CARD-01 Contributed Talks

Alexander Ginsberg

The University of Utah, Department of Mathematics

"A predictive propensity measure to enter REM sleep"

During sleep periods, most mammals alternate multiple times between rapid-eye-movement (REM) sleep and non-REM (NREM) sleep. A common theory proposes that these transitions are governed by an ``hourglass-like'' homeostatic need to enter REM sleep that accumulates during the inter-REM interval and partially discharges during REM sleep. However, markers or mechanisms for REM homeostatic pressure remain undetermined. Recently, an analysis of sleep in mice demonstrated that the cumulative distribution function (CDF) of the amount of NREM sleep between REM bouts correlates with REM bout duration, suggesting that time in NREM sleep influences REM sleep need. Here, we build on those results and construct a predictive measure for the propensity to enter REM sleep as a function of time in NREM sleep since the previous REM episode. The REM propensity measure is precisely defined as the probability to enter REM sleep before the accumulation of an additional pre-specified amount of NREM sleep. Analyzing spontaneous sleep in mice, we find that, as NREM sleep accumulates between REM bouts, the REM propensity exhibits a peak value and then decays to zero with further NREM accumulation. We show that the REM propensity at REM onset predicts features of the subsequent REM bout under certain conditions. Specifically, during the light phase and for REM propensities occurring before the peak in propensity, the REM propensity at REM onset is correlated with REM bout duration, and with the probability of the occurrence of a short REM cycle called a sequential REM cycle. Further, we also find that proportionally more REM sleep occurs during sequential REM cycles, supporting a correlation between high values of our REM propensity measure and high REM sleep need. These results support the theory that a homeostatic need to enter REM sleep accrues during NREM sleep, but only for a limited range of NREM sleep accumulation. Time permitting, we will discuss current research directions.

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Jared Barber

IU Indianapolis

"Mathematical model of blood flow in the brain after a major arterial occlusion."

Blood vessel adaptation plays an important role in maintaining healthy and well-oxygenated tissue throughout the body. This is especially true for the brain. To better characterize how blood flow changes when the brain suffers a major arterial occlusion (e.g. during a stroke) and to identify major factors that may affect flow restoration to downstream regions, we created a mathematical model of blood flow in the brain. The network is modeled as multiple larger vessels interconnected with multiple compartments of smaller vessels with each compartment consisting of identical vessels situated in parallel. The model further includes vessel adaptation in response to changes in pressure (myogenic response), wall shear stress (shear response), and oxygen saturation (metabolic response). By varying tissue oxygen demand and incoming pressure, we are able to identify that the number of collateral vessels moving flow from unobstructed to obstructed regions is a major factor. We also predicted a loss of normal function particularly reflected by a shift in the “autoregulation curve”, a curve that reflects the ability of vessels to reasonably respond to increases in pressure. Such results were consistent with experiment and reinforce the appropriateness of treatments that raise flow and oxygenation by minimizing tissue oxygen demand and raising vascular pressure.

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Cory Brunson

University of Florida

"Testing hypotheses of glomerular capillary development with geometric and topological data analysis"

Blood filtration occurs in renal capillary tufts called glomeruli, the internal structure of which bears on questions of function, development, evolution, and pathology. Due to the low resolution and labor-intensity of imaging technology, only a handful of studies reaching back decades have examined the spatial structure of glomerular capillaries. Several common features have been described, including lobular topology, plausibly associated with robustness to vascular damage, and circuitous geometry, hypothesized to ensure consistent filtration. However, these properties have been neither mathematically defined nor statistically confirmed. Recent developments in serial scanning electron microscopy and virtual reality enabled us to reconstruct the capillary networks of twelve murine glomeruli and trace spatial graph models. We used circuit analysis to represent these as Reeb graphs, the fundamental theorem of calculus to describe a mean trajectory and its curvature, and principal components analysis to reveal lateral and transverse symmetry. Separately, we built a non-spatial random graph growth model based on two mechanisms, angiogenesis and intussusception, which provided evidence that both contribute to development. We then introduced several topological measures of lobularity and found, surprisingly, that empirical glomeruli tend to be less lobular than those generated by our model. Ongoing work focuses on simulation-based attack tolerance and the development of a spatial growth model.

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Brendan Fry

Metropolitan State University of Denver

"Modeling the effects of vascular impairments on blood flow autoregulation in the retinal microcirculation"

The retinal microcirculation supplies blood and oxygen to the cells responsible for vision, and vascular impairments – including compromised flow regulation, reduced capillary density, and elevated intraocular pressure – are involved in the progression of eye diseases such as glaucoma. Here, an established theoretical model of a retinal microvascular network will be presented and extended to investigate the effects of these impairments on retinal blood flow and oxygenation as intraluminal pressure is varied. A heterogeneous description of the arterioles based on confocal microscopy images is combined with a compartmental representation of the downstream capillaries and venules. A Green’s function method is used to simulate oxygen transport in the arterioles, and a Krogh cylinder model is used in the capillary and venular compartments. Acute blood flow autoregulation is simulated in response to changes in pressure, shear stress, and metabolism. The model predicts that impaired flow regulation mechanisms, decreased capillary density, and increased intraocular pressure all cause a loss in the autoregulation plateau over the baseline range of intraluminal pressures (meaning that blood flow is not maintained constant over those pressures), leading to a corresponding decrease in oxygenation in that range. Small impairments in capillary density or intraocular pressure are predicted to mostly be offset by functional flow regulation; however, larger changes and/or combinations of vascular impairments lead to a significant decrease in oxygenation. Clinically, since poor retinal tissue oxygenation could lead to vision loss in advanced glaucoma, model results suggest early identification of vascular changes to prevent these impairments from progressing.

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