Maintenance Dose Equation, Derivation & Rearrangements

Core idea

Overview

The maintenance dose equation is used to determine the mass of drug administered per dosing interval to sustain a specific steady-state plasma concentration. It balances the rate of drug delivery into the systemic circulation against the rate of drug clearance from the body.

When to use: This equation is applied when designing a long-term treatment regimen or adjusting dosages for chronic conditions once a steady state is desired. It assumes the drug follows first-order kinetics, meaning the rate of elimination is directly proportional to the plasma concentration.

Why it matters: Calculating the correct maintenance dose ensures that the drug concentration remains within the therapeutic window, preventing ineffective sub-therapeutic levels or dangerous toxicity. This is especially critical for medications with a narrow therapeutic index, such as digoxin, theophylline, or certain antibiotics.

Symbols

Variables

MD = Maintenance Dose, C_p = Target Concentration, CL = Clearance, \tau = Dosing Interval, F = Bioavailability

M D

Maintenance Dose

m g

C_{p}

Target Concentration

m g / L

C L

Clearance

L / h

τ

Dosing Interval

h

F

Bioavailability

0 - 1

Walkthrough

Derivation

Derivation of Maintenance Dose

Calculates the dose required to maintain a steady-state plasma concentration: MD = Css × CL × τ.

Steady state has been reached (rate in = rate out).
Drug elimination follows first-order kinetics.
Dosing interval τ is constant.

1

At Steady State, Rate In = Rate Out:

The dosing rate (dose per interval) equals the rate of elimination (concentration × clearance) at steady state.

\frac{Dose}{τ} = C_{ss} \times C L

2

Solve for Maintenance Dose:

Rearranging gives the dose needed each interval to maintain Css.

M D = C_{ss} \times C L \times τ

3

Account for Bioavailability:

For oral drugs with bioavailability F, increase the dose accordingly.

M D = \frac{C _{ss} \times C L \times τ}{F}

Result

M D = \frac{C _{ss} \times C L \times τ}{F}

Source: Goodman & Gilman's — The Pharmacological Basis of Therapeutics

Free formulas

Rearrangements

Solve for $M D$

Make result the subject

M D = \frac{C _{p} C Lτ}{F}

Exact symbolic rearrangement generated deterministically for result.

Difficulty: 3/5

Solve for $C_{p}$

Make cp the subject

C_{p} = \frac{M D F}{C Lτ}

Exact symbolic rearrangement generated deterministically for cp.

Difficulty: 3/5

Solve for $C L$

Make cl the subject

C L = \frac{M D F}{C _{p} τ}

Exact symbolic rearrangement generated deterministically for cl.

Difficulty: 3/5

Solve for $τ$

Make tau the subject

τ = \frac{M D F}{C _{p} C L}

Exact symbolic rearrangement generated deterministically for tau.

Difficulty: 3/5

Solve for $F$

Make f the subject

F = \frac{C _{p} C Lτ}{M D}

Exact symbolic rearrangement generated deterministically for f.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

The graph is a straight line passing through the origin, representing a linear relationship between the independent variable and the maintenance dose. Because the maintenance dose is directly proportional to the chosen independent variable, the slope of the line is determined by the remaining constant parameters in the formula.

Graph type: linear

Why it behaves this way

Intuition

Imagine the body's plasma as a reservoir where drug is continuously flowing in (from the dose) and flowing out (due to clearance); the maintenance dose is adjusted to keep the drug level (plasma concentration)

MD

The total mass of drug to be administered in a single dose.

This is the actual amount of medication (e.g., in milligrams) a patient takes at each scheduled time to keep drug levels stable.

C_{p}

The desired average concentration of the drug in the blood plasma at steady state.

The target level of drug in the bloodstream that provides the best therapeutic effect without causing toxicity.

CL

The volume of plasma cleared of drug per unit time, representing the body's efficiency in eliminating the drug.

How quickly the body removes the drug from the blood. If clearance is high, the body gets rid of the drug faster, so more drug is needed.

τ

The duration of the dosing interval, i.e., the time between successive drug administrations.

How often the drug is given. A longer time between doses means more drug is eliminated, so a larger dose is needed each time to maintain the target level.

F

The fraction of the administered dose that reaches the systemic circulation unchanged.

The proportion of the drug that actually gets into the bloodstream and is available to work. If only half the drug gets in (F=0.5), you need to administer twice the amount to achieve the desired effective dose.

