Arterial branching network diameter and transit time calculations

[johnq2k7]

Model the arterial tree as a simple branching network in which each junction is composed of a parent vessel and daughter vessels, each having a diameter related to the parent's via the cube law (D_n-1)^3= (D_n*a)^3 + (D_n*b)^3 which is approx. equal to 2*(D_n)^3

a.) Show that D_n/D_0= 2^(-n/3)

b.) Show that 35 generations are required to model the arterial tree from aorta (D_0=2.6 cm) to the capillaries (D_34=10 um)


c.) Derive a formula for the mean transit time (ie vessel length divded by mean velociity) for an individual vessel. Based on this formula, how long does the model suggest it would take for blood to go from the aorta to the capillaries?



My work:


a.) since

(D_n-1)^3 is approximately equal to 2*(D_n)^3

therefore,

(D_n-1/D_n)^3= 2

therefore i Dn/Do= 2^(-n/3)

I'm not sure

i need help with the other questions, I'm confused how use this equation to solve the problem?

 

[berkeman]

Model the arterial tree as a simple branching network in which each junction is composed of a parent vessel and daughter vessels, each having a diameter related to the parent's via the cube law (D_n-1)^3= (D_n*a)^3 + (D_n*b)^3 which is approx. equal to 2*(D_n)^3

a.) Show that D_n/D_0= 2^(-n/3)

b.) Show that 35 generations are required to model the arterial tree from aorta (D_0=2.6 cm) to the capillaries (D_34=10 um)


c.) Derive a formula for the mean transit time (ie vessel length divded by mean velociity) for an individual vessel. Based on this formula, how long does the model suggest it would take for blood to go from the aorta to the capillaries?



My work:


a.) since

(D_n-1)^3 is approximately equal to 2*(D_n)^3

therefore,

(D_n-1/D_n)^3= 2

therefore i Dn/Do= 2^(-n/3)

I'm not sure

i need help with the other questions, I'm confused how use this equation to solve the problem?

I'd guess that you just plug in n=34, n=35, n=36 into the equation, and show that you need at least n=35 in order to get the large ratio in vessel sizes that they give in the question.

 

[johnq2k7]

How do you prove part a.) properly though, and for part c.) how do u derive the mean transit time equation from the information?

please help

 

[berkeman]

How do you prove part a.) properly though, and for part c.) how do u derive the mean transit time equation from the information?

please help

What are a and b in the original equation? Are they somehow constrained to be close to 1 each, and that's why the simplified approximation works?

For the proof, I'd try doing something like this...

$${D_{n-1}}^3 = 2 {D_n}^3$$

So

$${D_0}^3 = 2 {D_1}^3$$

$${D_1}^3 = 2 {D_2}^3$$

$${D_2}^3 = 2 {D_3}^3$$

etc., So

$${D_0}^3 = 2 {D_1}^3 = 2^2 {D_2}^3 = 2^3 {D_3}^3 = ... = 2^n {D_n}^3$$

The rest of the proof should follow. Is that what you mean?

And what does your text say about flow rate equations. There must be more information that you use to solve that part?

 

[johnq2k7]

How about part b.)

Do you simply

use, D_0= 2.6 cm

D(34)= 10 um

and substitute

(D_0)^3 / (2^34)= 10 um

How do you prove it?

I'm sort of confused?

 

[berkeman]

How about part b.)

Do you simply

use, D_0= 2.6 cm

D(34)= 10 um

and substitute

(D_0)^3 / (2^34)= 10 um

How do you prove it?

I'm sort of confused?

I'd said:

I'd guess that you just plug in n=34, n=35, n=36 into the equation, and show that you need at least n=35 in order to get the large ratio in vessel sizes that they give in the question.

So what is the ratio of the 2.6cm and 10um? What is the ratio of the following?

$$\frac{{D_0}^3}{{D_{33}}^3}$$

$$\frac{{D_0}^3}{{D_{34}}^3}$$

$$\frac{{D_0}^3}{{D_{35}}^3}$$