Compartmental Modeling - Introduction to Biomedical Engineering - page 403

7.7 THREE-COMPARTMENT MODELING

The general form of the three-compartment model is shown in Figure 7.20. As before, we
begin with the general form of the three-compartment model and then examine special
cases. To analyze the system in Figure 7.20, conservation of mass is used to write a differ-
ential equation for each compartment describing the rate of change of the quantity of solute
in the compartment, given as accumulation

¼ input - output, where

Compartment 1
Accumulation

Compartment 2
Accumulation

Compartment 3
Accumulation ¼ q 1 ¼ q 2 ¼ q 3

Input ¼ f 1 ð t Þþ K 21 q 2 þ K 31 q 3

Ouput

Input ¼ f 2 ð t Þþ K 12 q 1 þ K 32 q 3

Ouput

Input ¼ f 3 ð t Þþ K 13 q 1 þ K 23 q 2

Ouput ¼ð K 10 þ K 12 þ K 13 Þ q 1 ¼ð K 20 þ K 21 þ K 23 Þ q 2 ¼ð K 30 þ K 31 þ K 32 Þ q 3

Therefore,

q 1 ¼ f ð t Þþ K q þ K q ð K

10

þ K þ K Þ q

1
q 2 ¼ f 2 ð t Þþ K 12 q 1 þ K 32 q 3 ð K 20 þ K 21 þ K 23 Þ q 2
q 3 ¼ f 3 ð t Þþ K 13 q 1 þ K 23 q 2 ð K 30 þ K 31 þ K 32 Þ q 3

1 21 2 31 3 12 13

ð 7 : 78 Þ

The D-Operator is used to simplify the system, where Eq. (7.78) is written in matrix form as

D IQ ¼ AQ þ F ð 7 : 79 Þ

where

2 3 2 3 2 3

q 1
q 2
q 2

ð K 10 þ K 12 þ K 13 Þ K 21 K 31 f 1 ð t Þ

f 2 ð t Þ
f 2 ð t Þ

4 5 , A 4 5 , F 4 5

Q ¼ ¼ K 12 ð K 20 þ K 21 þ K 23 Þ K 32 ¼

K 13 K 23 ð K 30 þ K 31 þ K 32 Þ

f 1 (t) f 2 (t)

K 12

K 10 K 20

q 1 q 2

K 21

f 3 (t)

K 23

K 13

K 31

K 32

q 3

K 30

FIGURE 7.20 A general three-compartment model. Compartment 1 has volume V 1 , compartment 2 has volume
V 2 , and compartment 3 has volume V 3 .