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\begin{document}

%%%%%% Appendix %%%%%%%%%%%%%%%%%
\section*{Supporting Information}
\subsection*{The coagulation cascade}\label{sec:appendix}
As mentioned in the main text, we use the coagulation model from Anand \eal \cite{anand2008model}, where both the extrinsic or TF pathway and intrinsic or contact pathway are considered. 
The intrinsic pathway is initiated when XII is activated to XIIa. Subsequently, XIIa activates the zymogen XI to its active enzyme form XIa, which further activates IX to IXa in the presence of $\mathrm{Ca^{2+}}$. The role of intrinsic pathway on the propagation of coagulation under flow conditions is not quite known, but has been included here for the sake of completeness of the biochemical model (with the exception of factor XII). The list of reactants and their normal initial concentration along with their diffusion coefficients in blood plasma are given in \tab{list}.\\

\begin{table}[th]
\caption{List of reactants in the coagulation cascade along with their initial concentrations $c^0_i$ and diffusion constants $D_i$ \cite{anand2008model}.}
\label{tab:list}
\begin{center}
\begin{tabular}{ccp{2.5in}cc}
  \hline\hline
  Index     & Species   & Name                  & $c^0_i \ (nM)$            & $D_i \times10^{-7} \ (cm^2/s)$   \\
  \hline
  1         & IXa       & enzyme IXa             & $90\times10^{-5} $               & $6.25$          \\
  2         & IX        & zymogen IX            & $90 $               & $5.63$          \\
  3         & VIIIa     & enzyme VIIIa           & $0.7\times10^{-5} $               & $3.92$          \\
  4         & VIII      & zymogen VIII          & $0.7 $               & $3.12$          \\
  5         & Va        & enzyme Va              & $20\times10^{-5} $               & $3.82$          \\
  6         & V         & zymogen V             & $20 $               & $3.12$          \\
  7         & Xa        & enzyme Xa              & $170\times10^{-5} $               & $7.37$          \\
  8         & X         & zymogen X             & $170 $               & $5.63$          \\
  9         & IIa       & thrombin              & $1400\times10^{-5} $               & $6.47$          \\
  10         & II       & prothrombin          & $1400 $               & $5.21$          \\
  11        & Ia        & fibrin                & $7000\times10^{-5} $               & $2.47$          \\
  12        & I         & fibrinogen            & $7000 $               & $3.1$          \\
  13        & XIa       & enzyme XIa             & $30\times10^{-5} $               & $5.0$          \\
  14        & XI        & zymogen XI            & $30 $               & $3.97$          \\
  15        & ATIII     & anti-thrombin-III      & $2410 $               & $5.57$          \\
  16        & TFPI      & tissue factor pathway inhibitor  & $2.5 $    & $6.30$          \\
  17        & APC       & active protein C      & $60\times10^{-5} $               & $5.50$          \\
  18        & PC        & protein C             & $60 $               & $5.44$          \\
  19        & $\alpha_1$AT& $\alpha_1$-AntiTrypsin    & $45000 $           & $5.82$          \\
  20        & ADP       & adenosine diphosphate  & $0.0 $       & $25.70$          \\
  21        & Z         & tenase                & $1.125\times10^{-4}$                & $-$ \\
  22        & W         & prothrombinase        & $0.034$                & $-$ \\

  \hline\hline
\end{tabular}
\end{center}
\end{table}

The equations governing the generation and depletion of the species ($S_i$ in Eq.\ (11)) are formulated based on experimental data for the reaction kinetics, and are listed in \tab{reaction}. The kinetic constants, also obtained from experimental data, are given in the table's caption. Further, concentrations of two other chemical species tenase (Z) and prothrombinase (W) are computed through the relations [Z]=[VIIIa][IXa]/$K_{dZ}$ and [W]=[Va][Xa]/$K_{dW}$, respectively \cite{anand2008model}.\\

\begin{table}[tbhp]
\caption{Reaction equations for source terms. Kinetic constants are given as follows: $k_9=11 \ min^{-1}$, $K_{9M}=160 \ nM$, $h_9=0.0162 \ nM^{-1}min^{-1}$, $k_8=194.4 \ min^{-1}$, $K_{8M}=112,000 \ nM$, $h_8=0.222 \ min^{-1}$, $h_{C8}=10.2 \ min^{-1}$, $H_{C8M}=14.6 \ nM$, $k_5=27.0 \ min^{-1}$, $K_{5M}=140.5 \ nM$, $h_{5}=0.17 \ min^{-1}$, $h_{C5}=10.2 \ min^{-1}$, $H_{C5M}=14.6 \ nM$, $k_{10}=2391 \ min^{-1}$, $K_{10M}=160 \ nM$, $h_{10}=0.347 \ nM^{-1}min^{-1}$, $h_{TFPI}=0.48 \ nM^{-1}min^{-1}$, $k_2=1344 \ min^{-1}$, $K_{2M}=1060 \ nM$, $h_2=0.714 \ nM^{-1}min^{-1}$, $k_1=3540 \ min^{-1}$, $K_{1M}=3160 \ nM$, $h_1=1500 \ min^{-1}$, $H_{1M}=250,000 \ nM$, $k_{11}=0.0078 \ min^{-1}$, $K_{11M}=50 \ nM$, $h_{11}^{A3}=1.6\times10^{-3} \ nM^{-1}min^{-1}$, $h_{11}^{L1}=1.3\times10^{-5} \ nM^{-1}min^{-1}$, $k_{PC}=39 \ min^{-1}$, $K_{PCM}=3190 \ nM$, $h_{PC}=6.6\times10^{-7} \ nM^{-1}min^{-1}$, $K_{dZ}=0.56 \ nM$, $K_{dW}=0.1 \ nM$.}
\label{tab:reaction}
\begin{center}
\begin{tabular}{l}
  \hline\hline
  $S_1$ = ($k_9$[XIa][IX])/($K_{9M}$+[IX]) - $h_9$[IXa][ATIII] \\

