Page 31 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

8 BIOMEDICAL SYSTEMS ANALYSIS

λ 13
1 3
Gel layer GIT
λ
λ 12 14
λ 21
λ
λ 41 43
2 λ 42 4
Sol layer Macrophages
λ 24

λ 52 λ 25 λ 54 λ 45

5 Epithelium λ
subep. tissue 46
λ 56

6
LN Blood

FIGURE 1.3 A multicompartmental model for the clearance of inhaled insoluble particles
from the lung. [Reproduced with permission from Sturm (2007).]

coordinates) across the cross section results in the following expression (Reddy, 1986; Reddy and
Kesavan, 1989):
ρdQ/dt =πa ΔP/ 2aτ (1.5)
2
w
where ρ is the fluid density, Q is the flow rate, a is the wall radius, P is the pressure, is the length,
and τ is the fluid shear stress at the wall. If we assume that the wall shear stress can be expressed
w
3
using quasi-steady analysis, then the wall shear stress can be estimated by τ = 4μQ/a . Upon substi-
w
tuting for the wall stress and rearranging, the results are
4
2
[ρ /(πa )]dQ/dt =ΔP − [8μ /(πa )]Q (1.6)
The above equation can be rewritten as
LdQ/dt =ΔP − RQ (1.7)
4
2
where L =ρ /(πa ) and R = 8μ /(πa ).
It can be easily observed that flow rate Q is analogous to electrical current i, and ΔP is analogous
to the electrical potential drop (voltage) ΔE. In the above equation. L is the inductance (inertance)
and R is the resistance to flow. Therefore, Eq. (1.5) can be rewritten as
−
Ldi/dt =ΔE − Ri (1.8)
Fluid continuity equation, when integrated across the cross section, can be expressed as
dV/dt =ΔQ = Q − Q (1.9)
in out