Angiogenesis is the generation of mature vascular networks from pre-existing vessels. Angiogenesis is crucial during the organism' development, for wound healing and for the female reproductive cycle. Several murine experimental systems are well suited for studying developmental and pathological angiogenesis. They include the embryonic hindbrain, the post-natal retina and allantois explants. In these systems vascular networks are visualised by appropriate staining procedures followed by microscopical analysis. Nevertheless, quantitative assessment of angiogenesis is hampered by the lack of readily available, standardized metrics and software analysis tools. Non-automated protocols are being used widely and they are, in general, time - and labour intensive, prone to human error and do not permit computation of complex spatial metrics. We have developed a light-weight, user friendly software, AngioTool, which allows for quick, hands-off and reproducible quantification of vascular networks in microscopic images. AngioTool computes several morphological and spatial parameters including the area covered by a vascular network, the number of vessels, vessel length, vascular density and lacunarity. In addition, AngioTool calculates the so-called “branching index” (branch points / unit area), providing a measurement of the sprouting activity of a specimen of interest. We have validated AngioTool using images of embryonic murine hindbrains, post-natal retinas and allantois explants. AngioTool is open source and can be downloaded free of charge.

The formal representation of the quasi-periodic Helmholtz Green's function obtained by the method of images is only conditionally convergent and, thus, requires an appropriate summation convention for its evaluation. Instead of using this formal sum, we derive a candidate Green's function as a sum of two rapidly convergent series, one to be applied in the spatial domain and the other in the Fourier domain (as in Ewald's method). We prove that this representation of Green's function satisfies the Helmholtz equation with the quasi-periodic condition and, furthermore, leads to a fast algorithm for its application as an operator.We approximate the spatial series by a short sum of separable functions given by Gaussians in each variable. For the series in the Fourier domain, we exploit the exponential decay of its terms to truncate it. We use fast and accurate algorithms for convolving functions with this approximation of the quasi-periodic Green's function. The resulting method yields a fast solver for the Helmholtz equation with the quasi-periodic boundary condition. The algorithm is adaptive in the spatial domain and its performance does not significantly deteriorate when Green's function is applied to discontinuous functions or potentials with singularities. We also construct Helmholtz Green's functions with Dirichlet, Neumann or mixed boundary conditions on simple domains and use a modification of the fast algorithm for the quasi-periodic Green's function to apply them.The complexity, in dimension dR2, of these algorithms is O(k d log kCC(log e K1 ) d ), where e is the desired accuracy, k is proportional to the number of wavelengths contained in the computational domain and C is a constant. We illustrate our approach with examples.

We develop fast discrete Fourier transforms (and their adjoints) from a square in space to a disk in the Fourier domain. Since our new transforms are not unitary, we develop a fast inversion algorithm and derive corresponding estimates that allow us to avoid iterative methods typically used for inversion. We consider the eigenfunctions of the corresponding band-limiting and space-limiting operator to describe spaces on which these new transforms can be inverted and made useful. In the process, we construct polar grids which provide quadratures and interpolation with controlled accuracy for functions band-limited within a disk. For rapid computation of the involved trigonometric sums we use the unequally spaced fast Fourier transform, thus yielding fast algorithms for all new transforms. We also introduce polar grids motivated by linearized scattering problems which are obtained by discretizing a family of circles. These circles are generated by using a single circle passing through the origin and rotating this circle with the origin as a pivot. For such grids, we provide a fast algorithm for interpolation to a near optimal grid in the disk, yielding an accurate adjoint transform and inversion algorithm.