Twisted scroll wave dynamics: partially pinned waves in excitable chemical media.

We present an investigation of the dynamics of scroll waves that are partially pinned to inert cylindrical obstacles of varying lengths and diameters in three-dimensional Belousov-Zhabotinsky excitable media. Experiments are carried out in which a scroll wave is initiated with a special orientation to be partially pinned to the obstacle. Numerical simulations with the Oregonator model are also carried out, where the obstacle is placed in the region of the core of a preexisting freely rotating scroll wave. In both cases, the effect of the obstacle on the wave dynamics is almost immediately observable, such that after the first revolution of the wave, the pinned region of the scroll wave has a longer period than that of the freely rotating scroll wave. The dependence of the scroll wave period on the obstacle position gives rise to a transition from a straight scroll wave to a twisted scroll wave in the pinned region, while the form of the freely rotating wave remains unchanged. The twisted scroll wave arises from the interaction of the freely rotating scroll wave with the obstacle, giving rise to a pinned twisted wave with the same period. The twisted scroll wave gradually advances, displacing the slower untwisted scroll wave until the scroll wave helically wraps around the entire obstacle. At this point, the period of the entire wave has a single value equal to that of the freely rotating scroll wave. The time for the transition to the twisted wave structure increases when either the obstacle length is increased or the obstacle diameter is decreased, while the average speed of the development increases with both the obstacle length and diameter. After the transition, the twisted wave remains stable, with its structure depending on the obstacle diameter - the larger the diameter, the shorter the helical pitch but the higher the twist rate.

Correction of B 0-induced geometric distortion variations in prospective motion correction for 7T MRI.

OBJECTIVE: Prospective motion correction can effectively fix the imaging volume of interest. For large motion, this can lead to relative motion of coil sensitivities, distortions associated with imaging gradients and B 0 field variations. This work accounts for the B 0 field change due to subject movement, and proposes a method for correcting tissue magnetic susceptibility-related distortion in prospective motion correction. MATERIALS AND METHODS: The B 0 field shifts at the different head orientations were characterized. A volunteer performed large motion with prospective motion correction enabled. The acquired data were divided into multiple groups according to the object positions. The correction of B 0-related distortion was applied to each group of data individually via augmented sensitivity encoding with additionally integrated gradient nonlinearity correction. RESULTS: The relative motion of the gradients, B 0 field and coil sensitivities in prospective motion correction results in residual spatial distortion, blurring, and coil artifacts. These errors can be mitigated by the proposed method. Moreover, iterative conjugate gradient optimization with regularization provided superior results with smaller RMSE in comparison to standard conjugate gradient. CONCLUSION: The combined correction of B 0-related distortion and gradient nonlinearity leads to a reduction of residual motion artifacts in prospective motion correction data.

Correction of gradient nonlinearity artifacts in prospective motion correction for 7T MRI.

PURPOSE: To demonstrate the effect of gradient nonlinearity and develop a method for correction of gradient nonlinearity artifacts in prospective motion correction (Mo-Co). METHODS: Nonlinear gradients can induce geometric distortions in magnetic resonance imaging, leading to pixel shifts with errors of up to several millimeters, thereby interfering with precise localization of anatomical structures. Prospective Mo-Co has been extended by conventional gradient warp correction applied to individual phase encoding steps/groups during the reconstruction. The gradient-related displacements are approximated using spherical harmonic functions. In addition, the combination of this method with a retrospective correction of the changes in the coil sensitivity profiles relative to the object (augmented sensitivity encoding (SENSE) reconstruction) was evaluated in simulation and experimental data. RESULTS: Prospective Mo-Co under gradient fields and coils sensitivity inconsistencies results in residual blurring, spatial distortion, and coil sensitivity mismatch artifacts. These errors can be considerably mitigated by the proposed method. High image quality with very little remaining artifacts was achieved after a few iterations. The relative image errors decreased from 25.7% to below 17.3% after 10 iterations. CONCLUSION: The combined correction of gradient nonlinearity and sensitivity map variation leads to a pronounced reduction of residual motion artifacts in prospectively motion-corrected data.

Effect of excitability on partially pinned scroll waves in excitable chemical media.

