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.