Biomechanical influence of disk properties on the load transfer of healthy and degenerated disks using a poroelastic finite element model

Spine degeneration is a pathology that will affect 80% of the population. Since the intervertebral disks play an important role in transmitting loads through the spine, the aim of this study was to evaluate the biomechanical impact of disk properties on the load carried by healthy (Thompson grade I) and degenerated (Thompson grades III and IV) disks. A three-dimensional parametric poroelastic finite element model of the L4/L5 motion segment was developed. Grade I, grade II, and grade IV disks were modeled by altering the biomechanical properties of both the annulus and nucleus. Models were validated using published creep experiments, in which a constant compressive axial stress of 0.35 MPa was applied for 4 h. Pore pressure (PP) and effective stress (S(E)) were analyzed as a function of time following loading application (1 min, 5 min, 45 min, 125 min, and 245 min) and discal region along the midsagittal profile for each disk grade. A design of experiments was further implemented to analyze the influence of six disk parameters (disk height (H), fiber proportion (%F), drained Young's modulus (E(a),E(n)), and initial permeability (k(a),k(n)) of both the annulus and nucleus) on load-sharing for disk grades I and IV. Simulations of grade I, grade III, and grade IV disks agreed well with the available published experimental data. Disk height (H) had a significant influence (p<0.05) on the PP and S(E) during the entire loading history for both healthy and degenerated disk models. Young's modulus of the annulus (E(a)) significantly affected not only S(E) in the annular region for both disk grades in the initial creep response but also S(E) in the nucleus zone for degenerated disks with further creep response. The nucleus and annulus permeabilities had a significant influence on the PP distribution for both disk grades, but this effect occurred at earlier stages of loading for degenerated than for healthy disk models. This is the first study that investigates the biomechanical influence of both geometrical and material disk properties on the load transfer of healthy and degenerated disks. Disk height is a significant parameter for both healthy and degenerated disks during the entire loading. Changes in the annulus stiffness, as well as in the annulus and nucleus permeability, control load-sharing in different ways for healthy and degenerated disks.     

Direct perfusion measurements of cancellous bone anisotropic permeability

More extensive characterization of trabecular connectivity and intertrabecular space will be instrumental in understanding disease states and designing engineered bone. This project presents an experimental protocol to define the directional dependence of transport properties as measured from healthy cancellous bone when considered as a biologic, porous medium. In the initial design phases, mature bovine bone was harvested from the femoral neck (n=6 cylinders) and distal condyle (n=4 cubes) regions and used for "proof of concept" experimentation. A power study on those results led to the presented work on 20 cubic samples (mean volume=4.09cm(3)) harvested from a single bovine distal femur. Anisotropic intrinsic permeabilities (k(i)) were quantified along the orthogonal anatomic axes (i=medial-lateral, anterior-posterior, and superior-inferior) from each individual cubic bone sample. Using direct perfusion measurements, permeability was calculated based upon Darcy's Law describing flow through porous media. The maximum mean value was associated with the superior-inferior orientation (4.65x10(-10)m(2)) in comparison with the mean anterior-posterior (4.52x10(-10)m(2)) and medial-lateral (2.33x10(-10)m(2)) direction values. The results demonstrate the anisotropic (p=0.0143) and heterogeneous (p=0.0002) nature of the tissue and encourage the ongoing quantification of parameters within the established poroelastic models.     

Hepatocyte membrane water permeability measured by silicone layer filtering centrifugation

We previously found that hepatocytes are able to control their osmotic membrane water permeability (P(f)) by regulating the number of surface aquaporin water channels. Hepatocyte P(f) has been assessed by phase-contrast microscopy and cell image analysis, an established but relatively laborious procedure. We report here an alternative method to assess hepatocyte P(f) based on a single silicone layer filtering centrifugation system. Isolated rat hepatocytes were incubated in hypotonic or isotonic buffers containing (3)H(2)O as a tracer and, then, were filtered by rapid centrifugation through a silicone layer down to a lysis layer. Osmotically driven radioactivity (i.e., (3)H(2)O) within hepatocytes was calculated as the difference between the dpm in lysis media measured under hypotonic and isotonic conditions. The P(f) calculated from the initial slope of the radioactivity-versus-time curve was 18 microm/s at 4 degrees C. Hepatocytes treated with dibutyryl cyclic AMP, to increase P(f) through the plasma membrane insertion of aquaporins, showed an increased P(f) value of 37 microm/s. The aquaporin blocker dimethyl sulfoxide selectively prevented the agonist-induced hepatocyte P(f). These data are in good agreement with the corresponding values determined by quantitative phase-contrast microscopy; thus, the method developed allows the rapid and reliable measurement of hepatocyte P(f).     

