio between tissue and plasma:

\( K_p = \frac{C_{tissue}}{C_{plasma}} = \frac{f_{u,P}}{f_{u,T}} \cdot \frac{V_T}{V_{water,T}} \)

The overall V_d is the sum of contributions from all tissues: V_d = V_P + sum(V_T,i * K_p,i).

Drug Distribution Compartments

Python Simulation: Drug Distribution

Drug Distribution — Vd, Protein Binding, BBB Permeability & Tissue Kinetics

Python

script.py126 lines

import numpy as np

# ── Panel 1: Vd vs Tissue Binding (fu_P / fu_T ratio) ──
fu_P_values = np.linspace(0.01, 1.0, 100)
VP = 3.0  # L
VT = 39.0  # L (rest of body water)
fu_T_values = [0.01, 0.05, 0.1, 0.5, 1.0]

print("CHART1_TITLE: Volume of Distribution vs Plasma Unbound Fraction")
print("CHART1_XLABEL: f_u,P (fraction unbound in plasma)")
print("CHART1_YLABEL: V_d (L)")
for i in range(0, len(fu_P_values), 3):
    vals = []
    for ft in fu_T_values:
        vd = VP + VT * fu_P_values[i] / ft
        if vd < 5000:
            vals.append(f"fuT_{ft:.2f}={vd:.1f}")
    print(f"fuP={fu_P_values[i]:.3f} " + " ".join(vals))

# ── Panel 2: Protein Binding — Fraction Unbound vs Concentration ──
Kd = 50  # uM (binding affinity)
Pt = 600  # uM (albumin approximately 600 uM)
n_sites = 2
drug_concs = np.logspace(-1, 3, 200)  # uM

print("\nCHART2_TITLE: Protein Binding: Fraction Unbound vs Drug Concentration")
print("CHART2_XLABEL: [Drug] (uM)")
print("CHART2_YLABEL: Fraction Unbound (f_u)")
for i in range(0, len(drug_concs), 5):
    D = drug_concs[i]
    bound = n_sites * Pt * D / (Kd + D)
    total = D + bound
    fu = D / total if total > 0 else 1
    print(f"D={D:.2f} fu={fu:.4f}")

# ── Panel 3: BBB Permeability vs logP ──
logP_range = np.linspace(-2, 6, 200)
a, b, c = 0.7, -1.0, 0.001
MW_values = [200, 350, 500]

print("\nCHART3_TITLE: BBB Permeability vs Lipophilicity (logP)")
print("CHART3_XLABEL: logP (octanol-water)")
print("CHART3_YLABEL: log(BBB Permeability)")
for i in range(0, len(logP_range), 5):
    vals = []
    for mw in MW_values:
        log_bbb = a * logP_range[i] + b - c * mw
        vals.append(f"MW_{mw}={log_bbb:.3f}")
    print(f"logP={logP_range[i]:.2f} " + " ".join(vals))

# ── Panel 4: Tissue Distribution Over Time (perfusion-limited) ──
t = np.linspace(0, 12, 300)
C_plasma = 10 * np.exp(-0.2 * t)  # mg/L

# Different tissues
tissues = [
    ("Brain", 0.7, 1.5, 0.8),    # (name, Q/V, Kp, V_tissue)
    ("Muscle", 0.1, 2.0, 30.0),
    ("Fat", 0.03, 10.0, 15.0),
]

print("\nCHART4_TITLE: Tissue Concentration-Time Profiles")
print("CHART4_XLABEL: Time (h)")
print("CHART4_YLABEL: Concentration (mg/L)")
tissue_curves = {}
for name, qv, kp, vt in tissues:
    C_T = np.zeros(len(t))
    dt = t[1] - t[0]
    for j in range(1, len(t)):
        dCT = qv * (C_plasma[j-1] - C_T[j-1] / kp) * dt
        C_T[j] = C_T[j-1] + dCT
    tissue_curves[name] = C_T

for i in range(0, len(t), 6):
    vals = [f"plasma={C_plasma[i]:.3f}"]
    for name in ["Brain", "Muscle", "Fat"]:
        vals.append(f"{name}={tissue_curves[name][i]:.3f}")
    print(f"t={t[i]:.2f} " + " ".join(vals))

