Welcome SURP researchers!
Mentors: Andrea Dziubek and Edmond Rusjan
Summary: The Mathematical Modeling Lab at SUNY Poly, Utica, specializes in the development, analysis and verification of mathematical models and the current focus is on modeling the blood flow in the retina of the eye. This research has the potential to advance our understanding of various eye pathologies and help improve existing treatments and discover new treatments. For example, our physically based modeling, based on first principles, coupled with the most advanced analytical and numerical solution techniques, has predicted that changes in the curvature of the retina of the eye lead to significant changes in the blood flow, which in turn may play a significant role in primary open-angle glaucoma.
The Summer Undergraduate Student Project will involve the student in the current improvements to the model. The blood flow in the retina of the eye is modeled as a Darcy flow through a hierarchical porous medium and is described by the parameterized Darcy equation. This equation is similar to the traditional Darcy equation, which can be used for example to model the flow of water or oil through sand, but it is extended by an additional variable, which represents the various blood vessels: large arteries, small arteries, arterioles, capillaries, and the various size veins. In other words, the model describes not only the spatial flow, but also the hierarchical flow, from arteries, through capillaries, to veins.
The student will have the opportunity to participate and to contribute to all aspects of the project and to focus on one particular area of their choice, appropriate to their level. The prerequisites are a solid background in mathematics, minimally at the level of calculus, and preferably including linear algebra, differential equations and multi-variable calculus, familiarity with a programming language, preferably Python, and an interest in applied mathematics, including mathematical modeling and scientific programming.
References:
Slides from the talk at
Vanderbilt
Effect of ocular shape and vascular geometry on retinal
hemodynamics: a computational model
Collaborators: Giovanna Guidoboni, Anil Hirani, William Thistleton, Michael Reale
Everybody interested in advancing the frontiers of knowledge in applied mathematics, mathematical physics, mathematical biology, and in particular mathematical physiology and ophthalmology, is welcome to join our team at any time - there is always plenty to do! Welcome! Having said this, the SUNY Poly Summer Internship Opportunity program is very competitive. Self starters, who will prove their interest and ability in advance, will be given priority. We encourage you to come up to speed using the resources provided below, to check your knowledge by doing the exercises, and to start working on any of the current tasks listed. Please feel welcome to contact us, if you have questions, you need help, would like to share ideas or suggestions, or would simply like to reach out to us. Send an email to both dziubea@sunyit.edu and edmond@sunyit.edu and if needed we can skype, zoom, or meet.
Tools we use:
Linux / Ubuntu,
Git,
Bitbucket,
Python,
NumPy,
SciPy,
PyDEC,
Matplotlib,
Mayavi,
Gmsh.
Documentation:
all tools have documentation and some also have tutorials.
Deciding how much time to spend on learning, before starting working
on a specific task, is a natural part of research and can be difficult.
Most people find it easiest to be productive by doing
learning and working in parallel, so that
learning comes in small iterations and is naturally
guided by the work. You learn what you need to learn.
Understanding typically lags slightly behind and you are frequently
able to do something before you fully understand why it works.
The goal is, of course, to eventually understand everything we do.
Exercises:
Please take a look at the
Darcy Flow Example
and the paper listed there. Work through the program line by line
and identify libraries and functions being used. Then find the
necessary documentation in the tools section and do some exercises
listed in the documentation/tutorials for the tools, to help you
get familiar with the tools and understand the example.
The referenced paper is not easy to read, but will help you to get
some idea about the mathematics behind the program.
Current tasks:
- refactor the program into testable functions performing a single task
- write tests for the functions you just created
- change boundary conditions, so that the pressure equals 0 on the
left boundary and 1 on the right boundary.