Osmotic water permeability of plasma and vacuolar membranes in protoplasts I. High osmotic water permeability in radish (<Emphasis Type="Italic">Raphanus sativus</Emphasis>) root cells as measured by a new method

Intra- and transcellular water movements in plants are regulated by the water permeability of the plasma membrane (PM) and vacuolar membrane (VM) in plant cells. In the present study, we investigated the osmotic water permeability of both PM (P ( f1)) and VM (P ( f2)), as well as the bulk osmotic water permeability of a protoplast (P ( f(bulk))) isolated from radish (Raphanus sativus) roots. The values of P ( f(bulk)) and P ( f2) were determined from the swelling/shrinking rate of protoplasts and isolated vacuoles under hypo- or hypertonic conditions. In order to minimize the effect of unstirred layer, we monitored dropping or rising protoplasts (vacuoles) in sorbitol solutions as they swelled or shrunk. P ( f1) was calculated from P ( f(bulk)) and P ( f2) by using the 'three-compartment model', which describes the theoretical relationship between P ( f1), P ( f2) and P ( f(bulk)) (Kuwagata and Murai-Hatano in J Plant Res, 2007). The time-dependent changes in the volume of protoplasts and isolated vacuoles fitted well to the theoretical curves, and solute permeation of PM and VM was able to be neglected for measuring the osmotic water permeability. High osmotic water permeability of more than 500 mum s(-1), indicating high activity of aquaporins (water channels), was observed in both PM and VM in radish root cells. This method has the advantage that P ( f1) and P ( f2) can be measured accurately in individual higher plant cells.     

Ab initio prediction of the solution structures and populations of a cyclic pentapeptide in DMSO based on an implicit solvation model

Using a recently developed statistical mechanics methodology, the solution structures and populations of the cyclic pentapeptide cyclo(D-Pro(1)-Ala(2)-Ala(3)-Ala(4)-Ala(5)) in DMSO are obtained ab initio, i.e., without using experimental restraints. An important ingredient of this methodology is a novel optimization of implicit solvation parameters, which in our previous publication [Baysal, C.; Meirovitch, H. J Am Chem Soc 1998, 120, 800-812] has been applied to a cyclic hexapeptide in DMSO. The molecule has been described by the simplified energy function E(tot) = E(GRO) + summation operator(k) sigma(k)A(k), where E(GRO) is the GROMOS force-field energy, sigma(k) and A(k) are the atomic solvation parameter (ASP) and the solvent accessible surface area of atom k. This methodology, which relies on an extensive conformational search, Monte Carlo simulations, and free energy calculations, is applied here with E(tot) based on the ASPs derived in our previous work, and for comparison also with E(GRO) alone. For both models, entropy effects are found to be significant. For E(tot), the theoretical values of proton-proton distances and (3)J coupling constants agree very well with the NMR results [Mierke, D. F.; Kurz, M.; Kessler, H. J Am Chem Soc 1994, 116, 1042-1049], while the results for E(GRO) are significantly worse. This suggests that our ASPs might be transferrable to other cyclic peptides in DMSO as well, making our methodology a reliable tool for an ab initio structure prediction; obviously, if necessary, parts of this methodology can also be incorporated in a best-fit analysis where experimental restraints are used.     

