Critical time scales for advection–diffusion–reaction processes

The concept of local accumulation time (LAT) was introduced by Berezhkovskii and coworkers in 2010–2011 to give a finite measure of the time required for the transient solution of a reaction–diffusion equation to approach the steady–state solution (Biophys J. 99, L59 (2010); Phys Rev E. 83, 051906 (2011)). Such a measure is referred to as a critical time. Here, we show that LAT is, in fact, identical to the concept of mean action time (MAT) that was first introduced by McNabb in 1991 (IMA J Appl Math. 47, 193 (1991)). Although McNabb’s initial argument was motivated by considering the mean particle lifetime (MPLT) for a linear death process, he applied the ideas to study diffusion. We extend the work of these authors by deriving expressions for the MAT for a general one–dimensional linear advection–diffusion–reaction problem. Using a combination of continuum and discrete approaches, we show that MAT and MPLT are equivalent for certain uniform–to-uniform transitions; these results provide a practical interpretation for MAT, by directly linking the stochastic microscopic processes to a meaningful macroscopic timescale. We find that for more general transitions, the equivalence between MAT and MPLT does not hold. Unlike other critical time definitions, we show that it is possible to evaluate the MAT without solving the underlying partial differential equation (pde). This makes MAT a simple and attractive quantity for practical situations. Finally, our work explores the accuracy of certain approximations derived using the MAT, showing that useful approximations for nonlinear kinetic processes can be obtained, again without treating the governing pde directly.

Investigating the change in three dimensional deformity for idiopathic scoliosis using axially loaded MRI

Background: Adolescent idiopathic scoliosis is a complex three-dimensional deformity, involving a lateral deformity in the coronal plane and axial rotation of the vertebrae in the transverse plane. Gravitational loading plays an important biomechanical role in governing the coronal deformity, however, less is known about how they influence the axial deformity. This study investigates the change in three-dimensional deformity of a series of scoliosis patients due to compressive axial loading. Methods: Magnetic resonance imaging scans were obtained and coronal deformity (measured using the coronal Cobb angle) and axial rotations measured for a group of 18 scoliosis patients (Mean major Cobb angle was 43.4 o). Each patient was scanned in an unloaded and loaded condition while compressive loads equivalent to 50% body mass were applied using a custom developed compressive device. Findings: The mean increase in major Cobb angle due to compressive loading was 7.4 o (SD 3.5 o). The most axially rotated vertebra was observed at the apex of the structural curve and the largest average intravertebral rotations were observed toward the limits of the coronal deformity. A level-wise comparison showed no significant difference between the average loaded and unloaded vertebral axial rotations (intra-observer error = 2.56 o) or intravertebral rotations at each spinal level. Interpretation: This study suggests that the biomechanical effects of axial loading primarily influence the coronal deformity, with no significant change in vertebral axial rotation or intravertebral rotation observed between the unloaded and loaded condition. However, the magnitude of changes in vertebral rotation with compressive loading may have been too small to detect given the resolution of the current technique.

133Cs relaxation times in rat tissues

133Cs relaxation-time studies of tissues from rats into which cesium has been incorporated by dietary loading have been carried out in vivo and in vitro. Whereas tissue T1 values are on the order of seconds, T2 values are as low as a few tens of milliseconds, 133Cs tissue relaxation times are analogous to those of 39K in the same tissues, but are more readily measured because of the greater sensitivity of 133Cs compared with 39K, T1 and T2 data of excised tissue at two resonance frequencies (65.60 and 39.37 MHz) and temperatures (302 and 278 K) have been analyzed in terms of a general description of spin- relaxation. The results are consistent with most of the cesium ions being in a free state, undergoing fast exchange with bound ions having long correlation times located in one or more intracellular compartments,

NMR measurement of 39K detectability and relaxation constants in rat tissue

Differences in the NMR detectability of 39K in various excised rat tissues (liver, brain, kidney, muscle, and testes) have been observed. The lowest NMR detectability occurs for liver (61 ± 3% of potassium as measured by flame photometry) and highest for erythrocytes (100 ± 7%). These differences in detectability correlate with differences in the measured 39K NMR relaxation constants in the same tissues. 39K detectabilities were also found to correlate inversely with the mitochondrial content of the tissues. Mitochondria prepared from liver showed greatly reduced 39K NMR detectability when compared with the tissue from which it was derived, 31.6 ± 9% of potassium measured by flame photometry compared to 61 ± 3%. The detectability of potassium in mitochondria was too low to enable the measurement of relaxation constants. This study indicates that differences in tissue structure, particularly mitochondrial content are important in determining 39K detectability and measured relaxation rates.

