Research Article: A mathematical model of aortic aneurysm formation

Date Published: February 17, 2017

Publisher: Public Library of Science

Author(s): Wenrui Hao, Shihua Gong, Shuonan Wu, Jinchao Xu, Michael R. Go, Avner Friedman, Dai Zhu, Luigi F. Rodella.

http://doi.org/10.1371/journal.pone.0170807

Abstract

Abdominal aortic aneurysm (AAA) is a localized enlargement of the abdominal aorta, such that the diameter exceeds 3 cm. The natural history of AAA is progressive growth leading to rupture, an event that carries up to 90% risk of mortality. Hence there is a need to predict the growth of the diameter of the aorta based on the diameter of a patient’s aneurysm at initial screening and aided by non-invasive biomarkers. IL-6 is overexpressed in AAA and was suggested as a prognostic marker for the risk in AAA. The present paper develops a mathematical model which relates the growth of the abdominal aorta to the serum concentration of IL-6. Given the initial diameter of the aorta and the serum concentration of IL-6, the model predicts the growth of the diameter at subsequent times. Such a prediction can provide guidance to how closely the patient’s abdominal aorta should be monitored. The mathematical model is represented by a system of partial differential equations taking place in the aortic wall, where the media is assumed to have the constituency of an hyperelastic material.

Partial Text

AAA is an abnormal dilatation most commonly of the infrarenal aorta. One definition of AAA is a diameter greater than 3 cm [1]. The clinical significance of AAA stems from the high mortality associated with rupture. Approximately 60% of patients with ruptured AAA die before reaching the hospital [2] and mortality rates of emergency surgical repair are as high as 35–70% [3]. The risk of rupture increases with AAA diameter. The pathogenesis of AAA is largely unknown and likely multifactorial. Diameter remains the only clinically useful and available marker for risk of rupture. Surgical repair is recommended for aneurysms measuring greater than 5.5 cm [4, 5] and there is insufficient evidence to recommend surgery for all patients with smaller AAA [2]. However, even elective surgical repair of AAA remains a major operation, incurring significant morbidity and mortality particularly in older, sicker patients where there is a higher prevalence of AAA. Furthermore, patients die every year from rupture of aneurysms smaller than 5.5 cm, and up to 60% of AAA larger than 5 cm remain stable [6]. Thus, some patients with smaller AAA are denied lifesaving surgery and others with larger AAA undergo unnecessary major surgery. The reasons why some smaller aneurysms go on to rupture while some larger ones remain stable are not understood. Techniques that provide early identification of small AAA with increased risk for rupture and large AAA with low risk for rupture will improve overall mortality by prompting personalized treatment plans for AAA.

In this section we develop a mathematical model of aneurysm based on the diagram shown in Fig 2. The model, represented by a system of PDEs, includes the variables listed in Table 1. We assume that all cells are moving with a common velocity v; the velocity is the result of movement of macrophages, T cells and SMCs into the media and adventitia. We also assume that all species are diffusing with appropriate diffusion coefficients. The equation for each species of cells, X, has a form
dXdt+∇·(vX)-DXΔX=FX,
where the expression on the left-hand side includes advection and diffusion, and FX accounts for various biochemical reactions, and chemotaxis. The equations for the chemical species are the same as for cells, but without the advection term, which is relatively very small compared to their large diffusion coefficients. Fig 3 shows a 2D projection of a blood vessel ΩB with an aortic aneurysm ΩM ∪ ΩA.

We set Ω = ΩM ∪ ΓM ∪ ΩA and refer to this region, briefly, as the aortic wall. Aortic wall is assumed to have nonlinear, hyperelastic material properties [27, 41, 42]. Specifically, it is an ‘almost’ incompressible, homogeneous, and isotropic material with energy density function W of the form
W=β1(IB-3)+β2(IB-3)2,(14)
where β1 and β2 are forces that represent elastic coefficients; IB is the first invariant of the Left Cauchy-Green tensor B (namely IB = tr(B)), B = FFT, while F is the deformation gradient tensor. In terms of the displacement vector u = (ui), we can write B=[(I-∇u)T(I-∇u)]-1=(δij-∂uj∂xi-∂ui∂xj+∂uk∂xi∂uk∂xj)-1≐(aij)-1, where I is the identity tensor.

Fig 4 shows the profile of average of cell densities and cytokines concentrations in the first 500 days, and Fig 5 shows how the aneurysm deforms at day 500.

Abdominal aortic aneurysm is a localized enlargement of the abdominal aorta, which may lead to rupture of the aorta. The disease is asymptomatic until rupture, which is nearly always fatal. The risk of rupture associated with the increased diameter of the aorta varies among people. Hence it is difficult to determine, on the basis of just the current diameter (R0) of a patient’s aortic wall, how closely this patient’s abdominal aorta should be monitored. In order to make such a determination, we need address the following questions:
how fast the diameter will grow over time?how does the risk of rupture depend on the growing diameter?

All the parameter values are listed in Tables 2 and 3. Most parameters are taken from the literature while some are estimated below. A few remaining parameters are fitted.

 

Source:

http://doi.org/10.1371/journal.pone.0170807

 

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