# Analysis of a Partial Differential Equation Model for Necrotizing Enterocolitis.

Tronzo, Mark (2014) Analysis of a Partial Differential Equation Model for Necrotizing Enterocolitis. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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## Abstract

This thesis presents and analyzes a mathematical model for necrotizing enterocolitis (NEC), a devastating disease that attacks the gastrointestinal tract of pre-term infants. Mathematical models for NEC have been developed in the past. These modes are extremely valuable and provide important insights into the disease.
However, all of the models developed previously are one dimensional, ordinary differential equation models and, therefore, simulate only the transient effects of NEC but do not fully model its spatial effects. The mathematical model presented here is a three dimensional model in the form of a system of nonlinear partial differential equations. A three dimensional model is needed to accurately simulate diffusion and advection of the major factors in NEC, to account for the different effects of NEC in the different regions in the body, and to fully integrate all the effects of such mechanisms as epithelial cell degradation and migration.

This thesis presents medical research regarding NEC, constructs inflammatory cascades related to the disease, and develops the system of partial differential equation system. Also, full mathematical analysis of the system of equations. The mathematical analysis of the system of partial differential equations and the associated a mixed finite element analysis are, perhaps, the most important parts of the thesis. The results of this analysis have significance for the NEC system and have significance independent of the NEC system. For example, existence, uniqueness, and regularity analysis is presented in the weak mixed form for coupled nonlinear partial differential equations.
Furthermore, finite element analysis (using the mixed method) is done on this coupled system and convergence is proven, a new and very important result. No mixed method finite element analysis has previously been published for this system. Similar analysis is done on the rest of the partial differential equations in the system. At the end of the thesis, computer simulations are done using the mathematical model. These simulations demonstrate that the NEC mathematical model presented here produces realistic results consistent with the actual progression of the disease.

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## Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
Tronzo, Markmarktronzo@gmail.com
ETD Committee:
Committee ChairYotov, Ivanyotov@math.pitt.eduYOTOV
Committee MemberTrenchea, Catalintrenchea@pitt.eduTRENCHEA
Committee MemberSwigon, Davidswigon@pitt.eduSWIGON
Committee MemberRubin, Jonathan Ejonrubin@pitt.eduJONRUBIN
Committee MemberVodovotz, Yoramvodovotzy@upmc.eduVODOVOTZ
Date: 30 May 2014
Date Type: Publication
Defense Date: 14 March 2014
Approval Date: 30 May 2014
Submission Date: 20 January 2014
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 303
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: Nonlinear Partial Differential Equations, Mixed Finite Element Method, Necrotizing Enterocolitis
Date Deposited: 30 May 2014 13:40