Eugene C. Eckstein

Self-diffusion of particles in shear flow of a suspension

Self-diffusion coefficients were determined experimentally for lateral dispersion of spherical and disk-like particles in linear shear flow of a slurry at very low Reynolds number. Using a concentric-cylinder Couette apparatus, recurrent observations were made of the lateral position of a particular radioactively labelled particle. The self-diffusion coefficient D was calculated by means of random-walk theory, using the ergodic hypothesis. Owing to great experimental difficulties, the calculated values of D are not of high accuracy, but are correct to within a factor of two. In the range 0 D/a 2 ω increases from zero linearly with ϕ up to D/a 2 ω ≅ 0·02 (where ϕ = volumetric concentration of particles, a = particle radius, ω = mean shear rate of suspending fluid). In the range 0·2 D/a 2 ω is not clear because of experimental scatter, but in this range D/a 2 ω ≅ 0·025 to within a factor of two. Within the experimental accuracy, spheres and disks have the same value of D/a 2 ω.

The near-wall excess of platelet-sized particles in blood flow: Its dependence on hematocrit and wall shear rate

Methods involving microscopy were used to obtain concentration profiles of platelet-sized beads during flow through glass channels. Suspensions of fluorescent latex beads (2.38 microns diam) and washed red blood cells were made from an isotonic albumin-dextrose solution. A syringe pump regulated the suspension flow through glass channels, which were either 50 or 100 microns wide; most experiments used a wall shear rate of 1630 sec-1. Via stroboscopic epifluorescence microscopy, photographs were collected on image planes parallel to the channel wall. Profiles of the bead concentration in the narrow channel direction were made by assembling counts of the focused bead images in the photographs. The results showed that a near-wall excess of the beads occurred when the suspension contained a significant fraction of red cells (over 7%). For hematocrits of 15 to 45% (the highest studied), the excess was above five times the concentration in the central region. Experiments with channels of both widths showed the region of excess beads was 5 to 8 micron thick. A series of experiments with 50-micron channels, with a suspension hematocrit of 15%, and with wall shear rates from 50 to 3180 sec-1 showed that near-wall excesses were large only for wall shear rates of 430 sec-1 and above. This work demonstrated the effects of wall shear rate (flow rate) and hematocrit on the number of platelet-sized beads near a surface and hence illustrated physical (rheological) factors that act in blood-surface interaction.

Model of platelet transport in flowing blood with drift and diffusion terms.

A drift term is added to the convective diffusion equation for platelet transport so that situations with near-wall excesses of platelets can be described. The mathematical relationship between the drift and the fully developed, steady-state platelet concentration profile is shown and a functional form of the drift that leads to concentration profiles similar to experimentally determined profiles is provided. The transport equation is numerically integrated to determine concentration profiles in the developing region of a tube flow. With the approximate drift function and typical values of augmented diffusion constant, the calculated concentration profiles have near-wall excesses that mimic experimental results, thus implying the extended equation is a valid description of rheological events. Stochastic differential equations that are equivalent to the convective diffusion transport equation are shown, and simulations with them are used to illustrate the impact of the drift term on platelet concentration profiles during deposition in a tube flow.

Conditions for the occurrence of large near-wall excesses of small particles during blood flow

Prior work showed that the near-wall concentration of platelet-sized latex beads (2.38 microns diam) in flowing blood suspensions can be greater than three times the concentration in the central region of the flow. Similar methods were used to explore the dependence of the near-wall excess (NWE) of beads on the channel height and suspension composition. The bead diameter, suspending fluid viscosity, and red blood cell deformability were varied; the hematocrit was fixed at 15%. Results showed that NWEs greater than or equal to three times the central concentration were associated with shear stress, rather than with strain rate, required red cell deformability, and occurred with bead diameters of 2.2 microns or larger. The amplitude of NWEs observed in the 30- and 50-microns channels changed sharply from small to large as the wall shear rate (WSR) was increased, while those observed in 100-microns channels exhibited a more gradual dependence on WSR.

