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

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
= - kX^{2}

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

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

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
= C_{i}A_{i} e ^{(}^{l}^{it)}

Where
i = 1 and 2