Coronary Artery Disease (CAD) results from plaque deposit in a coronary artery. Early diagnosis is imperative, so a non-invasive detection method is being developed to identify acoustic signals caused by partial occlusions in the artery. The blood flow in the artery is disturbed and imposes oscillatory stresses on the artery wall. The deformations caused by the stresses can be detected at the chest surface. Therefore, by using data simulating these surface signals, which arise from randomly assigned source positions, machine learning (ML) can be utilised to predict the source of the occlusion. Seven ML algorithms were investigated, and the results from this study found that an ensemble model combining k-Nearest Neighbours and Random Forest had the best performance. The metrics used to evaluate this was the mean squared error and Euclidean distance.

Linear viscoelasticity can be characterized by a stress relaxation function. We consider a power-law type stress relaxation to yield a fractional order viscoelasticity model. The governing equation is a Volterra integral problem of the second kind with a weakly singular kernel. We employ spatially discontinuous Galerkin methods, symmetric interior penalty Galerkin method (SIPG) for spatial discretization, and the implicit finite difference schemes in time, Crank-Nicolson method. Further, in order to manage the weak singularity in the Volterra kernel, we use a linear interpolation technique. We present a priori stability and error analyses without relying on Gr & ouml;nwall's inequality, and so provide high quality bounds that do not increase exponentially in time. This indicates that our numerical scheme is well-suited for long-time simulations. Despite the limited regularity in time, we establish suboptimal fractional order accuracy in time as well as optimal convergence of SIPG. We carry out numerical experiments with varying regularity of exact solutions to validate our error estimates. Finally, we present numerical simulations based on real material data.

Abstract The stress-strain constitutive law for viscoelastic materials such as soft tissues, metals at high temperature, and polymers can be written as a Volterra integral equation of the second kind with a fading memory kernel. This integral relationship yields current stress for a given strain history and can be used in the momentum balance law to derive a mathematical model for the resulting deformation. We consider such a dynamic linear viscoelastic model problem resulting from using a Dirichlet–Prony series of decaying exponentials to provide the fading memory in the Volterra kernel. We introduce two types of internal variable to replace the Volterra integral with a system of auxiliary ordinary differential equations and then use a spatially discontinuous symmetric interior penalty Galerkin (SIPG) finite element method and – in time – a Crank–Nicolson method to formulate the fully discrete problems: one for each type of internal variable. We present a priori stability and error analyses without using Grönwall’s inequality and with the result that the constants in our estimates grow linearly with time rather than exponentially. In this sense, the schemes are therefore suited to simulating long time viscoelastic response, and this (to our knowledge) is the first time that such high quality estimates have been presented for SIPG finite element approximation of dynamic viscoelasticity problems. We also carry out a number of numerical experiments using the FEniCS environment (https://fenicsproject.org), describe a simulation using “real” material data, and explain how the codes can be obtained and all of the results reproduced.

We consider linear scalar wave equations with a hereditary integral term of the kind used to model viscoelastic solids. The kernel in this Volterra integral is a sum of decaying exponentials (The so-called Maxwell, or Zener model) and this allows the introduction of one of two types of families of internal variables, each of which evolve according to an ordinary differential equation (ODE). There is one such ODE for each decaying exponential, and the introduction of these ODEs means that the Volterra integral can be removed from the governing equation. The two types of internal variable are distinguished by whether the unknown appears in the Volterra integral, or whether its time derivative appears; we call the resulting problems the displacement and velocity forms. We define fully discrete formulations for each of these forms by using continuous Galerkin finite element approximations in space and an implicit ‘Crank–Nicolson’ type of finite difference method in time. We prove stability and a priori bounds, and using the FEniCS environment, https://fenicsproject.org/ (The FEniCS project version 1.5, Archive of Numerical Software, 3 (100), 9–23, 2015.) give some numerical results. These bounds do not require Grönwall’s inequality and so can be regarded to be of high quality, allowing confidence in long time integration without an a priori exponential build up of error. As far as we are aware this is the first time that these two formulations have been described together with accompanying proofs of such high quality stability and error bounds. The extension of the results to vector-valued viscoelasticity problems is straightforward and summarised at the end. The numerical results are reproducible by acquiring the python sources from https://github.com/Yongseok7717 , or by running a custom built docker container (instructions are given).

