Communications in Information and Systems

Volume 23 (2023)

Number 3

Special issue dedicated Professor Avner Friedman in celebration of his 90th birthday

Guest Editors: Jong-Shenq Guo, Bei Hu, Robert Jensen, and Stephen S.-T. Yau

Periodic solution for a free boundary problem modeling small plaques

Pages: 263 – 287

DOI: https://dx.doi.org/10.4310/CIS.2023.v23.n3.a3

Authors

Yaodan Huang (School of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang, China)

Bei Hu (Dept. of Applied & Computational Mathematics & Statistics, University of Notre Dame, Indiana, U.S.A.)

Abstract

Plaque formation within arteries is one of the leading causes of death in USA and worldwide. Mathematical models describing the growth of plaque in the arteries (e.g., $[\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3350421}{1}, \href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3345849}{2}, \href{https://doi.org/10.1371/journal.pone.0090497}{3}, \href{https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=c2541f3405cfea343e8f069a49b18b73725871f7}{5}, \href{https://doi.org/10.1007/s40808-019-00643-6}{6}])$ were introduced. All of these models include the interaction of the “bad” cholesterols, low density lipoprotein (LDL), and the “good” cholesterols, high density lipoprotein (HDL), in triggering whether plaque will grow or shrink.

Because the blood vessels tend to be circular, 2D cross section model is a good approximation, and the 2D models are studied in $[\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3350421}{2}, \href{https://mathscinet.ams.org/mathscinet/relay-station?mr=4473794}{7}, \href{https://mathscinet.ams.org/mathscinet/relay-station?mr=4246155}{8}, \href{https://mathscinet.ams.org/mathscinet/relay-station?mr=4420644}{9}]$. A bifurcation into a 3D plaque was recently studied in $[\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=4499492}{4}]$. All of these models assume a constant supply of LDL and HDL from the blood vessel.

In reality, nutrient concentration changes with the intake of food, which happens very often in a periodic manner. In this paper, we shall establish a periodic solution when the LDL and HDL supplies from the blood vessel are periodic.

Keywords

free boundary problem, periodic solution, atherosclerosis

2010 Mathematics Subject Classification

Primary 35B10, 35R35, 92B05. Secondary 35Q92.

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Huang’s research is supported in part by Guangdong Provincial Natural Science Foundation Grant No. 2021A1515111004.

Received 30 November 2022

Published 12 October 2023