## 1 Introduction  
Out-of-hospital cardiac arrest is a significant medical emergency, affecting more than 350,000 individuals annually in the United States. Despite extensive advancements in emergency care, the survival rates for these events remain low: less than 23% of patients who experience out-of-hospital cardiac arrest survive long enough to be admitted to the hospital, and only about 10% ultimately survive to hospital discharge [1]. Among those who do survive, brain injury remains a leading cause of morbidity and mortality, with over two-thirds of survivors enduring irreversible neurological damage during their hospital stay. These drive outcomes highlight the urgent need for a deeper understanding of the physiological changes in the brain during and after cardiac arrest. Recent reviews of clinical trials registered on ClinicalTrials.gov highlight that most of the out-of-hospital cardiac arrest (OHCA) studies focus on interventional procedures, devices, and drugs, with a growing emphasis on neurological monitoring strategies [2]. The brain's vulnerability to oxygen deprivation during cardiac arrest makes it particularly susceptible to ischemic injury [3]. Any disruption in oxygen delivery can lead to severe consequences, including secondary brain injury in conditions such as ischemia, severe traumatic brain injury, and sub-arachnoid hemorrhage [4]. Post-cardiac arrest brain injury is the leading cause of mortality in patients revived from cardiac arrest and the primary cause of long-term disability in survivors [5]. Cardiac arrest (CA) and cardiopulmonary resuscitation (CPR) introduce significant fluctuations in brain oxygen metabolism, posing a severe challenge to preserving cerebral function during these critical events. Brain function relies heavily on the precise balance between its metabolic needs, the adequate delivery of oxygen and nutrients, and the efficient removal of cellular waste. Achieving this balance necessitates the constant regulation of the cerebral blood flow (_CBF_) [6]. The rapid and precise restoration of the _CBF_ and oxygenation is crucial for preserving brain function, yet the complex interactions between cerebral perfusion, oxygen metabolism, and neuronal survival during these critical moments are not fully understood. Monitoring cerebral perfusion parameters--such as the cerebral tissue oxygen saturation, the cerebral blood volume, the _CBF_, the cerebral metabolic rate of oxygen (_CMRO_2), and redox cytochrome c oxidase (_rCCO_) activity--is essential for improving patient outcomes. These metrics offer vital insights into the brain's oxygen metabolism, a process critical for maintaining cerebral health, especially under the stress conditions of cardiac arrest. Understanding the relationship between the _CMRO_2, mitochondrial activity, and oxygen delivery is crucial for assessing cerebral health and function. Mitochondrial activity, particularly involving _rCCO_, plays a key role in oxygen consumption within the electron transport chain, with its oxidation state influenced by factors such as the oxygen availability and electron flow [4; 7; 8; 9]. This understanding provides insight into neurovascular coupling and how these processes are altered in pathological conditions. It is a key step in neurocritical care, as it can guide interventions aimed at preserving brain function and reducing neurological damage. Advancements in measurement techniques have enhanced our ability to quantify _rCCO_, with hyperspectral near-infrared spectroscopy (hNIRS) providing detailed in-sights into its activity. This advancement is critical for understanding how _rCCO_ activity correlates with oxygen delivery and consumption. However, not everyone has access to these sorts of technologies. To overcome this issue, mathematical models play a huge role. Mathematical models have been employed to study the interactions among _CBF_, _CMRO_2, and _rCCO_, providing a framework to understand these relationships more comprehensively.