ON FORWARD AND INVERSE PROBLEMS FOR A DCIS MODEL WITH FREE BOUNDARIES IN MATHEMATICAL BIOLOGY

Hongyu Liu; Keji Liu

doi:10.3934/ipi.2024012

ON FORWARD AND INVERSE PROBLEMS FOR A DCIS MODEL WITH FREE BOUNDARIES IN MATHEMATICAL BIOLOGY

Hongyu Liu, Keji Liu^*

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

Abstract

We are concerned with the mathematical study of a DCIS model which arises in characterizing the biological development of breast cancer. A salient feature of a DCIS model is the presence of free boundaries for describ-ing the tumor growth which is not known in advance. In this paper, we are particularly interested in the case with general free boundaries which may be asymmetric. We first propose an iterative finite difference method for the forward problem and show that the method is of 2nd order in both space and time. Then we propose to study an inverse problem of recovering the nutrient consumption rate by the incisional biopsy data. We establish the unique iden-tifiability of the inverse problem and develop a novel reconstruction scheme based on a certain integral formulation. Extensive numerical experiments are conducted to corroborate the theoretical findings. Our study opens up a new direction of research in mathematical biology with many potential extensions and developments. © 2024, American Institute of Mathematical Sciences. All rights reserved.

Original language	English
Pages (from-to)	1223-1242
Journal	Inverse Problems and Imaging
Volume	18
Issue number	5
Online published	Mar 2024
DOIs	https://doi.org/10.3934/ipi.2024012
Publication status	Published - Oct 2024

Funding

Keji Liu is supported by [the NNSF of China under grant No. 12071275] and Hongyu Liu is supported by [the NSFC/RGC Joint Research Scheme, N_CityU101/21, ANR/RGC Joint Research Scheme, A-CityU203/19, and the Hong Kong RGC General Research Funds (projects 11311122, 12301420 and 11300821)]

Research Keywords

asym-metric
Ductal carcinoma in situ
finite difference method
free boundaries
inverse problem
mathematical biology
reconstruction
unique identifiability

UN SDGs

This output contributes to the following UN Sustainable Development Goals (SDGs)

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title = "ON FORWARD AND INVERSE PROBLEMS FOR A DCIS MODEL WITH FREE BOUNDARIES IN MATHEMATICAL BIOLOGY",

abstract = "We are concerned with the mathematical study of a DCIS model which arises in characterizing the biological development of breast cancer. A salient feature of a DCIS model is the presence of free boundaries for describ-ing the tumor growth which is not known in advance. In this paper, we are particularly interested in the case with general free boundaries which may be asymmetric. We first propose an iterative finite difference method for the forward problem and show that the method is of 2nd order in both space and time. Then we propose to study an inverse problem of recovering the nutrient consumption rate by the incisional biopsy data. We establish the unique iden-tifiability of the inverse problem and develop a novel reconstruction scheme based on a certain integral formulation. Extensive numerical experiments are conducted to corroborate the theoretical findings. Our study opens up a new direction of research in mathematical biology with many potential extensions and developments. {\textcopyright} 2024, American Institute of Mathematical Sciences. All rights reserved.",

keywords = "asym-metric, Ductal carcinoma in situ, finite difference method, free boundaries, inverse problem, mathematical biology, reconstruction, unique identifiability",

author = "Hongyu Liu and Keji Liu",

year = "2024",

month = oct,

doi = "10.3934/ipi.2024012",

language = "English",

volume = "18",

pages = "1223--1242",

journal = "Inverse Problems and Imaging",

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T1 - ON FORWARD AND INVERSE PROBLEMS FOR A DCIS MODEL WITH FREE BOUNDARIES IN MATHEMATICAL BIOLOGY

AU - Liu, Hongyu

AU - Liu, Keji

PY - 2024/10

Y1 - 2024/10

N2 - We are concerned with the mathematical study of a DCIS model which arises in characterizing the biological development of breast cancer. A salient feature of a DCIS model is the presence of free boundaries for describ-ing the tumor growth which is not known in advance. In this paper, we are particularly interested in the case with general free boundaries which may be asymmetric. We first propose an iterative finite difference method for the forward problem and show that the method is of 2nd order in both space and time. Then we propose to study an inverse problem of recovering the nutrient consumption rate by the incisional biopsy data. We establish the unique iden-tifiability of the inverse problem and develop a novel reconstruction scheme based on a certain integral formulation. Extensive numerical experiments are conducted to corroborate the theoretical findings. Our study opens up a new direction of research in mathematical biology with many potential extensions and developments. © 2024, American Institute of Mathematical Sciences. All rights reserved.

AB - We are concerned with the mathematical study of a DCIS model which arises in characterizing the biological development of breast cancer. A salient feature of a DCIS model is the presence of free boundaries for describ-ing the tumor growth which is not known in advance. In this paper, we are particularly interested in the case with general free boundaries which may be asymmetric. We first propose an iterative finite difference method for the forward problem and show that the method is of 2nd order in both space and time. Then we propose to study an inverse problem of recovering the nutrient consumption rate by the incisional biopsy data. We establish the unique iden-tifiability of the inverse problem and develop a novel reconstruction scheme based on a certain integral formulation. Extensive numerical experiments are conducted to corroborate the theoretical findings. Our study opens up a new direction of research in mathematical biology with many potential extensions and developments. © 2024, American Institute of Mathematical Sciences. All rights reserved.

KW - asym-metric

KW - Ductal carcinoma in situ

KW - finite difference method

KW - free boundaries

KW - inverse problem

KW - mathematical biology

KW - reconstruction

KW - unique identifiability

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DO - 10.3934/ipi.2024012

M3 - RGC 21 - Publication in refereed journal

SN - 1930-8337

VL - 18

SP - 1223

EP - 1242

JO - Inverse Problems and Imaging

JF - Inverse Problems and Imaging

IS - 5

ER -

ON FORWARD AND INVERSE PROBLEMS FOR A DCIS MODEL WITH FREE BOUNDARIES IN MATHEMATICAL BIOLOGY

Abstract

Funding

Research Keywords

UN SDGs

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GRF: Mathematical Study of Fields Concentration and Gradient Estimates with Applications

GRF: Mathematical Studies of Surface-localized Transmission Eigenstates and Applications

NSFC: A Mathematical Theory of Subwavelength Resonances in Elasticity with Applications and Beyond

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ON FORWARD AND INVERSE PROBLEMS FOR A DCIS MODEL WITH FREE BOUNDARIES IN MATHEMATICAL BIOLOGY

Abstract

Funding

Research Keywords

UN SDGs

Access Output

Other Access Options

Permanent Link

Other Files and Links

Fingerprint

Projects

GRF: Mathematical Study of Fields Concentration and Gradient Estimates with Applications

GRF: Mathematical Studies of Surface-localized Transmission Eigenstates and Applications

NSFC: A Mathematical Theory of Subwavelength Resonances in Elasticity with Applications and Beyond

Cite this