The workforce of the medical specialty of Rehabilitation Medicine (RM) in the UK is 10 times less than the European average for the specialty of Physical and Rehabilitation Medicine (PRM). This can be explained partly by the difference in the scope of practice within the specialty between the UK and other European countries and USA. This opinion paper aims to compare the rehabilitation needs in chronic medical conditions and compare the scope of practice between countries within Europe and other regions of the world. The potential advantages of a broader remit specialty to improve rehabilitation care for patients by involving rehabilitation physicians in various medical conditions is explored. Recommendations have been put forward in the Rehabilitation Medicine Expansion Proposal (RMEP), which is likely to make the medical specialty of RM/ PRM more satisfying for the doctors working in the specialty and a more attractive career choice for those entering training in the specialty. There is a need for an international universal framework for the scope of the specialty to have a greater impact on improving the lives of those with chronic medical conditions.

Fractional calculus constitutive models are of wide applicability. They have been used, for example, for: xanthan gum, bread dough, synthetic polymers (e.g. nylon), single collagen fibrils, semi-hard zero-fat cheese, pure ice at -35 degrees C, asphalt sealants, epithelial cells, gels, polymers, concrete, asphalt, rock mass, waxy crude oil, breast tissue cells, lung parenchyma, and red blood cell membranes (e.g. Bonfanti et al. (2020)). In this note we are motivated by applications to dispersive media such as lossy dielectrics and viscoelastic polymers, both of which involve constitutive laws with fading memory convolution Volterra integrals, and both of which have been modelled by fractional calculus - which implies a weakly singular Volterra kernel. To step forward in time with such a model the entire hereditary Volterra integral needs to be numerically re-computed at each time step. Done naively with standard quadrature this involves O(N2) operations and O(N) storage to compute over N time steps, and this severely limits the usefulness of such simple methods in large scale computations. Several alternatives addressing these shortcomings of this naive approach have been offered in the literature, all of which can provide remedies. Here we propose another method, but ours is rather different in that we use a Fourier series proxy over much of the integration range. This allows for a recursive update to the 'history integral', with a much smaller effort expended in standard quadrature near the singularity. The recursive update is basically the same as that used when the Volterra kernel comprises decaying exponentials, a method which is already employed in some commercial codes. Given that this partial implementation is already in use, we intend that our method therefore has a lower 'barrier to entry' for incorporating fractional calculus models into commercial codes: it requires only minor surgery on the weakly singular kernel in order to apply the Fourier series proxy. This study is presented in the spirit of a discussion piece - there are several directions one could take from the basic observations presented here. We have aimed for simplicity and economy of presentation, and included just enough numerical results to give evidence that the method works. In summary, our method mitigates the O(N) storage and O(N2) storage issues using an easily implemented proxy. It can also be implemented for variable time steps. The code is available from https://github.com/variatio nalform/fouvol and https://hub.docker.com/u/variationalform.

The workforce of the medical specialty of Rehabilitation Medicine (RM) in the UK is 10 times less than the European average for the specialty of Physical and Rehabilitation Medicine (PRM). This can be explained partly by the difference in the scope of practice within the specialty between UK and other European countries. This opinion paper aims to compare the rehabilitation needs in chronic conditions and compare the scope of practice between countries within Europe and other regions of the world. The potential advantages of a broader remit specialty to improve rehabilitation care for patients across a wide range of conditions in explored. Recommendations have been put forward in the Rehabilitation Medicine Expansion Proposal (RMEP), which is likely to make the medical specialty more satisfying for the doctors working in the specialty and a more attractive career choice for those entering training in the medical specialty.

