Section 10.11

Problem 43Suppose you measure the arrival time of an action potential wave front at four points (1-4) in a diamond pattern, each a distance b from the central point (red). Calculate the wave front speed, direction, and curvature from these four measurements.

a) Assume the wave front is circular and propagates outward from the origin. Use the law of cosines to write r_{1}, r_{2}, r_{3}, and r_{4}(the distance of each electrode to the origin) in terms ofr, b, and the angle θ._{0}

b) Pull a factor ofoutside of the square root in each of your four expressions from part a).r_{0}

c) Assumer_{0}is much greater than b, and perform a Taylor expansion of each of the four expressions in terms of the small parameter ε = b/. Include terms that are constant, linear, and quadratic in ε.r_{0}

d) Write the arrival time at each electrode (n=1, 2, 3, and 4) as t_{n}=r/v, where v is the wave speed._{n}

e) Let Δt_{ij}=t_{i}– t_{j}. Find expressions for Δt_{13}and Δt_{24}in terms of b, θ, and v. Solve these expressions to determine v and θ in terms of Δt_{13}, Δt_{24}, and b.

f) Find expressions for Δt_{14}and Δt_{23}in terms of b, θ, and v. Now (and this is the most difficult step), find an expression for the radius of curvature, r_{0}, in terms of b, Δt_{13}, Δt_{24}, Δt_{14}, and Δt_{23}.

There are several advantages and several disadvantages to the expressions you will derive. The advantages are that the calculations require only four measurements of arrival time, and they provide not only the speed and direction but also (somewhat unexpectedly--at least to me) the radius of curvature,

*r*. The radius of curvature is important for propagation, because highly curved wave fronts propagate more slowly than nearly flat wave fronts. The radius of curvature at the core of a spiral wave is highly curved, and this curvature influences properties of the spiral wave such as how fast it rotates. There are some important limitations. First, a close examination of your expression for the radius of curvature will reveal that the method gives an indeterminate expression for propagation at angles of

_{0}*θ*= 45, 135, 225, and 315°. Second, the expressions contain the differences of activation times. In fact, the radius of curvature depends on the difference of a difference of activation times. If these activation times are all similar, then they need to be known precisely for the calculation of their differences to be accurate. The calculation assumes the wave front is circular, although really the wave front only needs to be circular locally, so this should not be too bad an approximation. The method also is based on the assumption that

*b*is much less than

*r*.

_{0}Despite these limitations, I think the expressions should be useful for characterizing properties of wave fronts in the heart. It may be particularly useful for obtaining wave front speed, direction, and curvature in computer simulations, where the calculation is computed over a regular two-dimensional Cartesian grid and where noise in the activation times may not be a big concern.

Have you tested your methods, comparing your approximations to experimental data? It would be interesting to identify the constraints associated with real tissue under which your simulations agree with measurements. If so, what experimental arrangement did you choose? Details, Please!

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