Modeling global cerebral ischemia and reperfusion injury
Massachusetts College of Liberal Arts
Sudden cardiac arrest accounts for approximately 500,000 deaths in the United States each year, with associated mortality largely mediated by global cerebral ischemia, a reduction or complete cessation of blood flow to the brain. Within seconds of cardiac arrest, inadequate delivery of blood oxygen and glucose initiates the ischemic cascade, a series of reactions that can lead to irreversible brain damage and ultimately death. Paradoxically, this process is worsened by the delivery of oxygenated blood, known as reperfusion injury. While much progress has been made on mathematical models of cerebral blood flow and tissue hypoxia in the context of focal ischemia (stroke), the relative complexity of global cerebral ischemia and reperfusion injury has hindered progress. This report attempts to summarize the state of mathematical modeling work on cerebral blood flow, brain ischemia, and reperfusion injury. Possible future directions for such research are offered within the context of underlying pathophysiology as well as evolving therapeutic technologies.
Mathematical models help fill the cardiac arrest research funding gap
Sudden cardiac arrest is a major public health problem, representing one of the leading causes of death and disability in the United States. According the American Heart Association (AHA), death certificate data suggest that a minimum of approximately 1 of every 7.4 Americans will die of sudden cardiac arrest each year . Furthermore, the AHA estimates that over 350,000 people suffered an out-of-hospital cardiac arrest (OHCA), and over 200,000 people suffered in-hospital cardiac arrest (IHCA) in the United States in 2015. Survival rates are low, with survival to hospital discharge at 10.8% and 25.8% for OHCA and IHCA respectively; most survivors suffer severe physiological and psychological impairment .
Yet, National Institutes of Health (NIH) funding for cardiac arrest research is relatively low and has actually declined over the past decade. Per annual death, the NIH invests approximately 90 for cardiac arrest despite the fact that both result in similar numbers of deaths and disability-adjusted life years (the sum of the number of years of life lost due to premature mortality and years lived with disability) [3, 4]; see Fig. 1. The disproportionate availability of funding limits progress in translational research, and there are few clinical trials relative to other leading causes of death. Faced with these limitations, mathematical models of relevant hemodynamics, cerebral blood flow, and attendant ischemia are of critical importance in helping to advance understanding of cardiac arrest, potentially paving the way for new life-saving therapies.
The pathophysiology of global cerebral ischemia and reperfusion injury
Although the causes of sudden cardiac arrest are multifactorial, its lethality ultimately derives from the lack of blood flow to the brain when the heart ceases to pump effectively. This lack of perfusion (the flow of blood to tissue) causes ischemia (a restriction in adequate blood supply). The brain is especially sensitive to such ischemia since it has relatively high metabolic demands yet lacks energy storage. Cardiac arrest results in global cerebral ischemia (cessation or extreme reduction of blood flow to the whole brain), whereas stroke results in focal cerebral ischemia (cessation or reduction of circulation in a particular region of the brain). Unless reversed, global cerebral ischemia ultimately leads to irreversible, fatal brain damage.
The pathophysiology of cerebral ischemia involves the initiation of the ischemic cascade, a process that unfolds in both global and focal cerebral ischemia, possibly as little as 20 seconds after cessation of blood flow to brain tissue . The ischemic cascade consists of a number of biochemical events that occur simultaneously and non-sequentially within ischemic tissue (“cascade” is a misnomer), including acidosis, inflammation, neuronal excitotoxicity, oxidative stress, and mitochondrial dysfunction leading to both necrosis (uncontrolled cell death initiated by factors external to the cell) and apoptosis (“programmed” cell death, initiated by intrinsic or extrinsic cellular factors, ostensibly to enhance survival of surrounding tissue and/or the whole organism). This process occurs alongside distinct morphological changes to the cerebral vasculature including narrowing of capillaries, increased vascular permeability, hyperviscosity, as well as decreased deformability and increased aggregation of red blood cells [7, 8, 9].
While the ischemic cascade unfolds over the course of hours, additional damage processes are rapidly initiated when oxygenated blood is reperfused to the ischemic tissue, collectively referred to as reperfusion injury. This damage is thought to be mediated by a sudden and drastic increase in the levels of free-radical species upon re-oxygenation. Neutrophils, white blood cells that are normally part of the innate immune system, are also believed to play a role in reperfusion injury . Neutrophils mediate what would otherwise be healthy inflammatory responses at the site of injured tissue; during ischemia reperfusion injury, hydrogen peroxide generated by neutrophils exacerbates oxidative damage, destroying endothelial cells that comprise the inner surface of the blood vessel lumen. Neutrophils may also play a major role in the no-reflow phenomenon, causing the blockage of a considerable portion of the cerebral microvasculature .
The no-reflow phenomenon refers to the observation that prolonged ischemia prevents restoration of normal blood flow, even when vessel obstructions are relieved . In addition to uncontrolled accumulation of neutrophils, previously mentioned microvascular and hemorheological changes also underlie the no-reflow phenomenon. These changes are associated with both vasogenic and cytotoxic edema of neurons, glia, and endothelial cells, further worsening the no-reflow phenomenon [13, 14, 15].
