Blood flow and oxygen delivery to human brain during functional activity: Theoretical modeling and experimental data

Coupling of cerebral blood flow (CBF) and cerebral metabolic rate for oxygen (CMRO2) in physiologically activated brain states remains the subject of debates. Recently it was suggested that CBF is tightly coupled to oxidative metabolism in a nonlinear fashion. As part of this hypothesis, mathematical models of oxygen delivery to the brain have been described in which disproportionately large increases in CBF are necessary to sustain even small increases in CMRO2 during activation. We have explored the coupling of CBF and oxygen delivery by using two complementary methods. First, a more complex mathematical model was tested that differs from those recently described in that no assumptions were made regarding tissue oxygen level. Second, [15O] water CBF positron emission tomography (PET) studies in nine healthy subjects were conducted during states of visual activation and hypoxia to examine the relationship of CBF and oxygen delivery. In contrast to previous reports, our model showed adequate tissue levels of oxygen could be maintained without the need for increased CBF or oxygen delivery. Similarly, the PET studies demonstrated that the regional increase in CBF during visual activation was not affected by hypoxia. These findings strongly indicate that the increase in CBF associated with physiological activation is regulated by factors other than local requirements in oxygen.

The problem of the spreading of a liquid film along a solid surface: A new mathematical formulation

A new mathematical model is proposed for the spreading of a liquid film on a solid surface. The model is based on the standard lubrication approximation for gently sloping films (with the no-slip condition for the fluid at the solid surface) in the major part of the film where it is not too thin. In the remaining and relatively small regions near the contact lines it is assumed that the so-called autonomy principle holds—i.e., given the material components, the external conditions, and the velocity of the contact lines along the surface, the behavior of the fluid is identical for all films. The resulting mathematical model is formulated as a free boundary problem for the classical fourth-order equation for the film thickness. A class of self-similar solutions to this free boundary problem is considered.

A model of excitation and adaptation in bacterial chemotaxis

Bacterial chemotaxis is widely studied because of its accessibility and because it incorporates processes that are important in a number of sensory systems: signal transduction, excitation, adaptation, and a change in behavior, all in response to stimuli. Quantitative data on the change in behavior are available for this system, and the major biochemical steps in the signal transduction/processing pathway have been identified. We have incorporated recent biochemical data into a mathematical model that can reproduce many of the major features of the intracellular response, including the change in the level of chemotactic proteins to step and ramp stimuli such as those used in experimental protocols. The interaction of the chemotactic proteins with the motor is not modeled, but we can estimate the degree of cooperativity needed to produce the observed gain under the assumption that the chemotactic proteins interact directly with the motor proteins.

Self-similar intermediate asymptotics for a degenerate parabolic filtration-absorption equation

The equation ∂tu = u∂xx2u − (c − 1)(∂xu)2 is known in literature as a qualitative mathematical model of some biological phenomena. Here this equation is derived as a model of the groundwater flow in a water-absorbing fissurized porous rock; therefore, we refer to this equation as a filtration-absorption equation. A family of self-similar solutions to this equation is constructed. Numerical investigation of the evolution of non-self-similar solutions to the Cauchy problems having compactly supported initial conditions is performed. Numerical experiments indicate that the self-similar solutions obtained represent intermediate asymptotics of a wider class of solutions when the influence of details of the initial conditions disappears but the solution is still far from the ultimate state: identical zero. An open problem caused by the nonuniqueness of the solution of the Cauchy problem is discussed.

The mechanical properties of Saccharomyces cerevisiae

Cell-wall mechanical properties play an integral part in the growth and form of Saccharomyces cerevisiae. In contrast to the tremendous knowledge on the genetics of S. cerevisiae, almost nothing is known about its mechanical properties. We have developed a micromanipulation technique to measure the force required to burst single cells and have recently established a mathematical model to extract the mechanical properties of the cell wall from such data. Here we determine the average surface modulus of the S. cerevisiae cell wall to be 11.1 ± 0.6 N/m and 12.9 ± 0.7 N/m in exponential and stationary phases, respectively, giving corresponding Young's moduli of 112 ± 6 MPa and 107 ± 6 MPa. This result demonstrates that yeast cell populations strengthen as they enter stationary phase by increasing wall thickness and hence the surface modulus, without altering the average elastic properties of the cell-wall material. We also determined the average breaking strain of the cell wall to be 82% ± 3% in exponential phase and 80% ± 3% in stationary phase. This finding provides a failure criterion that can be used to predict when applied stresses (e.g., because of fluid flow) will lead to wall rupture. This work analyzes yeast compression experiments in different growth phases by using engineering methodology.

A model for amplification of hair-bundle motion by cyclical binding of Ca2+ to mechanoelectrical-transduction channels

Amplification of auditory stimuli by hair cells augments the sensitivity of the vertebrate inner ear. Cell-body contractions of outer hair cells are thought to mediate amplification in the mammalian cochlea. In vertebrates that lack these cells, and perhaps in mammals as well, active movements of hair bundles may underlie amplification. We have evaluated a mathematical model in which amplification stems from the activity of mechanoelectrical-transduction channels. The intracellular binding of Ca2+ to channels is posited to promote their closure, which increases the tension in gating springs and exerts a negative force on the hair bundle. By enhancing bundle motion, this force partially compensates for viscous damping by cochlear fluids. Linear stability analysis of a six-state kinetic model reveals Hopf bifurcations for parameter values in the physiological range. These bifurcations signal conditions under which the system’s behavior changes from a damped oscillatory response to spontaneous limit-cycle oscillation. By varying the number of stereocilia in a bundle and the rate constant for Ca2+ binding, we calculate bifurcation frequencies spanning the observed range of auditory sensitivity for a representative receptor organ, the chicken’s cochlea. Simulations using prebifurcation parameter values demonstrate frequency-selective amplification with a striking compressive nonlinearity. Because transduction channels occur universally in hair cells, this active-channel model describes a mechanism of auditory amplification potentially applicable across species and hair-cell types.

Increased tricarboxylic acid cycle flux in rat brain during forepaw stimulation detected with 1H[13C]NMR.

NMR spectroscopy was used to test recent proposals that the additional energy required for brain activation is provided through nonoxidative glycolysis. Using localized NMR spectroscopic methods, the rate of C4-glutamate isotopic turnover from infused [1-(13)C]glucose was measured in the somatosensory cortex of rat brain both at rest and during forepaw stimulation. Analysis of the glutamate turnover data using a mathematical model of cerebral glucose metabolism showed that the tricarboxylic acid cycle flux [(V(TCA)] increased from 0.49 +/- 0.03 at rest to 1.48 +/- 0.82 micromol/g/min during stimulation (P < 0.01). The minimum fraction of C4-glutamate derived from C1-glucose was approximately 75%, and this fraction was found in both the resting and stimulated rats. Hence, the percentage increase in oxidative cerebral metabolic rate of glucose use (CMRglc) equals the percentage increases in V(TCA) and cerebral metabolic rate of oxygen consumption (CMRO2). Comparison with previous work for the same rat model, which measured total CMRglc [Ueki, M., Linn, F. & Hossman, K. A. (1988) J. Cereb. Blood Flow Metab. 8, 486-4941, indicates that oxidative CMRglc supplies the majority of energy during sustained brain activation.