Project
Summary:

The project is aimed at finding the real,
physically grounded way to model and simulate the transport of a multiphase healthy
blood medium in a muscle tissue blood vessel network. Since no one model exists
at the present time that could account for multiscaling and consider actual
blood vessels morphology, the suggested development will treat the blood flow
in four scale heterogeneous systems, including: 1- separate cells (red and
white) and dispersed cell medium modeling as well as capillary and arteriole
wall scale modeling; 2 - single blood vessel models including previous
multiphase scale governing equation statements; 3 - single muscle fiber
capillary network scale modeling; 4 -
finally the three phase blood transport modeling in muscle fiber bundle
capillary network. To create these kinds of models the nonlinear multiscale morphological modeling
approach that has been developing by the principal investigator must be used
and significantly improved. Modeling
procedures, it is believed, will provide more accurate and physical model
output while considering the transport of blood constituents on each level of
the hierarchy. After being developed, these models would be significantly
advantageous in comparison with existing one-phase-one-vessel models, due to
their multilevel description and direct dependence on given specific muscle
morphology. Suggested achievements should bring the biomechanical science of
blood circulation modeling on the next level of knowledge and understanding.

** **

Copyright © 1994-2011 V.S.Travkin All Rights Reserved

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TABLE OF CONTENTS
**

**1.
PHASE
1 RESEARCH PLAN
..................................................................................... **6

1.1. Specific Aims............................................................................................
6

1.2. Significance.............................................................................................
6

1.2.1. Short
Review of Existing Models in Blood Flow Modeling and

Specifically in Muscle Blood Supply System Modeling............................
6

1.2.2.
New Methodology to be Applied to the Muscle Blood Supply

System Modeling........................................................................... 7

1.3.
Relevant Experience
.................................................................................. 8

1.4.
Theoretical Methods and Numerical Experimentations
....................................... 9

1.4.1.
Multiphase Blood System Modeling of Red, White Cells as Corpuscles and

Non-Newtonian Blood Plasma
9

1.4.2.
The Arteriole and Capillary Blood Vessel Walls Level
............................ 12

1.4.3.
Single Capillary or Arteriole Blood Stream Multiphase Modeling
.............. 13

1.4.4. Development of Muscle Fiber Capillary Network
Basic Element List......... 14

1.4.5.
Muscle Fiber Capillary Network Simulation
......................................... 15

1.4.6.
Simulation of the Major Role of Blood Capillary, Arteriole and Venule Flow

Network in the Muscle Fiber Bundle
18

1.4.7.
Conclusions
................................................................................ 19

1.5.
Literature Cited
....................................................................................... 19

** **

**1. PHASE
1 RESEARCH PLAN**

**1
SPECIFIC AIMS **

At present time, blood system circulation modeling has no development of even a
one-vessel (on an arterial level as well as on a capillary level) comprehensive
correct biomechanical system simulation. The lack of corresponding theory is
the evident cause of the situation. The aims of the current work are to develop
a hierarchical multiphase system to model and research the flow of a
three-phase blood medium in the network of a single straight muscle fiber
bundle. This comprehensive approach at the I-st and II-nd Phase development will include in the modeling system
all of the major biomechanical
multiphase phenomena, except deformability and movement of the vessel
walls' soft tissue and mechanical control of the given part of circulation
system. Mathematical modeling of physical processes occurring in strongly
heterogeneous media results on the whole in the necessity for medium
characteristics development, and accordingly, for process equations.

The first
level of consideration is a separate cell (red and white) and dispersed cell
medium flow modeling in the plasma constituents. Also the first level of
consideration is modeling of phenomena in the capillary or arteriole wall as a
permeable and morphologically complex 3-phase structure entity.

The
second hierarchical level includes the processes of mass and momentum
multiphase transport in a single blood vessel.

At the
third level, the scope of problems extends toward the lowest level of the
capillary network around a single muscle fiber. Among the processes involved
for consideration, are the processes of gas exchange mechanisms and blood cell
permeation and transformation during the course in the blood fiber network.

The
fourth level represents the kind of problem which describes the transport of
blood in the blood vessel network of a separate muscle fiber bundle. This
includes consideration of all lower levels of the hierarchy - combining them
into a complex specific spatial structure which has no analogy among the
non-living natural and engineering systems.

