1. Modeling Blood Flow in the Cardiovascular System
Mette S Olufsen
MA 432
Spring 2017
Department of Mathematics,
North Carolina State University
NC STATE University

2. Compartment Model

3. Compartment Model 𝑞= 𝑝 1 −𝑝 𝑅 1 𝑉− 𝑉 𝑑 =𝐶 𝑝− 𝑝 𝑒𝑥𝑡 =𝐶𝑝 q V p p1 pext
𝑞= 𝑝 1 −𝑝 𝑅 1
𝑉− 𝑉 𝑑 =𝐶 𝑝− 𝑝 𝑒𝑥𝑡 =𝐶𝑝
Assuming that 𝑝 𝑒𝑥𝑡 =0
Dead space volume: 𝑉 𝑑

4. Solution Total volume Flow where
𝑉 𝑡𝑜𝑡 = 𝑉 𝑆𝐴 + 𝑉 𝑆𝑉 + 𝑉 𝑃𝐴 + 𝑉 𝑃𝑉 = 𝑉 0 (constant)
𝑄= 𝑉 0 𝑇 𝑆𝐴 + 𝑇 𝑆𝑉 + 𝑇 𝑃𝐴 + 𝑇 𝑃𝑉
𝑇 𝑆𝐴 = 𝐶 𝑆𝐴 𝐾 𝑅 + 𝐶 𝑆𝐴 𝑅 𝑆
𝑇 𝑆𝑉 = 𝐶 𝑆𝑉 𝐾 𝑅
𝑇 𝑃𝐴 = 𝐶 𝑃𝐴 𝐾 𝑅 + 𝐶 𝑃𝐴 𝑅 𝑃
𝑇 𝑃𝑉 = 𝐶 𝑃𝑉 𝐾 𝐿

5. Solution Flow Volume 𝑄= 𝑉 0 𝑇 𝑆𝐴 + 𝑇 𝑆𝑉 + 𝑇 𝑃𝐴 + 𝑇 𝑃𝑉
𝑄= 𝑉 0 𝑇 𝑆𝐴 + 𝑇 𝑆𝑉 + 𝑇 𝑃𝐴 + 𝑇 𝑃𝑉
𝑉 𝑖 = 𝑇 𝑖 𝑄 = 𝑇 𝑖 𝑉 0 𝑇 𝑆𝐴 + 𝑇 𝑆𝑉 + 𝑇 𝑃𝐴 + 𝑇 𝑃𝑉
𝑇 𝑆𝐴 = 𝐶 𝑆𝐴 𝐾 𝑅 + 𝐶 𝑆𝐴 𝑅 𝑆 ,
𝑇 𝑃𝐴 = 𝐶 𝑃𝐴 𝐾 𝑅 + 𝐶 𝑃𝐴 𝑅 𝑃 ,
𝑇 𝑃𝑉 = 𝐶 𝑃𝑉 𝐾 𝐿 ,
𝑇 𝑆𝑉 = 𝐶 𝑆𝑉 𝐾 𝑅

6. Solution Flow Pressure 𝑄= 𝑉 0 𝑇 𝑆𝐴 + 𝑇 𝑆𝑉 + 𝑇 𝑃𝐴 + 𝑇 𝑃𝑉
𝑄= 𝑉 0 𝑇 𝑆𝐴 + 𝑇 𝑆𝑉 + 𝑇 𝑃𝐴 + 𝑇 𝑃𝑉
𝑃 𝑖 = 𝑉 𝑖 𝐶 𝑖 = 𝑇 𝑖 𝑉 0 𝐶 𝑖 𝑇 𝑆𝐴 + 𝑇 𝑆𝑉 + 𝑇 𝑃𝐴 + 𝑇 𝑃𝑉
𝑇 𝑆𝐴 = 𝐶 𝑆𝐴 𝐾 𝑅 + 𝐶 𝑆𝐴 𝑅 𝑆 ,
𝑇 𝑃𝐴 = 𝐶 𝑃𝐴 𝐾 𝑅 + 𝐶 𝑃𝐴 𝑅 𝑃 ,
𝑇 𝑃𝑉 = 𝐶 𝑃𝑉 𝐾 𝐿 ,
𝑇 𝑆𝑉 = 𝐶 𝑆𝑉 𝐾 𝑅

7. Normal Parameter Values
Systemic
Pulmonary
Units R Rs
17.5 Rp
1.79
mmHg min/liter C
Csa
0.01
Cpa
.00667
liter/mmHg
Csv
1.75
Cpv
0.08
Right
Left
Unit K KR
2.8 KL
1.2
liter/min/mmHg V V0
5.0
liter

8. Values of Variables SA: Systemic Arteries SV: Systemic Veins
PA: Pulmonary Arteries
PV: Pulmonary Veins
P (mmHg)
V (liter) SA
100
1.0 SV 2
3.5 PA 15
0.1 PV 5
0.4
Stroke volume
70 ml/beat
Heart rate
60 beats/min
Cardiac output
5.6 l/min

9. Balancing the two sides of the heart
The flow is conserved
Systemic arterial pressure is higher 𝑃 𝑆𝐴 > 𝑃 𝑃𝐴
The left heart pumps harder (bigger muscle) 𝐾 𝐿 < 𝐾 𝑅
The systemic resistance is higher 𝑅 𝑆 > 𝑅 𝑃
Systemic compliance is higher 𝐶 𝑆𝐴 < 𝐶 𝑃𝐴 20 5 2
100

