2. Plasma Concentration vs. Time Curves (Cp vs. t) & PK Parameters
Following oral administration of a solid dosage form, the drug undergoes dissolution, gastrointestinal absorption, distribution throughout body tissues, and elimination via hepatic metabolism and renal excretion. A plot of Plasma Concentration ($C_p$) against Time ($t$) displays three distinct phases: (1) Absorption Phase where rate of absorption exceeds elimination, (2) Peak Concentration ($C_{max}$) at time $T_{max}$ where rate of absorption equals elimination ($dC/dt = 0$), and (3) Elimination Phase where drug clearance dominates. The Therapeutic Window is bounded by the Minimum Effective Concentration (MEC) and the Minimum Toxic Concentration (MTC).
Extracting Cmax, Tmax, elimination rate constant (Ke), and elimination half-life (t1/2) from clinical sampling data.
import numpy as np
import math
time_hrs = np.array([0.0, 0.5, 1.0, 1.5, 2.0, 3.0, 4.0, 6.0, 8.0, 12.0])
cp_ug_ml = np.array([0.0, 3.8, 7.9, 11.4, 12.5, 9.8, 6.7, 3.1, 1.4, 0.3])
# 1. Cmax and Tmax
c_max = np.max(cp_ug_ml)
t_max = time_hrs[np.argmax(cp_ug_ml)]
# 2. Elimination Rate Constant (Ke) from terminal elimination phase (hours 6, 8, 12)
# Slope of ln(Cp) vs time = -Ke
term_t = time_hrs[-3:]
term_cp = cp_ug_ml[-3:]
slope, _ = np.polyfit(term_t, np.log(term_cp), 1)
ke = -slope
half_life = 0.693 / ke
print(f"Peak Concentration (Cmax): {c_max:.2f} ug/mL")
print(f"Time to Peak (Tmax): {t_max:.2f} hours")
print(f"Elimination Rate Constant (Ke): {ke:.4f} hr^-1")
print(f"Elimination Half-Life (t1/2): {half_life:.2f} hours")| PK Parameter | Symbol | Standard Units | Clinical Significance |
|---|---|---|---|
| Peak Plasma Concentration | Cmax | μg/mL or mg/L | Determines therapeutic efficacy and risk of acute toxicity |
| Time to Peak | Tmax | Hours | Reflects rate of gastrointestinal absorption |
| Elimination Rate Constant | Ke | hr⁻¹ | Fraction of drug removed from body per unit time |
| Elimination Half-Life | t1/2 | Hours | Determines dosing frequency and time to steady state (4-5 half-lives) |
| Area Under the Curve | AUC | μg·hr/mL | Measures total systemic extent of drug absorption / bioavailability |
- $C_{max}$: Maximum drug concentration achieved in systemic circulation.
- $T_{max}$: Time required to reach $C_{max}$ (indicator of absorption rate).
- Therapeutic Range: Drug concentration interval between MEC and MTC.
- Elimination Half-life ($t_{1/2}$): Time required for plasma concentration to decrease by 50% ($t_{1/2} = 0.693 / K_e$).
दवा खाने के बाद खून में उसका स्तर कैसे बढ़ता और घटता है, इसका ग्राफ (Cp vs t Curve) कहलाता है। Cmax अधिकतम स्तर है, Tmax चरम पर पहुँचने का समय है, और t1/2 वह समय है जब दवा आधी रह जाती है।