Development of the new integro-differential transport equations in the heterogeneous
( and living) media and an application of non-classical equation types, are
being considered as the present agenda in this application. Whereas, in many
cases, quasihomogeneous and quasistochastic approaches used almost exclusively
now are not sufficient for description of the physical process peculiarities in
the heterogeneous media and especially in living systems. Those systems are
often quite nonlinear and sensitive to disturbances of any kind. Among major aims of the application are the
following:

1) Provide
theoretical derivation of main sets of governing equations (GE) on each level
of the hierarchy and estimate feasibility of closure developments.

2) Develop
theoretical mechanisms and varieties of closure approaches for each specific
blood system hierarchical level.

3) Estimate
the feasibility of theoretical procedures for the development of numerical
convergent algorithms for solution evaluations of the equations governing
transport.

**1.2.1 Some
of Existing Models in Blood Flow Modeling and Specifically in Muscle Blood **

** Supply
System Modeling **

The
proposed research will conduct development and computer simulation of
practically all of the dominant circulation mechanisms of the muscle blood
supply system (MBSS) comprising the main objective of the Phase II. These
achievements primarily will give the ability
to simulate and investigate the critical phenomena of muscle functioning
and provide the researches with the important tool. The targeted product of
Phase II is the theoretical models and computer software that will be capable
of attracting additional funding for further development of commercial software
to benefit the medical industry.

The
muscle blood circulation modeling equation forms can be very different. The parabolic, hyperbolic or integro-differential kind of equations allowed,
liner or nonlinear - because the assumptions and initial assumed depth of
considered physico-chemical phenomena can vary. The types of transport
equations usually applied for blood flow
modeling are incorrect in accounting for many important features.

The flow
of blood matter as three-phase heterogeneous multidispersed media with
extremely intricate white cell behavior has not been attempted to be explained
and implemented in correct mathematical forms. Research usually takes the most
important features as, for example, erythrocyte aggregation kinetics (Deng et
al., 1994; Shehada et al., 1994), or non-Newtonian viscosity of the blood taken
as a whole fluid (Perktold et al., 1989; Sagayamary and Devanathan, 1989;
Thurston, 1972, 1994), or erythrocyte sedimentation (Kuo et al., 1994; Oka, 1983, 1985 ) and etc.

The blood
vessel's wall structural complexity only recently got attention in constructing
constitutive relations see, for example, Humphrey et al. (1989). The authors
admit that "there has been no attempt to quantify the possible
heterogeneity within a layer of the wall, and this seems extremely difficult at
present." The nonhomogeneity of a blood vessel's form is being studied
primarily from the point of view of fundamental understanding of arterial
stenosis, Sagayamary and Devanathan (1989), Young (1979).

The
modeling of network morphologies in the blood supply system is actually
constrained due to the shortage or absence
of applicable methodologies and approaches. The effects of blood stream
division or bifurcation being investigated use almost no knowledge base from
large information background accumulated in mechanical engineering (Matsuo et
al., 1989; Pinchak and Ostrich, 1976).

**1.2.2 Methodology
Applicable for the Muscle Blood Supply System Modeling **

There are
four basic theories are usually admitted to derive transport equations for
heterogeneous media: 1) averaging
theories, 2) stochastic approaches, 3) homogenization theories and 4) classical
continuum theories (approaches). Meanwhile, an interception of above theories
might be found in several theories. To obtain averaged equations and model or
medium characteristics, different methods are being applied for periodic or
almost periodic microstructures, porous media and media with two- and three
phase movement. Depending on the investigated problem's peculiarities, goals
and problem statements either the quasihomogeneous approach or a stochastic interpretation
of it could be applied. The later could be called in analogy, a quasistochastic
one. The entire essence of quasihomogeneous approach results in getting
effective transport coefficients for deterministic classical equations of
mathematical physics.

A problem
formulation in a quasistochastic statement has parameters characterizing medium
properties, considered as stochastic quantities or stochastic coordinate
functions. These functions are assumed to be known or assigned, and because of
this the problem solution, depending on medium stochastic characteristics, will
have stochastic character as well. Determination or calculation of effective
parameters for equations is usually based on some medium morphology model and
could count tens of methods for quasihomogeneous and quasistochastic
approaches.

Meanwhile,
the strict mathematical averaging theory has been sufficiently developed mainly
for classical differential equations of mathematical physics. That means, first
of all, that those used now like the second order equations are taken without
junior terms or for the cases, when their influence is insignificant.

At the
same time, the new heterogeneous theory started by Whitaker, Slattery, Gray, and others in 1967 allow the
description of transport processes in a
heterogeneous media much more precisely and definitely. There are few
applications, which have practically benefited from these theoretical
achievements. At present proposal the
new approach is being presented.