Fractional telegraph equations

We investigate several aspects of the fractional telegraph equations, in an effort to better understand the anomalous diffusion processes observed in blood flow experiments. In the earlier work Eckstein et al. [Electron. J. Differential Equations Conf. 03 (1999) 39–50], the telegraph equation D 2 u + 2aDu + Au = 0 was used, where D = d/dt ,a nd it was shown that, as t tends to infinity, u is approximated by v ,w here 2aDv + Av = 0; here A =− d 2 /dx 2 on L 2 (R) ,o rA can be a more general nonnegative selfadjoint operator. In this paper the concern is with the fractional telegraph equation E 2 u + 2aEu + Au = 0, where E = D γ and 0

Anatomic and biomechanical properties of human lumbar dura mater

Determinants of dural defects subsequent to deliberate or accidental dural puncture include the equipment, techniques, and the inherent anatomic and biomechanical properties of dura mater. These properties were studied in specimens of human and canine lumbar dura mater in an attempt to delineate the structure of the tissue and to characterize its behavior in biomechanical terms. Human dura had a longitudinal orientation on gross appearance, and was confirmed microscopically to be composed of longitudinal lamella of collagen and elastin fibers. Longitudinal tensile strength and stiffness were greater than transverse tensile strength and stiffness, which is consistent with the duras apparent anatomic structure and functional requirements. Additional biomechanical testing of the dura demonstrated the property of relaxation which is a characteristic of a viscoelastic material. Significant differences were observed between human and canine dural properties, suggesting limited value of this animal model. Integration of these observed anatomic and biomechanical properties of the lumbar dura provides a greater understanding of dural puncture and may explain previous and often confusing clinical and experimental findings.

An experimental study of peristaltic pumping

An apparatus that approximates a two-dimensional, infinite train of peristaltic waves yields measurements of mean flow, of mean pressure rise, and of pressure-time pulses at fixed locations. In addition, visual observations of ‘reflux’ and ‘trapping’, using dyed fluid, are shown. The inertia-free range extends up to a Reynolds number of about 1. In this range, the theory of Shapiro, Jaffrin & Weinberg (1969) is confirmed with respect to mean pressure vs. mean flow, pressure vs. time, reflux, and trapping. The controversy regarding the criterion of material reflux is settled in favour of the Lagrangian time-mean velocity rather than the Eulerian time-mean velocity. Experiments at higher Reynolds numbers show that the second-order expansion theory of Jaffrin (1971) is valid up to a Reynolds number of about 10.

Transient lateral transport of platelet-sized particles in flowing blood suspensions.

Concentration profiles of 2.5 microns latex beads were measured to demonstrate lateral transport of platelet-sized objects in flows of blood suspensions; the flows had equivalent Poiseuille wall shear rates (WSRs) from 250 to 1220 s-1. Each experimental trial began with a steady flow of suspension without beads in a thin-walled capillary tube (219 microns ID; 10.2 microns SD). The tube entrance was then switched to a reservoir containing suspension of equal hematocrit, but with beads, for a short interval of flow at the same WSR. This process established a paraboloidal tongue of labeled suspension with a transient concentration gradient at its surface. The tube and contents were rapidly frozen to fix the suspended particles in flow-determined locations. Segments of frozen tube were collected at distances from the entrance corresponding to 13%, 39%, and 65% of the axial extent of the ideal paraboloidal tongue. Concentration profiles were estimated from distances measured on fluorescence microscope images of cross-cut tube segments. Experiments used tubes either 40 or 50 cm long, suspension hematocrits of 0, 15, or 40%, and bead concentrations in the range of 1.5-2.2 x 10(5)/mm3. Profiles for 0% hematocrit suspension, a dilute, single-component suspension, had features expected in normal diffusive mixing in a flow. Distinctly different profiles and more lateral transport occurred when the suspensions contained red cells; then, all profiles for 13% extent had regions of excess bead concentration near the wall. Suspension flows with 40% hematocrit exhibited the largest amount of lateral transport. A case is made that, to a first approximation, the rate of lateral transport grew linearly with WSR; however, statistical analysis showed that for 40% hematocrit, less lateral transport occurred when the WSR was 250 s-1 or 1220 s-1 than 560 s-1, thus indicating that the rate behavior is more complex.