Post-Polio Syndrome (PPS) may affect patients long after their initial viral infection. There are limited data reporting the impact of PPS on sleep. We investigated the prevalence of sleep disruption and sleep disorders in a cohort of PPS patients. We analysed the data of a large tertiary national service at Guy’s & St Thomas’ NHS Foundation Trust, London, UK (Registration: GSTT/2020/100030). Data were collected from electronic patient records including age, gender, diagnoses, comorbidities, respiratory support, medications, and rehabilitation support. Spirometry, oxygen saturations, respiratory polygraphy and Epworth Sleepiness Scale were recorded where available. Sleep disruption and disorders were categorised into sleep-disordered breathing, sleep fragmentation due to pain, insomnia, parasomnias, circadian rhythm disorders, hypersomnia, and others. 356 patients with PPS were included in the analysis (160 male/44.9%; 246 >50 years). Pain was the most common comorbidity (60.4%), while muscle weakness and wasting were reported by 48.9% and 16.6%, respectively. General fatigue was reported by 35.7% and arthropathy by 21.3%. 145 patients (40.1%) reported sleep disruption and/or sleep disorders of any type, the most frequent being sleep-disordered breathing (27.2%); 13.8% had obstructive sleep apnoea. Scoliosis was present in 12.1%. Sleep disruption due to pain was reported by 11.8%. Patients with PPS commonly experience sleep disruption and sleep disorders. A holistic approach towards these patients should include screening for sleep-related conditions, as sufficient treatment of sleep-related problems may improve health-related quality of life.

BACKGROUND:Post-polio syndrome is characterised by symptoms of fatigue, pain and new-onset neuromuscular weakness, and emerges decades after the initial poliovirus infection. We sought to evaluate the only post-polio syndrome specific self-management programme in the United Kingdom.METHODS:This was a retrospective study of patients who had completed a residential self-management programme led by a multi-disciplinary clinical team. Following a confirmed diagnosis of post-polio syndrome by rehabilitation and neurology specialists, patients were offered to participate in the programme. Although group-based, patients also received individually tailored support on physical exercise and fatigue management. Physical effects, physical function, psychosocial well-being measures were assessed at baseline and 6 months follow-up. Knowledge was tested at baseline and immediately following the programme. Statistical comparisons were made using paired t-test and Wilcoxon signed rank test according to the data distribution.RESULTS:Over a period of 17 years, 214 participants (median age 61.3 years old, 63% female) attended 31 programmes. At 6 months the following post-polio syndrome specific symptoms improved significantly: fatigue, as measured by the Multidimensional Assessment of Fatigue scale [37.6 (7.1) vs. 34.2 (9.3), P=0.005]; and pain [15.0 (6.1) vs. 13.1 (6.7), P=0.001], atrophy [10.0 (8.0-12.0) vs. 9.0 (7.0-11.0), P=0.002] and bulbar symptoms [3.0 (1.0-5.0) vs. 2.0 (0-4.0), P=0.003] as measured by the Index of Post-polio Sequelae scale. Knowledge related to post-polio syndrome also significantly increased [14.0 (11.0-16.0) vs. 17.0 (16.0-19.0), P=0.001]. Participants were able to walk at a faster speed over 10 meters [0.77 (0.59-1.00) vs. 0.83 (0.67-1.10) m/s, P=0.003] and walked longer distances during the 2-minute walk test [76.9 (31.7) vs. 82.0 (38.4) m, P=0.029]. Depression and anxiety scores did not change over time [PHQ-9, 2.0 (0.3-10.8) vs. 2.0 (0.3-6.8), P=0.450; GAD-7, 2.0 (0-7.0) vs. 1.0 (0-3.0), P=0.460] nor was there change in self-reported quality of life {60 [50-70] vs. 60 [55-70], P=0.200}.CONCLUSIONS:This study suggests that a post-polio syndrome self-management programme led to improvement in symptoms, knowledge and walking speed, but not quality of life. Anxiety and depression scores remained low.

