1. Biomedical background and objectives
The research in this proposal
has been motivated by a study of blood flow through compliant vessels
after endovascular repair. The PI and cardiologist Dr. Krajcer at
St. Luke's Episcopal Hospital in Houston have begun a study of
blood flow through the abdominal aorta after the insertion of a
prosthesis (a spring-like device called stent)
used to treat abdominal aneurysm (sac-like protrusion
of weakened sections of the aorta).
See Figure 1.
The procedure entails inserting a stent inside the aneurysm where it
serves to hold open the weakened artery and to exclude
the aneurysm from circulation.
The main goal of this collaborative study is to understand the behavior
of the prosthesis and of the walls of the aorta subject to the
pressures induced by pulsatile blood flow.
A sketch of abdominal aneurysm (left) and a sketch of a stent
inserted in the abdominal aorta (right), not showing the aneurysm sac
equations are obtained using asymptotic reduction of the
axi-symmetric Navier-Stokes equations in narrow and long channels. The
equations have first been derived in ,
and a mathematically rigorous asymptotic reduction is presented in .
The equations are in the form of a quasilinear hyperbolic system of
partial differential equations
describing conservation of mass and momentum
Here A(x,t) is the
cross-sectional area of the
vessel, m(x,t) = AV
is the momentum,
V=V(x,t) is the average axial velocity
and p is the transmural pressure.
for the fact that averaged quantities (momentum)
The source term includes the
effects of viscosity
To close the system, the
pressure term needs to be specified 
G0 describes tissue stiffness
(pressure-strain elastic modulus),
behavior of the channel wall (pressure-strain relationship).
Versions of this model
have been used by many authors
to model fluid flow in compliant tubes
[1, 2, 4, 5, 6, 7, 8, 11, 15, 18].
A novel approach in this study is to assume that the coefficient G0
is a piecewise constant function of x to reflect the fact that
the stent and the aorta
have different elastic properties [20, 14].
As a consequence, stent and the aorta respond differently
to the pressure induced by the pulsatile blood flow
Various complications associated with this phenomenon
have been reported in patient studies .
The main objective of the present study is to analyze the abrupt
changes in the cross-sectional area due to the abrupt changes in the
elasticity coefficient G0,
understand their consequences on the blood flow,
and study optimal design of multiple overlapping stents
in order to minimize harmful effects caused by the perturbed channel
In order to study discontinuities in the cross-sectional area we need
to be able to define a weak solution for a quasilinear system of
hyperbolic equations with discontinuous coefficients.
There is no general theory that deals with systems of hyperbolic
partial differential equations with discontinuous coefficients. The
literature is sparse even for hyperbolic equations in conservation form
allowing discontinuous coefficients.
There are various difficulties associated with the solutions
of such systems.
For example, physically reasonable solutions exhibit jump
discontinuities that are not compressive
because the jump discontinuity in the solution is induced by the
discontinuity in the coefficients,
and not by the compressive nature of the characteristics.
Deriving the ``correct'' entropy criterion for such jump
and capturing such solutions numerically, is a problem.
In the case when the equations are not in conservation form, as in the
case of our model,
even the basic issue such as the meaning of the weak form of the
in unclear. The main difficulty stems from the fact that the weak form
of the equations contains the products of the
Dirac measure by the Heaviside function which cannot be
defined in the sense of distributions.
In the case
when the equations are linear and in
there is a theory developed in 
that deals with numerical approximation of one-dimensional
linear conservation equations with discontinuous coefficients.
Unfortunately, the techniques used in 
do not generalize to nonlinear problems. Quasilinear scalar
hyperbolic problems have been studied in [16, 17] where
variational methods were used
to study existence and uniqueness of solutions.
The methods heavily rely on the fact that there is only one hyperbolic
equation under consideration and it is unlikely that the approach can
be generalized to systems. Finally, the most relevant to this proposal
are the work of A.I Volpert 
work by Dal Maso, P.G. LeFloch and F. Murat
 where issues
related to the
product of the Dirac measure by the Heaviside function are discussed.
Although these works do not deal with hyperbolic problems with
the ideas about the meaning of the ambiguous products
are very much related to the problems that arise in hyperbolic
equations with discontinuous coefficients.
In the work proposed here one of the objectives
is to generalize those ideas
to the systems with discontinuous coefficients.
For the past
ten years the PI has had continuous research support by
the National Science Foundation to study properties of solutions to
multi-dimensional systems of conservation laws and to study uniqueness
and properties of viscosity solutions to one-dimensional conservation
with smooth coefficients. The project described in this proposal opens
a new direction in the PI's research program.
Various aspects of this project, however, rely heavily on the knowledge
and the experience gained from the work supported by the NSF.
1. Theoretical approach
To study weak solutions of equations (1)
we begin with the standard approach:
multiply the equations by a test function
integrate by parts to obtain
It was shown in 
at the points where the coefficient G0 of p(x,A)
discontinuous, the cross-sectional area Acannot be continuous.
This implies that
the right hand-side contains the product of the Dirac measure Ax
by the Heaviside function p(x,A).
This difficulty cannot be avoided if
system (1), (2) is written in conservation form.
Converting the system into a conservation law gives rise to a source
term which is singular;
it is the product of a function of A, call it q(A),
and the derivative of the discontinuous coefficient, G0'(x),
. The weak
form of the equations
then contains the term
which is not defined in the sense of distributions.
To get around
this difficulty we take the following approach.
Motivated by the underlying application in which the Young's modulus
G0 is actually continuous, with high gradients at the
anchoring cites of the stent, we regularize the problem by smoothing
out the Dirac measure term G0'.
