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\title{In-silico calibration of intracranial aneurysm thrombosis models based on clinical data}

\author{Qiongyao Liu, Ali Sarrami-Foroushani, Yongxing Wang, Michael MacRaild, Christopher Kelly, Nishant Ravikumar, Zeike A. Taylor, Tufail Patankar, Toni Lassila, Alejandro F. Frangi}
\date{}


\begin{document}
% \linenumbers
\maketitle




\section{Governing equations}
For our in-silico observational study of spontaneous thrombosis (ST), we model residence time (RT) as a tracer passively transported with the blood flow \citep{sarrami2019computational}. Three-dimensional momentum equations for incompressible and Newtonian fluid, the Navier-Stokes equations, are used to describe blood flow. In addition, we assume the vessel wall is rigid and the no-slip wall boundary condition is imposed. 

\begin{equation}
\rho\frac{\partial \boldsymbol{u} }{\partial t} + \rho(\boldsymbol{u} \cdot\nabla)\boldsymbol{u} =-\nabla p + \mu\nabla^2 \boldsymbol{u} 
\label{NS}
\end{equation}

where $\boldsymbol{u}$ is the velocity vector, $t$ is time and $p$ represents the pressure. Blood flow is regarded as an incompressible and Newtonian fluid with a constant density $\rho$ of \SI{1060}{kg/{m}^3} and viscosity $\mu$ of \SI{0.004}{Pa \cdot s} \citep{villa2011neurist,geers2014approximating}.

RT is modeled as a tracer passively transported with the blood flow by solving the following advection-diffusion-reaction (ADR) equation:

\begin{equation}
\frac{\partial C_{RT}}{\partial t} + \boldsymbol{u} \cdot \nabla C_{RT}= D_{RT}\nabla^2C_{RT} + 1
\label{C_RT}
\end{equation}

Where $t$ is time, $\boldsymbol{u}$ is the velocity vector, $C_{RT}$ is the local concentration of RT, $D_{RT}$ represents the self-diffusivity of the flow ($D_{RT}=1.14 \times 10^{-11} \, m^2 s^{-1}$) \citep{harrison2007application,menichini2016mathematical} and the source term considers a unit increase in the concentration of RT for each unit increase in time \citep{menichini2016mathematical}. Shear rate (SR) was calculated automatically by using an associated built-in variable in ANSYS CFX \citep{sarrami2019computational}. To reduce the uncertainty caused by the unknown boundary conditions of a patient-specific vascular flow, time-dependent, and patient-specific inlet flow waveforms are used in this transient blood flow simulation. This transient blood flow simulation will run for three cardiac cycles and each cardiac cycle is equally discretized into 200 steps. The time-averaged RT and SR will be calculated from the last cardiac cycle. 
% When RT is greater than the RT threshold and SR is smaller than the SR threshold in the aneurysm sac, the thrombosis formation process will be triggered. 



\section{Mesh convergence test}
We used ANSYS ICEM CFD v19.3 (Ansys Inc. Canonsburg, PA, USA) to generate unstructured volumetric meshes. To discretize the computational domain including vascular region and aneurysm sac, tetrahedral elements with a maximum edge size of 0.2 mm and five layers of prismatic elements with a maximum edge size of 0.1 mm were used. \citet{sarrami2019computational} performed the mesh convergence tests based on the inflow rate at the aneurysm neck and the sac-averaged concentrations of the fibrin and platelets, and according to their tests, mesh independence was obtained for the above-mentioned element sizes. In addition to tests by \citet{sarrami2019computational}, we performed a mesh independence test based on the maximum RT in the aneurysm sac. When using a fine mesh (tetrahedral elements with a maximum edge size of 0.1 mm and five layers of prismatic elements with a maximum edge size of 0.1 mm) as the reference in the test, the maximum RT obtained on the above-mentioned element sizes (tetrahedral elements with maximum edge size of 0.2 mm and five layers of prismatic elements with a maximum edge size of 0.1 mm) differed from the maximum RT of the reference mesh by 2.5\%.