  $S_2$ = -($k_9$[XIa][IX])/($K_{9M}$+[IX])    \\

  $S_3$ = ($k_8$[IIa][VIII])/($K_{8M}$+[VIII]) - $h_8$[VIIIa] - ($h_{C8}$[APC][VIIIa])/($H_{C8M}$+[VIIIa])     \\

  $S_4$ = ($k_8$[IIa][VIII])/($K_{8M}$+[VIII])     \\

  $S_5$ = ($k_5$[IIa][V])/($K_{5M}$+[V]) - $h_5$[Va] - ($h_{C5}$[APC][Va])/($H_{C5M}$+[Va])     \\

  $S_6$ = -($k_5$[IIa][V])/($K_{5M}$+[V])     \\

  $S_7$ =  ($k_{10}$[Z][X])/($K_{10M}$+[X]) - $h_{10}$[Xa][ATIII] - $h_{TFPI}$[TFPI][Xa]      \\

  $S_8$ =  -($k_{10}$[Z][X])/($K_{10M}$+[X])     \\

  $S_9$ =  ($k_2$[W][II])/($K_{2M}$+[II]) - $h_2$[IIa][ATIII]     \\

  $S_{10}$ =  -($k_2$[W][II])/($K_{2M}$+[II])     \\

  $S_{11}$ =  ($k_1$[IIa][I])/($K_{1M}$+[I])    \\

  $S_{12}$ =  -($k_1$[IIa][I])/($K_{1M}$+[I])     \\

  $S_{13}$ =  ($k_{11}$[IIa][XI])/($K_{11M}$+[XI]) - $h_{11}^{A3}$[XIa][ATIII] - $h_{11}^{L1}$[XIa][$\alpha_1$AT]     \\

  $S_{14}$ =  -($k_{11}$[IIa][XI])/($K_{11M}$+[XI])     \\

  $S_{15}$ =  -($h_{9}$[IXa]+$h_{10}$[Xa] + $h_{2}$[IIa] + $h_{11}^{A3}$[XIa])[ATIII] \\

  $S_{16}$ =  -$h_{TFPI}$[TFPI][Xa]  \\

  $S_{17}$ =  ($k_{PC}$[IIa][PC])/($K_{PCM}$+[PC]) - $h_{PC}$[APC][$\alpha_1$AT]     \\

  $S_{18}$ =  -($k_{PC}$[IIa][PC])/($K_{PCM}$+[PC])    \\

  $S_{19}$ =  -$h_{PC}$[APC][$\alpha_1$AT] - $h_{11}^{L1}$[XIa][$\alpha_1$AT] \\

  $S_{20}$ =  $S_{rel}$ as in \eq{rel} \\
  \hline\hline
\end{tabular}
\end{center}
\end{table}

At the site of injury, we assume that the subendothelium-bound TF-VIIa complex drives the extrinsic pathway of the coagulation cascade through the subendothelium reactions that are represented by Neumann boundary conditions in the form of $-D_j \partial c_j / \partial n = B_j$. Surface reactions $B_j$ along with their kinetic constants are given in \tab{BC}. Further, [TF-VIIa]$^W$ is prescribed at the wall as a decreasing cubic function of fibrin concentration given the fact that fibrin deposition leads to fewer subendothelium binding sites available for the complex. The initial value of the concentration is considered to be in the physiologic range [TF-VIIa]$^0 \approx 1 \ (nM)$ \cite{kuharsky2001surface}.\\

\begin{table}[t]
\caption{Flux boundary conditions imposed at the injured vessel wall. The kinetic constants are given as $k_{7,9}=32.4 \ min^{-1}$, $K_{7,9M}=24 \ nM$, $k_{7,10}=103.0 \ min^{-1}$, $K_{7,10M}=240 \ nM$.}
\label{tab:BC}
\begin{center}
\begin{tabular}{p{0.75in} p{3in}}
  \hline\hline
  Index $j$      & Boundary flux terms $B_j$          \\
  \hline
  1          & ($k_{7,9}$[IX]$^W$[TF-VIIa]$^W$)/($K_{7,9M}$+[IX]$^W$) \\

  2          & -($k_{7,9}$[IX]$^W$[TF-VIIa]$^W$)/($K_{7,9M}$+[IX]$^W$) \\

  7          & ($k_{7,10}$[X]$^W$[TF-VIIa]$^W$)/($K_{7,10M}$+[X]$^W$) \\

  8          & -($k_{7,10}$[X]$^W$[TF-VIIa]$^W$)/($K_{7,10M}$+[X]$^W$) \\
  \hline\hline
\end{tabular}
\end{center}
\end{table}

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\end{document}  





