Haemoglobin polymorphism in atlantic cod(Gadus morhua): Allele frequency variation between yearclasses in a Norwegian fjord stock

A total of 262 specimens (0-, and 1-group) of Atlantic cod(Gadus morhua) representing 4 different yearclasses were caught in Trondheimsfjorden, Norway, during 1977–1984. They were genotyped by agar gel electrophoresis for the polymorphic haemoglobin locusHbI (Sick, 1961). The analyses revealed a highly significant (P=0.0003) heterogeneity ofHbI allele frequencies between yearclasses. The difference in the frequency of theHbI-1 allele between the first (1977) and the last (1983) yearclass amounted to 0.18 (±0.07). The results appear to support recent reports on considerable selection effects atHbI, and stress the unreliability of allele frequencies at this locus for use in studies of the genetic population structure of cod.Contribution No. 229, Biological Station, N-7000 Trondheim, Norway     

A mixed finite element formulation for a non-linear,transversely isotropic material model for the cardiac tissue

In this paper we present a mixed finite element method for modeling the passive properties of the myocardium. The passive properties are described by a non-linear, transversely isotropic, hyperelastic material model, and the myocardium is assumed to be almost incompressible. Single-field, pure displacement-based formulations are known to cause numerical difficulties when applied to incompressible or slightly compressible material cases. This paper presents an alternative approach in the form of a mixed formulation, where a separately interpolated pressure field is introduced as a primary unknown in addition to the displacement field. Moreover, a constraint term is included in the formulation to enforce (almost) incompressibility. Numerical results presented in the paper demonstrate the difficulties related to employing a pure displacement-based method, applying a set of physically relevant material parameter values for the cardiac tissue. The same problems are not experienced for the proposed mixed method. We show that the mixed formulation provides reasonable numerical results for compressible as well as nearly incompressible cases, also in situations of large fiber stretches. There is good agreement between the numerical results and the underlying analytical models.     

Multigrid Block Preconditioning for a Coupled System of Partial Differential Equations Modeling the Electrical Activity in the Heart

The electrical activity of the heart may be modeled with a system of partial differential equations (PDEs) known as the bidomain model. Computer simulations based on these equations may become a helpful tool to understand the relationship between changes in the electrical field and various heart diseases. Because of the rapid variations in the electrical field, sufficiently accurate simulations require a fine-scale discretization of the equations. For realistic geometries this leads to a large number of grid points and consequently large linear systems to be solved for each time step. In this paper, we present a fully coupled discretization of the bidomain model, leading to a block structured linear system. We take advantage of the block structure to construct an efficient preconditioner for the linear system, by combining multigrid with an operator splitting technique.     

A mixed finite element formulation for a non-linear, transversely isotropic material model for the cardiac tissue

In this paper we present a mixed finite element method for modeling the passive properties of the myocardium. The passive properties are described by a non-linear, transversely isotropic, hyperelastic material model, and the myocardium is assumed to be almost incompressible. Single-field, pure displacement-based formulations are known to cause numerical difficulties when applied to incompressible or slightly compressible material cases. This paper presents an alternative approach in the form of a mixed formulation, where a separately interpolated pressure field is introduced as a primary unknown in addition to the displacement field. Moreover, a constraint term is included in the formulation to enforce (almost) incompressibility. Numerical results presented in the paper demonstrate the difficulties related to employing a pure displacement-based method, applying a set of physically relevant material parameter values for the cardiac tissue. The same problems are not experienced for the proposed mixed method. We show that the mixed formulation provides reasonable numerical results for compressible as well as nearly incompressible cases, also in situations of large fiber stretches. There is good agreement between the numerical results and the underlying analytical models.     

0-group cod(Gadus morhua) in captivity: Differential survival of certain genotypes

Genotypes atLDH-3 (lactate dehydrogenase) andPGI-1 (phosphoglucoisomerase) in 263 0-group specimens of Atlantic cod(Gadus morhua) caught in Trondheimsfjorden, Norway, in fall 1983 showed significantly different survival rates during 72 days in captivity. The heterozygote was nominally superior at both loci, and there was a significant accumulation of double heterozygotes among the survivors. Apparently,LDH-3 andPGI-1 are not selectively neutral, and allele frequency differences at these two loci between groups of cod in nature should not be interpreted as markers of reproductive isolation. The results are related to current concepts of cod population structure.Contribution No. 231, Biological Station, N-7000 Trondheim, Norway     

Improved discretisation and linearisation of active tension in strongly coupled cardiac electro-mechanics simulations

Mathematical models of cardiac electro-mechanics typically consist of three tightly coupled parts: systems of ordinary differential equations describing electro-chemical reactions and cross-bridge dynamics in the muscle cells, a system of partial differential equations modelling the propagation of the electrical activation through the tissue and a nonlinear elasticity problem describing the mechanical deformations of the heart muscle. The complexity of the mathematical model motivates numerical methods based on operator splitting, but simple explicit splitting schemes have been shown to give severe stability problems for realistic models of cardiac electro-mechanical coupling. The stability may be improved by adopting semi-implicit schemes, but these give rise to challenges in updating and linearising the active tension. In this paper we present an operator splitting framework for strongly coupled electro-mechanical simulations and discuss alternative strategies for updating and linearising the active stress component. Numerical experiments demonstrate considerable performance increases from an update method based on a generalised Rush–Larsen scheme and a consistent linearisation of active stress based on the first elasticity tensor.     

