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