Hyperosmotically induced volume change and calcium signaling in intervertebral disk cells: the role of the actin cytoskeleton

Loading of the spine alters the osmotic environment in the intervertebral disk (IVD) as interstitial water is expressed from the tissue. Cells from the three zones of the IVD, the anulus fibrosus (AF), transition zone (TZ), and nucleus pulposus (NP), respond to osmotic stress with altered biosynthesis through a pathway that may involve calcium (Ca(2+)) as a second messenger. We examined the hypothesis that IVD cells respond to hyperosmotic stress by increasing the concentration of intracellular calcium ([Ca(2+)](i)) through a mechanism involving F-actin. In response to hyperosmotic stress, control cells from all zones decreased in volume and cells from the AF and TZ exhibited [Ca(2+)](i) transients, while cells from the NP did not. Extracellular Ca(2+) was necessary to initiate [Ca(2+)](i) transients. Stabilization of F-actin with phalloidin prevented the Ca(2+) response in AF and TZ cells and decreased the rate of volume change in cells from all zones, coupled with an increase in the elastic moduli and apparent viscosity. Conversely, actin breakdown with cytochalasin D facilitated Ca(2+) signaling while decreasing the elastic moduli and apparent viscosity for NP cells. These results suggest that hyperosmotic stress induces volume change in IVD cells and may initiate [Ca(2+)](i) transients through an actin-dependent mechanism.     

Intervertebral disc creep behavior assessment through an open source finite element solver

Degenerative Disc Disease (DDD) is one of the largest health problems faced worldwide, based on lost working time and associated costs. By means of this motivation, this work aims to evaluate a biomimetic Finite Element (FE) model of the Intervertebral Disc (IVD). Recent studies have emphasized the importance of an accurate biomechanical modeling of the IVD, as it is a highly complex multiphasic medium. Poroelastic models of the disc are mostly implemented in commercial finite element packages with limited access to the algorithms. Therefore, a novel poroelastic formulation implemented on a home-developed open source FE solver is briefly addressed throughout this paper. The combination of this formulation with biphasic osmotic swelling behavior is also taken into account. Numerical simulations were devoted to the analysis of the non-degenerated human lumbar IVD time-dependent behavior. The results of the tests performed for creep assessment were inside the scope of the experimental data, with a remarkable improvement of the numerical accuracy when compared with previously published results obtained with ABAQUS^®. In brief, this in-development open-source FE solver was validated with literature experimental data and aims to be a valuable tool to study the IVD biomechanics and DDD mechanisms.     

Unstirred layer effects in osmotic water flow across gallbladder epithelium

Summary The standard one-dimensional model of the unstirred layer is applied in a re-examination of the experimental results of Wright, Smulders and Tormey (Wright, E.M., Smulders, A.P., Tormey, J. McD., 1972,J. Membrane Biol. 7:198) who reported large transients in the osmotic flux of water from the serosal to the mucosal side of rabbit gallbladder epithelium. They initiated osmosis by the addition of sucrose to the mucosal bathing solution (initially, approximately 300mOsm NaCl) and observed that the initial flux was more than ten times its eventual steady-state value; they interpreted this as a consequence of the piling-up of NaCl in the unstirred tissue layer on the serosal side of the epithelium. The present analysis (both steady-state and unsteady) shows that if measured values of layer thickness are used, together with reasonable values of the reduced diffusivity of NaCl in the tissue and of the fraction of tissue available for water flow, then one would predict a discrepancy of only about 10%, not tenfold, between the initial and final values of the flux. Thus the standard model is inconsistent with the observations. Furthermore, Wright et al's results cannot be used to infer that the osmotic permeability of epithelial cell membranes is much larger than steadystate measurements on whole epithelia would indicate. Mucosal-to-serosal flow is also analyzed, and in this case a considerably greater osmotic permeability is predicted; this result is consistent with the observed changes in structure of the lateral intercellular spaces when the direction of flow is reversed.     

Age-related accumulation of pentosidine in aggrecan and collagen from normal and degenerate human intervertebral discs