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% \section{Tables and Figures}


%=============Aneurysm characteristics=================
\begin{table}[H]
\centering
\small
\caption{\label{all aneurysm}Aneurysm Characteristics.}
%{\begin{tabular}[l]{@{}l|ll}
\begin{threeparttable}
\begin{tabular}{m{7cm}p{7cm}}
\hline
Location & Maximum diameter:d(mm) \\ \hline
 ICA/PCoA – sidewall: 68\% & Small (d $\leq$ 10 mm ): 80\%   \\
MCA/Sylvian – bifurcation: 23\% & Large (10 mm \textless d \textless 25 mm ): 18.2\% \\
  BA – bifurcation: 9\% & Giant (d $\geq$ 25 mm ):1.8\% \\ \hline
  \\
%\tabnote{ICA, internal carotid artery;}\\
%\tabnote{PCoA, posterior communicating artery;}\\
%\tabnote{MCA, middle cerebral artery;}\\
%\tabnote{BA, basilar artery.}\\
%\tabnote{ICA, internal carotid artery; PCoA, posterior communicating artery; MCA, middle cerebral artery; BA, basilar artery.}
\end{tabular}

\begin{tablenotes}
    \footnotesize
    \item ICA, internal carotid artery; PCoA, posterior communicating artery; MCA, middle cerebral artery; BA, basilar artery.
\end{tablenotes}

\end{threeparttable}
\end{table}

%============Large and giant aneurysms===================

\begin{table}[H]
\centering
\caption{\label{LG}Positions of large and giant aneurysms.}
\begin{tabular}{m{3cm}p{1.5cm}p{1.5cm}p{1.5cm}p{1.5cm}p{1.5cm}}
\toprule
Location &ICA & PCoA & BA & MCA &other\\
\midrule
Included group & 7 & 7 & 4 & 4 &0\\
Omitted group & 11 & 4 & 3 & 2 &1 \\
All cases & 18 & 11 & 7& 6&1 \\
\bottomrule
\end{tabular}



\end{table}

%==========included and omitted group============
\renewcommand{\arraystretch}{1.5} %控制行高  
\begin{table}[H]
  
  \centering  
  %\fontsize{6.5}{8}\selectfont  
  \begin{threeparttable}  
  \caption{Age and gender information of included group and omitted group.}  
  \label{all_LG_small}  
    \begin{tabular}{ccccccc}  
    \toprule  
    \multirow{2}{*}{Characteristics}&  
    \multicolumn{2}{c}{ All Cases}&\multicolumn{2}{c}{ Included  Group}&\multicolumn{2}{c}{ Omitted group}\cr  
    \cmidrule(lr){2-3} \cmidrule(lr){4-5}  \cmidrule(lr){6-7}  
      &LG IAs&Small IAs&LG IAs&Small IAs&LG IAs&Small IAs\cr
      \midrule  
    Average Age&51.60&51.16&50.45&51.19&52.73&51.15\cr  
    Female:Male &27 : 16&107 : 52&14 : 8&46 : 21&13 : 8&61 : 31\cr  
    \bottomrule  
    \end{tabular}  
    \end{threeparttable}  
\end{table}  

% %============Scale factors of hypertension ===================

% \begin{table}[H]
% \centering
% \caption{\label{Scale factors} Scale factors for generating hypertensive waveforms.}
% \begin{tabular}{m{3cm}p{1.5cm}p{1.5cm}p{1.5cm}p{1.5cm}p{1.5cm}}
% \toprule
% Scale factors &ICA & PCoA & BA & MCA &other\\
% \midrule
% Included group & 7 & 7 & 4 & 4 &0\\
% Omitted group & 11 & 4 & 3 & 2 &1 \\
% % All cases & 18 & 11 & 7& 6&1 \\
% \bottomrule
% \end{tabular}



% \end{table}



%===============Statistic LG ST incidence=======
\begin{table}[H]
\small
\centering
\caption{\label{statistic ST}Statistical large and giant ST incidence.}
%\begin{tabular}{|m{2cm}|p{2cm}|p{2cm}|p{2cm}||p{2cm}|p{2cm}|p{2cm}|p{2cm}} %
\begin{tabular}{l|l|l|l|l|l|l}
\hline
 Cohort  & Age  &Femal : Male& Total IAs &LG IAs & ST IAs& Reference \\
\hline
 cohort1 & 14-70 (mean. 45)  &59 : 46& 139 &10/139 & 6/10& \citet{schubiger1980computed} \\ \hline
cohort2 & 21-66 (mean. 49) & 8 : 4 &302&22/302&12/22 &\citet{whittle1982spontaneous} \\ \hline
 cohort3  & 15-76 (mean. 52) & 16 : 13 &30 & 15/30 & 5/15& \citet{baumgartner1994transcranial} \\ \hline
 cohort4 &6-81 (mean. 43.7)&31 : 11&46&33/46&11/33& \citet{saatci2003ct} \\ \hline
 cohort5 &5-85 (mean. 68)&66 : 59& 129 &129/129&51/129& \citet{nurminen2014anatomy} \\ \hline
\end{tabular}
\end{table}

