Date: Thu, 28 Mar 2024 19:09:47 -0700 (PDT)
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Introduction
In the context of biology and medicine, electron tomography is a means o=
f estimating the internal structure of an object from measurements of the i=
ntensity of a high-voltage electron beam impinging upon it. The object is a=
ssumed to possess a degree of opacity with respect to individual electrons =
that results in an attenuation of the beam via effects such as scattering a=
nd absorption. The device that generates the electron beam, positions and o=
rients the object in the beam, and measures the beam's intensity is called =
a transmission electron microscope (TEM), with the label =E2=80=9Ctransmiss=
ion=E2=80=9D referring to the passage of the electron beam through the obje=
ct being investigated. A measurement of the intensity of the electron beam =
usually takes the form of a one- or two-dimensional grey-scale image referr=
ed to by various names, with =E2=80=9Ctilt=E2=80=9D, =E2=80=9Cmicrograph=E2=
=80=9D, =E2=80=9Celectron micrograph=E2=80=9D, =E2=80=9Ctransmission electr=
on micrograph=E2=80=9D, =E2=80=9CTEM micrograph=E2=80=9D, and =E2=80=9Cproj=
ection=E2=80=9D being among the most common. In the present document, we wi=
ll refer to all such measurements using the umbrella term projectio=
n.
Light=
microscopy versus electron microscopy
Theoretically speaking, the resolution limit of an imaging system depend=
s directly on the range of energy wavelengths it is capable of detecting. S=
ince the wavelength of visible light is between 390 to 750 nanometres (10-9m), standard light microscopes are incapable of resolving featur=
es smaller than 1/10th of a micrometre (10-7m). Because of the i=
nverse relationship between the relativistic momentum of an electron and it=
s wavelength, electrons that are accelerated to an appreciable fraction (e.=
g., 70%) of the speed of light possess wavelengths on the order of picometr=
es (10-12m). In terms of the electromagnetic spectrum, this is t=
he wavelength of gamma rays. Modern TEMs can therefore achieve magnificatio=
ns 1,000X that of standard light microscopes, easily resolving features in =
the micro- to nanometre range (10-6m--10-9m), with th=
e highest achieved resolutions (ca. 2005) in the 1/100ths of nanometres. Bi=
ological structures falling within this resolution range include mammalian =
cell nuclei (~6=CE=BCm in diameter), human red blood cells (~6--8=CE=BCm in=
diameter), Caulobacter crescentus bacteria (4--6=CE=BCm long), mi=
tochrondria (0.5--10=CE=BCm), Polio virus capsids (30nm in diameter), and t=
ypical cell membranes (6--10nm thick).
Projection acquisition<=
/h3>
When producing projections intended to be used for tomographic reconstru=
ction (a term defined below), the object being imaged is usually required t=
o be a thin section of tissue between 150--500nm thick. The tissue is often=
stained to increase its electron opacity, most commonly with compounds con=
taining heavy metals such as osmium, lead, or uranium (e.g., osmium tetroxi=
de and uranyl acetate). The high voltages used to accelerate the electrons =
in the TEM column require that the column be evacuated to prevent electrica=
l arcing and undesirable interactions between the electrons and atmospheric=
gas molecules. To withstand the vacuum of the TEM column, biological mater=
ial must also be fixated prior to imaging. Both osmium tetroxide and uranyl=
acetate, in addition to their staining properties, behave as biological fi=
xatives; plastic embedding provides alternative means of fixation. At some =
point during sample preparation, the object is marked with small particles,=
such as ~100nm diameter colloidal gold, which act as fiducial mark=
ers (or simply fiducials). If this marking occurs=
after fixation and any subsequent shaving down of the section block, only =
the exterior surfaces of the block will be marked. Following the sample pre=
paration, the object is placed on a specimen stage, which is the part of th=
e TEM that positions and orients the object in the electron beam. Projectio=
ns of the object are taken at a series of orientations, typically 1- or 2-d=
egree incremental rotations about a single axis (=E2=80=9Csingle tilt=E2=80=
=9D) or two orthogonal axes (=E2=80=9Cdouble tilt=E2=80=9D) -- in either ca=
se, the current rotation axis is often referred to as the tilt axis=
. Note that these two projection geometries are the most common on=
ly because they are the most mechanically convenient.
