# Using Coherent Hemodynamic Spectroscopy Model to Investigate Cardiac Arrest
## 4 Discussion  
We developed a novel algorithm for the analysis of the effects of the baseline fluctuations, large-scale blood flow modulation, and cardiac arrest in the brain. Our algorithm uses the non-linear CHS hemodynamic model previously developed in [1], to which we added the analytical solutions to the partial differential equations for the capillary and venule blood oxygen saturations, and the MATLAB-based algorithm for the inverse problem solution to find the physiological parameters of the model. While the previous implementations of the CHS model used solutions limited to small [19] or periodic [2; 20] perturbations at fixed values of the parameters, using the full analytical solutions with time-dependent arterial oxygen saturation and capillary oxygen diffusion rate allows for the applications of our algorithm to the full-range scenarios, such as cardiac arrest, when _CBF_ changes by 100% from the baseline value to zero. Using analytical solutions provides an advantage over the numeric solution in terms of the algorithm speed and accuracy, which is important for real-time applications. Our numeric simulation results show that our algorithm provides reasonable predictions for the responses of the hemodynamic variables to _CBF_ and _CBV_ changes. Our results on the application of our algorithm to the real data measured during the animal cardiac arrest experiment show similarity with the results of the numeric simulation and the feasibility of our method to measure hemodynamic parameters. Using our model, we found that both in simulated and real experiments, the capillary oxygen saturation can drop to zero % during cardiac arrest, while the tissue oxygen saturation measured by NIRS drops roughly by 50%. This finding provides additional important physiological information on the levels of brain tissue hypoxia during CA. Another notable finding is the quantitative discrepancy between the venule blood oxygen saturation values and time courses predicated by CHS and Equation (10). This equation has often been used by researchers [10] because it allows for estimating _CMRO\({}_{2}\)_ without measuring or calculating \(\langle S^{(v)}\rangle(t)\) using the CHS model. However, as we have shown in this study, the overestimation of \(\langle S^{(v)}\rangle(t)\) using Equation (10) leads to the underestimation of _CMRO\({}_{2}\)_ using [PERSON]'s principle Equation (9). The values of the physiological parameters obtained using our algorithm (Table 2) mostly fall into their ranges taken from other studies [1; 2; 3]. In particular, our absolute microvascular _CBF_ baseline values around 40 mL/100 g/min (Figure 3b), which were estimated though the model parameter _cBFf_ (capillary blood flow velocity factor, mm/s), were in agreement with the absolute _CBF_ values measured by the calibrated diffuse correlation spectroscopy (DCS) in piglets [10]. The same range of _CBF_ values is considered to be typical in healthy humans [21]. Our algorithm can recover not only the fixed values of the physiological parameters, but also their changes, which is achieved using the sweeping inverse problem solution over 30 s intervals. While arterial oxygen saturation at normal or baseline conditions can be measured by pulse oximetry, it cannot work reliably when the pulse is unstable, absent, or weak, which happens during ventricular fibrillations, CA, and CPR [22]. Our method allows for estimating \(S^{(a)}(t)\) without relying on the heartbeat. The changes in the arterial oxygen saturation shown in Figure 4b correspond to the \(S^{(a)}(t)\) values obtained from the blood samples. Since the capillary oxygen diffusion rate depends on the plasma, tissue, and hemoglobin oxygen concentrations according to the equation [1; 2]  
\[\alpha_{O}~{}\sim~{}(\left[O_{2}\right]^{plasma}-\left[O_{2}\right]^{tissue})/[ HbO_{2}], \tag{11}\]  
the increase in \(\alpha_{O}\) during and after CA shown in Figure 4c can be due to the critical decrease in the capillary \([HbO_{2}]\).