During aging and degeneration, many changes occur in the structure and composition of human cartilaginous tissues, which include the accumulation of the AGE (advanced glycation end-product), pentosidine, in long-lived proteins. In the present study, we investigated the accumulation of pentosidine in constituents of the human IVD (intervertebral disc), i.e. collagen, aggrecan-derived PG (proteoglycan) (A1) and its fractions (A1D1-A1D6) in health and pathology. We found that, after maturity, pentosidine accumulates with age. Over the age range studied, a linear 6-fold increase was observed in pentosidine accumulation for A1 and collagen with respective rates of 0.12 and 0.66 nmol x (g of protein)(-1) x year(-1). Using previously reported protein turnover rate constants (k(T)) obtained from measurements of the D-isomer of aspartic residue in collagen and aggrecan of human IVD, we could calculate the pentosidine formation rate constants (k(F)) for these constituents [Sivan, Tsitron, Wachtel, Roughley, Sakkee, van der Ham, DeGroot, Roberts and Maroudas (2006) J. Biol. Chem. 281, 13009-13014; Tsitron (2006) MSc Thesis, Technion-Israel Institute of Technology, Haifa, Israel]. In spite of the comparable formation rate constants obtained for A1D1 and collagen [1.81+/-0.25 compared with 3.71+/-0.26 micromol of pentosidine x (mol of lysine)(-1) x year(-1) respectively], the higher pentosidine accumulation in collagen is consistent with its slower turnover (0.005 year(-1) compared with 0.134 year(-1) for A1D1). Pentosidine accumulation increased with decreasing buoyant density and decreasing turnover of the proteins from the most glycosaminoglycan-rich PG components (A1D1) to the least (A1D6), with respective k(F) values of 1.81+/-0.25 and 3.18+/-0.37 micromol of pentosidine.(mol of lysine)(-1) x year(-1). We concluded that protein turnover is an important determinant of pentosidine accumulation in aggrecan and collagen of human IVD, as was found for articular cartilage. Correlation of pentosidine accumulation with protein half-life in both normal and degenerate discs further supports this finding.     

Structural mechanisms of acute VEGF effect on microvessel permeability

To investigate the ultrastructural mechanisms of acute microvessel hyperpermeability by vascular endothelial growth factor (VEGF), we combined a mathematical model (J Biomech Eng 116: 502-513, 1994) with experimental data of the effect of VEGF on microvessel hydraulic conductivity (L(p)) and permeability of various-sized solutes. We examined the effect of VEGF on microvessel permeability to a small solute (sodium fluorescein, Stokes radius 0.45 nm), an intermediate solute (alpha-lactalbumin, Stokes radius 2.01 nm), and a large solute [albumin (BSA), Stokes radius 3.5 nm]. Exposure to 1 nM VEGF transiently increased apparent permeability to 2.3, 3.3, and 6.2 times their baseline values for sodium fluorescein, alpha-lactalbumin, and BSA, respectively, within 30 s, and all returned to control within 2 min. On the basis of L(p) (DO Bates and FE Curry. Am J Physiol Heart Circ Physiol 271: H2520-H2528, 1996) and permeability data, the prediction from the model suggested that the most likely structural changes in the interendothelial cleft induced by VEGF would be a approximately 2.5-fold increase in its opening width and partial degradation of the surface glycocalyx.     

Inhibition of wild-type and mutant human immunodeficiency virus type 1 proteases by GW0385 and other arylsulfonamides

The arylsulfonamide derivatives described herein were such potent inhibitors of human immunodeficiency virus type 1 (HIV-1) protease (enzyme, E) that values for the inhibition constants (K(i)) could not be determined by conventional steady-state kinetic techniques (i.e., the minimal enzyme concentration usable for the activity assay was much greater than the value of the dissociation constant). Consequently, two alternative methods were developed for estimation of K(i) values. The first method employed kinetic determinations of values for k(1) and k(-1), from which K(i) was determined (k(-1)/k(1)). The second method was a competitive displacement assay used to determine binding affinities of other inhibitors relative to that of GW0385. In these assays, the inhibitor of unknown affinity was used to displace [(3)H]GW0385 from E.[(3)H]GW0385. From the concentration of E.[(3)H]GW0385 at equilibrium, the concentrations of enzyme-bound and free inhibitors were calculated, and the ratio of the K(i) value of the unknown to that of GW0385 was determined (K(i,unknown)/K(i,GW0385)). The values of k(1) were calculated from data in which changes in the intrinsic protein fluorescence of the enzyme associated with inhibitor binding were directly or indirectly monitored. In the case of saquinavir, the fluorescence changes associated with complex formation were large enough to monitor directly. The value of k(1) for saquinavir was 62 +/- 2 microM(-1) s(-1). In the case of GW0385, the fluorescence changes associated with complex formation were too small to monitor directly. Consequently, the value of k(1) was estimated from a competition experiment in which the effect of GW0385 on the binding of E to saquinavir was determined. The value of k(1) for GW0385 was estimated from these experiments to be 137 +/- 4 microM(-1) s(-1). Because E.[(3)H]GW0385 was stable in the standard buffer at room temperature for greater than 33 days, the value of the first-order rate constant for dissociation of E.[(3)H]GW0385 (k(-1)) could be estimated from the time-course for exchange of E.[(3)H]GW0385 with excess unlabeled GW0385. The value of k(-1) calculated from these data was (2.1 +/- 0.1) x10(-6) s(-1) (t(1/2) = 91 h). The K(i) value of wild-type HIV-1 protease for GW0385, calculated from these values for k(1) and k(-1), was 15 +/- 1 fM. Three multidrug resistant enzymes had K(i) values for GW0385 that were less than 5 pM.     