We regularize Dirac measure by using, the so called, admissible
averaging kernels 
Consider the weak form that corresponds to the regularized coefficient G0',
call it ,
and define the weak form of the original, singular problem
to be the one obtained in the limit, as ,
of the regularized forms. In order for the limit to be defined, we need
to specify the term
Following the ideas presented in [12,
we define (6) to be the mean
with respect to the measure G0'
approximated by the
admissible averaging kernels ,
It can be shown that if the kernel is symmetric,
i.e., if ,
then the resulting symmetric mean does not depend on the choice
of the symmetric kernel [4,
The choice of the kernel
will depend on the particular problem at hand,
and here is where the ``physics'' of the problem needs to be taken into
account in order to recover the ``physically''
or ``physiologically reasonable solution''.
Although this is similar to the viscosity solutions approach in
the theory of conservation laws,
the underlying issues of convergence and stability require different
techniques. One of the goals of this project is to generalize the
techniques presented in 
to study convergence and stability of the products like (6)
arising in systems with discontinuous coefficients.
2. Numerical simulations
The PI has studied the performance of two numerical methods on the
calculation of the solutions of equations (1)
with discontinuous G0: the Lax-Friedrichs method
(LF) and the Richmyer two-step Lax-Wendroff method (LW), .
Unusual observations have been obtained.
produces the solution which would have been recovered if symmetric
kernels were used to approximate the Dirac measure G0'
in the quasilinear form of the problem.
The LF-method, on the other hand, produces a different solution from
the LW-method, regardless whether one solves the conserved or the
nonconserved form of the equations.
A group of undergraduate students
who participated in the NSF-sponsored REU program this summer, studied
the behavior of the LF method.
They have shown
that the modified equation contains large (artificial) source terms due
to the large gradients
of G0 and the error contaminates
the solution everywhere downstream from the locus of discontinuity.
Even when G0 is smoothed out,
careful analysis is needed prior to the numerical simulation
in order to keep the effects of numerical diffusion below the effects
of blood viscosity, described by the source term ,
and below the smoothing effects of the discontinuous coefficient G0.
Similar analysis needs to be performed before using the second-order
LW-method. The issues here are related to balancing the effects of
numerical dispersion, numerical diffusion and large artificial source
with the regularization of G0 and with the source
project the PI plans to investigate
the effects that need to be taken into account when hyperbolic
equations with discontinuous coefficients and singular source terms are
simulated in order to obtain
the physically reasonable solutions.
Error analysis will be performed by generalizing the ideas presented in
 to the
Although the development of a general existence and uniqueness
theory for quasilinear hyperbolic problems with
discontinuous coefficients is one of the goals of the PI's research,
it is unrealistic to expect that within the two years of the duration
grant the PI and the students will be able to complete the
theory. A realistic goal is to study equations
(1) and (2)
and gain an insight into the properties
of the solutions that can be generalized to a larger class of
The PI on
the project is Canic.
She has extensive research experience in hyperbolic conservation
laws, in conservation laws that change type, and in modeling of
physical processes by conservation laws.
Her accomplishments have been recently recognized by the
UH 2001 Associate Professor Research and Scholarship Award.
The PI has experience in mentoring undergraduate students (3), graduate
students (2) and post-doctoral visitors (1). In addition, this summer
she ran a Research for Undergraduates program at UH.
One of the students (Burns)
who participated in the program is expected to graduate in the Fall
and continue working on a problem related to this project
as graduate student in Mathematics.
Another student (Sharma), currently a graduate student in Mechanical
is expected to change majors and begin working with the PI on the
numerical simulation and error analysis of the model equations with
commitment and sources of
Mathematics Department employs graduate students as Teaching
and this would be the normal source of support for students
engaged in the project.
The requested budget will partially support two graduate students
providing reduced teaching load to expedite the time to degree.
The students have already been recruited.
The PI has directed students who are now employed in academia
and in industry.
the Mathematics Department is planning to submit a VIGRE
(Vertical Integration of Research and Education) proposal in Fall 2001.
If awarded, the VIGRE support would
provide supplemental funds for two graduate students and one
The PI has
external support which expires this summer. A renewal for
the grant will be submitted in the Fall.
If awarded, the grant will provide
partial travel support and partial summer salary.
involvement and training opportunities
in science and engineering
very beginning of this project students and postdoctoral
have been involved in numerical simulation, visualization,
and the development of a new theory to study the underlying model
Two undergraduate students graduated (Pritts, Roy) and continued their
studies at the first rate universities.
Post-doctoral visitor Kim is moving
to a tenure-track position to continue her career in teaching and
a group of the most talented undergraduate students from all over the
US has been involved this summer in the NSF-sponsored REU program.
The topics of their projects originated from the blood-flow problem
described here. All of them will continue their studies as graduate
Student problems that cover a broad range of interests
have been formulated and offered by the PI to the graduate student
community at UH
(two students have already been recruited).
Student project 1. Analysis of
weak solutions to conservation
with discontinuous coefficients and a study of stability of the related
singular products with the focus on the blood flow problem.
Student project 2.
Numerical simulation of equations (1)
with discontinuous coefficient G0,
using different numerical methods. Analysis of the influence of the
artifical numerical effects on the calculation of the solution. Error
Student project 3.
Modeling on the behavior of the prostheses.
Development of a more sophisticated model and comparison with the
measurements obtained by Dr. Chandar at UT Austin.
Inclusion of the porosity of the stent in the model equations.
Student project 4.
Development of a simplified two-dimensional
model through curved vessels.
This entails studying fluid-structure interactions between the
stent/aorta and the blood flow.