|
Figure 1: This diagram serves as an illustration of the initial projecti=
on geometry inside an electron microscope. In this and subsequent diagrams,=
the electron source is located at the origin, but here the beam is turned =
off. The object being imaged is the opaque grey block in the center of the =
diagram; note the black spheres embedded in the surface of the block -- the=
se represent fiducial markers. There are also fiducial markers embedded on =
the far side of the object. The line parallel to the y-axis passin=
g through the middle of the block is the tilt axis. The white square to the=
right of the object is the projection screen: the plane o=
nto which the object is projected. |
|
Figure 2: In this diagram the electron beam has been turned on; the dash=
ed line running along the negative z-axis to the middle of the pro=
jection is the optical axis. In this example, the material comprising most =
of the object is assumed to be perfectly electron-transparent; on the other=
hand, the fiducial markers and any structures contained within the object =
are assumed to be perfectly electron-opaque. As discussed further below, ne=
ither assumption holds in practice. Judging by the projection, there appear=
to be two or three structures inside the object. Note the projections of t=
he fiducial markers embedded on the far side of the object. |
|
Figure 3: In this diagram, the opaque material of the object has been re=
ndered transparent to give the viewer a better idea of what is going on. Th=
ere are actually four structures inside the object: a rectangular solid, a =
small sphere, an elliptical solid, and a bowl-shaped structure. Note that t=
he projection of the rectangular solid at this orientation more-or-less com=
pletely shadows the small sphere. A mathematical description of this orient=
ation would be something like =CF=89 =3D (y-axis, 0.0). |
|
Figure 4: In this animated diagram, the object is rotated about the =
y-axis in 15-degree increments between 0.0 and 60.0 degrees. |
|
Figure 5: In this animated diagram, the specimen stage holding the objec=
t has been rotated 90.0 degrees about the z-axis and the object is=
rotated about the y-axis in 15-degree increments between 0.0 and =
60.0 degrees. Note that having a rotation axis parallel to the x-a=
xis would generate the same series of projections provided the projection s=
creen were rotated (or were assumed to be rotated) by -90.0 degrees about t=
he z-axis as well. |
A careful reader may have noticed a number of additional peculiarities i=
n the preceding diagrams; for further discussion, click here. (A link to a =
page containing further discussion of the figures.)
In the animated diagrams above, the rotation angles about the tilt axis =
range in magnitude from 0.0 to 60.0 degrees. This matches what is usually f=
ound in practice: assuming that the object at orientation =CF=89 =3D (tilt-=
axis =E2=89=A0 z-axis, 0.0) has a thickness along the z-a=
xis that is much less than its extents along the x- and y=
-axes, as the object's rotation about the tilt axis approaches +/-90.0 degr=
ees, the thickness of the object along the z-axis becomes very lar=
ge. For most biological samples oriented at high degree, the object's thick=
ness along the z-axis becomes too great for the electron beam to e=
ffectively penetrate.
Tomographic reconst=
ruction
A single projection of an object gives some indication of its internal s=
tructure, but because of shadowing, particularly in the case of perfectly e=
lectron-opaque structures, much of the detail is obscured. Furthermore -- e=
specially in the case of perfectly electron-opaque structures -- a single p=
rojection leaves spatial arrangement along the z-axis completely a=
mbiguous. Multiple projections of an object remove some of this spatial amb=
iguity, but the shadowing problem remains. (It would be instructive at this=
point to consider why the geometries of only three of the four structures =
in the object imaged in the diagrams above can be completely determined fro=
m their projections.) What is desired is the calculation of a 3D scalar fun=
ction, a tomogram, representing the density of the object =
with respect to the electron beam. The word =E2=80=9Ctomogram=E2=80=9D is a=
modern combination of two words from Attic Greek: =E2=80=9Ct=C3=B3mos=E2=
=80=9D, a cut or section, and =E2=80=9Cgr=C3=A1mma=E2=80=9D, something writ=
ten or drawn. Indeed, once calculated, a tomogram can be viewed as a stack =
of 2D slices, but there exist other techniques for viewing a tomogram as a =
whole volume. The technique of reconstructing an object's tomogram from its=
projections is called tomographic reconstruction or, simp=
ly, tomography.
