September 2, 2000
A simple shape is a contiguous set of points in three-space. Along the distinguished axis, each simple shape is vertically delimited by upper and lower border singularities. Between those singularities, the simple shape is delimited by a closed polygon composed of one or more border fences, which serve to separate this simple shape from adjacent ones or the infinite medium.
For the purposes of cardiac modeling, simple shapes serve to provide 3-D geometry to simulated tissue walls (e.g. the right ventricular free wall) and polyhedra in the torso model.
A Compound Shape is a collection of simple shapes that share common borders, which combine to represent objects with complex nested regions in space (e.g. the heart), plus any associated curves and loci.
The shared wall that separates any two simple shapes (or a simple shape from the infinite medium) is called a border. Borders provide geometry to the shapes that share those borders.
Curves are one-dimensional entities, consisting of a connected set of points in three-space.
For the purposes of cardiac modeling, curves are typically used to provide geometry for simulated fibers (e.g. atrioventricular bypass tracts).
A locus is a zero-dimensional entity, representing a point in three- space. Each locus always resides on exactly one explicit slice.
For the purposes of cardiac modeling, loci are typically used to position tissue nodes (e.g. the SA node foci) and interspatial junctions.
A string representing the name of the tissue type.
A flag that indicates whether cells of this tissue type are to be considered neuromuscular in nature. Only neuromuscular cells are capable of electrophysiological activation.
A flag which indicates whether the associated subtissues should be modelled as being composed of fibers oriented according to the relevant subtissue parameters (e.g. inner and outer fiber angles for tissue walls), particularly for the purposes of modelling anisotropic conduction and recovery gradients. If this flag is not set, any per-subtissue fiber orientation parameters are ignored for the associated subtissues; propagation and recovery will be isotropic in nature.
A piecewise linear representation of the basic action potential waveform (potential over time) for simulated cells of this tissue type. This base waveform is usually altered by interval/duration or interval/potential effects, or by prevailing pharmacology, to yield the actual per-cell waveform employed during electrophysiological simulation trials.
Though always piece-wise linear, the action potential waveform may be arbitrarily complex, but always comprises a distinct sequence of activation phases.
A temporal interval used as the propagation delay from cells of this tissue type to cells in other cell sets (e.g. from Purkinje to myocardial cells).
The isotropic bulk electrical conductivity for cells of this tissue type, used in dipole source generation.
A flag indicating whether the bench-wide vertical phase 2 duration gradient (measured in seconds/meter) should be applied to cells of this tissue type, typically to provide ventricular repolarization gradients.
The scalar ratio of longitudinal (along simulated fiber axes) and transverse (perpendicular to fiber axes) conduction speeds, used to compute propagation delays in various directions during simulated conduction.
For tissue types that are specified as having no fiber orientations, this parameter is ignored.
The name of the subtissue.
The name of the tissue type associated with this subtissue, from which it inherits many characteristics.
A flag which indicates whether cells of this subtissue are to be treated as excitable. An excitable cell will be activated by its inherent automaticity, or if excited by activation of a suitable neighboring cell, and exhibit its action potential waveform as it moves over simulated time through the fixed activation phases. In CESLab, only interfaces between neighboring excitable cells are capable of generating current dipole.
A flag which indicates whether cells of this subtissue are to be considered automatic. Each automatic cell will depolarize a certain fixed delay (known as the coupling interval) after the start of each cardiac cycle (unless it is activated earlier by a neighboring cell), and, if automatic cell retriggering has been enabled for the associated preparation, will be activated regularly thereafter based on its specified automatic period (again unless activated earlier).
For subtissues for which the "Is Automatic" flag is enabled, this specifies the temporal interval between successive automatic activations in a given trial (note that the first activation is based solely upon the coupling interval).
The automatic period of any cell may be influenced by prevailing pharmacology, in accord with the appropriate Modulator Effect Modality.
For automatic cells, the temporal interval after the start of the cardiac cycle at which this cell will depolarize. Note that successive automatic activations (if any) in any given trial will be timed in accordance with the specified automatic period, and not this value.
A flag which indicates whether cells of this subtissue will generate current dipoles. This flag permits the user to selectively ignore dipole generation on a per-subtissue basis, e.g. to observe only the dipole generated by a single subtissue.
A numerical factor that is multiplied into the square root of the phase 0 upstroke velocity (measured in Volts/second) to yield the axial and transverse conduction speeds of cells of this subtissue (see Roberts, 1979, and Lorange and Gulrajani, 1993).
A scalar potential which, if/when exceeded by the action potential of a neighboring cell, will initiate (after the appropriate conduction delay) activation of this cell (assuming it is excitable, i.e. in activation phase 3 RRP or 4).