We consider a fractional order viscoelasticity problem modelled by a power-law type stress relaxation function. This viscoelastic problem is a Volterra integral equation of the second kind with a weakly singular kernel where the convolution integral corresponds to fractional order differentiation/integration. We use a spatial finite element method and a finite difference scheme in time. Due to the weak singularity, fractional order integration in time is managed approximately by linear interpolation so that we can formulate a fully discrete problem. In this paper, we present a stability bound as well as a priori error estimates. Furthermore, we carry out numerical experiments with varying regularity of exact solutions at the end.

Introduction Post-polio syndrome (PPS) can affect patients decades after initial polio-virus infection and is characterised by new slow onset neuromuscular weakness, fatigue and pain. Hypoventilation is a feature when respiratory muscle weakness occurs. We assess the relationship between symptoms of PPS to diagnosed sleep disordered breathing (SDB) in form of obstructive sleep apnoea (OSA) and hypercapnic respiratory failure (HRF) in a cohort of patients referred to PPS specific self-management education programme. Methods Retrospective analysis of a cohort of patients with completed PPS in a national tertiary referral centre who completed the PPS self-management programme from 2006 to 2019.1 Physical symptoms were assessed via the Index of Post-Polio Sequelae (IPPS) questionnaire. Multi-nomial logistic regression was used to assess IPPS sub-domains of pain, atrophy, bulbar and temperature changes in relation to SDB diagnoses (no treatment; OSA; HRF) using HRF as reference group. Age and symptom score presented with mean and range. Results 168 participants (108 female) were included: 136 required no treatment for SDB (age: 62.8; 34–85), 20 (11.9%) with OSA (60.0; 43–73), 12 (7.1%) with HRF (64.7; 47–79). HRF suffered greater bulbar symptoms (score out of 10) compared to no treatment group (4.5(2–8) vs 2.7(0–10)) but was no different to OSA (4.5 vs 4.3(0–8)). No significant relationship was found between pain, atrophy and temperature sub-domains to SDB diagnoses in multi-nomial regression analysis (table 1). Conclusion Bulbar symptom domain in IPPS (including respiratory symptoms) predicted diagnosis of SDB independent of other symptom domains including general muscle atrophy. However, it does not differentiate between OSA and HRF. Patients with any bulbar or respiratory symptoms should be referred for SDB assessment with particular focus on hypercapnic respiratory failure. Reference Curtis A, Lee JS, Kaltsakas G, et al. JTD 2020 'The value of a post-polio syndrome self-management programme'.

We extend the formulation and a priori error analysis given by Claes Johnson (1993) from the acoustic wave equation to a Voigt and Maxwell–Zener viscodynamic system incorporating Rayleigh damping. The elastic term in the Rayleigh damping introduces a multiplicative T1∕2 growth in the constant but otherwise the error bound is consistent with that obtained by Johnson, with a constant that grows a priori with T1∕2 and also with norms of the solution. Gronwall’s inequality is not used and so we can expect that this bound is of high enough quality to afford confidence in long-time integration. The viscoelasticity is modelled by internal variables that evolve according to ordinary differential equations and so the system shares similarities with dispersive Debye and Drude metamaterial models currently being studied in electromagnetism, as well as to acoustic metamaterial systems. This appears to be the first time an a priori error analysis has been given for DG-in-time treatment of dispersive problems of this type.

