Compartmental Modeling - Introduction to Biomedical Engineering - page 406

between compartments 1 and 3. Each compartment can have an input and an output to the
environment.
The mammillary three-compartment model is given by the following set of equations:

q 1 ¼ f ð t Þþ K ð K þ K Þ q

q

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

1 21 2 10 12

ð 7 : 86 Þ

With

2
4

3
5 ¼

2
4

3
5

a a a ð K þ K Þ K 0

11 12 13 10 12 21

A ¼ a a a K ð K þ K þ K Þ K

21 22 23 12 20 21 23 32

a a a 0 K ð K þ K Þ

31 32 33 23 30 32

and Eq. (7.81), we have

Q

3 2

D a ð þ a þ a Þ D þ a ð a þ a a þ a a a a a a Þ D

11 22 33 11 22 22 33 33 11 12 21 23 32

¼

a 11 a a þ a a a þ a a a a a a

22 33 11 23 32 33 12 21 12 23 31

2
4

3
5

2

D a ð þ a Þ D þ a a a a a D a a a a

33 22 22 33 23 32 12 12 33 12 23

2

a 21 D a 21 a 33 D a 33 þ a 11 ð Þ D þ a 11 a 33 a 23 D a 23 a 11 F

2

a 21 a 32 a 32 D a 32 a 11 D a 22 þ a 11 ð Þ D þ a 11 a 22 a 12 a 21

ð 7 : 87 Þ

Returning to the time domain gives the following independent differential equations:

Þ q 1 þ a Þ q 1

__ q 1 a ð þ a þ a ð a þ a a þ a a a a a a

11 22 33 11 22 22 33 33 11 12 21 23 32

þ a ð a a þ a a a a a a Þ q

ð 7 : 88 Þ

11 23 32 33 12 21 11 22 33 1

¼ f 1 a 33 þ a 22 Þ f 1 þ a 22 a 33 a 23 a 32 Þ f 1 þ a 12 f 2 a 12 a 33 f 2 ð ð

Þ q 2 þ a Þ q 2 __ q 2 a ð þ a þ a ð a þ a a þ a a a a a a

11 22 33 11 22 22 33 33 11 12 21 23 32

þ a ð a a þ a a a a a a Þ q

ð 7 : 89 Þ 11 23 32 33 12 21 11 22 33 2

¼ a 21 f 1 a 21 a 33 a 23 a 31 Þ f 1 þ f 2 a 33 þ a 11 Þ f 2 þ a 11 a 33 f 2 þ a 23 f 3 a 23 a 11 f 3

ð ð

__ q 3 a 11 þ a 22 þ a 33 ð Þ q 3 þ a 11 a 22 þ a 22 a 33 þ a 33 a 11 a 12 a 21 a 23 a 32 ð Þ q 3

þ a 11 a 23 a 32 þ a 33 a 12 a 21 a 11 a 22 a 33 ð Þ q 3

ð 7 : 90 Þ

32 f 2 a

þ f Þ f 3 þ a ¼ a þ a a ð þ a ð a Þ f

a f a f a a

21 32 1 32 11 2 3 22 11 11 22 12 21 3

The roots of a mammillary three-compartment model are all real and determined from the
characteristic equation

3 2

s a ð þ a þ a Þ s þ a ð a þ a a þ a a a a a a Þ s

11 22 33 11 22 22 33 33 11 12 21 23 32

þ a a a þ a a a a a a ¼ 0

11 23 32 33 12 21 11 22 33