%=========== RT and SR in the aneurysm sac of LG cases ======
\begin{table}[H]
\small
\centering
\caption{\label{RT and SR of all LG IAs}Numerical results of Max RT and Min SR in aneurysm sac for LG IAs.}
\begin{tabular}{|m{2cm}|p{2cm}|p{2cm}|p{2cm}|p{2cm}|p{2cm}|p{2cm}} 
%\begin{tabular}{l|l|l|l|l|l}
\hline
 Case  & Max RT (s) &Min SR (\SI{}{s^{-1}}) & Size (mm) &Age &Gender \\
\hline
 1 & 1.56  &13.05& 11.2 &51 &F\\ \hline
 2 & 2.16 & 8.46 & 13.2&52 &F\\ \hline
 3 & 2.47 & 2.71 &17.9 &51 &F \\ \hline
 4 &2.31&6.02&17.3 &66 &M\\ \hline
 5 &0.50&42.08& 15.9 &48 &M\\ \hline
 6 & 2.27  &0.33& 20.6  &51 &F\\ \hline
 7 & 1.10 & 13.31 &15.2 &51 &F\\ \hline
 8  & 2.25 & 0.21 &10.6 &50 &M \\ \hline
 9 &2.83&0.41&11.1 &43 &F\\ \hline
 10 &0.50&19.30& 10.2 &54 & M\\ \hline
 11 & 1.59  &40.19& 11.7  &51 &F\\ \hline
12 & 2.39 & 9.18 &10.4 &51 &F\\ \hline
 13  & 2.29 & 18.74&25.5 &52 &M \\ \hline
 14 &0.86&29.52&10.5 &44 &F\\ \hline
 15 &1.63&30.82& 10.9 &51 &M\\ \hline
 16 & 2.43  &6.21& 20.2  &54 &F\\ \hline
17 & 1.89 & 21.53 &11.5 &44 &F\\ \hline
 18  & 2.37 & 12.51 &10.2 &52 &F \\ \hline
 19 &0.48&69.34&16.0 &51 &F\\ \hline
 20 &2.12&12.85& 11.5 &51 &M\\ \hline 
 21 &1.77&30.64& 11.2 &43 &F\\ \hline
 22 &2.24 &16.71& 12.3 &46 &M\\ \hline
\end{tabular}
\end{table}


%======== Normo V.S. Hyper in ST incidence=====
% \renewcommand{\arraystretch}{1.5} %控制行高  
% \begin{table}[H]

%  \centering  
%   %\fontsize{6.5}{8}\selectfont  
%   \begin{threeparttable}  
%   \caption{ST incidences comparison between normotension and hypertension.}  
%   \label{trigger_mechanisms_ST_incidences}  
%     \begin{tabular}{ccccccc}  
%     \toprule  
%     \multirow{2}{*}{Clinical ST incidence}& 
%     \multicolumn{3}{c}{ normotensive}&\multicolumn{3}{c}{ hypertensive}\cr  
%     \cmidrule(lr){2-4} \cmidrule(lr){5-7}  
%     &RT alone&SR alone &RT and SR&RT alone &SR alone&RT and SR\cr  
%     \midrule  
%     40.67\% &54.55\%&59.09\%&40.91\%&0&40.91\%&0\cr  
%     %Average inlet velocity ($cm\cdot s^{-1}$) & 23.39&23.46&23.40&22.69&22.56&22.96\cr  
%     \bottomrule  
%     \end{tabular}  
%     \end{threeparttable}  
% \end{table}  


%====== case analysis in terms of RT and SR =======
\renewcommand{\arraystretch}{1.5} %控制行高  
\begin{table}[H]
  
