Whenever the system responsible for elimination is saturated by the given substance, then:

# The eliminated amount is constant and not proportional to its concentration, thus the reaction is called Zero-Order reaction

and,

dX/dt = -k

for X = x0 at t = 0

X = x0 – kt

This is the equation of a straight Line

Or

X = x0 ( 1 – k/x0 t)

Elimination can also be proportional to the square of the original amount. This is called a Second-Order Reaction

dX/dt = - kX2

X = x0 /  ( 1 + x0 kt )

2) Two Compartments Systems

Only one case will be studied in details

Distribution Curves

Usually complicated systems

For simplicity assume that all compartments in the system are considered homogeneous and represented by one volume called the distribution volume

## Tow Open Compartments

For example, let

First compartment = plasma

Second compartment = liver

Then,

What is in the plasma (compartment one) will leave the compartment and enter compartment two.

For compartment two, the amount obtained from compartment one will increase the contents of the second compartment until elimination from this compartment starts reducing the amount to zero

Or

dX1 / dt = -k1X1

dX2 /dt = k1X1 – k2X2

By integrating both sides of first equation

X1 = x0 exp(-k1t)

By replacing the value of X1 in the second equation

dx2/ dt + k2X2 = k1x0 exp(-k1t)

By integrating both sides

X2 = x0 k1 / k2 – k1 ( e (-k1t) -  e (-k2t) )

Difference between two terms

Use the graphical techniques to calculate the values of K1 and K2

For X1

At t = 0

X1 = X0

At t = ∞

X1 = 0

For X2

At t = 0

X2 = 0

At t = ∞

X2 = 0

For any two or more compartmental system

For example two reversible strongly connected open system

The two differential equations obtained from the above system:

dX1/dt = - K12 X1 + K21X2 + I

dX2/dt = K12 X1 – (K21+K20) X2

In Linear Algebra form

X1              -K12            K21                  X1

=

X2               K12      -(K21+K20)           X2

Thus

-K12           K21

A =

K12      -(K21+K20)

The next step is to find the matrix

(lI - A)

This is a 2 X 2 matrix where I is the identity matrix

1          0

I =

0           1

The next step is to find the determinant of the (lI - A)

The determinant is called the characteristic polynomial

By putting the characteristic polynomial = 0

The equation that results is called the characteristic equation

The roots of this equation are called Eigenvalues

From the Eigenvalues the Eigenvectors are calculated

The General solution is in the form of

Xi = CiAi e (lit)

Where i = 1 and 2