Chapter 22: Electron Tomography

Tomography works far outside medicine too. Seeing the three-dimensional structure of the molecular machines inside a cell, intact and at nanometer resolution — that is electron tomography (cryo-ET). It uses an electron beam instead of X-rays and takes projections in a transmission electron microscope. The principle is the same as CT: an inverse problem recovering a volume from projections. But this world has a harsh constraint CT does not: the specimen cannot be tilted far. The limited angle of Chapter 20's tomosynthesis returns here as a three-dimensional missing wedge.

A specimen that cannot be tilted

In an electron microscope, the electron beam travels in a fixed direction from top to bottom through the column. To rotate the subject (a patient in CT; here a thin cell section or a protein), you tilt the whole specimen holder. But as you tilt, the holder eventually hits neighboring structure or the column and can tilt no further. The practical limit is about ±60–70°. Where CT could rotate past 180°, electron tomography records only about 140° worth.

This acquisition, called a tilt series, is essentially limited-angle tomography. It shares its essence with Chapter 20's tomosynthesis — some angles cannot be measured. The difference is that this is a 3D reconstruction, and the gap becomes a solid "wedge."

The missing wedge

Recall the Fourier slice theorem of Chapter 4 once more. A projection at tilt angle $\theta$ fills the central plane (a central line in 2D) at angle $\theta$ in Fourier space. If the specimen can tilt to $\pm 70°$, Fourier space fills only over that range, and the remaining $\pm 20°$ — around the beam direction — is missing as a wedge. This is the missing wedge.

Simulation: the missing wedge and elongation

Isolated particles (mimicking gold colloids) are 2D Fourier transformed, and the upper and lower wedge that the maximum tilt cannot fill is zeroed before inverse transforming. The center panel is the k-space with the wedge removed. Lower the maximum-tilt slider and the missing wedge widens above and below, stretching the particles on the right vertically — the beam (depth) direction. If you could tilt to $\pm 90°$ the wedge would vanish and the particles return to points, but in reality you cannot, so the elongation always remains.

How to fill the wedge

The missing wedge is a physical constraint, so measurement alone cannot remove it. So the "fill the gap with prior knowledge" techniques from earlier chapters are all brought to bear: tilting the specimen about two axes to shrink the wedge into a cone (dual-axis tomography), assuming smoothness via total variation (Chapter 10) or sparsity, averaging many copies of the same particle so their gaps complement each other (subtomogram averaging), and learning how to fill the gap with deep learning (Chapter 12). The tools that faced low dose and sparse views in CT are used unchanged in the nano world. The subject, the scale, and the instrument are entirely different, yet the way of fighting the inverse problem is remarkably the same.

Key points

Electron tomography images the 3D structure of cells and proteins at nanometer resolution with an electron beam. Because the specimen holder tilts only to ±60–70°, a missing wedge remains in Fourier space, producing an elongation artifact along the beam (depth) direction. This is the 3D version of Chapter 20's limited angle and depth blur, understood the same way through the Fourier slice theorem. Filling the gap uses the very techniques of this textbook: dual-axis tilting, compressed sensing, subtomogram averaging, and deep learning. With this, the journey that began with CT's line integrals has reached MRI's Fourier measurement, nuclear medicine's emission counting, limited angle, photoacoustic arcs, and the electron nanotomography here. The physics measured differs, but the skeleton of the inverse problem — recovering a volume from limited data — has been the same throughout. Tomography is one piece of mathematics wearing a great many faces.

References

• Frank J (ed.). Electron Tomography: Methods for Three-Dimensional Visualization of Structures in the Cell, 2nd ed. Springer (2006).
• Lučić V, Rigort A, Baumeister W. Cryo-electron tomography: the challenge of doing structural biology in situ. Journal of Cell Biology 202, 407–419 (2013).
• Mastronarde DN. Dual-axis tomography: an approach with alignment methods that preserve resolution. Journal of Structural Biology 120, 343–352 (1997).