NMR relaxation behavior and quadrupole coupling constants of 39K and 23Na ions in glycerol. Comparisons with 39K tissue data

The quadrupole coupling constants (qcc) for39K and23Na ions in glycerol have been calculated from linewidths measured as a function of temperature (which in turn results in changes in solution viscosity). The qcc of39K in glycerol is found to be 1.7 MHz, and that of23Na is 1.6 MHz. The relaxation behavior of39K and23Na ions in glycerol shows magnetic field and temperature dependence consistent with the equations for transverse relaxation more commonly used to describe the reorientation of nuclei in a molecular framework with intramolecular field gradients. It is shown, however, that τc is not simply proportional to the ratio of viscosity/temperature (ηT). The 39K qcc in glycerol and the value of 1.3 MHz estimated for this nucleus in aqueous solution are much greater than values of 0.075 to 0.12 MHz calculated from T2 measurements of39K in freshly excised rat tissues. This indicates that, in biological samples, processes such as exchange of potassium between intracellular compartments or diffusion of ions through locally ordered regions play a significant role in determining the effective quadrupole coupling constant and correlation time governing39K relaxation. T1 and T2 measurements of rat muscle at two magnetic fields also indicate that a more complex correlation function may be required to describe the relaxation of39K in tissue. Similar results and conclusions are found for23Na.

Factors affecting 39K NMR detectability in rat tissue

In this study we have found that NMR detectability of 39K in rat thigh muscle may be substantially higher (up to 100% oftotal tissue potassium) than values previously reported of around 40%. The signal was found to consist of two superimposed components, one broad and one narrow, of approximately equal area. Investigations involving improvements in spectral parameters such as signal-to-noise ratio and baseline roll, together with computer simulations of spectra, show that the quality of the spectra has a major effect on the amount of signal detected, which is largely due to the loss of detectability of the broad signal component. In particular, lower-field spectrometers using conventional probes and detection methods generally have poorer signal-to-noise and worse baseline roll artifacts, which make detection of a broad component of the muscle signal difficult.

Effect of magnesium depletion and potassium and chlorothiazide on intracellular pH in the rat, studied by 31P NMR

1. Both dietary magnesium depletion and potassium depletion (confirmed by tissue analysis) were induced in rats which were then compared with rats treated with chlorothiazide (250 mg/kg diet) and rats on a control synthetic diet. 2. Brain and muscle intracellular pH was measured by using a surface coil and [31P]-NMR to measure the chemical shift of inorganic phosphate. pH was also measured in isolated perfused hearts from control and magnesium-deficient rats. Intracellular magnesium status was assessed by measuring the chemical shift of β-ATP in brain. 3. There was no evidence for magnesium deficiency in the chlorothiazide-treated rats on tissue analysis or on chemical shift of β-ATP in brain. Both magnesium and potassium deficiency, but not chlorothiazide treatment, were associated with an extracellular alkalosis. 4. Magnesium deficiency led to an intracellular alkalosis in brain, muscle and heart. Chlorothiazide treatment led to an alkalosis in brain. Potassium deficiency was associated with a normal intracellular pH in brain and muscle. 5. Magnesium depletion and chlorothiazide treatment produce intracellular alkalosis by unknown mechanism(s).

Problems in the assessment of magnesium depletion in the rat by in vivo 31P NMR

Prior in vitro studies, utilizing 31Pn uclear magnetic resonance (31PN MR) to measure the chemical shift (CT) of 0-ATP and lengthening of the phosphocreatine spin-spin (7"') relaxation time, suggested an assessment of their efficacy in measuring magnesium depletion in vivo. Dietary magnesium depletion (Me$) produced markedly lower magnesium in plasma (0.44 vs 1. I3 mmol/liter) and bone (1 30 vs 190 pmol/g) but much smaller changes in muscle (41 vs 45 pmol/g, P < 0.01), heart (42.5 vs 44.6 prnol/g), and brain (30 vs 32 pmollg). NMR experiments in anesthetized rats in a Bruker 7-T vertical bore magnet showed that in M e $ rats there was a significant change in brain j3-ATP shift (16.15 vs 16.03 ppm, P < 0.05). These chemical shifts gave a calculated free [Mg"] of 0.71 mM (control) and 0.48 mM (MgZ+$). In muscle the change in j3-ATP shift was not significant (Me$ 15.99 ppm, controls 15.96 ppm), corresponding to a calculated free M P of 0.83 and 0.95 mM, respectively. Phosphccreatine Tz (Carr-Purcell, spin-echo pulse sequence) was no different with M e $ in muscle in vivo (surface coil) (M$+$ 136, control 142 ms) or in isolated perfused hearts (Helmholtz coil) (control 83, M e $ 92 ms). 3'P NMR is severely limited in its ability to detect dietary magnesium depletion in vivo. Measurement of j3-ATP shift in brain may allow studies of the effects of interaction in group studies but does not allow prediction of an individual magnesium status.

