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13 Modeling Biological Systems
Let
α k =
N k+1 πr 2
k+1
N k πr 2
k
= ν k ρ
2
k
ϕ k =
4
3 πN k+1 l 3
k+1
4
3 πN k l 3
k
= ν k λ
3
k
and define the sums
S 1 =
C
k=1
1
C−1
i=k ν i ρ 2
i λ i
S 2 =
C
k=1
1
N
1/3
k
C−1
i=k α i ϕ
1/3
i
S 3 =
C
k=1
1
N
1/3
k
The sums S 1 and S 2 are related by
S 1 = N
1/3
c S 2 .
Now the total volume of blood contained at branching level k is
V b,k = N k πr
2
k l k
=
N C πr 2
C l C
C−1
i=k ν i ρ 2
i λ i
so that the total blood volume is
V b =
N C πr 2
C l C
C−1
i=1 ν i ρ 2
i λ i
= N C πr
2
C l C S 1
= N
4/3
C πr
2
C l C S 2 .
Thus, the total number of capillaries can be expressed as
N C =
V b
πr 2
C l C S 2
3/4
.
13 Modeling Biological Systems
Let
α k =
N k+1 πr 2
k+1
N k πr 2
k
= ν k ρ
2
k
ϕ k =
4
3 πN k+1 l 3
k+1
4
3 πN k l 3
k
= ν k λ
3
k
and define the sums
S 1 =
C
k=1
1
C−1
i=k ν i ρ 2
i λ i
S 2 =
C
k=1
1
N
1/3
k
C−1
i=k α i ϕ
1/3
i
S 3 =
C
k=1
1
N
1/3
k
The sums S 1 and S 2 are related by
S 1 = N
1/3
c S 2 .
Now the total volume of blood contained at branching level k is
V b,k = N k πr
2
k l k
=
N C πr 2
C l C
C−1
i=k ν i ρ 2
i λ i
so that the total blood volume is
V b =
N C πr 2
C l C
C−1
i=1 ν i ρ 2
i λ i
= N C πr
2
C l C S 1
= N
4/3
C πr
2
C l C S 2 .
Thus, the total number of capillaries can be expressed as
N C =
V b
πr 2
C l C S 2
3/4
.