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






dX/dt = -k



for X = x0 at t = 0



X = x0 kt



This is the equation of a straight Line





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




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



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






-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