Theoretical study of the fibrous capsule tissue growth around a disk-shaped implant

Rabia Djellouli; Shant Mahserejian; A. Mokrane; Mohand Moussaoui; Taous-Meriem Laleg-Kirati

doi:10.1007/s00285-012-0566-1

Theoretical study of the fibrous capsule tissue growth around a disk-shaped implant

Rabia Djellouli, Shant Mahserejian, A. Mokrane, Mohand Moussaoui, Taous-Meriem Laleg-Kirati

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2 Scopus citations

Abstract

We analyze the mathematical properties of the fibrous capsule tissue concentration around a disk-shaped implant. We establish stability estimates as well as monotonicity results that illustrate the sensitivity of this growth to the biocompatibility index parameters of the implant. In addition, we prove that the growth of the tissue increases exponentially in time toward an asymptotic regime. We also study the mathematical properties of the solution of the inverse problem consisting in the determination of the values of the biocompatibility index parameters from the knowledge of some fibrous capsule tissue measurements. We prove that this model calibration problem admits a unique solution, and establish a characterization of the index parameters. Furthermore, we demonstrate analytically that such a solution is not continuous with respect to the data, and therefore the considered inverse problem is ill-posed due to the lack of the stability requirement. © 2012 Springer-Verlag.

Original language	English (US)
Pages (from-to)	833-867
Number of pages	35
Journal	Journal of Mathematical Biology
Volume	67
Issue number	4
DOIs	https://doi.org/10.1007/s00285-012-0566-1
State	Published - Aug 19 2012

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The first author acknowledges the support of MiniMed/Medtronic and the office of Graduate Studies at CSUN under grant# E1515. Some of this work was conducted while the first author was visiting the Mathematical and Computer Sciences and Engineering Division at the King Abdullah University of Science and Technology. He also would like to thank KAUST for its generous hospitality. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of MiniMed/Medtronic or of the office of Graduate Studies at CSUN, or KAUST.

ASJC Scopus subject areas

Modeling and Simulation
Applied Mathematics
Agricultural and Biological Sciences (miscellaneous)

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title = "Theoretical study of the fibrous capsule tissue growth around a disk-shaped implant",

abstract = "We analyze the mathematical properties of the fibrous capsule tissue concentration around a disk-shaped implant. We establish stability estimates as well as monotonicity results that illustrate the sensitivity of this growth to the biocompatibility index parameters of the implant. In addition, we prove that the growth of the tissue increases exponentially in time toward an asymptotic regime. We also study the mathematical properties of the solution of the inverse problem consisting in the determination of the values of the biocompatibility index parameters from the knowledge of some fibrous capsule tissue measurements. We prove that this model calibration problem admits a unique solution, and establish a characterization of the index parameters. Furthermore, we demonstrate analytically that such a solution is not continuous with respect to the data, and therefore the considered inverse problem is ill-posed due to the lack of the stability requirement. {\textcopyright} 2012 Springer-Verlag.",

author = "Rabia Djellouli and Shant Mahserejian and A. Mokrane and Mohand Moussaoui and Taous-Meriem Laleg-Kirati",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: The first author acknowledges the support of MiniMed/Medtronic and the office of Graduate Studies at CSUN under grant# E1515. Some of this work was conducted while the first author was visiting the Mathematical and Computer Sciences and Engineering Division at the King Abdullah University of Science and Technology. He also would like to thank KAUST for its generous hospitality. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of MiniMed/Medtronic or of the office of Graduate Studies at CSUN, or KAUST.",

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doi = "10.1007/s00285-012-0566-1",

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AU - Mahserejian, Shant

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AU - Moussaoui, Mohand

AU - Laleg-Kirati, Taous-Meriem

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: The first author acknowledges the support of MiniMed/Medtronic and the office of Graduate Studies at CSUN under grant# E1515. Some of this work was conducted while the first author was visiting the Mathematical and Computer Sciences and Engineering Division at the King Abdullah University of Science and Technology. He also would like to thank KAUST for its generous hospitality. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of MiniMed/Medtronic or of the office of Graduate Studies at CSUN, or KAUST.

PY - 2012/8/19

Y1 - 2012/8/19

N2 - We analyze the mathematical properties of the fibrous capsule tissue concentration around a disk-shaped implant. We establish stability estimates as well as monotonicity results that illustrate the sensitivity of this growth to the biocompatibility index parameters of the implant. In addition, we prove that the growth of the tissue increases exponentially in time toward an asymptotic regime. We also study the mathematical properties of the solution of the inverse problem consisting in the determination of the values of the biocompatibility index parameters from the knowledge of some fibrous capsule tissue measurements. We prove that this model calibration problem admits a unique solution, and establish a characterization of the index parameters. Furthermore, we demonstrate analytically that such a solution is not continuous with respect to the data, and therefore the considered inverse problem is ill-posed due to the lack of the stability requirement. © 2012 Springer-Verlag.

AB - We analyze the mathematical properties of the fibrous capsule tissue concentration around a disk-shaped implant. We establish stability estimates as well as monotonicity results that illustrate the sensitivity of this growth to the biocompatibility index parameters of the implant. In addition, we prove that the growth of the tissue increases exponentially in time toward an asymptotic regime. We also study the mathematical properties of the solution of the inverse problem consisting in the determination of the values of the biocompatibility index parameters from the knowledge of some fibrous capsule tissue measurements. We prove that this model calibration problem admits a unique solution, and establish a characterization of the index parameters. Furthermore, we demonstrate analytically that such a solution is not continuous with respect to the data, and therefore the considered inverse problem is ill-posed due to the lack of the stability requirement. © 2012 Springer-Verlag.

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Theoretical study of the fibrous capsule tissue growth around a disk-shaped implant

Abstract

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