1. Assessment of Embryo Health by Microscopy
Ersel Karbeyaz
Carey Rappaport
Charles A. DiMarzio
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 )
and by a grant from the W. M. Keck Foundation.

2. 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.

3. 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.

4. 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.

5. 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 μm
Mitochondria Ratio (2D = 3D)
~16.4%
Frequency
474.1 tera Hz
Number of Mitochondria (2D)
~1070
∆x (Space step) μm
nintercellular
1.33
∆t (Time step)
0.0792e-12 sec

9. Difficulty in Processing Pupil Plane Fieldx

10. 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.

11. 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.

12. 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 [ ] are given below.

14. 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.

15. 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.

16. 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.

17. 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 50m with a matched cytoplasm background is probed with 12 plane waves of distinct incidence angles and a common frequency such that diameter = 40b (free space  = 1712 nm, in the infrared region).

19. 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.

20. References
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microscopy in embryo imaging”, Optics Letters, Vol. 29, No. 19,
Oct 2004, P
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electromagnetics: linear inversion formulation and minimum energy
solution” IEEE Transactions on Antennas and Propagation Vol. 47
No. 2, Feb 1999, P
- Dunn A. K., and Richards-Kortum R., “Three-Dimensional
Computation of Light Scattering From Cells” IEEE Journal of
Special Topics in Quantum Electronics Vol. 2 No. 4, Dec 1996, P.898.
- Dunn A. K., Smithpeter C., Welch A. J., and Richards-Kortum R.,
“Finite-Difference Time-Domain Simulation of Light Scattering from
Single Cells” Journal of Biomedical Optics Vol. 2 No. 3 July 1997 P.262
- Drezek R., Dunn A. K., and Richards-Kortum R., “Light Scattering
form Cells: Finite-Difference Time-Domain Simulations and
Goniometric Simulations” Applied Optics Vol. 38 No. 16 June 1999
P. 3651
- Richmond, J. H., “Scattering by a dielectric cylinder of arbitrary cross
section shape”, IEEE Trans. Antennas and Propagation, May, 1965, pp