Effects of Ageing on Aortic Circulation During Atrial Fibrillation; a Numerical Study on Different Aortic Morphologies

Atrial fibrillation (AF) can alter intra-cardiac flow and cardiac output that subsequently affects aortic flow circulation. These changes may become more significant where they occur concomitantly with ageing. Aortic ageing is accompanied with morphological changes such as dilation, lengthening, and arch unfolding. While the recognition of AF mechanism has been the subject of numerous studies, less focus has been devoted to the aortic circulation during the AF and there is a lack of such investigation at different ages. The current work aims to address the present gap. First, we analyse aortic flow distribution in three configurations, which attribute to young, middle and old people, using geometries constructed via clinical data. We then introduce two transient inlet flow conditions representative of key AF-associated defects. Results demonstrate that both AF and ageing negatively affect flow circulation. The main consequence of concomitant occurrence is enhancement of endothelial cell activation potential (ECAP) throughout the vascular domain, mainly at aortic arch and descending thoracic aorta, which is consistent with some clinical observations. The outcome of the current study suggests that AF exacerbates the vascular defects occurred due to the ageing, which increases the possibility of cardiovascular diseases per se. Supplementary Information The online version contains supplementary material available at (10.1007/s10439-021-02744-9).

To understand the geometrical changes of aorta as a result of ageing, a comprehensive survey was carried out on healthy cases without any overt cardiovascular disease. The survey was focused to collect age-associated changes of the main conduit of aorta and the supra-aortic trunk (SAT) configuration.  Table 1 displays a summary of the main body of the articles from which the related geometrical specifications of aorta were picked. Table 2 represents the percentage of normal SAT separate origin of brachiocephalic artery (BCA), left common carotid artery (LCCA) and left subclavian artery (LSCA) from different ethnicities, genders, and regions across the world. The summary emphasises the prevalence of the SAT standard type worldwide. One of the main difficulties in collecting geometrical data of aorta from different sources is an inconsistency in naming of different parts and thus the associated dimensions. This issue becomes more challenging when the curvilinear or rectilinear length is reported. Figure 1 exhibits the curvilinear length between defined cross sections reported by different sources.
The reported values are minimum and maximum of the average values for various age groups, irrespective of the gender and ethnicity. In fact, the minimum values are associated to the younger people, and the maximum values refer to the older groups. The majority of studies were performed on people in an age range between 20 and 80. In this study, to associate the geometries to different age groups, the collected data are classified as shown in Figure 1 in the main text. Thereafter, starting from the minimum average values for each segment, and based on the mean values of growth per decade of life, new datasets are estimated. Initially, the groups are categorised based on the decade of life as 20-30, 30-40, 40-50, 50-60, 60-70, and 70-80. Then the geometries are constructed for the young, middle age and old groups, who fall between 20-30, 40-60, and above 70, respectively.

Three-element Windkessel (RCR WK) model parameter estimation
To mimic the downstream flow, the RCR WK model is employed. Then, to set a proper lumped-3D coupling. Equation (1) defines the RCR model through a first order ordinary differential equation: (1) (1), Rp and Rd are proximal and distal resistances, respectively, and C is the capacitance of each branch; furthermore, P(t) and Q(t) are the outlet pressure and flow rate, respectively.
To tune the parameter for different age groups, a set of systolic-diastolic pressures was opted for the young, middle age and old groups [29]. The mean pressures for the young, middle age and old groups are 93.33, 101.11, and 108.85 mmHg respectively. Furthermore, considering an identical value for the mean flow rate of different ages 79.2 mL/s the total resistance was estimated from Eq. (2) as follows: In which mean arterial pressure and flow rates were obtained through Eq. (3) and (4): Thereafter, to control a normal blood perfusion through different branches, using the percentage values used by [30] 1.4% to the LCA, 3.6% to the RCA, 7.5% to each branch of SAT, and 65% to the DTA the total resistance for each branch is calculated as follows: To obtain proximal and distal resistance for each branch, they can be estimated by knowing that a proximal resistance (Rp) is around 0.09Ri and a distal resistance (Rd) is about 0.91Ri. The next step is to find the total capacitance. It has been proven that for the afterload condition, two-element Windkessel model predicts pressure appropriately [31], therefore using the two-element Windkessel model, during the diastole, the pressure can be estimated as: (6) Employing two time points, one just at the end of systole ( ) and another one at the end of diastole, just before the start of systole ( ), the total capacitance can be obtained through the following equation: (7) In which, P0 and P1 are the pressures attributed to and , respectively. In a normal pressure waveform, P0 is around 90% of systolic pressure. Once the total capacitance is found, the capacitance of each branch can be estimated through Eq. (8):

RCR WK implementation
Eq. (1) was discretised implicitly, using first order backward Euler method as follows: In Eq. (9), denotes timestep size, while n and n-1 superscripts define two consecutive time points at the current and previous moments, respectively. Thereafter, the discretised model was implemented by writing several User Defined Functions (UDF). The codes were scripted in FORTRAN programming language environment by employing relevant macros for CFX Expression Language (CEL). In particular, eight UDFs were written, one as the master code, which was controlling the seven other codes for each branch. The algorithm of the coding is shown below:

Element size and mesh independency
To mesh the computational domain, ANSYS-Meshing toolbox used to obtain a reliable grid network. Given the average inlet velocity, diameter, density and dynamic viscosity and Y + = 10, 0.1 of mean aortic diameter was chosen as the total thickness of the prism layer. To have a consistent mesh for different geometries of age classifications, identical element sizes were used for each region as shown below: In Table 3 meshes. Furthermore, as presented in Table 4, the prism layer thickness for each region and different age categories are defined. Therefore, as a result between 3.2 and 4.5 million cells were obtained for the young, middle age and old groups, respectively. The result independence from the chosen grid networks are displayed in Figure 5 for seven cross sections.

Results flow waveforms along the aorta
Since different age classifications have different geometries, therefore, it is worthy to compare how flow develops from the aortic root towards the descending thoracic aorta. Figure 6 is depicted to show the flow waveforms along the aorta for the normal, LVSD, and HFF. Despite all the morphological differences of aorta at different ages, it is observed that negligible variations can be seen in different phases of a cardiac cycle.