Aromatic l-amino acid decarboxylase : histochemical localization in rat kidney and lack of effect of dietary potassium or sodium loading on enzyme distribution

Utilizing a mono-specific antiserum produced in rabbits to hog kidney aromatic L-amino acid decarboxylase (AADC), the enzyme was localized in rat kidney by immunoperoxidase staining. AADC was located predominantly in the proximal convoluted tubules; there was also weak staining in the distal convoluted tubules and collecting ducts. An increase in dietary potassium or sodium intake produced no change in density or distribution of AADC staining in kidney. An assay of AADC enzyme activity showed no difference in cortex or medulla with chronic potassium loading. A change in distribution or activity of renal AADC does not explain the postulated dopaminergic modulation of renal function that occurs with potassium or sodium loading.

Effect Of changes in sodium intake on cell transport parameters

Changing sodium intake from 70-200 mmol/day elevates blood pressure in normotensive volunteers by 6/4 mmHg. Older people, people with reduced renal function on a low sodium diet and people with a family history of hypertension are more likely to show this effect. The rise in blood pressure was associated with a fall in plasma volume suggesting that plasma volume changes do not initiate hypertension. In normotensive individuals the most common abnormality in membrane sodium transport induced by an extra sodium load was an increased permeability of the red cell to sodium. Some normotensive individuals also had an increase in the level of a plasma inhibitor that inhibited Na-K ATPase. These individuals also appeared to have a rise in blood pressure. Sodium intake and blood pressure are related. The relationship differs in different people and is probably controlled by the genetically inherited capacity of systems involved in membrane sodium transport.

Inhibition of guinea-pig renal [Na + K]-ATPase by normotensive human plasma : effects of a high sodium diet

The effect of plasma taken from normotensive humans, while on a low and high sodium diet, on [Na + K]-ATPase and 3H-ouabain binding was measured in tubules from guinea-pig kidneys. Plasma from the high sodium, compared to the low sodium, diet period: (a) inhibited [Na + K]-ATPase activity; (b) decreased 3H-ouabain affinity for binding sites; (c) increased the number of available 3H-ouabain binding sites; (d) decreased [Na + K]-ATPase turnover (activity/3H-ouabain binding sites). The inhibition of [Na + K]-ATPase suggests an increase in a (possible) natriuretic factor. The decreased affinity of 3H-ouabain binding suggests an endogenous ouabainoid, which may be the natriuretic factor.

Tooth fracture risk analysis based on a new finite element dental structure models using micro-CT data

The finite element (FE) analysis is an effective method to study the strength and predict the fracture risk of endodontically-treated teeth. This paper presents a rapid method developed to generate a comprehensive tooth FE model using data retrieved from micro-computed tomography (μCT). With this method, the inhomogeneity of material properties of teeth was included into the model without dividing the tooth model into different regions. The material properties of the tooth were assumed to be related to the mineral density. The fracture risk at different tooth portions was assessed for root canal treatments. The micro-CT images of a tooth were processed by a Matlab software programme and the CT numbers were retrieved. The tooth contours were obtained with thresholding segmentation using Amira. The inner and outer surfaces of the tooth were imported into Solidworks and a three-dimensional (3D) tooth model was constructed. An assembly of the tooth model with the periodontal ligament (PDL) layer and surrounding bone was imported into ABAQUS. The material properties of the tooth were calculated from the retrieved CT numbers via ABAQUS user's subroutines. Three root canal geometries (original and two enlargements) were investigated. The proposed method in this study can generate detailed 3D finite element models of a tooth with different root canal enlargements and filling materials, and would be very useful for the assessment of the fracture risk at different tooth portions after root canal treatments.

Towards determining soft tissue properties for modelling spine surgery : current progress and challenges

Current complication rates for adolescent scoliosis surgery necessitate the development of better surgical planning tools to improve outcomes. Here we present our approach to developing finite element models of the thoracolumbar spine for deformity surgery simulation, with patient-specific model anatomy based on low-dose pre-operative computed tomography scans. In a first step towards defining patient-specific tissue properties, an initial 'benchmark' set of properties were used to simulate a clinically performed pre-operative spinal flexibility assessment, the fulcrum bending radiograph. Clinical data for ten patients were compared with the simulated results for this assessment and in cases where these data differed by more than 10%, soft tissue properties for the costo-vertebral joint (CVJt) were altered to achieve better agreement. Results from these analyses showed that changing the CVJt stiffness resulted in acceptable agreement between clinical and simulated flexibility in two of the six cases. In light of these results and those of our previous studies in this area, it is suggested that spinal flexibility in the fulcrum bending test is not governed by any single soft tissue structure acting in isolation. More detailed biomechanical characterisation of the fulcrum bending test is required to provide better data for determination of patient-specific soft tissue properties.