As a first step towards an acoustic localisation device for coronary stenosis to provide a non-invasive means of diagnosing arterial disease, measurements are reported for an agar-based tissue mimicking material (TMM) of the shear wave propagation velocity, attenuation and viscoelastic constants, together with one dimensional quasi-static elastic moduli and Poisson's ratio. Phase velocity and attenuation coefficients, determined by generating and detecting shear waves piezo-electrically in the range 300 Hz-2 kHz, were 3.2-7.5 ms(-1) and 320 dBm(-1). Quasi-static Young's modulus, shear modulus and Poisson's ratio, obtained by compressive or shear loading of cylindrical specimens were 150-160 kPa; 54-56 kPa and 0.37-0.44. The dynamic Young's and shear moduli, derived from fitting viscoelastic internal variables by an iterative statistical inverse solver to freely oscillating specimens were 230 and 33 kPa and the corresponding relaxation times, 0.046 and 0.036 s. The results were self-consistent, repeatable and provide baseline data required for the computational modelling of wave propagation in a phantom.

We consider time domain formulations of Maxwell's equations for the Lorentz model for metamaterials. The field equations are considered in two different forms which have either six or four unknown vector fields. In each case we use arguments tuned to the physical laws to derive data-stability estimates which do not require Gronwall's inequality. The resulting estimates are, in this sense, sharp. We also give fully discrete formulations for each case and extend the sharp data-stability to these. Since the physical problem is linear it follows (and we show this with examples) that this stability property is also reflected in the constants appearing in the a priori error bounds. By removing the exponential growth in time from these estimates we conclude that these schemes can be used with confidence for the long-time numerical simulation of Lorentz metamaterials.

This work was supported in part by NSFC Project 11271310, NSF grant DMS-1416742, and a grant from the Simons Foundation (#281296 to Li), in part by scheme 4 London Mathematical Society funding and in part by the Engineering and Physical Sciences Research Council (EP/H011072/1 to Shaw).

This article defines a novel spatial-temporal modelling and analysis methodology applied to a systems biology case study, namely phase variation patterning in bacterial colony growth. We employ coloured stochastic Petri nets to construct the model and run stochastic simulations to record the development of the circular colonies over time and space. The simulation output is visualised in 2D, and sector-like patterns are automatically detected and analysed. Space is modelled using 2.5 dimensions considering both a rectangular and circular geometry, and the effects of imposing different geometries on space are measured. We close by outlining an interpretation of the Petri net model in terms of finite difference approximations of partial differential equations (PDEs). One result is the derivation of the “best” nine-point diffusion model. Our multidimensional modelling and analysis approach is a precursor to potential future work on more complex multiscale modelling.

Objectives: Coronary artery disease (CAD) remains the leading cause of mortality in the Western world. Disturbed blood flow downstream of an atherosclerotic plaque gives rise to low-amplitude shear-waves that can be detected at the chest surface and thus provide the basis for a new low cost, non-invasive screening tool for the early detection of CAD.

In [ Comput. Methods Appl. Mech. Engrg. , 190 (2001),pp. 6685--6708] Werder et al. demonstrated that timediscretizations of the heat equation by a temporally discontinuousGalerkin finite element method could be decoupled by diagonalizingthe temporal Gram matrices. In this article we propose acompanion approach for the heat equation by using a continuous Galerkin time discretization. As a result, ifpiecewise polynomials of degree $d$ are used as the trialfunctions in time and the spatial discretization produces systemsof dimension $M$, then, after decoupling, $d$ systems of size $M$need to be solved rather than a single system of size $M d$. Thesedecoupled systems require complex arithmetic, as didthe Werder et al. technique, but are amenable to parallelsolutions on modern multicore architectures. We give numericaltests for temporal polynomial degrees up to six for threedifferent model test problems, using both Galerkin and spectralelement spatial discretizations, and show convergence and temporalsuperconvergence rates that accord with the bounds given by Azizand Monk [ Math. Comp. , 52 (1989), pp. 255--274]. Wealso interpret error as a function of computational time and seethat our high order schemes may offer greater efficiency than theCrank--Nicolson method in terms of accuracy per unit ofcomputational time---although in a multicore world, with highlytuned iterative solvers, one has to be cautious with such claims.We close with a speculation on the application of these ideas tothe Navier--Stokes equations for incompressible fluids.