17 May 2010 Concentrations of hemoglobin fractions calculation using modified Lambert-Beer law and solving of an ill-posed system of equations
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Based on the optical methods, it could be able to identify the concentrations of the hemoglobin fractions in a few seconds. An array of Light Emitting Diodes "LEDs" with different wavelengths and a photo detector were used to detect the transmitted or reflected light intensities. A system of equations was built according to the modified Lambert-Beer Law to be solved as an inverse problem. Typically practical inverse problems are all ill-posed. Even well-posed problems may be unstable or ill-conditioned when they digitalized. Also, applications in tissue spectroscopy and optical imaging result in an ill-posed problems. The difficulties involved in solving ill-posed systems are known, where large changes in the solution result from small perturbations in the right hand members of the system that can be caused by measurement tolerance or noise. A good numerical method to solve it may be beneficial in the applications to the optimization problems including linear programming and nonlinear programming. To account for the sensitivity to noise, a regularization method is usually applied to solve this sort of ill-posed problem, where a suitable regularized parameter is used to depress the bias in the computed solution by a better balance of approximation error and propagated data error. The Method has better computational efficiency and accuracy even for the highly ill-conditioned linear equations with a large disturbance on the given data. Comparing the results with that obtained from a direct solution of the system of equations, we prove well-posedness, stability and convergence of the method. The estimated hemoglobin fractions were highly correlated to the reference laboratory measurements.
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Omar Abdallah, Omar Abdallah, Mohammed Natsheh, Mohammed Natsheh, Kawther Abo Alam, Kawther Abo Alam, Qasem Qananwah, Qasem Qananwah, Ahmed Al Nabulsi, Ahmed Al Nabulsi, Armin Bolz, Armin Bolz, } "Concentrations of hemoglobin fractions calculation using modified Lambert-Beer law and solving of an ill-posed system of equations", Proc. SPIE 7715, Biophotonics: Photonic Solutions for Better Health Care II, 77151I (17 May 2010); doi: 10.1117/12.854603; https://doi.org/10.1117/12.854603

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