We present an investigation of excitability effects on the dynamics of scroll waves partially pinned to inert cylindrical obstacles in three-dimensional Belousov-Zhabotinsky excitable media. We also report on corresponding numerical simulations with the Oregonator model. The excitability varies according to the concentration of sulfuric acid [H_{2}SO_{4}] in the Belousov-Zhabotinsky (BZ) reaction and the parameter ɛ^{-1} in the Oregonator model. Initially, the freely rotating scroll segment rotates faster than the pinned one. The difference in the frequency of the two parts results in a transition from a straight pinned scroll wave to a twisted one, which helically wraps around the entire obstacle. The wave frequency in the whole volume is equal to that of the freely rotating scroll wave. When the excitability is increased, the time for the transition to the twisted wave structure decreases while the average speed s of the development increases. After the transition, the twisted wave remains stable. In media with higher excitability, the helical pitch is shorter but the twist rate ω is higher. Analysis presented in this study together with our previous findings of the effect of the cylindrical obstacle diameter on the wave dynamics results in common features: The average speed s and the twist rate ω of both studies fit well to functions of the difference in the initial frequency Δf of the freely rotating and untwisted pinned waves. We also demonstrate the robustness of the partially pinned scroll waves against perturbations from spontaneous waves emerging during the wave generation in the BZ medium with high [H_{2}SO_{4}]. Even though the scroll wave is partly disturbed at the beginning of the experiment, the spontaneous waves are gradually suppressed and the typical wave structure is finally developed.

Combined prospective and retrospective motion correction to relax navigator requirements.

Prospective motion correction can prevent motion artifacts in magnetic resonance imaging of the brain. However, for high-resolution imaging, the technique relies on precise tracking of head motion. This precision is often limited by tracking noise, which leads to residual errors in the prospectively-corrected k-space data and artifacts in the image. This work shows that it is possible to estimate these tracking errors, and hence the true k-space sample locations, by applying a two-sided filter to the tracking data after imaging. A conjugate gradient reconstruction is compared to gridding as a means of using this information to retrospectively correct for the effects of the residual errors.

Self-organization of multiarmed spiral waves in excitable media.

We present an investigation of self-organized multiarmed spiral waves pinned to unexcitable circular obstacles in a thin layer of the excitable Belousov-Zhabotinsky reaction and in simulations using the Oregonator model. The multiarmed waves are initiated by a series of wave stimuli. In the proximity of the obstacle boundary, the wave rotation around the obstacle causes damped oscillations of the wave periods of all spiral arms. The dynamics of wave periods cause the wave velocities as well as the angular displacements between the adjacent arms to oscillate with decaying amplitudes. Eventually, all displacements approach approximately the same stable value so that all arms are distributed evenly around the obstacle. A further theoretical analysis reveals that the temporal dynamics of the angular displacements can be interpreted as underdamped harmonic oscillations. Far from the obstacles, the wave dynamics are less pronounced. The wave period becomes stable very soon after the initiation. When the number of spiral arms increases, the rotation of individual arms slows down but the wave period of the multiarmed spiral waves decreases. Due to their short period, multiarmed spiral waves emerging in the heart potentially result in severe pathological conditions.

Stability of scroll ring orientation in an advective field.

The stability of the orientation of scroll rings in the excitable Belousov-Zhabotinsky reaction under an applied electrical current was investigated in experiments and simulations. The parallel and antiparallel orientations of the scroll ring unit vector with respect to the current are two stationary states, the first one unstable, the latter stable. For any other orientation, the scroll rings were forced to rotate by the current. At the stable stationary orientation, the scroll rings may contract or expand under the same applied current depending on the radius of the scroll rings. In simulations, delicate adjustments caused a scroll ring to propagate with a constant radius in an advective field.

Reorientation of scroll rings in an advective field.

When scroll rings in the excitable Belousov-Zhabotinsky reaction are subjected to an applied electrical current, a reorientation of the scroll ring is induced which is accompanied by a linear drift towards the cathode. The findings can be explained using a modified theory of local filament dynamics under parameter gradients. Numerical simulations using the Oregonator model with an additional advective term accounting for the applied electric field reproduce the experimental results and provide insights into the deformation of the structure of the filament during the reorientation.

Robustness of free and pinned spiral waves against breakup by electrical forcing in excitable chemical media.

We present an investigation on the breakup of free and pinned spiral waves under an applied electrical current in the Belousov-Zhabotinsky reaction. Spiral fronts propagating towards the negative electrode are decelerated. A breakup of the spiral waves occurs when some segments of the fronts are stopped by a sufficiently strong electrical current. In the absence of obstacles (i.e., free spiral waves), the critical value of the electrical current for the wave breakup increases with the excitability of the medium. For spiral waves pinned to circular obstacles, the critical electrical current increases with the obstacle diameter. Analysis of spiral dynamics shows that the enhancement of the robustness against the breakup of both free and pinned spiral waves is originated by the increment of wave speed when either the excitability is strengthened or the obstacle size is enlarged. The experimental findings are reproduced by numerical simulations using the Oregonator model. In addition, the simulations reveal that the robustness against the forced breakup increases with the activator level in both cases of free and pinned spiral waves.

