VIII Conference “Mathematical Models and Numerical Methods in Biomathematics” Computational model of two-phase transport of transmembrane peptides by blood flow Y.N. Soe Moscow Institute of Physics and Technology Under supervision of S.S. Simakov INM RAS, 03.11 2016

Motivation Blood flow model Two-Phase Transport Model Results

Motivation Some drugs cause toxic systemic side-effects. Administration and delivery of these drugs should be efficient and targeted to the limited area. Transmembrane peptides (CPPs) are short peptides that facilitate cellular intake/uptake of various molecular equipment. The function of the CPPs is to bound the useful cargo with the cells (e.g. uptake by erythrocytes) and to deliver it to the target region for intake (e.g. by cancer cells). Purpose of the work Development of a model of two-phase transport by blood on the basis of 1D global network model of circulation. Analysis of drugs intake by targeted tissue region.

Review of the CPPs targets and possible side-effects Rego De Figueiredo et al., 2014

1D Blood flow model 1) Mass balance 2) Momentum balance 3) Boundary conditions 3.1) Poiseuille's pressure drop k — vessel’s index, m — junction’s index 3.2) Mass balance 3.3) approximation of compatibility conditions A.S. Kholodov, 2001

1D Blood flow model: elasticity 4) Elasticity of the walls (wall-state equation) Physical experiment on collapsible tubes Pedley, Luo, 1996 A.S. Kholodov, 2001 Experimental data: Armentano, et.al., 1995, Studinger , et.al., 2003, Dobrin, et.al. 1988

Two-phase transport 1) 2) 3) Injection model:

Two-phase transport: boundary conditions 4) Transport through the heart: v — terminal vein a — aorta 5) Vessels junctions: Inflow Assumptions for the junction Discretization of the transport equation Outflow M — junction’s index Instant mixing Mass conservation

Relative velocity of the erythrocytes 1) For incompressible steady flow erythrocytes plasma Velocity profile 2) Boundary conditions 3) M. Sharan and A.S. Popel (2001)

Relative velocity of the erythrocytes 5) 4) 6) Assuming: we derive: Conditions for parabolic profile:

Results

Structure of the major systemic vessels Arteries Veins A.P. Avolio (2010) Total time – 60s; Injection time 5s :15s Tissue

Validation ᵟ =11,6% ᵟ =16,4% ᵟ =25,9% ᵟ =18.0% simulations Common carotid artery Descending Aorta [ml/sec] ᵟ =11,6% [cm/s] ᵟ =16,4% Ascending Aorta [cm/s] Артерия Common iliac artery ᵟ =25,9% [cm/s] ᵟ =18.0% simulations Data from Ph. Reymonda, et.al., 2013

Mass balance control [Volume, L] Time, seconds Pseudo stationary flow development [Volume, L] Time, seconds

Convection (WC) vs. Diffusion (WD) Name (Number) Arteries d (cm) Veins d(cm) Arc of aorta (5) 2,14 3,21 Common iliac (84) 1,04 1,56 Common carotid (22) 0,74 1,11 Brachial (42) 0,56 0,84 Hepatic (63) 0,44 0,66 Popliteal (111) 0,4 0,6 Vertebral (9) 0,38 0,57 Anterior tibial (125) 0,2 0,3 Interosseous (96) 0,18 0,27 Cerebral (47) 0,16 0,24

Sensitivity analysis of the coefficients and Time, seconds Time, seconds transmembrane diffusion coefficient ; erythrocytes plasma – tissue diffusion coefficient; tissue metabolization coefficient

Analysis of the relative consumption/propagation Сtissue Time, seconds

Analysis of the consumption/ propagation through targeted vessel Tissue

Conclusions On the basis of the model, developed a software package for the calculation of the two-phase transport of medicines by blood with tissue consumption. The method of assessment of relative erythrocytes velocity The method of assessment of the efficiency of targeted drugs delivery. Possible range of the diffusion and consumption constants for the drug was predicted.

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