Page 506 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 506
MULTISCALE MODELING AND METHODS: APPLICATION IN ENGINEERING,
BIOLOGY AND MEDICINE (MS-80)
Mathematical Modeling of Inflammatory Processes of Atherosclerosis
Ghada Abi Younes, abiyounes@math.univ-lyon1.fr
Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
Coauthors: Nader El Khatib, Vitaly Volpert
Atherosclerosis is a chronic disease which involves the build up of cholesterol and fatty deposits
within the inner lining of the artery. It is associated with a progressive thickening and hardening
of the arterial wall that result in narrowing of the vessel lumen and restriction of blood flow to
vital organs. These events may cause heart attack or stroke, the commonest causes of death
worldwide. We study the early stages of atherosclerosis via a mathematical model of partial
differential equations of reaction-diffusion type. The model includes several key species and
identifies endothelial hyperpermeability, believed to be a precursor on the onset of atheroscle-
rosis. For simplicity, we reduce the system to a monotone system and provide a biological
interpretation for the stability analysis according to endothelial functionality. The existence of
solutions of traveling waves type are as well investigated along with numerical simulations. The
results obtained are in good agreement with current biological knowledge. Likewise, they con-
firm and generalize results of mathematical models previously performed in literature. Then,
we study the non monotone reduced model and prove the existence of perturbed solutions and
perturbed waves, particularly in the bistable case. Finally, we extend the study by considering
the complete model proposed initially, perform numerical simulations and provide more spe-
cific results. We examine the consistency between the reduced and complete model analysis
for a certain range of parameters, we elaborate bifurcation diagrams showing the evolution of
inflammation upon endothelial permeability and LDL accumulation and we consider the effect
of anti-inflammatory process on the system behavior. The study of the model shows that the
regulation of atherosclerosis progression is mediated by anti-inflammatory responses that, up to
certain extent, lead to plaque regression.
Stokes Equations In An Infinite Strip With a Hole And transmission
Conditions
Olivier Bodart, olivier.bodart@univ-st-etienne.fr
Institut Camille Jordan, Université Jean Monnet, France
Let 0 < ai < bi < li, i = 1, 2 and S = (0, l1) × (0, l2), S = (a1, b1) × (a2, b2). Let also
Y ⊂ S×] − 1, 1[ be a convex open set with smooth boundary ∂Y . Let Λ be the infinite vertical
domain in R3 defined by
Λ = S×] − ∞, +∞[ \ Y .
We define the following subsets in Λ:
Λ− = y = (y , y3) ∈ R3 ; y ∈ S , y3 < −1 ,
Ω= S×] − ∞, +∞[ \ Y ,
Λ+ = y = (y , y3) ∈ R3 ; y ∈ S , y3 > 1 ,
Γ− = y = (y , y3) ∈ R3 ; y ∈ S , y3 = −1 ,
Γ− = y = (y , y3) ∈ R3 ; y ∈ S , y3 = 1 ,
where we denoted y = (y1, y2). Then we can decompose Λ as follows:
Λ = Λ− ∪ Ω ∪ Λ+.
504
BIOLOGY AND MEDICINE (MS-80)
Mathematical Modeling of Inflammatory Processes of Atherosclerosis
Ghada Abi Younes, abiyounes@math.univ-lyon1.fr
Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
Coauthors: Nader El Khatib, Vitaly Volpert
Atherosclerosis is a chronic disease which involves the build up of cholesterol and fatty deposits
within the inner lining of the artery. It is associated with a progressive thickening and hardening
of the arterial wall that result in narrowing of the vessel lumen and restriction of blood flow to
vital organs. These events may cause heart attack or stroke, the commonest causes of death
worldwide. We study the early stages of atherosclerosis via a mathematical model of partial
differential equations of reaction-diffusion type. The model includes several key species and
identifies endothelial hyperpermeability, believed to be a precursor on the onset of atheroscle-
rosis. For simplicity, we reduce the system to a monotone system and provide a biological
interpretation for the stability analysis according to endothelial functionality. The existence of
solutions of traveling waves type are as well investigated along with numerical simulations. The
results obtained are in good agreement with current biological knowledge. Likewise, they con-
firm and generalize results of mathematical models previously performed in literature. Then,
we study the non monotone reduced model and prove the existence of perturbed solutions and
perturbed waves, particularly in the bistable case. Finally, we extend the study by considering
the complete model proposed initially, perform numerical simulations and provide more spe-
cific results. We examine the consistency between the reduced and complete model analysis
for a certain range of parameters, we elaborate bifurcation diagrams showing the evolution of
inflammation upon endothelial permeability and LDL accumulation and we consider the effect
of anti-inflammatory process on the system behavior. The study of the model shows that the
regulation of atherosclerosis progression is mediated by anti-inflammatory responses that, up to
certain extent, lead to plaque regression.
Stokes Equations In An Infinite Strip With a Hole And transmission
Conditions
Olivier Bodart, olivier.bodart@univ-st-etienne.fr
Institut Camille Jordan, Université Jean Monnet, France
Let 0 < ai < bi < li, i = 1, 2 and S = (0, l1) × (0, l2), S = (a1, b1) × (a2, b2). Let also
Y ⊂ S×] − 1, 1[ be a convex open set with smooth boundary ∂Y . Let Λ be the infinite vertical
domain in R3 defined by
Λ = S×] − ∞, +∞[ \ Y .
We define the following subsets in Λ:
Λ− = y = (y , y3) ∈ R3 ; y ∈ S , y3 < −1 ,
Ω= S×] − ∞, +∞[ \ Y ,
Λ+ = y = (y , y3) ∈ R3 ; y ∈ S , y3 > 1 ,
Γ− = y = (y , y3) ∈ R3 ; y ∈ S , y3 = −1 ,
Γ− = y = (y , y3) ∈ R3 ; y ∈ S , y3 = 1 ,
where we denoted y = (y1, y2). Then we can decompose Λ as follows:
Λ = Λ− ∪ Ω ∪ Λ+.
504
