Two-Compartment IV Bolus

Two-compartment mammillary model with central and peripheral compartments, exhibiting bi-exponential concentration decline.

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Function Signature

neopkpd.simulate_pk_twocomp_iv_bolus(
    cl: float,
    v1: float,
    q: float,
    v2: float,
    doses: list[dict],
    t0: float,
    t1: float,
    saveat: list[float],
    alg: str = "Tsit5",
    reltol: float = 1e-10,
    abstol: float = 1e-12,
    maxiters: int = 10**7,
    alag: float | None = None,
    bioavailability: float | None = None,
) -> dict

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Parameters

Parameter	Type	Description
cl	float	Clearance from central (L/h)
v1	float	Central compartment volume (L)
q	float	Inter-compartmental clearance (L/h)
v2	float	Peripheral compartment volume (L)
doses	list[dict]	Dose events
t0, t1	float	Simulation time range
saveat	list[float]	Output time points

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Returns

{
    "t": [0.0, 1.0, ...],
    "states": {
        "A_central": [...],        # Amount in central
        "A_peripheral": [...]      # Amount in peripheral
    },
    "observations": {
        "conc": [...]              # Central concentration (A_central/V1)
    },
    "metadata": {...}
}

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Model Equations

\[\frac{dA_1}{dt} = -(k_{10} + k_{12}) \cdot A_1 + k_{21} \cdot A_2\]

\[\frac{dA_2}{dt} = k_{12} \cdot A_1 - k_{21} \cdot A_2\]

Micro-rate Constants

Constant	Formula
k10	CL / V1
k12	Q / V1
k21	Q / V2

Bi-exponential Solution

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

where \(\alpha > \beta\) (distribution and terminal phases).

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Basic Examples

Single IV Bolus

import neopkpd

result = neopkpd.simulate_pk_twocomp_iv_bolus(
    cl=10.0,      # Clearance (L/h)
    v1=20.0,      # Central volume (L)
    q=15.0,       # Inter-compartmental clearance (L/h)
    v2=50.0,      # Peripheral volume (L)
    doses=[{"time": 0.0, "amount": 500.0}],
    t0=0.0,
    t1=48.0,
    saveat=[i * 0.5 for i in range(97)]
)

# Initial concentration (C0)
print(f"C0: {result['observations']['conc'][0]:.1f} mg/L")  # 500/20 = 25 mg/L

IV Infusion

# 500 mg over 2 hours
result = neopkpd.simulate_pk_twocomp_iv_bolus(
    cl=10.0, v1=20.0, q=15.0, v2=50.0,
    doses=[{"time": 0.0, "amount": 500.0, "duration": 2.0}],
    t0=0.0, t1=48.0,
    saveat=[i * 0.25 for i in range(193)]
)

conc = result['observations']['conc']
cmax = max(conc)
print(f"Cmax: {cmax:.2f} mg/L (at end of infusion)")

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Distribution Phases

import neopkpd
import numpy as np

result = neopkpd.simulate_pk_twocomp_iv_bolus(
    cl=10.0, v1=20.0, q=15.0, v2=50.0,
    doses=[{"time": 0.0, "amount": 500.0}],
    t0=0.0, t1=24.0,
    saveat=[i * 0.1 for i in range(241)]
)

t = np.array(result['t'])
conc = np.array(result['observations']['conc'])

# Log-linear plot shows two phases
log_conc = np.log(conc)

# Distribution phase (early - fast decline)
print(f"C at 0h: {conc[0]:.2f} mg/L")
print(f"C at 1h: {conc[10]:.2f} mg/L (distribution phase)")

# Terminal phase (late - slow decline)
print(f"C at 6h: {conc[60]:.2f} mg/L")
print(f"C at 12h: {conc[120]:.2f} mg/L (terminal phase)")

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Calculate Rate Constants

import numpy as np

# Parameters
cl, v1, q, v2 = 10.0, 20.0, 15.0, 50.0

# Micro-rate constants
k10 = cl / v1      # 0.5 /h
k12 = q / v1       # 0.75 /h
k21 = q / v2       # 0.3 /h

# Hybrid constants (alpha and beta)
sum_k = k10 + k12 + k21
prod_k = k10 * k21
discriminant = np.sqrt(sum_k**2 - 4*prod_k)

alpha = (sum_k + discriminant) / 2
beta = (sum_k - discriminant) / 2

# Half-lives
t_half_alpha = 0.693 / alpha
t_half_beta = 0.693 / beta

print(f"Distribution t1/2 (alpha): {t_half_alpha:.2f} h")
print(f"Terminal t1/2 (beta): {t_half_beta:.2f} h")