Towards a computational method for imaging the extracellular potassium concentration during regional ischemia

We investigate the possibility of using body surface potential maps to image the extracellular potassium concentration during regional ischemia. The problem is formulated as an inverse problem based on a linear approximation of the bidomain model, where we minimize the difference between the results of the model and observations of body surface potentials. The minimization problem is solved by a one-shot technique, where the original PDE system, an adjoint problem, and the relation describing the minimum, are solved simultaneously. This formulation of the problem requires the solution of a 5 × 5 system of linear partial differential equations. The performance of the model is investigated by performing tests based on synthetic data. We find that the model will in many cases detect the correct position and approximate size of the ischemic regions, while some cases are more difficult to locate. It is observed that a simple post-processing of the results produces images that are qualitatively very similar to the true solution.     

Solving the heart mechanics equations with Newton and quasi Newton methods--a comparison

The non-linear elasticity equations of heart mechanics are solved while emulating the effects of a propagating activation wave. The dynamics of a 1 cm(3) slab of active cardiac tissue was simulated as the electrical wave traversed the muscular heart wall transmurally. The regular Newton (Newton-Raphson) method was compared to two modified Newton approaches, and also to a third approach that delayed update only of some selected Jacobian elements. In addition, the impact of changing the time step (0.01, 0.1 and 1 ms) and the relative non-linear convergence tolerance (10(-4), 10(-3) and 10(-2)) was investigated. Updating the Jacobian only when slow convergence occurred was by far the most efficient approach, giving time savings of 83-96%. For each of the four methods, CPU times were reduced by 48-90% when the time step was increased by a factor 10. Increasing the convergence tolerance by the same factor gave time savings of 3-71%. Different combinations of activation wave speed, stress rate and bulk modulus revealed that the fastest method became relatively even faster as stress rate and bulk modulus was decreased, while the activation speed had negligible influence in this respect.     

Electromechanical feedback with reduced cellular connectivity alters electrical activity in an infarct injured left ventricle: a finite element model study

Myocardial infarction (MI) significantly alters the structure and function of the heart. As abnormal strain may drive heart failure and the generation of arrhythmias, we used computational methods to simulate a left ventricle with an MI over the course of a heartbeat to investigate strains and their potential implications to electrophysiology. We created a fully coupled finite element model of myocardial electromechanics consisting of a cellular physiological model, a bidomain electrical diffusion solver, and a nonlinear mechanics solver. A geometric mesh built from magnetic resonance imaging (MRI) measurements of an ovine left ventricle suffering from a surgically induced anteroapical infarct was used in the model, cycled through the cardiac loop of inflation, isovolumic contraction, ejection, and isovolumic relaxation. Stretch-activated currents were added as a mechanism of mechanoelectric feedback. Elevated fiber and cross fiber strains were observed in the area immediately adjacent to the aneurysm throughout the cardiac cycle, with a more dramatic increase in cross fiber strain than fiber strain. Stretch-activated channels decreased action potential (AP) dispersion in the remote myocardium while increasing it in the border zone. Decreases in electrical connectivity dramatically increased the changes in AP dispersion. The role of cross fiber strain in MI-injured hearts should be investigated more closely, since results indicate that these are more highly elevated than fiber strain in the border of the infarct. Decreases in connectivity may play an important role in the development of altered electrophysiology in the high-stretch regions of the heart.     

Notes on the swim-bladder physiology of cod(Gadus morhua) investigated from the underwater laboratory “Helgoland”

In situ sampling of gas from cod swim-bladders took place during a fortnight's saturation mission with the underwater laboratory Helgoland in May–June 1975. These samples were compared to those done by the conventional method of transporting the fish to the surface for sampling. Based upon these in-situ measurements, the mean O₂-concentration was 55.7% in buoyant cod at 15 m depth. Repeated sampling of the same fish showed a change in gas composition. Compared to the conventional method of transporting fish for sampling to the surface, in-situ sampling gave results with less variation, and indicated that surface-sampling does not give the correct gas composition of buoyant fish at depth of catch.