Because the aforementioned pathophysiological processes are common to both stroke-associated focal cerebral ischemia and cardiac arrest-associated global cerebral ischemia (including reperfusion injury and no-reflow), mathematical models developed within the context of stroke are typically also applicable to global cerebral ischemia. Indeed, while most existing models of cerebral blood flow were developed to study stroke, findings obtained are adaptable to global cerebral ischemia. Even in cases where models are not scalable to the whole brain, data may be relevant to identifying particular regions especially vulnerable to damage and/or where neuroprotection should be prioritized following cardiac arrest, as well as ischemia-mediating factors that may be common to multiple regions.
2. The Navier-Stokes equations
The Navier-Stokes equations comprise the adaptation of Newton’s second law of motion (2.1) to fluid flow. What follows is a brief derivation of the Navier-Stokes equation applicable to an incompressible, isotropic Newtonian fluid, i.e. one in which viscosity remains constant across the fluid, and shear stress is proportional to the flow velocity gradient as related by (2.2), where is the shear tensor, is the shear viscosity of the fluid (a constant), and is the fluid flow velocity gradient.
By dividing the familiar form of Newton’s second law (2.1) by volume, we obtain (2.3), where is fluid density, is the sum of the surface forces (pressure and viscous forces) and body forces (e.g. gravity, electromagnetic, and centrifugal forces), and is the fluid flow velocity (as mentioned above).
Fluid flow velocity
is a vector field (2.4), and by the chain rule we obtain (2.5).
The right-hand side of (2.5) can be rewritten using the gradient of fluid flow velocity,
By the Euler equations for the flow of an incompressible fluid, the sum of the forces can be represented by (2.7), where
represents the body forces per unit volume.
Substituting (2.2) for ,
we obtain (2.8), one form of the Navier-Stokes equation for an incompressible Newtonian fluid, a nonlinear partial differential equation. In order to consider non-Newtonian fluids (in which viscosity is not constant), any of a number of viscosity models must be used (discussed in section 4).
Note that the conservation of mass is accounted for by the volume continuity equation (2.9).
3. Applications of Newtonian blood flow models
Mathematical models employing the Navier-Stokes equations via Computational Fluid Dynamics (CFD) commonly consider blood to be a Newtonian fluid; this assumption yields results consistent with physiological observations for large blood vessels such as the aorta and/or when the shear rate is above 100 s-1 . That is, blood flowing through relatively large diameter vessels with low wall shear stress (“the tangential drag force produced by blood moving across the endothelial surface” ) can be typically considered Newtonian, while blood flowing through small diameter vessels with high wall shear stress must be considered non-Newtonian.
The average density of blood is similar to that of pure water, approximately 1 g/mL. Although the number 1.060 g/mL is commonly cited, Vitello et al. (2015) recently determined that the average density of blood under normal physiologic conditions is 0.994 g/mL 0.032 g . When blood is modeled as a Newtonian fluid, the shear viscosity constant is taken to be [19, 20]. In a Newtonian fluid, shear stress is directly proportional to shear rate, by (10).
There is a considerable body of literature on the differences between Newtonian (i.e. viscosity is constant) and non-Newtonian fluid (i.e. viscosity is affected by shear stress) assumptions in the modeling of cerebral blood flow in the context of intracranial aneurysms (the rupture of which is the most common cause of hemorrhagic stroke). In the case of intracranial aneurysms, wall shear stress is a major contributing factor in eventual rupture. Recent findings using these models suggest that only viscosity-limited non-Newtonian models accurately predict blood flow and wall shear stress under conditions of high shear rate  such as those likely to be encountered in the microvascular stenosis (narrowing) and plugging characteristic of cerebral circulation in the post-cardiac arrest period.
During the peri-arrest period, resuscitative efforts include chest compressions, electrical cardioversion, and (increasingly) extracorporeal membrane oxygenation (ECMO). Attempts to adequately perfuse the brain during the onset of the ischemic cascade are especially dependent on blood flow through the common carotid and vertebral arteries (from which most of the cerebral blood supply is obtained).
Figure 3: CA = Carotid artery, SCA = Stenosed carotid artery. Newtonian blood flow model of the effects of stenosis on wall shear stress .
Rahman et al. (2017) used a Newtonian fluid model of blood flow to investigate the effects of carotid artery stenosis on wall shear stress during pulsatile flow . While their Newtonian model was intended to represent carotid stenosis in the context of ischemic stroke, their findings may be applicable to artery stenosis during global cerebral ischemia and pulsatile flow via resuscitative chest compressions. As can be seen in Fig. 3, considerable wall shear stress is encountered in the stenotic carotid artery during peak systole, which could be considered analogous to the peak arterial pressure generated during cardiopulmonary resuscitation.
