Assessment of Embryo Health by Microscopy Ersel Karbeyaz (ekarbeya@ece.neu.edu) Carey Rappaport (rappaport@ece.neu.edu) Charles A. DiMarzio (dimarzio@ece.neu.edu) CenSSIS is a National Science Foundation Engineering Research Center supported in part by the Engineering Research Centers Program of the National Science Foundation (Award # EEC-9986821) and by a grant from the W. M. Keck Foundation.
Abstract To assess the health of an embryo, we need a three dimensional understanding of its interior. This task is made complicated by the relatively denser substance contained inside the embryo’s membrane. All the information contained by the third dimension is hidden in the resulting image on the pupil plane. The goal of our research is to develop algorithms that will produce information about how the embryo varies with depth. One possible forward model utilizes Radar Cross Section (RCS) patterns of various simple models of cells. After comparing the resulting far-field patterns to a more rigorous model for a validity check we see that there are many similarities. This can then lead to an invertible forward cell model that can be used to discern the distribution of mitochondria. Another model creates a stack of images produced by simulating focusing a microscope to multiple image planes along an embryo’s depth. Constructing this data cube allows us to understand how light interacts with the embryo at all depths. This information can then be manipulated to develop a three dimensional understanding of the object. Another approach is to transform this scattering problem into a radiation problem, with equivalent (polarization) currents existing only where the scatterer is. These currents are proportional to the electrical properties of the scatterer, and determination of these currents using the far zone observations can give us structural information about the scatterer. Unlike the previous method, this approach does not suffer from the light scattered by the out-of-focus objects.
Methods for Diagnosis Mitochondria Distribution Literature Suggest that the distribution of mitochondria is related to embryo health. Specifically aggregation is a sign of bad health. Cell Counting An embryo’s developmental speed can indicate future health problems after birth. Rapid cell division is an indicator for a healthy child. It is difficult to discern the distribution of mitochondria in three space. Particularly when more than 16% of the cell’s volume is mitochondria. Cells inside of a developing embryo become tightly packed and are stacked on top of each other; which makes counting very difficult. Both determining the distribution of mitochondria (which are very weak scatterers) and the number of the cells packed and stacked in the developing embryo are challenging problems.
Finite Difference Time Domain (FDTD) Method and Radar Cross Section (RCS) Computations FDTD models EM fields sampled in time and space ( ≤ /10), in terms of similarly sampled media characteristics. Pros: Produces accurate scattering simulations for extremely non-uniform geometries Cons: Requires knowledge of each pixel in the computational grid, for mitochondria distribution prediction is more detail than necessary Large computational cost for realistic geometries makes reconstruction infeasible A good way to describe the electromagnetic far-field pattern of an object illuminated with an electromagnetic wave is to obtain its RCS, which is the normalized electric field intensity pattern at the far field. FDTD method is used to compute the RCS patterns of the structures investigated. Basic concepts related to FDTD method, such as PML (Perfectly Matched Layer) ABC (Absorbing Boundary Condition), total field – scattered field formulation, near to far-field transformation are used in the implementation of simulation programs.
Cell and Mitochondria Distribution Modeling Following mitochondria distributions have been investigated: Uniform (with / without a nucleus) Aggregated Perinuclear Cortical A simple effective medium for uniform mitochondria distribution (with nucleus) has also been investigated. Cell Radius 35 μm ncytoplasm 1.37 Nucleus Radius 12 μm nmitochondria 1.42 Mitochondria Dimensions 0.75 0.25μm nnucleus 1.40 Number of Mitochondria (3D) ~150,000 Wavelength 0.6328 μm Mitochondria Ratio (2D = 3D) ~16.4% Frequency 474.1 tera Hz Number of Mitochondria (2D) ~1070 ∆x (Space step) 0.0357 μm nintercellular 1.33 ∆t (Time step) 0.0792e-12 sec
Difficulty in Processing Pupil Plane Field x
Equivalent Sources Approach The problem of light scattering from a cell can be transformed into a radiation problem, with equivalent (polarization) currents existing only where the scatterer is, keeping the scattered fields the same. These currents are proportional to the electrical properties of the scatterer.
The relation between the observed far zone scattering pattern () and the equivalent current density distribution Jeq(x’,y’) is given by where Unfortunately, the solution to this inverse source problem is not unique unless additional constraints are imposed on Jeq(x’,y’), such as having minimum L2 norm (as explained in the article by Marengo and Devaney), which is not relevant to our case. Indeed, the observed far zone pattern () is the 2D Fourier transform of Jeq(x’,y’) evaluated only on the circle kx2 + ky2 = k2 and it is not possible to fill the whole kx - ky space with observations to uniquely specify Jeq(x’,y’) in the x’ - y’ space.
However, we may assume that the components not lying on the r = k circle are negligible and try to obtain Jeq (x’,y’) using only the available data. The resulting current density will not reveal much about the scatterer, but the outcomes of many experiments with corresponding incident plane waves having distinct incidence angles can be combined to solve the object function O(x’,y’), considering the inner scattering within the object via: with Results of a simulation involving a cell model illuminated with up to 24 distinct plane waves having incidence angles equally spaced in [00 3600] are given below.
From the simulation results, we conclude that using this method, we can extract limited information about the investigated cell, such as the size and location of the nucleus and a very rough idea about how the other organelles are distributed. To have meaningful results, a great number of experiments with distinct incidence angles and corresponding far zone scattering patterns are needed.
Mode Matching Approach Using the far zone approximation for the Hankel function and after some manipulations, the far zone electric field pattern due to a current density distribution Jz(r,) confined within a circle of radius r = a is given by where If the pattern () is known, then the coefficients an can be easily obtained as follows, using the orthogonality property of the functions ejn.
Now, suppose that the radiating sources are the equivalent sources of a scattering problem, existing only where the scatterer is and radiating the far zone scattering pattern. These sources would be of the form: with Assume further that the total field inside the scatterer can be approximated as where the mode dependent coefficient bn and wavenumber kn can be solved for, by imposing the continuity of the tangential field components at r =a.
Expanding the object function in terms of the Fourier-Bessel basis functions and substituting it in the radiation expression, after some manipulations, we obtain which can be solved for the unknown expansion coefficients cmn . To illustrate the method, a cell model of diameter 50m with a matched cytoplasm background is probed with 12 plane waves of distinct incidence angles and a common frequency such that diameter = 40b (free space = 1712 nm, in the infrared region).
Future Work Adapt RCS forward model for use in reconstruction algorithms Create a forward model for cell counting that can easily be inverted Create an algorithm that will allow mapping of mitochondria’s distribution from z stacks Include the effects of larger associated sub-cellular structures in the model. Establish detectability criteria for clumped versus dispersed microscatterers.
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