1. 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,

2. Motivation
Blood flow model
Two-Phase Transport Model
Results

3. 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.

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

5. 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

6. 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

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

8. 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

9. Relative velocity of the erythrocytes1)
For incompressible steady flow
erythrocytes
plasma
Velocity profile 2)
Boundary conditions 3)
M. Sharan and A.S. Popel (2001)

10. Relative velocity of the erythrocytes5) 4) 6)
Assuming:
we derive:
Conditions for parabolic profile:

11. Results

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

13. 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

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

15. 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

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

17. Analysis of the relative consumption/propagation
Сtissue
Time, seconds

18. Analysis of the consumption/ propagation through targeted vessel
Tissue

19. 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.

20. Thanks a lot !  
be healthy & be happy