# ── Panel 5: Effect of Hypoalbuminemia on Free Drug ──
albumin_levels = np.linspace(0.5, 5.0, 100)  # g/dL
normal_albumin = 4.0
drugs_binding = [
    ("Warfarin", 0.99),
    ("Phenytoin", 0.90),
    ("Diclofenac", 0.995),
    ("Gentamicin", 0.10),
]

print("\nCHART5_TITLE: Hypoalbuminemia: Free Drug Fraction vs Albumin Level")
print("CHART5_XLABEL: Serum Albumin (g/dL)")
print("CHART5_YLABEL: Fold-Increase in Free Fraction")
for i in range(0, len(albumin_levels), 3):
    alb = albumin_levels[i]
    vals = []
    for name, fb_normal in drugs_binding:
        fu_normal = 1 - fb_normal
        fu_hypo = 1 - fb_normal * (alb / normal_albumin)
        fu_hypo = max(fu_normal, min(1, fu_hypo))
        fold = fu_hypo / fu_normal
        vals.append(f"{name}={fold:.2f}")
    print(f"albumin={alb:.2f} " + " ".join(vals))

# ── Panel 6: P-gp Efflux Effect on Brain Concentration ──
print("\nCHART6_TITLE: P-gp Efflux: Brain Concentration with/without P-gp")
print("CHART6_XLABEL: Plasma Concentration (mg/L)")
print("CHART6_YLABEL: Brain Concentration (mg/L)")
C_p_range = np.linspace(0, 20, 100)
PS = 0.5  # mL/min/g
Vmax_pgp = 5.0
Km_pgp = 2.0

for i in range(0, len(C_p_range), 3):
    cp = C_p_range[i]
    # Without P-gp
    c_brain_no_pgp = cp * 0.8  # Kp = 0.8
    # With P-gp (steady state: PS*Cp = PS*Cb/Kp + Pgp efflux)
    # Simplified: Cb = Kp * (PS*Cp - Vmax*Cb/(Km+Cb)) / PS
    # Iterative solution
    cb = c_brain_no_pgp
    for _ in range(20):
        efflux = Vmax_pgp * cb / (Km_pgp + cb) if cb > 0 else 0
        cb = max(0, 0.8 * (cp - efflux / PS * 0.5))
    print(f"Cp={cp:.2f} no_Pgp={c_brain_no_pgp:.3f} with_Pgp={cb:.3f}")

print("\nSimulation complete: 6 panels generated")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Clinical Applications

Phenytoin & Hypoalbuminemia

Phenytoin is 90% albumin-bound. In nephrotic syndrome (albumin approximately 2 g/dL), free fraction doubles from 10% to approximately 20%. Total phenytoin level appears "subtherapeutic" but free levels are adequate. The Sheiner-Tozer correction: C_adjusted = C_measured / (0.2 * albumin/4.4 + 0.1).

Thiopental Redistribution

Ultra-short action of thiopental is due to redistribution, not elimination. After IV bolus, it rapidly enters the well-perfused brain (onset in 30 seconds) but then redistributes to muscle and fat over minutes, causing rapid awakening despite a long elimination half-life (11 hours).

Loperamide vs Morphine (P-gp)

Both are mu-opioid agonists, but loperamide is a P-gp substrate actively effluxed from the brain. At normal doses, brain levels are negligible, giving peripheral antidiarrheal effects without central opioid effects (no sedation or addiction).

Chloroquine Tissue Sequestration

V_d approximately 13,000 L due to extensive binding to melanin-containing tissues, liver, and spleen. This massive tissue reservoir means chloroquine persists in the body for months after the last dose, enabling weekly dosing for malaria prophylaxis.

Key Takeaways

1.
V_d = Dose/C_0 is a hypothetical volume. V_d = V_P + V_T * (f_u,P / f_u,T). Extensive tissue binding produces V_d far exceeding body water.
2.
V_d connects clearance and half-life: t_1/2 = 0.693 * V_d / CL. V_d,ss from AUMC/AUC^2 is the most accurate estimate.
3.
Only unbound drug (f_u) is active. Albumin binds acidic drugs, AAG binds basic drugs. Hypoalbuminemia increases free fraction.
4.
BBB permeability depends on lipophilicity (optimal logP approximately 1.5-2.7), molecular weight, and P-gp efflux activity.
5.
Tissue distribution kinetics are perfusion-limited (small lipophilic drugs) or permeability-limited (large/polar drugs), with partition coefficient K_p determining equilibrium ratios.

← 2.1 Absorption 2.3 Metabolism →

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