Kiwoon Kwon

Born expansion and Fréchet derivatives in nonlinear Diffuse Optical Tomography

The nonlinear Diffuse Optical Tomography (DOT) problem involves the inversion of the associated coefficient-to-measurement operator, which maps the spatially varying optical coefficients of turbid medium to the boundary measurements. The inversion of the coefficient-to-measurement operator is approximated by using the Frechet derivative of the operator. In this work, we first analyze the Born expansion, show the conditions which ensure the existence and convergence of the Born expansion, and compute the error in the mth order Born approximation. Then, we derive the mth order Frechet derivatives of the coefficient-to-measurement operator using the relationship between the Frechet derivatives and the Born expansion.

Electrochemical detection of high-sensitivity CRP inside a microfluidic device by numerical and experimental studies

The concentration of C-reactive protein (CRP), a classic acute phase plasma protein, increases rapidly in response to tissue infection or inflammation, especially in cases of cardiovascular disease and stroke. Thus, highly sensitive monitoring of the CRP concentration plays a pivotal role in detecting these diseases. Many researchers have studied methods for the detection of CRP concentrations such as optical, mechanical, and electrochemical techniques inside microfluidic devices. While significant progress has been made towards improving the resolution and sensitivity of detection, only a few studies have systematically analyzed the CRP concentration using both numerical and experimental approaches. Specifically, systematic analyses of the electrochemical detection of high-sensitivity CRP (hsCRP) using an enzyme-linked immunosorbant assay (ELISA) inside a microfluidic device have never been conducted. In this paper, we systematically analyzed the electrochemical detection of CRP modified through the attachment of an alkaline phosphatase (ALP-labeled CRP) using ELISA inside a chip. For this analysis, we developed a model based on antigen-antibody binding kinetics theory for the numerical quantification of the CRP concentration. We also experimentally measured the current value corresponding to the ALP-labeled CRP concentration inside the microfluidic chip. The measured value closely matched the calculated value obtained by numerical simulation using the developed model. Through this comparison, we validated the numerical simulation methods, and the calculated and measured values. Lastly, we examined the effects of various microfluidic parameters on electrochemical detection of the ALP-labeled CRP concentration using numerical simulations. The results of these simulations provide insight into the microfluidic electrochemical reactions used for protein detection. Furthermore, the results described in this study should be useful for the design and optimization of electrochemical immunoassay chips for the detection of target proteins.

Numerical Modeling of Compression-Controlled Low-level Laser Probe for Increasing Photon Density in Soft Tissue

Various methods have been investigated to increase photon density in soft tissue, an important factor in low-level laser therapy. Previously we developed a compression-controlled low-level laser probe (CCLLP) utilizing mechanical negative compression, and experimentally verified its efficacy. In this study, we used Bezier curves to numerically simulate the skin deformation and photon density variation generated by the CCLLP. In addition, we numerically modeled changes in optical coefficients due to skin deformation using a linearization technique with appropriate parameterization. The simulated results were consistent with both human in vivo and porcine ex vivo experimental results, confirming the efficacy of the CCLLP.

Uniqueness and Nonuniqueness in Inverse Problems for Elliptic Partial Differential Equations and Related Medical Imaging

Unique determination issues about inverse problems for elliptic partial differential equations in divergence form are summarized and discussed. The inverse problems include medical imaging problems including electrical impedance tomography (EIT), diffuse optical tomography (DOT), and inverse scattering problem (ISP) which is an elliptic inverse problem closely related with DOT and EIT. If the coefficient inside the divergence is isotropic, many uniqueness results are known. However, it is known that inverse problem with anisotropic coefficients has many possible coefficients giving the same measured data for the inverse problem. For anisotropic coefficient with anomaly with or without jumps from known or unknown background, nonuniqueness of the inverse problems is discussed and the relation to cloaking or illusion of the anomaly is explained. The uniqueness and nonuniqueness issues are discussed firstly for EIT and secondly for ISP in similar arguments. Arguing the relation between source-to-detector map and Dirichlet-to-Neumann map in DOT and the uniqueness and nonuniqueness of DOT are also explained.

The Second-Order Born Approximation in Diffuse Optical Tomography

Diffuse optical tomography is used to find the optical parameters of a turbid medium with infrared red light. The problem is mathematically formulated as a nonlinear problem to find the solution for the diffusion operator mapping the optical coefficients to the photon density distribution on the boundary of the region of interest, which is also represented by the Born expansion with respect to the unperturbed photon densities and perturbed optical coefficients. We suggest a new method of finding the solution by using the second-order Born approximation of the operator. The error analysis for the suggested method based on the second-order Born approximation is presented and compared with the conventional linearized method based on the first-order Born approximation. The suggested method has better convergence order than the linearized method, and this is verified in the numerical implementation.

Reconstruction Formula for Photoacoustic Tomography with Cylindrical Detectors

Photoacoustic tomography, the well-known example of a hybrid imaging method, is a biomedical imaging modality based on the photoacoustic effect. Assuming cylindrical-shaped detectors are used to record the acoustical data for the photoacoustic tomography, we propose a new Radon-type transform to reconstruct the initial density function. The reconstruction of the suggested Radon-type transform is based on the properties of Bessel functions and Fourier–Bessel series expansion. Numerical simulation results are presented to demonstrate our proposed algorithm.

Uniqueness, Born Approximation, and Numerical Methods for Diffuse Optical Tomography

Diffuse optical tomogrpahy (DOT) is to find optical coefficients of tissue using near infrared light. DOT as an inverse problem is described and the studies about unique determination of optical coefficients are summarized. If a priori information of the optical coefficient is known, DOT is reformulated to find a perturbation of the optical coefficients inverting the Born expansion which is an infinite series expansion with respect to the perturbation and the a priori information. Numerical methods for DOT are explained as methods inverting first- or second-order Born approximation or the Born expansion itself.