Accessing depth-resolved high spatial frequency content from the optical coherence tomography signal

Optical coherence tomography (OCT) is a rapidly evolving technology with a broad range of applications, including biomedical imaging and diagnosis. Conventional intensity-based OCT provides depth-resolved imaging with a typical resolution and sensitivity to structural alterations of about 5–10 microns. It would be desirable for functional biological imaging to detect smaller features in tissues due to the nature of pathological processes. In this article, we perform the analysis of the spatial frequency content of the OCT signal based on scattering theory. We demonstrate that the OCT signal, even at limited spectral bandwidth, contains information about high spatial frequencies present in the object which relates to the small, sub-wavelength size structures. Experimental single frame imaging of phantoms with well-known sub-micron internal structures confirms the theory. Examples of visualization of the nanoscale structural changes within mesenchymal stem cells (MSC), which are invisible using conventional OCT, are also shown. Presented results provide a theoretical and experimental basis for the extraction of high spatial frequency information to substantially improve the sensitivity of OCT to structural alterations at clinically relevant depths.


Theoretical analysis of the scattering optical wave in reflection configuration
It is well known that the object's structure can be described using 3D function, which is usually called the scattering potential 16 : or its 3D Fourier transform: where nrefractive index, K is the spatial frequency vector.
We restrict our consideration to the first Born approximation, do not show the dependence on time and ignore polarization effects. Illumination wave under the scalar representation could be written as Where k=2π /λwavenumber, k0x=2πvx, k0y=2πvy, k0z=2πv0z, v0x, v0y, v0z are the incident spatial frequencies, x, y and z are cartesian coordinates of a point r within the object space.
The scattered wave US(r`) at some point r` can be written as a volume integral 16,17 : represents the Green function: Under the Born approximation, we can use ⅈ instead of = ⅈ + to describe the scattered wave in Eq. (S4).
After some simplifications we get: where ̃( ) =̃( − 0 , − 0 , + 0 ) ̃ is the angular spectrum 18 of the complex wave scattered by the lateral cross-section of the object located at depth z. The angular spectrum is given by 2D Fourier transform of the scattering potential Eq. (S1). The integral in Eq. (S7) represents the superposition of the angular spectrum of the complex backscattered waves centred at (k0x, k0y) from all depths within the object. So, this equation provides information about the entire 3D structure of the object. The structure is described by spatial frequency vector, which can be written as: where s, s0 are unit vectors of scattered and illumination waves (Fig. S1a), x, y and zspatial frequencies of the object's structure along Cartesian coordinates.
Equation (S8) shows that the complex amplitude of the scattered wave at a given wavelength in the far zone for a given direction depends entirely on only one Fourier component (one spatial frequency) of the 3D scattering potential, labelled by the vector K. At a constant illumination angle, the end point of each vector Fourier component of the 3D scattering potential for given collection angle corresponds to a point on Ewald's sphere. Illumination and collection geometry is presented in Fig. S1a. If the object is illuminated by a plane wave with a certain spectral bandwidth, then the spatial frequencies distribution in K-space for all collection angles can be illustrated as multiple Ewald's spheres with different diameters. For an illumination beam at normal incidence and n = 1 the back scattered wave Eq. (S7) gives the 3D frequency distribution in K-space. Example of such distribution in 2D plane for spectral bandwidth 1220 nm -1400 nm is presented in Fig. S1b. Figure S1. Spatial frequency representation in K-space. (a) -Schematic of object illumination and collection; (b)spatial frequency representation in K-space depending on wavelength.
If we detect the scattered field in K-space (the complex amplitudes of all Fourier components), then we could synthesize the 3D Fourier transform of the scattering potential Eq. (S2). After that the scattering potential can be reconstructed via 3D inverse Fourier transform. However, even if all spatial frequencies will be captured, the scattering potential will be reconstructed under low-pass filtered approximation and the best possible resolution will be about half of wavelength 16 . The objective lens, used for collection, will further limit the accessible bandwidth of the spatial frequencies, depending on the numerical aperture (NA), and instead of spheres we will have NA-restricted Ewald sphere caps as it shown in Fig. S1b within the red rectangle.
Supplementary Figures S2 -S4 for samples at day 1 and day 4.