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Register a new userMembranes in Optic Nerve Models
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Membranes are present in all cells and tissues. Mathematical models of cells and tissues need a compact mathematical description of membranes with a resolution of about 1 nm. Membranes isolate cells because ions have difficulty penetrating the dielectric barrier they create. Here we introduce a dielectric mathematical membrane condition to replace a condition that did not include dielectric properties. Our mathematical membrane condition includes a dielectric lipid bilayer punctured by channels that conduct ions selectively.
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Complex fluids flow in complex ways in complex structures. Transport of water and various organic and inorganic molecules in the central nervous system are important in a wide range of biological and medical processes [C. Nicholson, and S. Hrabv{e}tova, Biophysical Journal, 113(10), 2133(2017)]. However, the exact driving mechanisms are often not known. In this paper, we investigate flows induced by action potentials in an optic nerve as a prototype of the central nervous system (CNS). Different from traditional fluid dynamics problems, flows in biological tissues such as the CNS are coupled with ion transport. It is driven by osmosis created by concentration gradient of ionic solutions, which in term influence the transport of ions. Our mathematical model is based on the known structural and biophysical properties of the experimental system used by the Harvard group Orkand et al [R.K. Orkand, J.G. Nicholls, S.W. Kuffler, Journal of Neurophysiology, 29(4), 788(1966)]. Asymptotic analysis and numerical computation show the significant role of water in convective ion transport. The full model (including water) and the electrodiffusion model (excluding water) are compared in detail to reveal an interesting interplay between water and ion transport. In the full model, convection due to water flow dominates inside the glial domain. This water flow in the glia contributes significantly to the spatial buffering of potassium in the extracellular space. Convection in the extracellular domain does not contribute significantly to spatial buffering. Electrodiffusion is the dominant mechanism for flows confined to the extracellular domain.
We present a novel mathematical approach to model noise in dynamical systems. We do so by considering dynamics of a chain of diffusively coupled Nagumo cells affected by noise. We show that the noise in transmembrane current can be effectively modelled as fluctuations in electric characteristics of the membrane. The proposed approach to model noise in a nerve fibre is different from the standard additive stochastic current perturbation (the Langevin type equations).
The accumulation of potassium in the narrow space outside nerve cells is a classical subject of biophysics that has received much attention recently. It may be involved in potassium accumulation textcolor{black}{including} spreading depression, perhaps migraine and some kinds of epilepsy, even (speculatively) learning. Quantitative analysis is likely to help evaluate the role of potassium clearance from the extracellular space after a train of action potentials. Clearance involves three structures that extend down the length of the nerve: glia, extracellular space, and axon and so need to be described as systems distributed in space in the tradition used for electrical potential in the `cable equations of nerve since the work of Hodgkin in 1937. A three-compartment model is proposed here for the optic nerve and is used to study the accumulation of potassium and its clearance. The model allows the convection, diffusion, and electrical migration of water and ions. We depend on the data of Orkand et al to ensure the relevance of our model and align its parameters with the anatomy and properties of membranes, channels, and transporters: our model fits their experimental data quite well. The aligned model shows that glia has an important role in buffering potassium, as expected. The model shows that potassium is cleared mostly by convective flow through the syncytia of glia driven by osmotic pressure differences. A simplified model might be possible, but it must involve flow down the length of the optic nerve. It is easy for compartment models to neglect this flow. Our model can be used for structures quite different from the optic nerve that might have different distributions of channels and transporters in its three compartments. It can be generalized to include a fourth (distributed) compartment representing blood vessels to deal with the glymphatic flow into the circulatory system.
The optic nerve head (ONH) typically experiences complex neural- and connective-tissue structural changes with the development and progression of glaucoma, and monitoring these changes could be critical for improved diagnosis and prognosis in the glaucoma clinic. The gold-standard technique to assess structural changes of the ONH clinically is optical coherence tomography (OCT). However, OCT is limited to the measurement of a few hand-engineered parameters, such as the thickness of the retinal nerve fiber layer (RNFL), and has not yet been qualified as a stand-alone device for glaucoma diagnosis and prognosis applications. We argue this is because the vast amount of information available in a 3D OCT scan of the ONH has not been fully exploited. In this study we propose a deep learning approach that can: textbf{(1)} fully exploit information from an OCT scan of the ONH; textbf{(2)} describe the structural phenotype of the glaucomatous ONH; and that can textbf{(3)} be used as a robust glaucoma diagnosis tool. Specifically, the structural features identified by our algorithm were found to be related to clinical observations of glaucoma. The diagnostic accuracy from these structural features was $92.0 pm 2.3 %$ with a sensitivity of $90.0 pm 2.4 % $ (at $95 %$ specificity). By changing their magnitudes in steps, we were able to reveal how the morphology of the ONH changes as one transitions from a `non-glaucoma to a `glaucoma condition. We believe our work may have strong clinical implication for our understanding of glaucoma pathogenesis, and could be improved in the future to also predict future loss of vision.
Purpose: To develop a deep learning approach to de-noise optical coherence tomography (OCT) B-scans of the optic nerve head (ONH). Methods: Volume scans consisting of 97 horizontal B-scans were acquired through the center of the ONH using a commercial OCT device (Spectralis) for both eyes of 20 subjects. For each eye, single-frame (without signal averaging), and multi-frame (75x signal averaging) volume scans were obtained. A custom deep learning network was then designed and trained with 2,328 clean B-scans (multi-frame B-scans), and their corresponding noisy B-scans (clean B-scans + gaussian noise) to de-noise the single-frame B-scans. The performance of the de-noising algorithm was assessed qualitatively, and quantitatively on 1,552 B-scans using the signal to noise ratio (SNR), contrast to noise ratio (CNR), and mean structural similarity index metrics (MSSIM). Results: The proposed algorithm successfully denoised unseen single-frame OCT B-scans. The denoised B-scans were qualitatively similar to their corresponding multi-frame B-scans, with enhanced visibility of the ONH tissues. The mean SNR increased from $4.02 pm 0.68$ dB (single-frame) to $8.14 pm 1.03$ dB (denoised). For all the ONH tissues, the mean CNR increased from $3.50 pm 0.56$ (single-frame) to $7.63 pm 1.81$ (denoised). The MSSIM increased from $0.13 pm 0.02$ (single frame) to $0.65 pm 0.03$ (denoised) when compared with the corresponding multi-frame B-scans. Conclusions: Our deep learning algorithm can denoise a single-frame OCT B-scan of the ONH in under 20 ms, thus offering a framework to obtain superior quality OCT B-scans with reduced scanning times and minimal patient discomfort.
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