  \centering  
  %\fontsize{6.5}{8}\selectfont  
  \begin{threeparttable}  
  \caption{RT and SR in normotension and hypertension.}  
  \label{RT_and_SR_in_rest_and_exercise}  
    \begin{tabular}{ccccccc}  
    \toprule  
    \multirow{2}{*}{Results}& 
    % \multirow{2}{*}{Threshold}&
    \multicolumn{3}{c}{ normotension}&\multicolumn{3}{c}{ hypertension}\cr  
    \cmidrule(lr){2-4} \cmidrule(lr){5-7}  
    &high&mean&low&high&mean&low\cr  
    \midrule  
    Max RT (s)&2.46&2.48&2.47&1.62&1.61&1.61\cr  
    Min SR (\SI{}{s^{-1}})&3.46&2.71&2.52&10.94&10.37&10.21\cr  
    %Average inlet velocity ($cm\cdot s^{-1}$) & 23.39&23.46&23.40&22.69&22.56&22.96\cr  
    \bottomrule  
    \end{tabular}  
    \end{threeparttable}  
\end{table}  


%&&&&&&&&&&&& Figures &&&&&&&&&&&&&&&&&&&&&&&&&

%==========inlet flow waveforms===================
\begin{figure}[H]
    \centerline{\includegraphics[width=10cm]{inlet_flow_waveforms.png}} 
    \caption{Patient-specific inlet flow waveforms.}
    \label{inflow}
\end{figure}

%============ numerical ST incidence =========
% \begin{figure}[H]
%     \centerline{\includegraphics[width=8cm]{Figures/ST_incidence.jpg}} 
%     \caption{Simulation ST incidence.}
%     \label{ST}
% \end{figure}

%========== case shape =============
\begin{figure}[H]
    \centerline{\includegraphics[width=6cm, trim={8cm 0cm 8cm 0cm}]{aneurysm_shape.png}} 
    \caption{Aneurysm shape.}
    \label{shape}
\end{figure}



%======= normo V.S. hyper ========

% \begin{figure}[H]
%   \centering
%   \begin{subfigure}[b]{0.3\textwidth}
%     \includegraphics[width=\textwidth, trim={6cm 0cm 6cm 0cm}]{Figures/high_rest_RT_labeled.jpg}
%     \caption{RT in highFlow normotension}
%     \label{high_rest_RT}
%   \end{subfigure}
% \quad
%   \begin{subfigure}[b]{0.3\textwidth}
%     \includegraphics[width=\textwidth, trim={6cm 0cm 6cm 0cm}]{Figures/mean_rest_RT_labeled.jpg}
%     \caption{RT in meanFlow normotension}
%     \label{mean_rest_RT}
%   \end{subfigure}
%   \quad  
%   \begin{subfigure}[b]{0.3\textwidth}
%     \includegraphics[width=\textwidth, trim={6cm 0cm 6cm 0cm}]{Figures/mean_hyper_RT_labeled.jpg}
%     \caption{RT in meanFlow hypertension}
%     \label{mean_exercise_RT}
%   \end{subfigure}
%   \begin{subfigure}[b]{0.3\textwidth}
%     \includegraphics[width=\textwidth, trim={7cm 0cm 7cm 0cm}]{Figures/high_rest_SR_labeled.jpg}
%     \caption{SR in highFlow normotension}
%     \label{high_rest_SR}
%   \end{subfigure}
% \quad  
% %\quad
%   \begin{subfigure}[b]{0.3\textwidth}
%     \includegraphics[width=\textwidth, trim={7cm 0cm 7cm 0cm}]{Figures/mean_rest_SR_labeled.jpg}
%     \caption{SR in meanFlow normotension}
%     \label{mean_rest_SR}
%   \end{subfigure}
% \quad  
% %\quad
%  \begin{subfigure}[b]{0.3\textwidth}
%     \includegraphics[width=\textwidth, trim={7cm 0cm 7cm 0cm}]{Figures/mean_hyper_SR_labeled.jpg}
%     \caption{SR in meanFlow hypertension}
%     \label{mean_exercise_SR}
%   \end{subfigure}
% \caption{Residence Time (RT) and Shear Rate (SR) distribution in normotension and hypertension.}
% \label{distribution}
% \end{figure}



\end{document}