# Volume at steady state
vss = v1 + v2
print(f"Vss: {vss} L")

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Multiple Dosing

# 500 mg every 8 hours for 3 days
doses = [{"time": i * 8.0, "amount": 500.0} for i in range(9)]

result = neopkpd.simulate_pk_twocomp_iv_bolus(
    cl=10.0, v1=20.0, q=15.0, v2=50.0,
    doses=doses,
    t0=0.0, t1=72.0,
    saveat=[i * 0.5 for i in range(145)]
)

# Steady-state metrics in last interval (64-72 h)
conc = result['observations']['conc']
t = result['t']

ss_start = next(i for i, x in enumerate(t) if x >= 64)
ss_conc = conc[ss_start:]

print(f"Cmax,ss: {max(ss_conc):.2f} mg/L")
print(f"Cmin,ss: {min(ss_conc):.2f} mg/L")

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Compartment Amounts

import neopkpd

result = neopkpd.simulate_pk_twocomp_iv_bolus(
    cl=10.0, v1=20.0, q=15.0, v2=50.0,
    doses=[{"time": 0.0, "amount": 500.0}],
    t0=0.0, t1=24.0,
    saveat=[0, 1, 4, 8, 12, 24]
)

# Track drug distribution over time
a_central = result['states']['A_central']
a_periph = result['states']['A_peripheral']
t = result['t']

print("Time | Central | Peripheral | % Central")
print("-" * 45)
for i, time in enumerate(t):
    total = a_central[i] + a_periph[i]
    if total > 0:
        pct_central = 100 * a_central[i] / total
        print(f"{time:4.0f}h | {a_central[i]:7.1f} | {a_periph[i]:10.1f} | {pct_central:6.1f}%")

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Clinical Example: Vancomycin

import neopkpd

# Typical vancomycin parameters
result = neopkpd.simulate_pk_twocomp_iv_bolus(
    cl=4.5,       # L/h
    v1=15.0,      # L (central)
    q=4.0,        # L/h
    v2=40.0,      # L (peripheral)
    doses=[{"time": 0.0, "amount": 1000.0, "duration": 1.0}],  # 1g over 1h
    t0=0.0,
    t1=24.0,
    saveat=[i * 0.25 for i in range(97)]
)

conc = result['observations']['conc']
t = result['t']

# TDM sampling times
idx_2h = next(i for i, x in enumerate(t) if x >= 2.0)
idx_trough = next(i for i, x in enumerate(t) if x >= 12.0)

print(f"Peak (2h): {conc[idx_2h]:.1f} mg/L")
print(f"Trough (12h): {conc[idx_trough]:.1f} mg/L")

# Target ranges: Peak 30-40 mg/L, Trough 15-20 mg/L

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Visualization

import neopkpd
from neopkpd.viz import plot_pk_profile

result = neopkpd.simulate_pk_twocomp_iv_bolus(
    cl=10.0, v1=20.0, q=15.0, v2=50.0,
    doses=[{"time": 0.0, "amount": 500.0}],
    t0=0.0, t1=24.0,
    saveat=[i * 0.25 for i in range(97)]
)

fig = plot_pk_profile(
    result['t'],
    result['observations']['conc'],
    title="Two-Compartment IV Bolus",
    xlabel="Time (h)",
    ylabel="Concentration (mg/L)",
    log_y=True  # Semi-log plot shows bi-exponential phases
)

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Equations Summary

Quantity	Formula
k10	CL / V1
k12, k21	Q/V1, Q/V2
alpha	\((k_{10}+k_{12}+k_{21}+\sqrt{\Delta})/2\)
beta	\((k_{10}+k_{12}+k_{21}-\sqrt{\Delta})/2\)
Vss	V1 + V2
Terminal t1/2	0.693 / beta
AUC	Dose / CL

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See Also

• One-Compartment IV - Simpler model
• Two-Compartment Oral - With absorption
• Three-Compartment IV - More compartments
• Population Simulation