1. Andrey I. Svitenkov*, Oleg V. Rekin, P. S. Zun,
Parameters of boundary conditions for detailed 1D arterial blood flow model
Andrey I. Svitenkov*, Oleg V. Rekin, P. S. Zun,
Alfons G. Hoekstra.
VIII-th conference
Mathematical models and numerical methods in biomathematics
2017, Moscow

2. 1D blood flow modeling U – average velocity; A – area of the lumen;
Blood flow simulations
1D blood flow modeling
U – average velocity; A – area of the lumen;
ν – visc. of blood; P – pressure;
ρ – blood density; α – some coeff.;
x – direction along the artery; t - time .
Boundary conditions:
Inlet: fixed dynamics of flux;
Outlet:
R1 , R2, C – three free parameters. How to adjust?

3. Three arterial networks
Original problem formulation included three arterial models of various level of detail.
Boundary conditions were known only for less detailed model.
I. 31 bifurcations
II. 109 bifurcations
III. 968 bifurcations
Boileau et al, 2015
ADAN56 model

4. Missed distal part of arterial network is compensated by windkessel BC
Missing of distal part of the arterial tree
Adjustment of outlet boundary conditions
Pressure
Flux
Missed distal part of arterial network is compensated by windkessel BC

5. Calibration procedure
Iteration process
Calibration procedure
Estimate parameters based on minimization of reflection at the boundary and fixed value of systolic (Ps) and diastolic pressure (Pd):
Ad – lumen of an artery for diastolic pressure;
cd – wave speed for diastolic pressure;
Qmin, Qmax – min. and max. flux;
Δt - time span between these maximum and minimum.
How to estimate Ps and Pd at the terminal vessels?

6. Results of calibration
Conditionality of flux
Results of calibration
Pressure in Superficial femoral artery:
Less then 5% difference for pressure
3x times difference for flux (average)

7. Conditionality of flux
Example
Let Q0 is fixed and delta P is fixed with finite accuracy. R1 and R2 – are resistance of parallel parts of arterial net, and r1 and r2 – total resistance of BC. Q0
Variation of pressure is of second order, while the variation of flux – first.
Calibration procedure, based on fluxes (not pressure only) is needed

8. Murray’s law Expression for vessels radii at bifurcation.
Minimize dissipation
Murray’s law
Expression for vessels radii at bifurcation.
Based on minimization expenditure of energy by the organism.
Cost function:
L – length of an artery, a – radius of an artery, γ – some cofficient.
If substitute Q from the Poiseuille's law ~a4, then:
0 – parent vessel, 1,2..n - daughter
L.p.: (a0)3 on X
R.p.: (ai)3 on X

9. Adjustment method Given: Find:
Minimize dissipation
Adjustment method
…which is based on minimization expenditure of energy.
R0 – effective resistance
of the BC:
Given:
Find:
R0 is known and fixed .
Find the set of parameters:
R12 , R22, C1, C2
which minimizes the cost function:
Boileau et al, 2015
ADAN 56
av – average value
and conserve R0.
R1, R2, C are given for each terminal vessel
New arteries taken from detailed model

10. Calibration procedure
Finally
Calibration procedure
Described problem formulation is solving level by level while adding of novel bifurcations (tri- and n-furctions)
Minimize:
Conserve R0:
Calculation of derivatives:
Q0 is fixed
Change parameter and simulate the downstream part of the tree
Do averaging
Find derivative Q0
After the terminal nodes of detailed arterial network are reached – the procedure can be repeated for better accuracy.

11. Average flux difference <10% for all control points
Simulations
Results of calculations
R. Ant. Tibial (3)
R. Sup. Femoral (1)
Average flux difference <10% for all control points

12. Finally
Conclusions
1. Sensitivity of flux to small variations of pressure at the boundary vessels.
2. Minimization expenditure of energy principle works pretty good even for most coarse estimations.
3. Adjustment procedure based on minimization expenditure of energy includes account of flux and conserve correct average value.

13. Questions
Thank you!