Cardiac Electromechanics: The effect of contraction model on the mathematical problem and accuracy of the numerical scheme

Models of cardiac electromechanics usually contain a contraction model determining the active tension induced at the cellular level, and the equations of nonlinear elasticity to determine tissue deformation in response to this active tension. All contraction models are dependent on cardiac electro-physiology, but can also be dependent on\ud the stretch and stretch-rate in the fibre direction. This fundamentally affects the mathematical problem being solved, through classification of the governing PDEs, which affects numerical schemes that can be used to solve the governing equations. We categorise contraction models into three types, and for each consider questions such as classification and the most appropriate choice from two numerical methods (the explicit and implicit schemes). In terms of mathematical classification, we consider the question of strong ellipticity of the total strain energy (important for precluding ‘unnatural’ material behaviour) for stretch-rate-independent contraction models; whereas for stretch-rate-dependent contraction models we introduce a corresponding third-order problem and explain how certain choices of boundary condition could lead to constraints on allowable initial condition. In terms of suitable numerical methods, we show that an explicit approach (where the contraction model is integrated in the timestep prior to the bulk deformation being computed) is: (i) appropriate for stretch-independent contraction models; (ii) only conditionally-stable, with the stability criterion independent of timestep, for contractions models which just depend on stretch (but not stretch-rate), and (iii) inappropriate for stretch-rate-dependent models

Compact Routing on Internet-Like Graphs

The Thorup-Zwick (TZ) routing scheme is the first generic stretch-3 routing scheme delivering a nearly optimal local memory upper bound. Using both direct analysis and simulation, we calculate the stretch distribution of this routing scheme on random graphs with power-law node degree distributions, $P_k \sim k^{-\gamma}$. We find that the average stretch is very low and virtually independent of $\gamma$. In particular, for the Internet interdomain graph, $\gamma \sim 2.1$, the average stretch is around 1.1, with up to 70% of paths being shortest. As the network grows, the average stretch slowly decreases. The routing table is very small, too. It is well below its upper bounds, and its size is around 50 records for $10^4$-node networks. Furthermore, we find that both the average shortest path length (i.e. distance) $\bar{d}$ and width of the distance distribution $\sigma$ observed in the real Internet inter-AS graph have values that are very close to the minimums of the average stretch in the $\bar{d}$- and $\sigma$-directions. This leads us to the discovery of a unique critical quasi-stationary point of the average TZ stretch as a function of $\bar{d}$ and $\sigma$. The Internet distance distribution is located in a close neighborhood of this point. This observation suggests the analytical structure of the average stretch function may be an indirect indicator of some hidden optimization criteria influencing the Internet's interdomain topology evolution.Comment: 29 pages, 16 figure

Does acute passive stretching increase muscle length in children with cerebral palsy?

This article has been made available through the Brunel Open Access Publishing Fund. Copyright @ The Authors. This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in anymedium, provided the original author(s) and the source are credited.Background: Children with spastic cerebral palsy experience increased muscle stiffness and reduced muscle length, which may prevent elongation of the muscle during stretch. Stretching performed either by the clinician, or children themselves is used as a treatment modality to increase/maintain joint range of motion. It is not clear whether the associated increases in muscle–tendon unit length are due to increases in muscle or tendon length. The purpose was to determine whether alterations in ankle range of motion in response to acute stretching were accompanied by increases in muscle length, and whether any effects would be dependent upon stretch technique. Methods: Eight children (6–14 y) with cerebral palsy received a passive dorsiflexion stretch for 5 × 20 s to each leg, which was applied by a physiotherapist or the children themselves. Maximum dorsiflexion angle, medial gastrocnemius muscle and fascicle lengths, and Achilles tendon length were calculated at a reference angle of 10° plantarflexion, and at maximum dorsiflexion in the pre- and post-stretch trials. Findings: All variables were significantly greater during pre- and post-stretch trials compared to the resting angle, and were independent of stretch technique. There was an approximate 10° increase in maximum dorsiflexion post-stretch, and this was accounted for by elongation of both muscle (0.8 cm) and tendon (1.0 cm). Muscle fascicle length increased significantly (0.6 cm) from pre- to post-stretch. Interpretation: The results provide evidence that commonly used stretching techniques can increase overall muscle, and fascicle lengths immediately post-stretch in children with cerebral palsy