At this point is necessary to touch briefly upon the mathematical theory=
underpinning tomographic reconstruction. Tomographic reconstruction is wha=
t physicists and mathematicians refer to as an inverse problem. An inverse problem is one that attempts to determine the values of a =
set of model parameters from a set of observations. Assume that the synthet=
ic TEM in the diagrams above is perfectly stable, and that its physical geo=
metry is completely known. Furthermore, assume that we have a sensible mode=
l for the physics of electrons, the relationship between the attenuation of=
an electron beam passing through matter and that matter's density, and the=
electron detector comprising the projection screen. The projections and th=
eir corresponding object orientations thus stand as a set of observations, =
and the model parameters we seek are therefore associations between the poi=
nts in the 3D space occupied by the object and scalar densities. Mathematic=
al models of image formation in a TEM are usually couched in the language o=
f line integrals: the intensity associated with a particular point on a giv=
en projection is the integral along a line passing through that point and t=
he electron source. Tomographic reconstruction can be understood as the inv=
erting of these line integrals. In theory, completely reconstructi=
ng an object's tomogram with perfect fidelity requires that the ob=
ject be of finite extent, that all line integrals not passing thro=
ugh the object equal zero, and that all line integrals passing thr=
ough the object be computed. If a set of line integrals are missing, they a=
re assumed to equal some value (usually zero), which will degrade the accur=
acy of the tomogram. As mentioned in the previous section, in the case of t=
ypical biological specimens there is an entire range of orientations at whi=
ch projections cannot be taken of the object. These missing projections and=
their associated orientations are known as the missing wedge and are a source of error in tomographic reconstructions.
Given a set of projections of an object, there are three common approach=
es to reconstructing its tomogram, only one of which concerns us at present=
: filtered back-projection. The filtered back-projection a=
lgorithm is comprised of three steps. The first step, alignment, calculates a geometrical association -- a set of parameters referred=
to as a projection map -- between the various orientation=
s of the object and the detector at a fixed reference orientation (alternat=
ively, the geometrical associations can be understood to be between project=
ions oriented with respect to the object at a fixed reference orientation).=
This is usually achieved via approximations that depend on tracked 2D proj=
ections of a set of fiducial markers, although, theoretically, anything tha=
t can be tracked with tolerable accuracy throughout the set of projections =
can be used. The second step, filtration, is too mathemati=
cally abstract to be discussed much here, though it can accurately be const=
rued as a sharpening of the projections. It is worth noting, however, that =
while in theory filtration depends directly on alignment, often in practice=
this dependence is assumed to take a very simplified form, which amounts t=
o a sharpening along the horizontal pixel rows in the projection. The third=
step, reconstruction, is typically implemented as follows=
: for each 3D point x in the tomogram, for each filtered projectio=
n indexed by orientation =CF=89, find the 2D point xp in the current projection that x projects to according to =
projection map indexed by =CF=89 and add the filtered intensity at xp to the intensity of the tomogram at x. The=
mapping from x to xp is called forward-projection, while the carrying of the filtered intensit=
y at xp back to x is called b=
ack-projection. Given some means of calculating the projection tra=
jectory that passes through point xp in the fi=
ltered projection indexed by =CF=89 (alternatively, given a simplified proj=
ection geometry), forward-projection is not a theoretical necessity, and th=
e filtered intensity at xp can be back-project=
ed at constant value along its trajectory through the tomogram. In practice=
, however, particularly when the projection trajectories are curvilinear, t=
his leads to sampling errors in the tomogram and foward-projection must be =
used.
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