1
|
Mukherjee A, Fok PW. A new approach to calculating fiber fields in 2D vessel cross sections using conformal maps. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:3610-3623. [PMID: 36899595 DOI: 10.3934/mbe.2023168] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
An arterial vessel has three layers, namely, the intima, the media and the adventitia. Each of these layers is modeled to have two families of strain-stiffening collagen fibers that are transversely helical. In an unloaded configuration, these fibers are coiled up. In the case of a pressurized lumen, these fibers stretch and start to resist further outward expansion. As the fibers elongate, they stiffen, affecting the mechanical response. Having a mathematical model of vessel expansion is crucial in cardiovascular applications such as predicting stenosis and simulating hemodynamics. Thus, to study the mechanics of the vessel wall under loading, it is important to calculate the fiber configurations in the unloaded configuration. The aim of this paper is to introduce a new technique of using conformal maps to numerically calculate the fiber field in a general arterial cross-section. The technique relies on finding a rational approximation of the conformal map. First, points on the physical cross section are mapped to points on a reference annulus using a rational approximation of the forward conformal map. Next, we find the angular unit vectors at the mapped points, and finally a rational approximation of the inverse conformal map is used to map the angular unit vectors back to vectors on the physical cross section. We have used MATLAB software packages to achieve these goals.
Collapse
Affiliation(s)
- Avishek Mukherjee
- Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA
| | - Pak-Wing Fok
- Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA
| |
Collapse
|
2
|
Schirm S, Haghikia A, Brack M, Ahnert P, Nouailles G, Suttorp N, Loeffler M, Witzenrath M, Scholz M. A biomathematical model of atherosclerosis in mice. PLoS One 2022; 17:e0272079. [PMID: 35921269 PMCID: PMC9348695 DOI: 10.1371/journal.pone.0272079] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2021] [Accepted: 07/12/2022] [Indexed: 11/18/2022] Open
Abstract
Atherosclerosis is one of the leading causes of death worldwide. Biomathematical modelling of the underlying disease and therapy processes might be a useful aid to develop and improve preventive and treatment concepts of atherosclerosis. We here propose a biomathematical model of murine atherosclerosis under different diet and treatment conditions including lipid modulating compound and antibiotics. The model is derived by translating known biological mechanisms into ordinary differential equations and by assuming appropriate response kinetics to the applied interventions. We explicitly describe the dynamics of relevant immune cells and lipid species in atherosclerotic lesions including the degree of blood vessel occlusion due to growing plaques. Unknown model parameters were determined by fitting the predictions of model simulations to time series data derived from mice experiments. Parameter fittings resulted in a good agreement of model and data for all 13 experimental scenarios considered. The model can be used to predict the outcome of alternative treatment schedules of combined antibiotic, immune modulating, and lipid lowering agents under high fat or normal diet. We conclude that we established a comprehensive biomathematical model of atherosclerosis in mice. We aim to validate the model on the basis of further experimental data.
Collapse
Affiliation(s)
- Sibylle Schirm
- Institute for Medical Informatics, Statistics and Epidemiology, University of Leipzig, Leipzig, Germany
| | - Arash Haghikia
- Department of Cardiology, Charité - Universitätsmedizin Berlin, Campus Benjamin Franklin, Berlin, Germany
- DZHK (German Center for Cardiovascular Research), Partner Site Berlin, Berlin, Germany
- Berlin Institute of Health (BIH), Berlin, Germany
| | - Markus Brack
- Charité - Universitätsmedizin Berlin, Corporate Member of Freie Universität Berlin and Humboldt-Universität zu Berlin, Division of Pulmonary Inflammation, Berlin, Germany
- Charité - Universitätsmedizin Berlin, Corporate Member of Freie Universität Berlin and Humboldt-Universität zu Berlin, Department of Infectious Diseases and Respiratory Medicine, Berlin, Germany
| | - Peter Ahnert
- Institute for Medical Informatics, Statistics and Epidemiology, University of Leipzig, Leipzig, Germany
| | - Geraldine Nouailles
- Charité - Universitätsmedizin Berlin, Corporate Member of Freie Universität Berlin and Humboldt-Universität zu Berlin, Division of Pulmonary Inflammation, Berlin, Germany
| | - Norbert Suttorp
- Charité - Universitätsmedizin Berlin, Corporate Member of Freie Universität Berlin and Humboldt-Universität zu Berlin, Department of Infectious Diseases and Respiratory Medicine, Berlin, Germany
| | - Markus Loeffler
- Institute for Medical Informatics, Statistics and Epidemiology, University of Leipzig, Leipzig, Germany
| | - Martin Witzenrath
- Charité - Universitätsmedizin Berlin, Corporate Member of Freie Universität Berlin and Humboldt-Universität zu Berlin, Division of Pulmonary Inflammation, Berlin, Germany
- Charité - Universitätsmedizin Berlin, Corporate Member of Freie Universität Berlin and Humboldt-Universität zu Berlin, Department of Infectious Diseases and Respiratory Medicine, Berlin, Germany
| | - Markus Scholz
- Institute for Medical Informatics, Statistics and Epidemiology, University of Leipzig, Leipzig, Germany
- LIFE Research Center of Civilization Diseases, University of Leipzig, Leipzig, Germany
| |
Collapse
|