Oxygen pathway in mammals:

Modeling of the passage from air to blood.

Benjamin Mauroy

Laboratory MSC, University Paris 7 / CNRS - France

Oxygen pathway in mammals: Modeling of the passage from air to blood.

Oxygen pathway

The red blood cell

Modeling of the red blood cell

Motion of red blood cells in a capillary

Results (2D-axi)

Adding hemoglobin and oxygen …

Oxygen pathway

Bronchial tree

O2

Direction of blood

circulation

alveolus capillary

vascular tree

The red blood cell (or erythrocyte)

Scheme of the membrane of the red blood cell (from Guillaume Lenormand)

Red blood cells in a capillary (Tsukada et al, 2000)

Modeling of the red blood cell

7.4 μm

2.4 μm

Membrane thickness: 0.2 μm

Red blood cell sizes:

F

F

The membrane is assumed to be an hyperelastic material.

Its energy of deformation is given by (Yeoh, 1990) :

W = C1(l12+ l22 +l32-3) + C2(l12+ l22 +l32-3)3

This energy formulation has already been used to model the red blood cell membrane by Mills et al in 2004.

In order to validate our code, a numerical reconstitution of optical tweezers action on red blood cells has been performed, to mimic Mills et al experiments:

The numerical results are in agreements with their experiments.

Motion of red blood cells in a capillary

Virtual section (= camera) of capillary moving at red blood cell(s) speed

Reference geometry for the numerical problem

φ

The transformation φ is built with the following properties:

- The walls of the reference frame must be deformed in order to coincide to the walls of the capillary at the corresponding position of the camera.

- The shape of the red blood cell(s) is the consequence of fluid-structure interactions between the fluids inside and outside the cell(s) and the membrane of the cell(s).

The consequences of this method are that:

- The boundary conditions for the fluid at each extremity of the section should be chosen carefully.

- The physical frame is moving with red blood cell(s) velocity and is thus accelerated, new terms could arise in equations.

Results (2D-axisymmetric)

Initial state of the red blood cell

In a capillary, vplasma(t=0)=0.5 mm/s

(capillary data from Jeong et al, 2006)

L

D

« aspect ratio » of the red blood cell

These results are in agreement

with Jeong et al observations

Three red blood cells …

Resistance of the capillary portion

Adding hemoglobin and oxygen …

Hemoglobin properties:

- It is a complex molecule which can carry up to four molecules of oxygen.

- There exists two main allosteric states of hemoglobin which reaction rates with oxygen are different.

Its chemical reaction with oxygen can be represented with the following scheme:

This chemistry will be implemented on the geometries obtained in the previous section. Note also that the spatial convection/diffusion of all reagents and products will be

integrated in the equations.

Diameter = 6.5 μm

Diameter = 7.5 μm

Conclusion & prospects

The shape of the red blood cells in the capillaries plays an important role for:

1- the hydrodynamic of the blood in small vessels which is the consequence of a complex interaction between plasma, red blood cells and capillary size.

2- the capacity of blood to catch oxygen in the capillaries which is dependant of the geometry of both capillaries and red blood cells but also of the duration of red blood cells exposure to oxygen source.

In order to develop a realistic model of oxygen transport in the body, it is thus necessary to take into account these results and to integrate them to the larger scale models of the vascular network we are developing.

Note that numerical simulations have been implemented in Comsol Multiphysics.