Compartmental Modeling - Introduction to Biomedical Engineering - page 392

f 1 (t) f 2 (t)

K 12

q 1 q 2

K 21

K 10 K 20

FIGURE 7.15 A general two-compartment model. Compartment 1 has volume V 1 and compartment 2 has

volume V 2 .

The easiest way to analyze the system in Eq. (7.47) is using the D-Operator or its equiva-
lent, the Laplace variable s , where we represent the system in matrix form as

D IQ ¼ AQ þ F ð 7 : 48 Þ

where

, A , F

q ð K þ K Þ K f ð t Þ

1
q

10 12 21 1

Q ¼ ¼ ¼

K ð K þ K Þ f ð t Þ

2 12 20 21 2

Equation (7.48) is solved as

1

Þ 1 F Q ¼ D ð I A ¼ adj ð D I A Þ F

det ð Þ

D I A

or

det ð D I A Þ Q ¼ adj ð D I A Þ F ð 7 : 49 Þ

and using MATLAB, we have

>>

syms D q1 q2 K10 K20 K12 K21

>> ¼ þ þ

A [-(K10 K12) K21;K12 -(K20 K21)]

>>

det(D*eye(2)-A)
ans

¼

^ þ þ þ þ þ þ þ

D 2 D*K20 D*K21 K10*D K10*K20 K10*K21 K12*D K12*K20

>> ¼

adj det(D*eye(2)-A)*inv(D*eye(2)-A)

¼

adj

þ þ

[D K20 K21, K21]

þ þ

[ K12, D K10 K12]