Biomechanical properties of human articular cartilage under compressive loads

The function of articular cartilage is to support and distribute loads and to provide lubrication in the diarthrodial joints. Cartilage function is described by proper mechanical and rheological properties, strain and depth-dependent, which are not completely assessed. Unconfined and confined compression are commonly used to evaluate the Young's modulus (E) and the aggregate modulus (H(A)), respectively. The Poisson's ratio (nu) can be calculated indirectly from the equilibrium compression data, or using the biphasic indentation technique; it has recently been optically evaluated by using video microscopy during unconfined compression. The transient response of articular cartilage during confined compression depends on its permeability k; a constant value of k can be easily identified by a simple analytical model of confined compression tests, whereas more complex models or direct measurements (permeation tests) are needed to study the permeability dependence on deformation. A poroelastic finite element model of articular cartilage was developed for this purpose. The elastic parameters (E,nu) of the model were evaluated performing unconfined compression creep tests on human articular cartilage disks, whereas k was identified from the confined test response. Our combined experimental and computational method can be used to identify the parameters that define the permeability dependence on deformation, as a function of depth from articular surface.     

fMLP-stimulated neutrophils increase endothelial [Ca2+]i and microvessel permeability in the absence of adhesion: role of reactive oxygen species

Our previous study demonstrated that firm attachment of leukocytes to microvessel walls does not necessarily increase microvessel permeability (Am J Physiol Heart Circ Physiol 283: H2420-H2430, 2002). To further understand the mechanisms of the permeability increase associated with leukocyte accumulation during acute inflammation, we investigated the direct relation of reactive oxygen species (ROS) release during neutrophil respiratory burst to changes in microvessel permeability and endothelial intracellular Ca(2+) concentration ([Ca(2+)](i)) in intact microvessels. ROS release from activated neutrophils was quantified by measuring changes in chemiluminescence. When isolated rat neutrophils (2 x 10(6)/ml) were exposed to formyl-Met-Leu-Phe-OH (fMLP, 10 microM), chemiluminescence transiently increased from 1.2 +/- 0.2 x 10(4) to a peak value of 6.7 +/- 1.0 x 10(4) cpm/min (n = 12). Correlatively, perfusing individual microvessels with fMLP-stimulated neutrophils in suspension (2 x 10(7)/ml) increased hydraulic conductivity (L(p)) to 3.7 +/- 0.4 times the control value (n = 5) and increased endothelial [Ca(2+)](i) from 84 +/- 7 nM to a mean peak value of 170 +/- 7 nM. In contrast, perfusing vessels with fMLP alone did not affect basal L(p). Application of antioxidant agents, superoxide dismutase, vitamin C, or an iron chelator, deferoxamine mesylate, attenuated ROS release in fMLP-stimulated neutrophils and abolished increases in L(p). These results indicate that release of ROS from fMLP-stimulated neutrophils increases microvessel permeability and endothelial [Ca(2+)](i) independently from leukocyte adhesion and the migration process.     

