Mathematical Methods of Optical Coherence Tomography

Abstract

In this chapter a general mathematical model of Optical Coherence Tomography (OCT) is presented on the basis of the electromagnetic theory. OCT produces high resolution images of the inner structure of biological tissues. Images are obtained by measuring the time delay and the intensity of the backscattered light from the sample considering also the coherence properties of light. The scattering problem is considered for a weakly scattering medium located far enough from the detector. The inverse problem is to reconstruct the susceptibility of the medium given the measurements for different positions of the mirror. Different approaches are addressed depending on the different assumptions made about the optical properties of the sample. This procedure is applied to a full field OCT system and an extension to standard (time and frequency domain) OCT is briefly presented.

Quantitative reconstructions in multi-modal photoacoustic and optical coherence tomography imaging

“…The Born approximation allows us to obtain an explicit form forÊ ν from the Lippmann-Schwinger equation (10). In the limiting caseχ → 0, we take the first order approximation of the electric field by replacingÊ ν withÊ (0) ν on the incident illumination we state that what we actually measure in OCT is the backscattered light at a detector placed far from the medium [12,Proposition 8].…”

“…Given a medium as in Definition 2.1 and incident fields E (0) ν of the form (6) for all ν > 0, the inverse problem is to recover the parametersχ and Γ given the internal PAT measurements p ν (x), for x ∈ Ω, and all ν > 0, given by (9), and the external OCT dataχ(ω, ω c (ϑ + e 3 )), ω ∈ R \ {0}, ϑ ∈ S 2 + , given by (12).…”

Maxwell's equations based 3D model of light scattering in the retina

“…As OCT standard techniques only provide structural information [4], it is necessary to expand OCT data analysis to account for both structural and functional information. This could be achieved by inverse modeling of OCT data, using the physical and optical properties of healthy and diseased human retinas and by developing a mathematical model of the respective OCT data [5]. A proper OCT reference for a certain pathology would further enable the identification of the cellular alterations responsible for the observed OCT scans from patients, contributing to an earlier diagnosis and, hopefully, a more efficient treatment.…”

“…For that purpose, the direct scattering problem needs to be assessed, which comprises two main steps (1) the study of the electromagnetic wave propagation and scattering as it travels through the sample and (2) the measurement off scattered light at the detectors [5]. This behaviour of light scattering through the sample can be described by a variety of methods, such as the radiative transfer theory, Lambert's Beer Law, Maxwell's equations and also some statistical approaches using Monte Carlo [5].…”

The inverse scattering problem for orthotropic media in polarization-sensitive optical coherence tomography

“…Standard OCT uses broadband, continuous wave light for illumination and the images are obtained by measuring the time delay and the intensity of the backscattered light from the sample. For a detailed description of OCT systems we refer to the books [2,12] and for a mathematical modeling we refer to [13].…”