Numerical Solution of an Applied Biophysics Inverse Problem (original) (raw)

The verification of an inverse problem in radiation therapy

International Journal of Radiation Oncology*Biology*Physics, 1990

The inverse problem in radiation therapy presents a solution for a fluence distribution based on the specification of a region of dos,e in a patient. We show results for one such solution based on the inversion of an integral over a function of the fluence profile of a rotating beam. We use Monte Carlo methods and numerical integrations to evaluate dose distributions obtained with the inverse method and show the limitations of this theoretical approach. Our results show that dose to a single circular region at an arbitrary position in a 2-dimensional volume can be calculated. Uniform dose to arbitrarily shaped regions cannot be calculated with this formalism, although practical solutions can still be obtained.

Solving an Inverse Diffusion Problem for Magnetic Resonance Dosimetry by a Fast Regularization Method

Real-Time Imaging, 2001

A n inverse diffusion problem that appears in Magnetic Resonance dosimetry is studied. The problem is shown to be equivalent to a deconvolution problem with a known kernel. To cope with the singularity of the kernel, nonlinear regularization functionals are considered which can provide regular solutions, reproduce steep gradients and impose positivity constraints. A fast deterministic algorithm for solving the involved non-convex minimization problem is used. Accurate restorations on real 2566256 images are obtained by the algorithm in a few minutes on a 266-MHz PC that allow to precisely quantitate the relative absorbed dose.

Inverse problems explicit and implicit formulations with applications in engineering, biophysics and biotechnology

Inverse Problems in Science and Engineering, 2007

In this work, we present a few explicit and implicit formulations used in the analysis and solution of inverse problems. These formulations have been developed and/or applied by the authors to the solution of different problems with applications in engineering, biophysics and biotechnology. Here emphasis is given to a matrix-based explicit formulation, and to the construction of a generalized family of regularization terms which is used when the inverse problem is formulated implicitly as an optimization problem. The efficacy of the methodologies developed is demonstrated with the results for a few test cases using noiseless and noisy synthetic data.

Numerical Treatment and Evaluation of Inverse Problems

All regularization methods for computing stable solutions to inverse problems, involve a trade-off between the "size" of the regularized solution and the quality of the fit that it provides to the given data. Though the appropriate choice of the regularization parameters is important, resolution and uncertainty analysis are as significant. Thus, we should also proceed to the resolution analysis in order to determine what scale features in the model can actually be resolved. In this work, we choose the proper values of damping and smoothing factors using two of the most well known regularization tools, the Picard condition and the L-curve [8], which generally provide a good estimation of the regularization parameters in combination with the standard maximum likelihood approach [3]. Additionally, the spread function as proposed by Menke [18] and a checkerboard test are applied in order to have an estimate of the resolution. The efficiency of the preferred methods is tested through a series of tests in real data from the area of Urals in Russia.

Efficient parameter estimation in multiresponse models measuring radioactivity retention

Radiation and Environmental Biophysics, 2019

After incorporation of radioactive substances, workers are routinely checked by bioassays (isotopic activity excreted via urine, measurements of radionuclides retained in the whole body or in the lungs, etc.). From the results, the isotopic activity incorporated by the worker is inferred, as well as the values of other parameters related to the metabolism of the incorporated substance, using the 'response function'. This function depends on several factors and it is usually obtained by solving a system of linear differential equations, resulting from the compartmental model which describes the human body (or a part of it). The possibility of using different types of bioassays from the same worker improves estimation of some of the parameters that characterize the solution of the system of equations, specially the unknown incorporated activity to the system. The transfer coefficients are usually considered to be known, using the values that are published in the corresponding International Commission of Radiological Protection (ICRP) publication. In the present study some practical cases will be presented, and optimal design criteria are developed that allow taking the bio-samples at the most informative times. The methodology presented here requires solving the models of element distribution in the human organism as a function of time, for which the recently updated models recommended by the ICRP have been used. Initially thought for workers in facilities dealing with radioactive substances, the study results, procedures and conclusions can be applied to other clinical or laboratory settings, and to the design of action protocols in case of environmental public exposure.

Mathematical modelling of nuclear medicine data

2020 IEEE 20th Mediterranean Electrotechnical Conference ( MELECON), 2020

Positron Emission Tomography using 2-[18F]-2deoxy-D-glucose as radiotracer (FDG-PET) is currently one of the most frequently applied functional imaging methods in clinical applications. The interpretation of FDG-PET data requires sophisticated mathematical approaches able to exploit the dynamical information contained in this kind of data. Most of these approaches are formulated within the framework of compartmental analysis, which connects the experimental nuclear data with unknown tracer coefficients measuring the effectiveness of the tracer metabolism by means of Cauchy systems of ordinary differential equations. This paper provides a coincise overview of linear compartmental methods, focusing on the analytical solution of the forward compartmental problem and on the specific issues concerning the corresponding compartmental inverse problem.

Numerical approximation of solutions of a nonlinear inverse problem arising in olfaction experimentation

Mathematical and Computer Modelling, 2006

Identification of detailed features of neuronal systems is an important challenge in the biosciences today. Cilia are long thin processes that extend from the olfactory receptor neurons into the nasal mucus. Transduction of an odor into an electrical signal occurs in the membranes of the cilia. The cyclicnucleotide-gated (CNG) channels which reside in the ciliary membrane and are activated by adenosine 3',5'-cyclic monophosphate (cAMP) allow a depolarizing influx of Ca 2+ and Na + and are thought to initiate the electrical signal. In this paper, a mathematical model consisting of two nonlinear differential equations and a constrained Fredholm integral equation of the first kind is developed to model experiments involving the diffusion of cAMP into cilia and the resulting electrical activity. The unknowns in the problem are the concentration of cAMP, the membrane potential and, the quantity of most interest in this work, the distribution of CNG channels along the length of a cilium. A simple numerical method is derived that can be used to obtain estimates of the spatial distribution of CNG ion channels along the length of a cilium. Certain computations indicate that this mathematical problem is ill-conditioned.

Mathematical Modeling of Radiography Experiments

An approach is presented to construct operators for transforming the characteristics of incident radiation to transmitted radiation, as well as operators for transforming the transmitted radiation to measured values. Simu-lating the radiation transport is based on Monte Carlo modeling of the interaction of X-ray photons and electrons with matter. The proposed method permits to construct, for instance, the operator connecting the initial radiation spectrum with the absorbed photon energy penetrating a given object. The elaborated approach provides the possibility of effective mathematical modeling of radiation techniques such as radiography, treating complex multi-component objects. Moreover, the method can be used to construct the operator equation for solving in-verse problems, e.g. the reconstruction of the initial radiation spectrum using simple experimental measure-ments. Comparison with some experimental measurements is presented.