Unimpaired osmotic water permeability and fluid secretion in bile duct epithelia of AQP1 null mice

The mechanisms by which fluid moves across the luminal membrane of cholangiocyte epithelia are uncertain. Previous studies suggested that aquaporin-1 (AQP1) is an important determinant of water movement in rat cholangiocytes and that cyclic AMP mediates the movement of these water channels from cytoplasm to apical membrane, thereby increasing the osmotic water permeability. To test this possibility we measured agonist-stimulated fluid secretion and osmotically driven water transport in isolated bile duct units (IBDUs) from AQP1 wild-type (+/+) and null (-/-) mice. AQP1 expression was confirmed in a mouse cholangiocyte cell line and +/+ liver. Forskolin-induced fluid secretion, measured from the kinetics of IBDU luminal expansion, was 0.05 fl/min and was not impaired in -/- mice. Osmotic water permeability (P(f)), measured from the initial rate of IBDU swelling in response to a 70-mosM osmotic gradient, was 11.1 x 10(-4) cm/s in +/+ mice and 11.5 x 10(-4) cm/s in -/- mice. P(f) values increased by approximately 50% in both +/+ and -/- mice following preincubation with forskolin. These findings provide direct evidence that AQP1 is not rate limiting for water movement in mouse cholangiocytes and does not appear to be regulated by cyclic AMP in this species.     

Quantitative investigations on the effects of exposure durations to the combined cryoprotective agents on mouse oocyte vitrification procedures

Vitrification by using two-step exposures to combined cryoprotective agents (CPAs) has become one of the most common methods for oocyte cryopreservation. By quantitatively examining the status of oocytes during CPA additions and dilutions, we can analyze the degree of the associated osmotic damages. The osmotic responses of mouse MII oocyte in the presence of the combined CPAs (ethylene glycol, EG, and dimethyl sulfoxide, DMSO) were recorded and analyzed. A two-parameter model was used in the curve-fitting calculation to determine the values of hydraulic conductivity (L(p)) and permeability (P(s)) to the combined CPAs at 25°C and 37°C. The effects of exposure durations and the exposure temperatures on the cryopreservation in terms of frozen-thawed cell survival rates and subsequent development were examined in a series of cryopreservation experiments. Mouse MII oocytes were exposed to pretreatment solution (PTS) and vitrification solution (VS) at specific temperatures. The PTS used in our experiment was 10% EG and 10% DMSO dissolved in modified PBS (mPBS), and the VS was EDFS30 (15% EG, 15% DMSO, 3 × 10(-3) M Ficoll, and 0.35 M sucrose in mPBS).The accumulative osmotic damage (AOD) and intracellular CPA concentrations were calculated under the different cryopreservation conditions, and for the first time, the quantitative interactions between survival rates, subsequent development rates, and values of AOD were investigated.     

Absence of uterokinetic effects of prostaglandin F 2 alpha on oxytocin-reactive uterus in the mare

In most cyclic females, prostaglandin F(2alpha) (PGF(2alpha)) triggers a uterine motility response resembling that of oxytocin (OT). To determine if PGF(2alpha) is a uterokinetic substance in the cycling mare, uterine motility was measured by intrauterine balloon technique in 12 conscious, normally cyclic mares. After 60 min of saline infusion, continuous intravenous (i.v.) infusion with OT (1 i.u./min) was followed by PGF(2alpha) (200 mug/min) for 60 min each. The experiment was repeated 3 wk later except with PGF(2alpha) preceeding OT. A second group of mares was administered OT (60 i.u.) either i.v., intramuscularly (i.m.), or intrauterinely (i.u.). Plasma samples were studied for progesterone concentration. Control uterine motility for the first group of mares was (mean +/- SEM) 545.83 +/- 45.10 mm(2). Significant (P<0.05) elevation in uterine motility was recorded for OT (1118.60 +/- 70.56 mm(2)) regardless if PGF(2alpha) preceded OT infusion or vice-versa. No significant difference (P>0.05) was seen in motility after PGF(2alpha) (423.33 +/- 31.12 mm(2)) infusion. The uterokinetic effect of OT was greatest when OT was administered i.v. (1696.50 +/- 195.46 mm(2)) followed by i.m. (819.82 +/- 39.96 mm(2)), and it was least effective when administered i.u. (607.83 +/- 21.56 mm(2)) as compared to control uterine motility (279.78 +/- 22.33 mm(2)). Skin electrical resistance values rose from 0 to 2000 ohms with PGF(2alpha) infusion (but not with OT), indicating that PGF(2alpha) was bioactive. It was concluded that PGF(2alpha) was not a uterokinetic substance in the cyclic mare.     

