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
= - 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
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
= CiAi e (lit)
Where
i = 1 and 2