The
current application has the intention for developing the multiscale muscle
blood supply system models (MBSSM) on the basis of the latest results obtained
in heterogeneous media transport theory.

** **

**1.4.1.
Multiphase Blood System Modeling of Red, White Cells as Corpuscles and
Non-Newtonian Blood Plasma **

The
deformable, soft substance of red and white cells represents a very distinct
kind of modeling problem comparing to the conveniently understood two-three
phase flow phenomena. Further, the nonlinear, non-Newtonian continuous fluid of
the plasma should dictate a quite different approach to model development, Fig.
1. How influential nonlinear medium characteristics would be on the transport
characteristics might be shown in the next example, more related to blood
plasma momentum transport description.
..

Many
different phenomena of blood rheological properties including non-Newtonian
viscosity are directly outlined by the multiphase nature of the substance.
However, there is no correct models exist to treat a blood medium with the
whole extent of it's major feature, see Thurston (1972, 1994), Deng et al. (1994), Oka (1983, 1985), Schmid-Schonbein (1987). Among many
quite unusual properties, the movement and form changing of the white blood
cells expose an outstanding difficulty for mathematical description. Despite a
smaller volume fracture, the influence of white blood cells on blood rheology
is significant, Chien et al. (1983), Bagge et al. (1980), Schmid-Schonbein
(1987). The multiphase medium of blood cells and nonlinear blood plasma
fluid will be treated as an actual
3-phase medium with the specific peculiarities of white cells.

**1.4.2. The
Arteriole and Capillary Blood Vessel Walls Level **

A very
complicated feature of internal blood vessel transport is the multiphase layer
medium near the vessel's wall, Fig. 2. Thus, the flow over the rough surface of
endothelial cells (Sato and Ohshima, 1994; Worthen et al. 1987), plus the larger objects with independent
determinism, assigned partially by external
mechanisms (meaning with no close or directly related affiliation) and
partially induced by current near-neighborhood interrelations with determined
functions of vessel's morphology see, for example, Gallik et al. (1989), Goetz
et al. (1994), Harlan (1985) constitute few of the difficulties. These objects
are leukocytes, stenosis, other defects and structured heterogeneities of blood
vessel's wall. Mechanisms developed in works by the principal investigator will
allow to deal with these morphology phenomena.

The size
of blood vessels and their wall's internal structure are determinant issues for
the blood constituents and nutrients transport through the vessel walls, see Figs.
3, 4. Nevertheless, in the approach by Humphrey et al. (1989), which is rather
comprehensive (still incorrect) it is admitted that "although the intima
plays an important role as a blood-vessel interface, it is probably not
structurally significant.... and will not be considered further".

Modeling
of two main transport mechanisms through the capillary wall: literally
diffusion of nutrient constituents and active transport through the diapedesis
of white cells is the challenging problem
due to an unusual combination of relatively well known physical mechanism and still unknown and
un-simulated physical phenomena of deformable white cell transport through the
vessel's wall.

**1.4.3. Single
Capillary or Arteriole Blood Vessel Multiphase Modeling **

Considering a heat- and mass transport mathematical modeling in a
separate blood vessel medium, Figs. 4, 5, the equations of mass balances and
heat transfer usually can be derived from the following forms of the equations

valid for
process description in a separately taken element (capillary) of a given
heterogeneous media Fig. 5, - the blood circulation network. The boundary conditions at the blood-wall
interphase would be

where *r _{s}* function corresponding
to the transformation, penetration or reaction rate on the unit surface,

The real
transport conditions on the vessel wall need to be depicted in quite different
forms of the boundary equations, which will be a topic of awareness in the
proposed study.

The
complexity of blood flow in a capillary seen in Fig. 5, where a multidisperse
self-organizing blood medium moves through the highly non-homogeneous channel
with stochastic obstacles on the channels wall.

Among the
main problems encountered on that level of hierarchy are:

a) Modeling of three-phase (red, white cells and
plasma) blood medium momentum transport in blood vessels;

b)
Non-smooth irregular channel walls with moving living obstacles on them;

c)
Irregular shape and longitudinal curvature of separate blood vessel;

d) Evaluation of the assumptions due to
non-Newtonian approximation of flow phenomena for the 3-phase blood medium.

Justifying the choice of calculation formulae used to determine the
transfer parameters is important and difficult. Various erroneous positions are
shown to exist in the literature for determining transfer parameters, i.e.,
equating the errors of the calculation formulae to the error of obtaining
transfer parameters; using integral values of the sought field to find
significantly varying transport parameters, etc.