Signs and relationships

CL: As clearance increases, the body removes the drug more rapidly. To maintain the target plasma concentration, a proportionally larger maintenance dose is required, hence `CL` is in the numerator.
τ: A longer dosing interval means more time passes between doses, allowing more drug to be eliminated from the body. To compensate for this greater loss and maintain the target concentration, a proportionally larger dose is
F: Bioavailability `F` represents the fraction of the administered dose that actually reaches the systemic circulation. If `F` is less than 1 (e.g., for oral drugs), the administered dose must be larger than the desired

Free study cues

Insight

Canonical usage

In pharmacokinetics, typical units involve mass in milligrams (mg), volume in liters (L), and time in hours (h), ensuring dimensional consistency for the maintenance dose in mass units.

Common confusion

The most frequent error is using inconsistent time units for clearance (CL) and the dosing interval (τ), such as clearance in L/h and interval in minutes, leading to incorrect dose calculations.

Unit systems

$M D$ mg · The resulting mass of drug to be administered per dose. Its unit will be determined by the mass unit in Cp.

$C_{p}$ mg/L · Commonly expressed as mass per liter (e.g., mg/L, μg/mL, ng/mL). Ensure consistency with dose mass units.

$C L$ L/h · Represents volume cleared per unit time (e.g., L/h, mL/min). The time unit must match the dosing interval (τ).

$τ$ h · The duration between doses. Its time unit must match the time unit of clearance (CL).

$F$ dimensionless · A fraction between 0 and 1, representing the proportion of the administered drug that reaches systemic circulation.

Ballpark figures

Quantity:

One free problem

Practice Problem

A physician wants to maintain a steady-state plasma concentration of 15 mg/L for an oral medication. The drug has a clearance of 2 L/h and a bioavailability of 0.5. If the drug is administered every 8 hours, what is the required maintenance dose?

Target Concentration15 mg/L

Clearance2 L/h

Dosing Interval8 h

Bioavailability0.5 0-1

Solve for: $r es u l t$

Hint: Multiply the target concentration, clearance, and dosing interval, then divide by the bioavailability.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

A patient requires a steady-state concentration of 10 mg/L of a drug with a clearance of 2 L/h, administered every 12 hours orally with a bioavailability of 0.5. MD = (10 * 2 * 12) / 0.5 = 480 mg.

Study smarter

Tips

Set the bioavailability (F) to 1.0 for all intravenous (IV) drug administrations.
Ensure that the time units for the clearance rate (CL) and the dosing interval (tau) are identical (e.g., both in hours).
Always adjust the clearance variable for patients with renal or hepatic impairment to prevent drug accumulation.

Avoid these traps

Common Mistakes

Failing to convert bioavailability percentage to a decimal (e.g., 90% = 0.9).
Using the equation for loading doses, which requires Volume of Distribution (Vd) rather than Clearance.
Not adjusting clearance for patients with renal or hepatic impairment.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Calculates the dose required to maintain a steady-state plasma concentration: MD = Css × CL × τ.

This equation is applied when designing a long-term treatment regimen or adjusting dosages for chronic conditions once a steady state is desired. It assumes the drug follows first-order kinetics, meaning the rate of elimination is directly proportional to the plasma concentration.

Calculating the correct maintenance dose ensures that the drug concentration remains within the therapeutic window, preventing ineffective sub-therapeutic levels or dangerous toxicity. This is especially critical for medications with a narrow therapeutic index, such as digoxin, theophylline, or certain antibiotics.

Failing to convert bioavailability percentage to a decimal (e.g., 90% = 0.9). Using the equation for loading doses, which requires Volume of Distribution (Vd) rather than Clearance. Not adjusting clearance for patients with renal or hepatic impairment.

A patient requires a steady-state concentration of 10 mg/L of a drug with a clearance of 2 L/h, administered every 12 hours orally with a bioavailability of 0.5. MD = (10 * 2 * 12) / 0.5 = 480 mg.

Set the bioavailability (F) to 1.0 for all intravenous (IV) drug administrations. Ensure that the time units for the clearance rate (CL) and the dosing interval (tau) are identical (e.g., both in hours). Always adjust the clearance variable for patients with renal or hepatic impairment to prevent drug accumulation.

References

Sources

Goodman & Gilman's The Pharmacological Basis of Therapeutics, 13th Edition
Basic and Clinical Pharmacology, 15th Edition, by Bertram G. Katzung
Applied Biopharmaceutics & Pharmacokinetics, 7th Edition, by Leon Shargel and Andrew B.C. Yu
Wikipedia: Pharmacokinetics
Wikipedia: Maintenance dose
Goodman & Gilman's The Pharmacological Basis of Therapeutics
Applied Biopharmaceutics & Pharmacokinetics (Shargel and Yu)
Basic & Clinical Pharmacology (Katzung)

Maintenance Dose

Overview

Variables

Derivation

At Steady State, Rate In = Rate Out:

Solve for Maintenance Dose:

Account for Bioavailability:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Loading Dose

Body Surface Area (Mosteller)

Frequently Asked Questions

Sources