5.4 Convolution

Definition

\[ f \ast g = \int_{0}^{t} f(\tau)g(t-\tau)d\tau \]

Properties

\[ f \ast g = g \ast f \] \[ L(f \ast g) = L(f) \cdot L(g) \] \[ L^{-1}(f \cdot g) = L^{-1}(f) \ast L^{-1}(g) \]

Underlying Basic Principle: Rule of superposition

Use

Let f(t) denote input function: Changing rate of amount in body at time t if there is no elimination and no disposition in body(confined in the measurement compartment). That is to say

\[ \frac{dA}{dt} = f(t) \]

A: amount in the body when no elimination, no disposition occurs

Let g(t) denote disposition function : Describe C(t) after bolus injection. That is to say

\[ C(t) = A_0 \cdot g(t) \]

A0: Amount in body at time 0

Then, for any input and disposition model, concentration can be described as follow,

\[ C(t) = f(t) \ast g(t) \]

Proof. If we divide the time into many tiny time intervals which is sufficient to ignore errors,

\[ \Delta t = \frac{t}{n} \]

The drug amount input in the body at i-th interval is approximately

\[ f(i \cdot \Delta t) \Delta t \]

And the drug concentration contributed by the above amount at time t is

\[ f(i \cdot \Delta t) \Delta t \cdot g(t - i \cdot \Delta t) \]

\(t - i \cdot \Delta t\): elapsed time

Total concentration at time t is the summation of each concentration by rule of superposition

\[ \sum_{i=0}^{n-1} f(i \cdot \Delta t) \Delta t \cdot g(t - i \cdot \Delta t) \]

If we divide time more and more the result will be more accurate. The expression follows

\[ \lim_{n \to \infty} \sum_{i=0}^{n-1} f(i \cdot \Delta t) \Delta t \cdot g(t - i \cdot \Delta t) \]

We can replace the following symbols without loss of generality.

\[ i \cdot \Delta t = \tau \] \[ \Delta t = \Delta \tau \]

Then above equation using Sigma() can be expressed using integral symbol.

\[ \int_{0}^{t} f(\tau)g(t-\tau)d\tau = f \ast g \]

For more stringent proof, the case of i from 1 to n also should be proved to be converging to above equation.