Фармакокінетика: Як двокомпонентна модель прогнозує рівні препаратів

Diffusion of Substances

The diffusion of substances is governed by Fick's Laws. Fick’s First Law describes the flux (J) of a substance as proportional to the concentration gradient (ΔC/Δx). Specifically, J = -D(∂C/∂x), where D is the diffusion coefficient and ΔC/Δx represents the change in concentration with respect to distance.

Fick’s Second Law states that the rate of change of concentration (∂C/∂t) at a point is proportional to the negative gradient of concentration. This means ∂C/∂t = -D(∂²C/∂x²), where D is again the diffusion coefficient. This law describes how the concentration changes over time due to diffusion.

The concept of diffusion is fundamental to understanding many phenomena, including the movement of gases and liquids through membranes, as well as the spreading of solutions in a medium. The rate of diffusion depends on several factors, including temperature, pressure, and the properties of the diffusing substance and the medium it's moving through.

dC1/dt = -(k10 + k12)·C1 + k21·(V2/V1)·C2 + dose_input(t)
dC2/dt = k12·(V1/V2)·C1 - k21·C2

  C1, C2 = concentration in central / peripheral compartment
  k10    = elimination rate constant (clearance from central compartment)
  k12,k21 = distribution rate constants between compartments
  V1, V2 = volumes of distribution of each compartment

IV bolus: the classic biexponential decline

An intravenous bolus deposits the whole dose into the central compartment instantly, so C1 starts at its maximum and falls immediately — no absorption phase to wait through. Plotted on a log scale, the concentration-time curve for a two-compartment model is not a single straight line but the sum of two exponential decays: a fast initial distribution phase as the drug equilibrates into the peripheral compartment, followed by a slower, shallower elimination phase once the two compartments are roughly in equilibrium and the decline is governed mainly by clearance. Mistaking the fast distribution phase for the true elimination half-life is a classic dosing-interval error.

Поедание: фаза абсорбції змінює все

Поедання додає третій процес: препарат повинен спочатку бути засвоєний з кишківника в центральний резервуар, зазвичай моделюється як його власна перша-порядкова константа ka. Це створює знайому криву «підйом і спад» — концентрація зростає, поки абсорбція переважає над виведенням, досягає піку на Cmax у час Tmax (момент, коли рівні абсорбції та виведення точно збалансовані, dC/dt = 0), а потім падає, коли виведення домінує, коли більша частина дози була засвоєна. Препарат із швидким засвоєнням (великий ka) відносно виведення створює різкий, ранній, високий пік; повільне засвоєння створює більш рівний, затриманий, нижчий пік для тієї ж загальної дози — основа для розширених форм випуску, які свідомо зменшують ka, щоб згладити криву.

IV bolus:   C1(0) = Dose/V1, then biexponential decline (distribution + elimination)
Oral dose:  dC1/dt includes + ka·A(t),  dA/dt = -ka·A(t)   (A = drug remaining in gut)
            Cmax, Tmax found where dC1/dt = 0
Infusion:   constant input rate R, approaches steady state  Css = R / CL
            (CL = clearance = k10 · V1)

The numbers a PK curve is boiled down to

Four summary numbers drive almost every real dosing decision. Cmax and Tmax describe the peak — how high the concentration gets and how fast — and matter most for drugs where toxicity or efficacy depends on a peak threshold. AUC (area under the concentration-time curve) is proportional to total drug exposure and to bioavailability; comparing AUC between an oral dose and an IV dose of the same drug is literally how bioavailability is measured. The elimination half-life, t½ = ln(2)/k10, sets the dosing interval — a drug is considered to reach steady state after roughly 4-5 half-lives of repeated dosing, and it clears the body to a negligible level after roughly the same number of half-lives once dosing stops.

Infusion and steady state

A continuous IV infusion at constant rate R does not spike and decay — concentration rises asymptotically toward a steady-state plateau Css = R / CL, where clearance CL is the product of the elimination rate constant and the central volume of distribution. This is the standard way critical-care drugs (sedatives, some antibiotics, vasopressors) are dosed when a stable, sustained concentration matters more than a peak — the infusion rate is chosen backward from the desired Css, and reaching it still takes the same 4-5 half-lives regardless of the rate, only the final plateau height changes.

Frequently asked questions

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