Bioflows

Numerical computation of blood flow through human arteries may save many experiments and may give information about the probable outcome of potential surgical intervention. It makes it possible to investigate many variants without such a risk as physical intervention. Beside that simulations help to understand the phenomena that take place inside human body. Computational Fluid Dynamics delivers more information than measurements and the results of computer simulation are more and more realistic due to continually increasing computer power and numerical methods evolution.

Physics of blood flow

Blood flow behaviour may be described by Navier-Stokes and continuity equations

(1)

The general form has too many unknown variables. We need another equation known as the Newtonian hypothesis. It describes a linear relation between the stress tensor and the strain rate tensor

(2)

where the strain rate tensor is given by

(3)

Having 10 scalar formulas (1 scalar + 1 vector + 1 symmetrical tensor) we still have 12 unknown variables: . We may treat blood as an incompressible medium then

(4)

The 12 scalar formulas can be rewritten now as follows

(5)

There are 4 equations and 5 unknown variables: . It is necessary to decide whether to treat blood as Newtonian or non-Newtonian. A non-Newtonian behaviour allows a relationship between viscosity and velocity gradients according to the following power law

(6)

where k and n are constants and units

(7)

for Newtonian fluids we have and for non-Newtonian . E.g. for blood

(8)

and shear strain rate for a general 3D case

(9)

Finally, the fifth non-linear equation is established, which describes the non-Newtonian behaviour of blood

(10)

Formulas (5) and (10) may be solved numerically using the Finite Volume Method or the Finite Element Method. Analytical solution of Navier-Stokes equations is only possible for very simple geometries such as tube and flat plates.

For the tube shown in the figure above equations (5) and (10) are simplified due to one dimensional velocity field and steady state conditions to

(11)

the solution of this steady state and non-linear differential equation describing non-Newtonian blood flow is presented below and can be used for code validation

(12)

There is also the solution of unsteady state flow in a tube known in literature as Womersley flow. That flow is forced by pressure gradient described by the sine function. In real life the geometry does not look like a tube, flow is unsteady state and boundary conditions are not similar to sine function. Realistic boundary condition forcing the flow is presented below.

Blood flow through realistic geometries

Magnetic Resonance Imaging (MRI) technique makes it possible to obtain realistic geometries of human arteries. However a set of digital pictures must be first post-processed using digital image procedure to separate contours of the arteries. Having contours one must ‘vectorise’ the geometry for further discretisation. An example of vectorisation is shown below.

Such prepared geometry is ready to disretisation and calculation using CFD software. As a result of the numerical solution we have among others velocity field and pressure that are used for further calculation such as wall shear stress or dissipation (presented below).

Stream lines and pressure distribution

Wall shear stress and dissipation intensity distributions

Shape optimisation of some biomedical devices

Stents are metallic cage-like structures that are inserted into an artery blocked by calcified plaque (stenosis). There are a lot of possible patterns of stents that present thousand of potential solutions. Thus stents differ significantly in shape, cross-section, and other details, which affects the haemodynamics of the blood flow through the treated region.

A simplification must be introduced to describe the richness of stent shapes concentrating on a few major features to cover a relatively large design space.

Numerical solution of blood flow around the stent is obtained using CFD. However CFD is time consuming. Therefore Genetic Algorithms were used as a method for finding improved stent design as they offered quick convergence. Another advantage of GA is that it does not need additional information about the objective fitness function. Such information is inaccessible in this stent application due to the complicated nature of performance measures.

One could imagine a lot of different fitness functions. It is believed that Wall Shear Stress plays the most important role in biomedical flows. However as WSS is distributed along the surface it cannot be directly used as a performance measure in driving a search algorithm that needs a number(s). Indeed two different WSS distributions cannot be compared directly, yet it is possible to define such a performance based on that distribution.

We define wall shear stress norm

(13)

where S is the speculative surface and average wall shear stress is given by

(14)

Such a definition yields a single number characterising the distribution of . It should be borne in mind that if a number is generated from the 3D distribution then information is always lost.

Dissipated power is introduced here as an alternative performance measure. Let us consider a form of the Gibbs equation that has the shape of an energy equation:

(15)

where T is temperature and is the intensity of entropy production. As blood is incompressible and we neglect the heat conductivity we have

(16)

In other words the dissipated energy causes an incremental change of internal energy. The effect of this energy dissipation is an increase of local temperature. Intensity of entropy production may be calculated from the velocity field (for an incompressible medium).

(17)

It can be proved that not all of the work in Equation 12 is converted into kinetic energy

(18)

The energy that is not converted is dissipated. Therefore dissipated power is defined as follows

(19)

or dissipated energy

(20)

Minimising such an objective fitness function helps us to search for a stent shape with the smallest contribution to energy losses. The solutions found using both these measures: the dissipated power and WSS distribution are compared below. Both results look more or less like the first stent presented in the above.

Nomenclature

density

velocity

gravity acceleration

stress tensor

p pressure

dynamic viscosity

identity tensor

strain rate tensor

wall shear stress

dissipation intensity

T temperature

e internal energy