IV Drip Concentration

A drip adds medicine at a steady rate while the body removes a fixed fraction of what is there. The level climbs and then flattens where input and removal balance.

How it works

First-order linear ODEs

In one sentence

With a constant infusion rate $r$ into volume $V$ and first-order elimination rate $k$ , the concentration solves $C' = r/V - kC$ , so $C(t) = \frac{r}{kV}\left(1 - e^{-kt}\right)$ , which approaches the steady state $r/(kV)$ .

General formula

{\htmlData{sym=0}{\frac{dC}{dt}}} = \frac{\htmlData{sym=2}{{r}}}{\htmlData{sym=3}{{V}}} - {\htmlData{sym=4}{k}}{\htmlData{sym=1}{C}} \;\;\Rightarrow\;\; {\htmlData{sym=1}{C}}({\htmlData{sym=5}{t}}) = \frac{\htmlData{sym=2}{{r}}}{{\htmlData{sym=4}{k}}{\htmlData{sym=3}{V}}}\left(1 - e^{-{\htmlData{sym=4}{k}}{\htmlData{sym=5}{t}}}\right)

$r/V$ is medicine flowing in each hour; $-kC$ is the body clearing a fixed share of what is there. When they balance, $C$ stops changing.

What the symbols mean

$\frac{dC}{dt}$
Rate of change
How fast the drug level in the blood is rising or falling right now.
$C$
Concentration
How much drug is in each liter of blood.
$r$
Infusion rate
How much drug the IV drips in per hour.
$V$
Blood volume
The volume the drug spreads through.
$k$
Elimination rate
The fraction of the drug the body clears each hour.
$t$
Time
Hours since the infusion started.

Input is constant, removal is proportional to the current concentration.
The steady state $C_\infty = r/(kV)$ is where $dC/dt = 0$ .
It takes about $4.6/k$ hours to get within 1% of steady state.

Learn the math

Increasing, Decreasing and the First Derivative TestCalc I · Lesson 30, Applications of Derivatives Linear EquationsDiff Eq · Lesson 4, First Order Differential Equations Modeling with First Order EquationsDiff Eq · Lesson 10, First Order Differential Equations IVPs with Step FunctionsDiff Eq · Lesson 27, Laplace Transforms

Learn the math

Increasing, Decreasing and the First Derivative TestCalc I · Lesson 30, Applications of Derivatives

Linear EquationsDiff Eq · Lesson 4, First Order Differential Equations

Modeling with First Order EquationsDiff Eq · Lesson 10, First Order Differential Equations

IVPs with Step FunctionsDiff Eq · Lesson 27, Laplace Transforms