10. Balancing the two sides of the heart
If 𝐾 𝐿 = 𝐾 𝑅 , 𝑅 𝑆 = 𝑅 𝑃 , … then
𝑃 𝑆𝐴 = 𝑃 𝑃𝐴
𝑉 𝑆𝐴 = 𝑉 𝑃𝐴 ….

11. Balancing the two sides of the heart
What happens if 𝐾 𝑅 is suddenly reduced (right heart failure)?
Temporarily 𝑄 𝑅 < 𝑄 𝐿
Net transfer of volume to the systemic circulation
Raise systemic venous pressure
Lower pulmonary venous pressure
Driver flow back to equality

12. Ratio pulmonary to systemic volume
Recall that the volume giving
Showing that decreasing 𝐾 𝑅 increases the denominator reducing the ratio of the volume in the pulmonary circuit to the systemic circuit.
𝑉 𝑖 = 𝑇 𝑖 𝑄
𝑉 𝑃 𝑉 𝑆 = 𝑉 𝑃𝐴 + 𝑉 𝑃𝑉 𝑉 𝑆𝐴 + 𝑉 𝑆𝑉 = 𝑇 𝑃𝐴 𝑄+ 𝑇 𝑃𝑉 𝑄 𝑇 𝑆𝐴 𝑄+ 𝑇 𝑆𝑉 𝑄 = 𝑇 𝑃𝐴 + 𝑇 𝑃𝑉 𝑇 𝑆𝐴 + 𝑇 𝑆𝑉
= 𝐶 𝑆𝐴 + 𝐶 𝑃𝑉 𝐾 𝐿 + 𝐶 𝑃𝐴 𝑅 𝑃 / 𝐶 𝑆𝐴 + 𝐶 𝑆𝑉 𝐾 𝑅 + 𝐶 𝑆𝐴 𝑅 𝑆

13. Why does this work?
Dependence of cardiac output on venous pressure 𝑄=𝐾 𝑃 𝑉

14. What if 𝑸 was a parameter?
Lose equations for the heart
Lose 2 equations
But only 1 variable
Necessary to add one relationship, e.g. replace
𝑉 𝑡𝑜𝑡 =𝑉 𝑆𝐴 + 𝑉 𝑆𝑉 + 𝑉 𝑃𝐴 + 𝑉 𝑃𝑉
with
𝑉 𝑆𝐴 + 𝑉 𝑆𝑉 = 𝑉 𝑆
𝑉 𝑃𝐴 + 𝑉 𝑃𝑉 = 𝑉 𝑃
The two sides become independent

15. Cardiovascular Control
What happens when you start running?
Heart rate goes up
More blood to the muscles in the legs
More blood is pumped out from the heart

16. Cardiovascular control
Arteries
Veins BR
afferent BR nerves
sympathetic
nerves
para-sympathetic

17. Cardiovascular control

18. Effects of changing the parameters
𝑄= 𝑉 0 𝑇 𝑆𝐴 + 𝑇 𝑆𝑉 + 𝑇 𝑃𝐴 + 𝑇 𝑃𝑉
𝑃 𝑆𝐴 = 𝑇 𝑆𝐴 𝑉 0 𝐶 𝑖 𝑇 𝑆𝐴 + 𝑇 𝑆𝑉 + 𝑇 𝑃𝐴 + 𝑇 𝑃𝑉
What is the effect on of reducing 𝑅 𝑆 by a factor of 2?
Normal
RS halved
Change % Q
5.6
6.2
0.6
11%
PSA
100 57
-43
-43%

19. Changing Parameters (HW: 1.3)
Norm KS KL RS RP V0
CSA
CSV
CPA
CPV Q
5.6
6.2
PSA
100 57
PSV 2
PPA 15
PPV 5
VSA 1
VSV
3.5
VPA
0.1
VPV
0.4

20. Sensitivity The sensitivity of 𝑌 with respect to 𝑋
𝜎 𝑌𝑋 = Δ log 𝑌 Δ log 𝑋
𝜎 𝑌𝑋 = log 𝑌 ′ −log⁡(𝑌) log X ′ −log(X)
𝜎 𝑌𝑋 = log( 𝑌 ′ /𝑌) log( X ′ /X)

21. Sensitivity Example (1) 𝜎 𝑄 𝑅 𝑆 = log( 𝑄 ′ /𝑄) log( 𝑅 𝑆 ′/ 𝑅 𝑆 )

22. Sensitivity Example (2) 𝜎 𝑃 𝑆𝐴 𝑅 𝑆 = log( 𝑃 ′ /𝑃) log( 𝑅 𝑆 ′/ 𝑅 𝑆 )

23. Sensitivity In the limit of 𝑋 and 𝑌 small 𝜎 𝑌𝑋 = 𝑑𝑌 𝑌 / 𝑑𝑋 𝑋
𝜎 𝑌𝑋 = 𝑑𝑌 𝑑𝑋 𝑋 𝑌

24. Sensitivity
It is possible to show (homework 1.10) that -( 𝜎 𝑄 𝑅 𝑆 )+ 𝜎 𝑃 𝑆𝐴 𝑅 𝑆 ≈1 Because 𝑃 𝑆𝐴 ≈𝑄 𝑅 𝑆 The latter holds because 𝑃 𝑆𝐴 ≫ 𝑃 𝑆𝑉

25. Sensitivities (use parameter change table) HW 1.4KR KL RS RP V0
CSA
CSV
CPA
CPV Q
𝜎 𝑄 𝐾 𝑅
PSA
𝜎 𝑃 𝑆𝐴 𝐾 𝑅
PSV
PPA
PPV
VSA
VSV
VPA
VPV