In general, tissue walls possess the following characteristics:
The linear thickness of this tissue wall (used only for tissue walls of uniform thickness generated from compound shapes, e.g. Purkinje layers).
The angles to be used in establishing the fiber orientation of cells located at the inside and outside surfaces (respectively) of this tissue wall. Cells lying within the tissue wall acquire fiber orientations by interpolation between these extrema based upon fractional depth and the prevailing intramural fractional depth mapping script interpreter function.
Fiber angles are measured and signed in the manner of Streeter 1979: a zero angle means the fibers are oriented completely horizontally, lying in a plane tangential to the tissue wall surface. Positive angles imply a counter-clockwise rotation of the fibers within that tangent plane as seen from outside the heart.
This specifies an intramural phase 2 duration gradient to be applied in accordance with per-cell outward depth, which is defined as the Euclidean distance between the cell and the nearest point on an endocardial surface.
Note that, for these purposes, the right side of the ventricular septum is treated as an epicardial surface after the data of Burgess et al, which indicate that the duration of phase 2 is longer on the right side of the septum than the left.
The position of the cell in three-space. As in most finite-element models, valid locations are strictly quantized in a rectangular 3-D grid or mesh. No two cells can occupy the same location.
A number that uniquely identifies the associated subtissue.
Fractional depth is relevant only for cells that are associated with a tissue wall, and indicates the intramural fraction at which the cell resides in the encompassing wall: zero indicates the cell resides at the epicardium, unity at the endocardium.
This specifies the line in three-space along which myocardial fibers would lie in a biological preparation at this cell location. The fiber orientation controls the speed at which the cell propagates activation to its neighbors. Typically, activation is propagated more rapidly to neighbors that are located in the direction of the fiber orientation.
For some small number of cells in a tissue wall, a fiber orientation may not be calculable (e.g. cells at the extreme apex of the heart have no obvious proper orientation). For such cells, the non-directional conduction velocity of the associated subtissue is used to compute activation propagation delays.
This is a fixed set of temporal intervals (in simulated time quanta) representing the time delay from the activation of this cell after which activation may be propagated to other cells in the immediate neighborhood (which may be either 18 or 26 locations).
These time delays are scaled by the Euclidean distance between the centroid of this cell and each particular neighbor. During electrophysiological simulation trials, propagation delays are typically adjusted in accordance with prevailing decremental conduction curves and electropharmacology.
Used to store the initial potential for the facet, which that induced on the facet by the test dipole.
For the first iteration of the G&S procedure, this field is set to the charge necessary to exactly cancel out any force along the facet outward unit normal caused by the test dipole. For succeeding iterations, it is the charge necessary to cancel out the total force along the facet outward unit normal evoked by the old charge deltas on all other facets.
At the end of the G&S procedure, this field is set to the initial potential plus the aggregate potential induced by the total charges at all other facets.
This field serves to accumulate all values applied to "New Charge Delta".
This field serves to hold the "New Charge Delta" value from the last iteration of the G&S procedure.
An array of computed transfer coefficients, containing one vector for each regional dipole source location. Instantaneous current dipole vectors collected at each regional dipole are multiplied into the corresponding transfer coefficient vector, and then algebraically summed to yield scalar surface potentials.
The electrodes are grouped as follows:
The three limb electrodes (left, right, and umbilical). Typically, they form an equilateral triangle about the heart location in the frontal plane of the torso.
The six precordial electrodes, positioned at locations fixed by convention.
These lead potentials are computed as the bipolar differences between the relevant limb electrode potentials.
The augmented lead potentials are one limb electrode potential relative to the average of the other two (with a resultant augmentation over the potential relative to the average of all three).
The precordial leads are unipolar potentials relative to the Wilson potential (the average of the three limb lead potentials). See Pilkington and Plonsey, p. 91 for more information on lead systems and relevant electrical theory and mathematics.
VX = 1/2 (V5 + V6) - V7
VY = V8 - V9
VZ = V4 - 1/3 (V1 + V2 + V3)
VX = 0.610VA + 0.171VC - 0.781VI
VY = 0.655VF + 0.345VM - 1.000VH
VZ = 0.736VM + 0.133VA - 0.264VI
-0.374VE - 0.231VC
The maximal conduction speeds generally found in biological neuromuscular tissue are on the order of 3 m/s. In a high-resolution preparation at 1 mm per cell, this produces a minimal intercell propagation delay of over 20 time quanta, which yields a high degree of temporal resolution even in this worst case scenario.
In CESLab, the cubic cells are arranged in a 3-D rectangular grid, so each cell has six locations with which it shares a common cubic face (monaxial neighbors), twelve locations with which it shares only an edge (diaxial neighbors), and eight locations with which is shares only a corner vertex (triaxial neighbors).
