Dipartimento di Matematica - Politecnico di Milano

April 20, 2015 – April 24, 2015





This course will be taught by three instructors who are faculty members at Indiana University (Indianapolis, IN, USA).
These instructors have complementary expertise spanning clinical and mathematical perspectives:

Dr. Julia Arciero, Department of Mathematical Sciences, Indiana University - Purdue University at Indianapolis (IUPUI).
Prof. Giovanna Guidoboni, Department of Mathematical Sciences, Indiana University - Purdue University at Indianapolis (IUPUI).
Prof. Alon Harris, Director of Clinical Research, Eugene and Marilyn Glick Eye Institute, Department of Ophthalmology, Indiana University School of Medicine.


Course Description

This course covers a variety of mathematical methods for modeling biological fluids and tissues. Theoretical concepts will be applied to various areas of anatomy and physiology, with a particular emphasis on the human eye since Drs. Arciero, Guidoboni and Harris have an established collaboration that combines their mathematical and clinical expertise to explain the relationship between eye mechanics and hemodynamics and glaucoma incidence and progression.


Main Features

The scientific strength and broad impact of the course are a direct result of the following three factors that govern the approach of the course.

- Interdisciplinary approach: The course material requires an in-depth study of topics in mathematics, engineering, physiology and medicine.  The material will be presented by experts in these different disciplines who have established a successful interdisciplinary collaboration on the human eye.

- International approach: The three instructors are faculty at Indiana University and they are not affiliated with any Italian institution.  This course developed naturally as a result of the international and interdisciplinary workshop entitled "Integrated Multidisciplinary Approaches in the Study and Care of the Human Eye," which was held in Milano, 26-27 June 2013 (web site: The workshop was attended by mathematicians, engineers, clinical scientists and medical doctors, and it attracted faculty as well as graduate and undergraduate students from Italy, Israel and the USA.

- Research approach:  The established collaborative research currently being conducted among the three instructors will naturally feed into the course material. Drs. Arciero, Guidoboni and Harris have  developed the first mathematical models capable of qualitatively and quantitatively describing the relationship between intraocular pressure (IOP), blood pressure, blood flow autoregulation and retinal blood flow. These mathematical models have been used to interpret clinical data and resolve a long-standing controversy in the field of ophthalmology related to whether or not IOP elevation would induce a reduction in retinal blood flow in every individual.  Their research is currently funded by the National Science Foundation (NSF). Drs. Arciero, Guidoboni and Harris have also recently started collaborating with Dr. Riccardo Sacco (Politecnico di Milano) on the development of mathematical models for the biophysics of retinal blood flow regulation. This international collaboration between IUPUI, the Indiana University School of Medicine and Politecnico di Milano has been officially stipulated funded by the NSF through an international supplement to their currently active NSF grant


Course Outline

The course consists of 25 hours of lectures, presentation and disucssion given by the main instructor and guest speakers. Covered topics include:

  • (2 hrs)  Guidoboni/Harris: Introduction.
  1. (1 hr) Guidoboni: Description of clinically-relevant questions in the study of biological fluids and tissues and the advantages and limitations of human and animal models using in vivo and in vitro studies; Challenges in mathematical modeling of biological fluids and tissues including classical mechanics and fluid dynamics, fluid-structure interactions, and the multiple time and length scales involved.
  2. (1 hr) Harris: Physiology of tissue perfusion and oxygenation in the human eye. Relevance to diagnosis, therapy and management of ocular, cerebral and systemic diseases.
  • (13 hrs)  Guidoboni: Mechanics of solids and fluids.
  1. (6 hrs) Solids. Stresses and strains; Lagrangian and Eulerian description of motion; balance of mass, momentum, and energy; constitutive equations; growth and remodeling. Mathematical, computational and experimental perspectives.
  2. (4 hrs) Fluids. From the electric analogy to the Navier-Stokes equations; homogeneous and non-homogeneous fluids; Newtonian and non-Newtonian fluids; rheology of biological fluids; constitutive equations for blood. Mathematical, computational and experimental perspectives.
  3. (3 hrs) Fluid-structure interaction. Fluid-structure interface conditions on velocity and stress; Lagrangian-Eulerian coupling; poro-elastic modeling of tissue perfusion. Mathematical, computational and experimental perspectives.
  • (8 hrs) Arciero:  Mass transport in a tissue.
  1. (4 hrs) Transport and diffusion of solutes in a fluid; oxygen delivery and exchange in a tissue; blood flow regulation according to pressure and metabolism; effect of nitric oxide and endothelin on vascular smooth muscle. Mathematical, computational and experimental perspectives.
  2. (4 hrs) Theoretical predictions of the relationship between cell function and blood flow are evaluated using clinical measures of nutrient and oxygen delivery;  Neuronal function in the retina is assessed using theoretical techniques.
  • (2 hrs)  Harris:  Integrating Mathematics and Physiology in the Clinic.
  1. (1 hr) Developments in nanotechnology: artificial retinas and beyond.
  2. (1 hr) Concluding remarks on the past, present and future of clinical research and medical practice and their relationship with mathematics.


Hands-on experience

(5 hrs) Students will be able to select projects of mathematical, engineering or clinical orientation, depending on their interests. Examples include, but are not limited to:

- Hyperbolic/parabolic nature of fluid-structure interaction problems
- Changes in mechanical tissue properties due to aging and clinical implications
- Influence of intraocular pressure on retinal circulation
- Analysis of blood velocity in color doppler imaging and clinical interpretation
- Modeling the relationship between intraocular pressure, blood pressure and intracranial pressure
- Retinal oxygenation: modeling, simulations and comparison with retinal oxymetry maps