A model for predicting growth responses in plants to changes in external water potential: Zea mays primary roots

In response to osmotic step changes, three distinct phases have been noted in the growth response of Zea mays primary roots. They are cessation or slowing of growth over a period of 15–20 minutes, tissue contraction, and a damped oscillatory return to nearly normal growth rate, all within a period of about one hour. A system model of the tissue response is presented to explain such behavior and to serve in a predictive capacity to govern future experiments.It is supposed that for turgor pressure in excess of a cell wall yield threshold, plastic flow is the major component of wall deformation, and that when turgor falls below yield threshold, elastic deformation is dominant. The equations of the model describe growth rate as a function of time in terms of the following properties; plastic flow, elastic deformation, permeability to water, and solute uptake. They are derived from basic equations of feedback interactions between internal osmotic pressure and growth rate, and between wall softening, turgor and growth rate.The model predicts oscillatory growth rate regulation, and phase and amplitude relationships between turgor pressure and growth rate. The simplest model which accounts for all observations is that of biphasic deformation, two modes of wall softening, and a dual feedback system involving osmotic and yield threshold control of growth rate.It should be noted that to predict the time course of turgor pressure, osmotic pressure, yield pressure, and growth rate, two initial conditions and six system parameter values are sufficient. So far only the initial values of growth rate and its derivative can be obtained for Zea mays primary roots. However, values for wall softening and hardening coefficients (including the strain and turgor independent component), plastic extensibility, water permeability and dilution rate coefficients have not been obtained as yet for Zea roots. Values for some of these parameters have been obtained for other roots, coleoptiles, and giant algal cells.Lest the reader despair, it should be pointed out that experimental observations coupled with simulation studies will help establish restricted ranges of values that the system parameters might assume. These can then be compared with known values in the literature and values experimentally obtained in the future.     

Experimental and numerical identification of cortical bone permeability

Bone is a complex system, and could be modeled as a poroelastic media. The aim of this paper is to identify the macroscopic value of the cortical bone permeability coefficient. A simple experimental method was designed in order to determine the permeability coefficient. Two bone samples taken from different ox femurs were filled with water, to place them under internal pressure. The measurements gave both the fluid flow through the lateral surfaces and the internal pressure. The originality of this work is the coupling between an experimental process and a structural computation performed with a finite element method. The mean cortical bone permeability coefficient identified was about k=1.1x10(-13)m(2). This value tends to confirm other values found in the literature, obtained by different methods and often at macroscopic scale. It confirms also the domination of vascular permeability (Haversian and Volkmann's canals).     

Influence of experimental protocols on the mechanical properties of the intervertebral disc in unconfined compression

The lack of standardization in experimental protocols for unconfined compression tests of intervertebral discs (IVD) tissues is a major issue in the quantification of their mechanical properties. Our hypothesis is that the experimental protocols influence the mechanical properties of both annulus fibrosus and nucleus pulposus. IVD extracted from bovine tails were tested in unconfined compression stress-relaxation experiments according to six different protocols, where for each protocol, the initial swelling of the samples and the applied preload were different. The Young's modulus was calculated from a viscoelastic model, and the permeability from a linear biphasic poroviscoelastic model. Important differences were observed in the prediction of the mechanical properties of the IVD according to the initial experimental conditions, in agreement with our hypothesis. The protocol including an initial swelling, a 5% strain preload, and a 5% strain ramp is the most relevant protocol to test the annulus fibrosus in unconfined compression, and provides a permeability of 5.0 ± 4.2e(-14)m(4)/N[middle dot]s and a Young's modulus of 7.6 ± 4.7 kPa. The protocol with semi confined swelling and a 5% strain ramp is the most relevant protocol for the nucleus pulposus and provides a permeability of 10.7 ± 3.1 e(-14)m(4)/N[middle dot]s and a Young's modulus of 6.0 ± 2.5 kPa.