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

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?

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

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

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?

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)

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

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

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

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.

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

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.

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