Two-Compartment Model Alpha and Beta Phase Calculator
Calculate the alpha and beta hybrid rate constants and plasma concentration for a two-compartment IV bolus pharmacokinetic model.
📊 What is the Two-Compartment Alpha and Beta Phase Model?
The two-compartment model describes drugs that don't distribute into tissue instantly, instead splitting the body into a central compartment (blood and rapidly-equilibrating organs) and a peripheral compartment (slower tissue), connected by transfer rate constants. Concentration in the central compartment follows a biphasic curve, C(t) = A e^(-alpha t) + B e^(-beta t), where alpha and beta are hybrid rate constants combining all three underlying micro-rate constants (k10, k12, k21).
Clinical pharmacologists use this model for drugs with a genuinely visible distribution phase, common examples include many general anesthetics, some cardiovascular drugs, and aminoglycoside antibiotics, where the early rapid decline (alpha phase) and later slower decline (beta phase) have real clinical consequences for dosing and monitoring. Pharmacokinetics researchers use the model to fit real concentration-time data using nonlinear regression, extracting alpha, beta, A, and B from measured plasma samples. Anesthesiologists specifically care about the alpha phase, since it governs how quickly a drug's effect-site concentration falls after stopping an infusion, directly affecting recovery time.
A common misconception is that alpha and beta are the same as the underlying transfer rate constants. They are not, alpha and beta are hybrid values derived mathematically from all three micro-rate constants together (via a quadratic equation), and neither one maps directly onto a single physiological process like "elimination" or "distribution" alone.
This calculator computes alpha, beta, the coefficients A and B, and the concentration at any time you specify, plus a log-scale chart that makes the two distinct linear decay phases visually obvious.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Moderate Distribution and Elimination
Dose = 1000 mg, V1 = 10 L, k10 = 0.3, k12 = 0.2, k21 = 0.1/hr, t = 1 hr
Example 2 — Faster Distribution, Larger Volume
Dose = 500 mg, V1 = 20 L, k10 = 0.4, k12 = 0.3, k21 = 0.15/hr, t = 2 hr
Example 3 — Rapid Distribution Phase
Dose = 800 mg, V1 = 15 L, k10 = 0.25, k12 = 0.5, k21 = 0.2/hr, t = 0.5 hr
❓ Frequently Asked Questions
🔗 Related Calculators
What are alpha and beta in the two-compartment model?
Alpha and beta are the two 'hybrid' rate constants that describe the two distinct exponential phases of a two-compartment drug's concentration-time curve. Alpha is always the larger value, governing the faster early distribution phase, and beta is always the smaller value, governing the slower terminal elimination phase.
How do you calculate alpha and beta from the micro-rate constants?
First compute Sum = k10+k12+k21 and Product = k10 x k21, then alpha and beta are the two roots of the quadratic lambda^2 - Sum*lambda + Product = 0, given by (Sum plus or minus the square root of (Sum^2 - 4*Product)) divided by 2, with the plus sign giving alpha and the minus sign giving beta.
What do the micro-rate constants k10, k12, and k21 represent?
k10 is the elimination rate constant from the central compartment (blood) out of the body, k12 is the transfer rate from the central compartment into the peripheral (tissue) compartment, and k21 is the transfer rate back from the peripheral compartment into the central compartment.
How do you calculate the coefficients A and B?
A = Dose(alpha - k21) / (V1(alpha - beta)) and B = Dose(k21 - beta) / (V1(alpha - beta)), where V1 is the central compartment volume. A useful check is that A plus B always equals exactly Dose/V1, the concentration immediately after the bolus dose.
Why does the concentration equation have two exponential terms instead of one?
Because drug is simultaneously distributing into peripheral tissue and being eliminated from the body, two coupled differential equations (one per compartment) govern the system, and their combined solution for central compartment concentration is the sum of two separate exponential decay terms, not just one.
What is the difference between the alpha phase and beta phase clinically?
The alpha (distribution) phase reflects the drug equilibrating between blood and tissue, often too fast to matter for dosing decisions. The beta (elimination) phase reflects the true, slower, terminal elimination rate and is what actually determines how long a drug stays in the body and how often it needs re-dosing.
How is beta different from the elimination rate constant in the simpler one-compartment model?
In a true two-compartment drug, beta plays a role similar to ke in a one-compartment model, since it governs the terminal decline, but it is a hybrid constant combining all three micro-rate constants, not simply the fractional clearance rate. Beta is always smaller than k10 alone would suggest, since some drug is temporarily 'hiding' in the peripheral compartment.
When should I use a two-compartment model instead of a one-compartment model?
Use the two-compartment model when a drug's real concentration-time data shows a visible bend on a semi-log plot, a clear steep early phase followed by a shallower late phase. If the data instead falls on a single straight line, the simpler one-compartment model already provides an accurate description.
What are the alpha and beta half-lives?
They are computed the same way as any first-order half-life: t1/2(alpha) = ln(2)/alpha for the fast distribution phase, and t1/2(beta) = ln(2)/beta for the slow terminal phase. Beta's half-life is always considerably longer than alpha's, since beta is always the smaller rate constant.
What units should I use for the inputs?
Enter dose in milligrams, central compartment volume (V1) in liters, and all three micro-rate constants (k10, k12, k21) in per hour, this combination yields concentration in mg/L and both half-lives in hours.