MIT Unit Affiliation:
Lab Affiliation(s):
Stochastic Analysis and Nonlinear Dynamics Lab
Post Doc Sponsor / Advisor:
Themis Sapsis
Areas of Expertise:
  • Uncertainty Quantification
  • Numerical Analysis
  • Cardiovascular System Modeling
Date PhD Completed:
August, 2013
Expected End Date of Post Doctoral Position:
June 1, 2015

Will Cousins

  • Post Doctoral

MIT Unit Affiliation: 

  • Mechanical Engineering

Lab Affiliation(s): 

Stochastic Analysis and Nonlinear Dynamics Lab

Post Doc Sponsor / Advisor: 

Themis Sapsis

Date PhD Completed: 

Aug, 2013

Top 3 Areas of Expertise: 

Uncertainty Quantification
Numerical Analysis
Cardiovascular System Modeling

Expected End Date of Post Doctoral Position: 

June 1, 2015


Research Projects: 

Uncertainty quantification and prediction of extreme ocean waves.

Thesis Title: 

Boundary Conditions and Uncertainty Quantification for Hemodynamics

Thesis Abstract: 

We address outflow boundary conditions for blood flow modeling. In particular, we consider a variety of fundamental issues in the structured tree boundary condition. We provide a theoretical analysis of the numerical implementation of the structured tree, showing that it is sensible but must be performed with great care. We also perform analytical and numerical studies on the sensitivity of model output on the structured tree's defining geometrical parameters. The most important component of this dissertation is the derivation of the new, generalized structured tree boundary condition. Unlike the original structured tree condition, the generalized structured tree does not contain a temporal periodicity assumption and is thus applicable to a much broader class of blood flow simulations. We describe a numerical implementation of this new boundary condition and show that the original structured tree is in fact a rough approximation of the new, generalized condition.


We also investigate parameter selection for outflow boundary conditions, and attempt to determine a set of strutured tree parameters that gives reasonable simulation results without requiring any calibration.  We are successful in doing so for a simulation of the systemic arterial tree, but the same paramter set yields physiologically unreasonable results in simulations of the Circle of Willis.  Finally, we investigate the extension of recently introduced PDF methods to smooth solutions to systems of hyperbolic balance laws subject to uncertain inputs.  These methods, currently available only for scalar equations, would provide a powerful tool for quantifying uncertainty in predictions of blood flow and other phenomena governed by first order hyperbolic systems.

Top 5 Awards and honors (name of award, date received): 

Rose-Winton Award, NC State University, 2013
Poster Prize Winner, Poster Session, SIAM Conference on the Life Sciences, 2012
NSF East Asia and Pacific Summer Institutes Fellow, 2010
Outstanding Mathematics Graduate, Pepperdine University, 2009
Poster Prize Winner, Undergraduate Poster Session, Joint Mathematics Meetings, 2009

5 Recent Papers: 

W. Cousins, P.A. Gremaud (Submitted -- Aug 2013). Impedance Boundary Conditions for General Transient Hemodynamics. Int. J. Numer. Meth. Biomed. Engng.--preprint available at

W. Cousins, P.A. Gremaud, D.M. Tartakovsky (2013). A New Physiological Boundary Condition for Hemodynamics. SIAM J. Appl. Math., 73(3), 1203-1223.

W. Cousins, P.A. Gremaud. Boundary Conditions for Hemodynamics: The Structured Tree Revisited (2012). Journal of Computational Physics, 231(18).

K. Anderson, A. Burt, W. Cousins, B. Hancock, D. Strong. A Sinkhorn-Knopp Fixed Point Problem (2011). Pi Mu Epsilon Journal, 13(5).

Contact Information:
77 Massachusetts Av