A Pill's Path from Gut to Blood

A swallowed pill dissolves in the gut, passes into the blood, and is cleared by the body. Track both amounts at once and every dose traces a path that slides into zero without circling. Two real, negative eigenvalues are why.

Click or drag anywhere to start a new trajectory there.

How it works

Systems with real eigenvalues

In one sentence

For $\mathbf x' = A\mathbf x$ with distinct real eigenvalues $\lambda_1, \lambda_2$ and eigenvectors $\mathbf v_1, \mathbf v_2$ , $\mathbf x = c_1e^{\lambda_1t}\mathbf v_1 + c_2e^{\lambda_2t}\mathbf v_2$ ; if both eigenvalues are negative the origin is a stable node.

General formula

{\htmlData{sym=0}{x}}' = -{\htmlData{sym=2}{k_a}}\,{\htmlData{sym=0}{x}}, \qquad {\htmlData{sym=1}{y}}' = {\htmlData{sym=2}{k_a}}\,{\htmlData{sym=0}{x}} - {\htmlData{sym=3}{k_e}}\,{\htmlData{sym=1}{y}}

Each eigenvector is a straight-line path that only shrinks. Every other path is a mix of the two, and the faster one dies first, leaving the slow direction.

What the symbols mean

$x$
Drug in gut
Milligrams not yet absorbed.
$y$
Drug in blood
Milligrams circulating, where the drug works.
$k_a$
Absorption rate
Fraction of the gut amount absorbed per hour.
$k_e$
Elimination rate
Fraction of the blood amount cleared per hour.

The pill dissolves in the gut ( $x$ ) and is absorbed into the blood ( $y$ ) at rate $k_ax$ ; the body clears the blood at rate $k_ey$ .
In matrix form $\mathbf x' = A\mathbf x$ with $A = \begin{bmatrix}-k_a & 0\\ k_a & -k_e\end{bmatrix}$ , whose eigenvalues are $-k_a$ and $-k_e$ .
Two negative real eigenvalues make the origin a stable node: every trajectory slides into it without circling.
Near the end paths run along the slow eigenvector, so the blood level's final decay is set by the slower rate.

Learn the math

Systems of Differential EquationsDiff Eq · Lesson 30, Systems of Differential Equations Eigenvalues and EigenvectorsDiff Eq · Lesson 31, Systems of Differential Equations Real EigenvaluesDiff Eq · Lesson 33, Systems of Differential Equations