Trkalian fields: ray transforms and mini-twistors

We study X-ray and Divergent beam transforms of Trkalian fields and their relation with Radon transform. We make use of four basic mathematical methods of tomography due to Grangeat, Smith, Tuy and Gelfand-Goncharov for an integral geometric view on them. We also make use of direct approaches which provide a faster but restricted view of the geometry of these transforms. These reduce to well known geometric integral transforms on a sphere of the Radon or the spherical Curl transform in Moses eigenbasis, which are members of an analytic family of integral operators. We also discuss their inversion. The X-ray (also Divergent beam) transform of a Trkalian field is Trkalian. Also the Trkalian subclass of X-ray transforms yields Trkalian fields in the physical space. The Riesz potential of a Trkalian field is proportional to the field. Hence, the spherical mean of the X-ray (also Divergent beam) transform of a Trkalian field over all lines passing through a point yields the field at this point. The pivotal point is the simplification of an intricate quantity: Hilbert transform of the derivative of Radon transform for a Trkalian field in the Moses basis. We also define the X-ray transform of the Riesz potential (of order 2) and Biot-Savart integrals. Then, we discuss a mini-twistor respresentation, presenting a mini-twistor solution for the Trkalian fields equation. This is based on a time-harmonic reduction of wave equation to Helmholtz equation. A Trkalian field is given in terms of a null vector in C3 with an arbitrary function and an exponential factor resulting from this reduction.Comment: 37 pages, http://dx.doi.org/10.1063/1.482610

Combination therapy with triamcinolone acetonide and bevacizumab for the treatment of severe radiation maculopathy in patients with posterior uveal melanoma

Nisha V Shah,1 Samuel K Houston,1 Arnold Markoe,2 Timothy G Murray1,2,3 1Department of Ophthalmology, Bascom Palmer Eye Institute, University of Miami Miller School of Medicine, Miami, FL, USA; 2Department of Radiation Oncology, University of Miami Miller School of Medicine, Miami, FL, USA; 3Murray Ocular Oncology and Retina, Miami, FL, USA Purpose: To evaluate the role of intravitreal triamcinolone acetonide in patients who developed severe, visually compromising radiation maculopathy or progressed despite anti-angiogenic treatments. Methods: An Institutional Review Board approved, consecutive, retrospective study from 2006 to 2009 of patients who developed severe, visually compromising radiation retinopathy manifesting as macular edema secondary to iodine-125 plaque brachytherapy for posterior uveal melanoma, were treated with a combination of intravitreal bevacizumab and intravitreal triamcinolone. Patients were evaluated with spectral domain optical coherence tomography (SD-OCT) at 2–4 month intervals following plaque removal. Treatment with intravitreal bevacizumab commenced at the first signs of visually compromising macular edema diagnosed with SD-OCT. Triamcinolone acetonide was administered to patients with severe maculopathy as consolidative therapy, or for patients that were refractory to repeated bevacizumab injections with persistent or worsening cystoid macular edema and lack of improvement or progressive worsening of best corrected visual acuity (BCVA). Results: Twenty-five patients were evaluated after receiving a combination of intravitreal bevacizumab and triamcinolone. Initial treatment commenced at a mean of 14.5 (range of 2–42) months after plaque brachytherapy. Patients were given a mean of two injections (range 1–6) of triamcinolone acetonide, and a mean of 8.8 bevacizumab injections (range of 1–26) with a mean follow-up of 31.2 months. Radiation maculopathy upon first detection had a mean SD-OCT grade of 3.6 (median = 4), with an associated mean entry level BCVA of 20/70. Visual acuity at time of first intravitreal triamcinolone was 20/138. At last follow-up (mean of 45.5 months after plaque brachytherapy) mean BCVA was 20/136; however, 9 of 25 (36%) patients who presented with severe radiation maculopathy demonstrated 20/50 or better vision at last follow-up. Conclusion: This case series suggests a beneficial role for intravitreal triamcinolone as a consolidation treatment for patients who present with severe radiation maculopathy or as an adjuvant to bevacizumab for refractory or progressive maculopathy. Keywords: triamcinolone acetonide, bevacizumab, radiation maculopathy, uveal melanom