Propagation of spiral waves pinned to circular and rectangular obstacles.

We present an investigation of spiral waves pinned to circular and rectangular obstacles with different circumferences in both thin layers of the Belousov-Zhabotinsky reaction and numerical simulations with the Oregonator model. For circular objects, the area always increases with the circumference. In contrast, we varied the circumference of rectangles with equal areas by adjusting their width w and height h. For both obstacle forms, the propagating parameters (i.e., wavelength, wave period, and velocity of pinned spiral waves) increase with the circumference, regardless of the obstacle area. Despite these common features of the parameters, the forms of pinned spiral waves depend on the obstacle shapes. The structures of spiral waves pinned to circles as well as rectangles with the ratio w/h∼1 are similar to Archimedean spirals. When w/h increases, deformations of the spiral shapes are observed. For extremely thin rectangles with w/h≫1, these shapes can be constructed by employing semicircles with different radii which relate to the obstacle width and the core diameter of free spirals.

Unpinning of spiral waves by electrical forcing in excitable chemical media.

We present experimental observations on the electrically forced release of spiral waves pinned to unexcitable circular obstacles in the Belosov-Zhabotinsky reaction. When the applied electric current density reaches the necessary current density J(unpin), the spiral tip is detached and subsequently drifts away from the obstacle. J(unpin) is found to increase with the obstacle diameter d. The growth rate ΔJ(unpin)/Δd is much higher for obstacles larger than the free spiral core compared to that for smaller obstacles. The experimental findings are confirmed by numerical simulations using the Oregonator model. The results imply that it is more difficult to release spiral waves pinned to larger obstacles, especially when the obstacle size exceeds that of the free spiral core.

Influence of excitability on unpinning and termination of spiral waves.

Application of electrical forcing to release pinned spiral waves from unexcitable obstacles and to terminate the rotation of free spiral waves at the boundary of excitable media has been investigated in thin layers of the Belousov-Zhabotinsky (BZ) reaction, prepared with different initial concentrations of H_{2}SO_{4}. Increasing [H_{2}SO_{4}] raises the excitability of the reaction and reduces the core diameter of free spiral waves as well as the wave period. An electric current with density stronger than a critical value Junpin causes a pinned spiral wave to drift away from the obstacle. For a given obstacle size, Junpin increases with [H_{2}SO_{4}]. Under an applied electrical current, the rotation center of a free spiral wave drifts along a straight path to the boundary. When the current density is stronger than a critical value Jterm, the spiral tip is forced to hit the boundary, where the spiral wave is terminated. Similar to Junpin for releasing a pinned spiral wave, Jterm also increases with [H_{2}SO_{4}]. These experimental findings were confirmed by numerical simulations using the Oregonator model, in which the excitability was adjusted via the ratio of the excitation rate to the recovery rate of the BZ reaction. Therefore, our investigation shows that decreasing the excitability can facilitate elimination of spiral waves by electrical forcing, either in the presence of obstacles or not.

Scroll wave instabilities in an excitable chemical medium.

Two kinds of scroll wave instabilities were studied experimentally in the excitable Belousov-Zhabotinsky reaction: three-dimensional meandering and negative line tension of the scroll wave filament. The filament displays a flat zigzag shape in the initial stages of the experiment. As the chemical medium ages, the filament assumes a wiggly shape while its length increases substantially. Numerical simulations underpin the experimental findings and their interpretation.

An elegant method to study an isolated spiral wave in a thin layer of a batch Belousov-Zhabotinsky reaction under oxygen-free conditions.

A method to prepare a uniform thin layer of a batch Belousov-Zhabotinsky (BZ) reaction under oxygen-free conditions for the study of an isolated spiral wave is presented. After a first layer of gel soaked with the BZ solution has been delivered into the reactor, a single spiral wave was initiated, and finally the remaining reactor volume was filled with gel and BZ medium. The completely filled reactor is sealed gas-tightly, yielding oxygen-free, and thus more controlled, reaction conditions. A systematic study of the behaviour of an isolated spiral wave in a ferroin-catalyzed BZ reaction under batch conditions has been performed. Recipes for BZ media that support a slowly rotating meandering spiral were developed. In cases of extremely low excitability (i.e., relative large stimuli are required to induce a propagating wave), the number of petals in the trajectory of a spiral tip decreased due to aging of the reaction system. Since oxygen-free conditions are necessary for the study of the dynamics in three-dimensional excitable media, and the wave velocities of a spiral are sufficiently low, the developed chemical recipes are suitable for studies of the behaviour of scroll waves in three-dimensional systems by optical tomography.