# Using Coherent Hemodynamic Spectroscopy Model to Investigate Cardiac Arrest
## Appendix A  
To calculate the oxygen saturation of blood in capillaries \(\langle S^{(c)}\rangle(t)\), we used the damped unidirectional wave equation [2], which describes the evolution of \(S^{(c)}(x,t)\) along the capillary over both space (\(x\)) and time (\(t\)):  
\[c^{(c)}(t)\ \frac{\partial S^{(c)}(x,t)}{\partial x}+\frac{\partial S^{(c)}(x,t)}{\partial t}=-\alpha_{O}S^{(c)}(x,t)\] (A1)  
where \(c^{(c)}(t)\) is the time-dependent blood velocity in capillaries and \(\alpha_{O}(t)\) is the time-dependent oxygen diffusion rate.  
The boundary condition is as follows:  
\[S^{(c)}(0,t)=S^{(a)}(t),\] (A2)  
where \(S^{(a)}(t)\) is the time-dependent arterial oxygen saturation.  
The initial condition is  
\[S^{(c)}(x,0)=S^{(a)}(t)(0)\exp(-\alpha_{O}(0)x/c^{(c)}(0)),\] (A3)which is the steady-state solution of Equation (A1). We used Laplace transform to find the solution of Equation (A1) with the conditions (A2) and (A3), which is  
\[\begin{split} S^{(c)}(x,t)=\\ S^{(a)}(t)\big{(}b_{c}^{-1}(b_{c}(t)-x)\big{)}e^{-\alpha_{0}(t)+ \widetilde{a}(b(t)-x)}+e^{-\alpha_{0}(t)}D(x-b(t)\cdot h(x-b(t))\,,\end{split}\] (A4)  
where \(h(\xi)=0(\xi<0)\), \(h(\xi)=1(\xi>0)\), \(\alpha_{0}(t)=\int_{0}^{t}\alpha_{O}(\tau)d\tau\), \(b_{c}(t)=\int_{0}^{t}c^{(c)}(\tau)d\tau\), \(\widetilde{a}(\xi)=a(b^{-1}(\xi))\), \(D(x)=S^{(c)}(0,x)-S^{(a)}(0)\) if \(x>0\) and \(D(x)=0\) if \(x\leqslant 0\).  
\(S^{(c)}(x,t)\), given by Equation (A4), satisfies Equations (A1)-(A3). Using Equation (A4), one can find the average capillary saturation:  
\[\begin{split}\langle S^{(c)}\rangle(t)=\frac{1}{\mathbb{C}}\int_ {0}^{L^{(c)}}\ S^{(c)}(x,t)dx=\\ \frac{c^{-\alpha_{0}(t)}}{L^{(c)}}\Big{[}\int_{0}^{t}\ e^{\alpha_{ 0}(\tau)}\Big{(}u(0,\tau)-u\Big{(}L^{(c)},\tau\Big{)}\Big{)}d\tau\Big{]}+e^{- \alpha_{0}(t)}\Big{\langle}S^{(c)}_{0}\Big{\rangle},\end{split}\] (A5)  
where \(\Big{\langle}S^{(c)}_{0}\Big{\rangle}=\frac{1}{L_{c}}\int_{0}^{L_{c}}\ u(x,0 )dx\).  
The oxygen saturation in venules \(\langle S^{(v)}\rangle(x,t)\) obeys the undamped unidirectional wave equation [2]:  
\[c^{(v)}(t)\ \frac{\partial S^{(v)}(x,t)}{\partial x}+\frac{\partial S^{(v)}(x,t) }{\partial t}=0\] (A6)  
where \(c^{(v)}(t)\) is the venule blood velocity. The difference between Equations (A1) and (A5) is that there is no oxygen diffusion in venules.  
The boundary condition for the Equation (A6) is  
\[S^{(v)}(0,t)=S^{(c)}\Big{(}L^{(c)},t\Big{)}.\] (A7)  
The initial condition is  
\[S^{(v)}(x,0)=S^{(c)}\Big{(}L^{(c)},0\Big{)},\] (A8)  
which is the steady-state solution of Equation (A6).  
Using Laplace transform, one can find the solution of Equation (A6) satisfying the conditions (A7) and (A8):  
\[S^{(v)}(x,t)=S^{(c)}\Big{(}L^{(c)},b_{v}^{-1}((t)-x)\Big{)}+D_{v}(x-b_{v}(t)) h(x-b_{v}(t))\Big{]},\] (A9)  
where \(b_{v}(t)=\int_{0}^{t}\ c^{(v)}(\tau)d\tau\), \(D_{v}(\xi)=S^{(v)}(\xi,0)-S^{(c)}\Big{(}L^{(c)},0\Big{)}\).  
\(S^{(v)}(x,t)\), given by Equation (A9), satisfies Equations (A6)-(A8). Using Equation (A9), the average venule saturation \(\Big{\langle}S^{(v)}\Big{\rangle}(t)=\frac{1}{\mathbb{C}}\int_{0}^{L^{(v)}}\ q(x,t)dx\) is  
\[\langle S^{(v)}\rangle(t)=\frac{1}{L^{(v)}}\int_{0}^{t}\ c^{(v)}(\tau)\Big{(} q\Big{(}L^{(v)},\tau\Big{)}-u\Big{(}L^{(c)},\tau\Big{)}\Big{)}d\tau+\Big{\langle}S^{ (v)}_{0}\Big{\rangle}.\] (A10)