4. Non-Newtonian blood flow models
Newtonian models generally don’t account for the rheological and material properties of blood, characteristics that are of critical importance in situations where the shear rate is low and blood exhibits non-Newtonian behavior (e.g. red blood cell deformability, and hematocrit, the blood volume percentage of red blood cells). Such situations are likely to be encountered during global cerebral ischemia in the peri and post-arrest period, as in the narrowing and plugging of the cerebral microvasculature (e.g. by aggregation of red blood cells and accumulation of neutrophils) that underlies the no-reflow phenomenon. Indeed, when the shear rate is sufficiently low, red blood cells aggregate into stacks called Rouleaux.
The viscosity of blood decreases as the shear rate increases, a non-Newtonian behavior called shear thinning (Fig. 4). At the low shear rates encountered during increased microvascular resistance (characteristic of global cerebral ischemia), aggregation of RBCs and Rouleaux formation can lead to considerable increases in viscosity. The behavior of blood under such conditions is strictly non-Newtonian. A number of non-Newtonian fluid models are used to account for the rheological properties of blood (Fig. 4).
Power-Law (Ostwald–de Waele) model
As the name suggests, this model relates shear stress to shear rate by a power law, with n as the flow behavior index ( indicates that blood is considered a shear thinning or pseudoplastic fluid) and K is the consistency coefficient . While power-law fluid models are popular due to their simplicity, they predict infinite viscosity as shear rate approaches zero, and zero viscosity as shear rate approaches infinity (see Fig. 5b).
The Sisko model produces fluid behavior similar to the Power-Law model at very low shear rates. This model is especially useful since it can predict the behavior of both Newtonian and non-Newtonian fluids .
Like the Sisko model, the Carreau also describes the behavior of both Newtonian (at low shear rate) and non-Newtonian (at high shear rate) fluids; this is governed by the relationship between shear rate and the reciprocal of the relaxation time constant, .
This complex model, which combines features of shear thinning and the Bingham plastic fluid model (e.g. toothpaste and mayonnaise can be considered Bingham plastics), requires parameters consistency coefficient , the power law index n, and the yield shear stress .
Other models commonly used to predict the behavior of blood include the Casson model and Quemada model. Turkeri et al. (2011) compared the predictions of Newtonian, Carreau, Casson, and Generalized Power-Law models using an anatomic model derived from CT (computer tomography) data. They found considerable differences between the models at low velocities; the Carreau and Generalized Power-Law models generated greater viscosity at lower shear rate while the Casson model predicted the lowest wall shear stress of the non-Newtonian models [25, 26].
Kuke et al. (2001) found that blood viscosity declines during one hour of global cerebral ischemia (23% lower than normal under low shear rates), but increases rapidly during reperfusion (41% higher than normal under low shear rates), only returning to normal 12 hours after reperfusion; similar patterns were seen for hematocrit and shear thinning parameters . These findings offer valuable guidance for future fluid modeling in the context of relatively scarce extant literature on the use of models in the context of global cerebral ischemia and reperfusion.
A modified Quemada model was employed by Sriram et al. (2014) for determining the effects of the rheological properties of blood on wall shear stress in the microcirculation, a context highly relevant to global cerebral ischemia and reperfusion . Blood was modeled as an inhomogeneous non-Newtonian fluid with a central core of red blood cells and a peripheral cell-free layer. Their model (validated against previously published data) allows for predictions of velocity, cell-free layer thickness, and wall shear stress under such conditions.
5. Modeling cerebrovascular hemodynamics
Another approach to mathematical models of cerebral blood flow involves modeling of hemodynamics, that is, cerebrovascular regulation in the context of tissue hypoxia. In what is arguably the most complex model so far described here, Ursino et al. describe a mathematical model of cerebral blood flow, focusing on the effects of oxygen-sensitive regulation of the large and small pial arteries . The pial vasculature includes the arteries and veins embedded within the pia mater (and subarachnoid space), the meningeal layer in direct contact with the surface of the brain. The regulation of blood flow in pial arteries is critical; the uninterrupted flow of oxygenated blood through these arteries and arterioles can mean the difference between life and death.
An update to previous work, more than a dozen parameters were added (along with additional equations). Among these new parameters, the authors included the carrying capacity and solubility of oxygen in the blood. An especially important modifiable parameter is oxygen saturation (SO2 or SaO2). The authors chose to use oxygen saturation in cerebral venous blood on the basis of research showing an indirect effect of SO2 of cerebral venous blood on the partial pressure of O2 in brain tissue.
The model successfully reproduced some experimental responses involving vasodilation and vasoconstriction (i.e. in cats) and changes in PaO2 and PaCO2 (partial pressures of oxygen and carbon dioxide respectively). The model was also able to illuminate the relationship between hypoxia and intracranial pressure. The use of these simulations may shed light on the relationship between blood viscosity and cerebral blood flow rise during hemodilution, a decrease in concentration of blood cells and other solutes due to an increase in fluid, a situation that underlies ischemia reperfusion injury and the no-reflow phenomenon.
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