There is
shortage of developed methods for finding out the effective one-concentration (
one-temperature) coefficients for mathematical models with the variable or
nonlinear coefficients and more of that, with additional integro-differential
terms in two- and three-phase statements. The biggest challenging problems here
are the consistent lack of new equations understanding and insufficient
development of closure theory, especially for integro-differential equations.

In the
suggested study the correct equations for three-phase blood medium transport
will be developed and the differences with the most often adopted non-Newtonian
blood models estimated.

**1.4.4. Muscle
Fiber Capillary Network Basic Element List
**

The
morphology of a MBSS capillary network features the infinite set of varieties
of junctions and branches, see Fig. 6. The problem in this situation is to
develop morphological classification and description of basic key morphological
elements. Some information on hydrodynamics and resistance coefficients of
arterial branching had been developed in series of works, for example, by
Pinchak and Ostrich (1976), Matsuo et al. (1989). Then the developed database
specific elements will be used in the theoretical development of closure
models, procedures and algorithms. That will be one of the key points in the
successful closure of complex additional terms in the governing equations.

**1.4.5. Muscle
Fiber Capillary Network Simulation **

The non-Darcian effects usually considered are
the no-slip and the inertia
effects. These effects normally
decrease the flow and heat transfer
rates while consideration of the porous medium nonhomogeneity enhance
the mass transfer. In the commonly accepted equation of momentum, all kinds of
nonlinearities in resistance, convective and diffusive phenomena included are
in the Darcy resistance term

Almost always in porous medium modeling the
diffusion term ( Brinkman term) in the
momentum equation is dropped

usually turning out to be significant near interphase
boundaries. What is much more interesting, that this term should be
sufficiently influential near inner boundaries between two heterogeneous
structures. What concerns the flow resistance terms of the first ( Darcian drag
term) and second power (called the
Forchheimer term)

then, as shown in works of Travkin and Catton (1992 a,b,c) is that all
of the forms of the Darcy term as well as the quadratic term depend directly on
the assumed version of the convective and diffusion terms, while the second and
more important thing is that both diffusion (Brinkman) and drag resistance
terms in the final forms of the flow equations are all directly connected together.

Meanwhile, the Darcy's law becomes inadequate, which is shown and in
other works, when the flow Reynolds number based on the mean pore diameter is
of the order of one or greater, due to the fact that the inertia force is no
longer negligible in comparison to the
viscous effect. Most non-Darcian studies are based on a model summarized by the
equation

where b is
a constant, determined either experimentally or analytically, and U is the
Darcian velocity.

Meanwhile, the correct transport equations in heterogeneous 2 scale
blood capillary network look very different. The 1-D momentum equation in the direction along
the longitudinal x axis, Fig. 6, with detailed description of the medium
morphology can be depicted as follows

where the sign
^ means fluctuation of velocity ( and further, others variables ) inside of the
representative volume. The next example is the generalized longitudinal 1-D
mass transport equation of non-linear blood constituent fluid in the capillary
network phase, including description of potential morpho-fluctuation
influences, for a medium morphology
with only 1‑D fluctuations it can be written in 2-scale morphology as
follows

while the corresponding mass transport equation in
the heterogeneous tissue of fiber and vessel's wall (motionless) phase might be simplified

There
are usually from 2 to 4 terms in each of the equations that need to be modeled.
The morphology of the blood capillary network governs closure of these very
influential and difficult to implement nonlinear terms. The modeling of
two-medium, namely blood capillary network medium and muscle tissue medium,
mass transfer exchange phenomena, of course, should not be limited and tied by
the Fick's law shortcomings - Wheatley and Malone (1993). Development of the
third scale capillary network governing equations along with the closure
modeling technique will be the aim of that step in the research.

**1.4.6. Simulation of Major Role of Blood
Capillary, Arteriole and Venule Flow Network in the Muscle Fiber Bundle **

More
complex problems arise while describing the flow in the more highly structured
heterogeneous medium, such as the muscle capillary bundle shown on Fig. 7.

The fourth scale modeling achievements for MBSS include derivation of
the governing equation set, identifying the specific goals for the muscle
bundle capillary network simulation, procedures to reach them, and development
of modeling algorithms and software.
Also the subsequent set of complementary experimental work to promote
the estimation of models validity will be formulated.

**1.4.7. SUMMARY **

** 1.5.
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