We study eigenvalues and eigenfunctions (vibration modes) on the class of self-similar symmetric finitely ramified fractals which includes 3n-gaskets. We consider such examples as the Sierpinski gasket, a non-p.c.f. analog of the Sierpinski gasket, the level-3 Sierpinski gasket, a fractal 3-tree, the hexagasket, and one dimensional fractals. We develop a theoretical matrix analysis, including analysis of singularities, which allows us to compute eigenvalues, eigenfunctions and their multiplicities exactly. We support our theoretical analysis by symbolic and numerical computations.

We show how to calculate the spectrum of the Laplacian operator on fully symmetric, finitely ramified fractals. We consider well-known examples, such as the unit interval and the Sierpiński gasket, and much more complicated ones, such as the hexagasket and a non-post critically finite self-similar fractal. We emphasize the low computational demands of our method. As a conclusion, we give exact formulas for the limiting distribution of eigenvalues (the integrated density of states), which is a purely atomic measure (except in the classical case of the interval), with atoms accumulating to the Julia set of a rational function. This paper is the continuation of the work published by the same authors in Ref. 1.

Abstract. Diffuse optical tomography has shown promising results as a tool for breast cancer screening and monitoring response to chemotherapy. Dynamic imaging of the transient response of the breast to an external stimulus, such as pressure or a respiratory maneuver, can provide additional information that can be used to detect tumors. We present a new digital continuous-wave optical tomography system designed to simultaneously image both breasts at fast frame rates and with a large number of sources and detectors. The system uses a master-slave digital signal processor-based detection architecture to achieve a dynamic range of 160 dB and a frame rate of 1.7 Hz with 32 sources, 64 detectors, and 4 wavelengths per breast. Included is a preliminary study of one healthy patient and two breast cancer patients showing the ability to identify an invasive carcinoma based on the hemodynamic response to a breath hold. C 2011 Society of Photo-Optical Instrumentation Engineers (SPIE).

Peripheral arterial disease (PAD) is the narrowing of arteries due to plaque accumulation in the vascular walls. This leads to insufficient blood supply to the extremities and can ultimately cause cell death. Currently available methods are ineffective in diagnosing PAD in patients with calcified arteries, such as those with diabetes. In this paper we investigate the potential of dynamic diffuse optical tomography (DDOT) as an alternative way to assess PAD in the lower extremities. DDOT is a non-invasive, non-ionizing imaging modality that uses near-infrared light to create spatio-temporal maps of oxy- and deoxy-hemoglobin in tissue. We present three case studies in which we used DDOT to visualize vascular perfusion of a healthy volunteer, a PAD patient and a diabetic PAD patient with calcified arteries. These preliminary results show significant differences in DDOT time-traces and images between all three cases, underscoring the potential of DDOT as a new diagnostic tool.

Objective Vascular optical tomographic imaging (VOTI) is a novel imaging modality that is capable of detecting hemoglobin concentrations in tissue. VOTI is non-invasive, non-ionizing and does not require contrast injection. We applied this technology to diagnosing peripheral arterial disease (PAD) within lower extremities of diabetic patients with calcified arteries. This could be of substantial benefit as these patients suffer from comorbidities such as arterial incompressibility, which complicates diagnosis and monitoring. Methods Forty subjects (10 non-diabetic patients with PAD, 10 diabetic patients with PAD, and 20 healthy volunteers) were enrolled in a diagnostic pilot study using the VOTI system. The patients were imaged during a thigh pressure cuff occlusion. Results The VOTI system was capable of quantifying the blood volume changes within the foot during the thigh cuff occlusion and outputting diagnostic parameters, such as change in hemoglobin concentration, enabling the assessment of foot perfusion. This study resulted in a statistically significant difference between the healthy cohort and both the non-diabetic and the diabetic PAD cohorts (P=0.006, P=0.006). Receiver operating characteristic (ROC) curve analysis showed that PAD diagnosis could be made with over 80% sensitivity or specificity depending on the characteristic cut-off point. In addition, VOTI was capable of providing the locations of under-perfused regions within the foot and evaluating the severity of arterial disease, even within diabetic patients with calcified arteries, which are traditionally difficult to diagnose. Conclusion VOTI can effectively diagnose PAD independently of arterial compressibility, making it very useful for assessing vascular disease in diabetic patients.