Localized elasticity measured in epithelial cells migrating at a wound edge using atomic force microscopy

Restoration of lung homeostasis following injury requires efficient wound healing by the epithelium. The mechanisms of lung epithelial wound healing include cell spreading and migration into the wounded area and later cell proliferation. We hypothesized that mechanical properties of cells vary near the wound edge, and this may provide cues to direct cell migration. To investigate this hypothesis, we measured variations in the stiffness of migrating human bronchial epithelial cells (16HBE cells) approximately 2 h after applying a scratch wound. We used atomic force microscopy (AFM) in contact mode to measure the cell stiffness in 1.5-microm square regions at different locations relative to the wound edge. In regions far from the wound edge (>2.75 mm), there was substantial variation in the elastic modulus in specific cellular regions, but the median values measured from multiple fields were consistently lower than 5 kPa. At the wound edge, cell stiffness was significantly lower within the first 5 microm but increased significantly between 10 and 15 microm before decreasing again below the median values away from the wound edge. When cells were infected with an adenovirus expressing a dominant negative form of RhoA, cell stiffness was significantly decreased compared with cells infected with a control adenovirus. In addition, expression of dominant negative RhoA abrogated the peak increase in stiffness near the wound edge. These results suggest that cells near the wound edge undergo localized changes in cellular stiffness that may provide signals for cell spreading and migration.

Matrix remodeling expression in anulus cells subjected to increased compressive load.

Study Design. Mechanobiology study of gene expression changes as a result of compressive overload of anular fibrochondrocytes. Objective. To test hypotheses regarding phenotype shift in genes coding for representative extracellular matrix (ECM) proteins and matrix modulators. Summary of the Background Data. In degenerative disc disease, the transfer of compressive load through the disc shifts largely from the nucleus onto the anulus. In vivo models simulating this condition have shown derangement of the collagenous ultrastructure in the anulus. In vitro models of cultured anulus cells subjected to static compressive stress generally suggest a down-regulation of synthesis. This study evaluated the expression of specific isomers of genes responsible for mechanical viability and metabolism of the disc under cyclic compressive loads. Methods. Fibrochondrocytes were digested from the anuli of 3, 2-week-old pigs, embedded in 1.5% alginate gel, and hydrostatically compressed at 0.5 Hz for 3 hours to amplitudes of 10 and 30 atm. These levels represented nominal load transfer through the healthy disc and high load transfer through the degenerative disc. Ribonucleic acid was isolated, reverse transcribed, and evaluated by real-time polymerase chain reaction for expression of type I (C-I) and type II (C-II) collagen, aggrecan, the matrix metalloproteinase (MMP-1), and the transforming growth factor beta (TGF&bgr;-1). Results were expressed at percentages of uncompressed controls. Results. The lower pressure of 10 atm resulted in up-regulation of all ECM protein genes. C-I and C-II both averaged 141%, and aggrecan 121% of controls (P < 0.05). MMP-1 and TGF&bgr;-1 were essentially unchanged. With the pressure increased to 30 atm, C-II remained approximately at the level expressed under lower pressure, but C-I was reduced to 42% of controls (P < 0.05), indicating a phenotype shift. MMP-1 and TGF&bgr;-1 also were down-regulated to 71% and 54% of controls, respectively (P < 0.05). Conclusions. The up-regulation of the ECM genes with nominal pressure highlights the mechanobiological importance of common activity in fibrocartilage homeostasis. Differential regulation of the 2 primary collagen types with high pressure indicates a capacity of the anulus to remodel according to pathomechanical conditions.