The neighborhood size for a given preparation is configurable by the user from the following choices:
Activation may be conducted to up to six axial neighbors and twelve diaxial neighbors.
Activation may be conducted to up to six axial neighbors, twelve diaxial neighbors, and eight triaxial neighbors.
Each activation phase may contain an arbitrary number of subphases, each of which represents a linear segment of the action potential waveform. By adding enough subphases to the phases above, waveforms of any degree of complexity can be modelled as closely as desired.
Cells remain in the resting state until they are activated by a neighbor (or, for automatic cells, their automatic period elapses).
Cells remain in phase 0 for a fixed time period, and then enter phase 1.
Cells remain in phase 1 for a fixed time period, after which they enter phase 0.
Cells remain in phase 3 ARP for a time duration dependent on the base action potential waveform specified for the tissue type, multiplied by any prevailing interval/duration factors, after which they enter phase 3 RRP.
Cells remain in phase 3 RRP until reactivated, or until a specific time duration elapses (the phase 3 RRP duration from the action potential waveform specified for the associated tissue type, multiplied by the interval/duration factor), after which they reenter phase 4.
The delay to be employed before propagation of activation to a neighboring cell is dependent upon the following:
For each of the three modelled cycle-length effects, the user may specify a single non-tissue type-specific curve and/or tissue type- specific curves for any/all tissue types. For activation of a given cell, if a tissue type-specific curve exists for the tissue type associated with the cell, it is employed; otherwise, the non-tissue type-specific curve is employed.
Each curve yields a scalar factor that is multiplied into the conduction speed, action potential (relative to the resting potential), or action duration to express the desired effect.
For the first activation of a given cell within a trial, the default interactivation interval for the preparation is used to look up the appropriate points on the relevant curves, but any succeeding activations within the trial will use the actual time elapsed between the activation and the one immediately preceding it.
For the first activation of a given cell within a trial, a decremental conduction factor of unity is always employed.
The scaling factor is multiplied into the base automatic period specified for the subtissue to compute the effective automatic period.
This effect modality causes the excitation threshold value specified in the subtissue to be linearly rescaled with respect to the modulated resting potential.
For example, if the unadjusted resting potential of a given cell is -80 mV, its unadjusted excitation threshold (listed in the subtissue window) is -10 mV, the adjusted resting potential is -70 mV, and pharmacology causes the net excitation threshold rescaling factor to be 1.1, the effective excitation threshold will be (-10 - -70) * 1.1 -70 = -4 mV.
This Modulator Effect Modality has the sole effect of inducing conduction block if the excitation threshold is raised so far that a cell cannot excite its neighbor (possibly of a differing tissue type and waveform). There is a clear relationship between raised excitation thresholds and reduced conduction speed in biological preparations, but this is assumed here to be a factor of increased phase 0 duration for barely superthreshold stimuli. In CESLab, to model decreases in conduction speeds produced by a given simulated drug due to increased excitation thresholds, the user must define a suitable curve to rescale the phase 0 duration in sync with the rescaled excitation threshold at differing serum concentrations.
This modality causes the duration of phase 0 of the action potential to be rescaled by the indicated amount. Since the square root of the duration of this phase is inversely correlated with the conduction speed (see Hunter), reduction of phase 0 duration causes the conduction speed to be automatically increased accordingly.
This modality causes the potential at the start of phase 1 (the end of phase 0) to be linearly rescaled with respect to the resting (phase 4) potential. Like the Rescale Phase 0 Duration modality, this modality influences the conduction speed.
This modality causes the duration of phase 1 of the action potential to be rescaled by the indicated amount.
This modality causes the potential at the start of phase 2 (the end of phase 1) to be linearly rescaled with respect to the resting (phase 4) potential.
This modality causes the duration of phase 2 of the action potential to be rescaled by the indicated amount.
This modality causes the potential at the start of phase 3 (the end of phase 2) to be linearly rescaled with respect to the resting (phase 4) potential.
This modality causes the duration of phase 3 of the action potential to be rescaled by the indicated amount. The relative proportion of phase 3 associated with the absolute refractory period (ARP) vs. the relative refractory period (RRP) is preserved.
This modality causes the phase 4 (resting) potential of affected tissue types to be linearly rescaled with respect to the zero potential. Note that, in CESLab, differences between resting potentials of adjacent cells do not produce current dipoles.
This modality causes the vertical phase 2 duration gradient for affected subtissues to be linearly rescaled by the appropriate factor. This permits the user to precisely compensate (if such is desired) for changes in observable ventricular gradients caused by other Modulator Effect Modalities.
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