Quantitative microscopy

Quantitative microscopy
Ilan Hammel DSc
One picture is worth ten thousand words
Problems of Quantitative Microscopy
The amount of material that is actually
examined in the microscope is often a tiny
fraction of the whole object of interest.
If a 100x magnification oil immersion lens is used a 3
dimensional (3D) sampling 'brick' with dimensions of
about 70µm x 90µm x 10µm can be used for counting
cells.
For a brain of volume 500 cm3 each brick thus
represents a fraction of about 1/8,000,000,000th
(an eight billionth) of the whole brain.
1
How can a meaningful and representative estimate of the
total number of neurons in the brain be made from this tiny
fraction of the brain?
BY USING STEREOLOGICAL METHODS
Stereology is related to the areas of stochastic
geometry and spatial statistics.
It can be described as geometric sampling theory or
a set of sampling methods for quantifying geometric
parameters such as the number of cells, the length of
a root, the area of a particular region in an
agricultural field, or the volume of an organ.
Mandarim -De-Lacerda CA. Stereological tools in biomedical research. An Acad
Bras Cienc.;75:469-86. 2003.
Problems of Quantitative Microscopy
The act of taking a cross-section
through an object, such as a polished
metallurgical section or a thin
histological section, causes the feature
of interest to be seen with reduced
dimensionality.
An unavoidable reduction in
dimensionality (from Dn to Dn-1)
has been introduced by the
sectioning process.
2
Sectioning features in a
3D space with a plane
showing the area
intersection with a
volume, the line
intersection with a
surface, and the point
intersection with a
linear feature.
Compression caused by the knife action in
microtomy direction.
Paraffin - or similar embedding media like Paraplast or
Paraplast Plus - causes an intrinsic distortion of
structures: the shrinkage is caused by fixation,
dehydration and paraffin infiltration.
In paraffin embedded material shrinkage can be
around 25% and compression around 10% relative to
the fresh material.
Vo /Vf = (lo /l f) 3
3
Geometric Probability - Stereology
• Both of these problems are simply resolved in practice by
applying a branch of applied mathematics and probability theory
known as stereology.
• It is the science of obtaining 3D information from 2D sections.
• It was founded in the early 1960's by a group of Mathematicians,
Biologists and Material Scientists who had a common interest in
quantifying 3D structure from 2D cross-sections.
• The founders of this new inter-disciplinary science realised that
there was a common generic problem. The 3D features they were
interested in could not be directly quantified from cross-sections.
• The insight provided by the founders of stereology was that they
realised that the features seen on a section were related to the
features in 3D space in a well-defined statistical way.
Sampling
• The problem of drawing sound conclusions
about a geometrical feature from microscopical
observations has a direct analogy in the world of
survey sampling .
For example, consider the task confronted by
opinion pollsters.
• By taking a small sample (perhaps 500 people)
from a large population (6,000,000), it is
possible for opinion pollsters to give surprisingly
accurate predictions of the outcome of elections.
Correct Sampling = All parts of the structure or
population have to have the same chance to contribute
to the sample.
Accurate forecasts rely upon two factors.
1. The sample must be representative of the
population.
2. The questions that are asked should
generate truthful answers.
If either of these principles are ignored then the
results of an opinion poll are likely to be unreliable,
that is to say, the results will have a systematic
error or bias.
4
Uniform Random Sample
Each and every member of the population must have
an equal chance of being selected for the sample
before the sampling begins.
If the opinion pollster took a sample of people coming
out of the Stock Exchange
he/she would almost definitely get a markedly different result
than
If he/she had sampled people coming out of an
unemployment advice centre.
The samples are not representative because the
selection of individuals is not of uniform random.
A uniform random sample of a population
is said to be free from selection bias.
In order to ensure that the questions get
truthful answers from the sampled
individuals:
1. the interviews and questionnaires should not
use leading questions
2. adopt a moralistic high tone
3. should be carried out in a neutral manner.
The quantification of micro-structures from samples
taken from a larger object can be considered as a
spatial variant of the opinion pollsters problem.
• The population in this context is the totality of
the geometrical feature of interest within the
object (i.e. volume, surface area, length or
number).
• The sample is the subset of material that is
actually imaged in the microscope.
• The questions that can be asked take the form
of geometrical probes such as points, lines,
planes and volumes.
5
To ensure that the conclusions drawn from a
microscopic analysis are representative of the
whole object
Care Must Be Used In Taking A (Spatial)
Sample And Asking Truthful
(Geometrical) Questions.
A representative sample of the object
of interest can be obtained
By taking a uniform random
sample from the object
(e.g lung).
• This means that each and
every portion of the object
should have an equal chance
of being seen in the
microscope field of view before
the sampling begins.
• In practice this sampling will
often have a nested structure.
The Nested Structure of
Tissue Sampling
• The object will be cut exhaustively into thick slabs.
• A random sample of these are further sectioned
into rectangular blocks.
• A random sub-sample of the blocks are embedded
in wax, resin or plastic.
• A random series of sections are taken with a
microtome and mounted on microscope slides.
• Randomly positioned fields of view on the slides
are then imaged and used for analysis.
6
Geometrical probes
• Geometrical probes (points, lines, planes or volumes)
are 'thrown' into the object of interest.
• The number of intersections is dependent both on the
amount of the probe and the amount of the feature of
interest.
• Providing that either the set of probes or set of features
are randomised properly there is a well defined
statistical relationship between the amount of the feature
and the number of intersections that are produced.
• These statistical relationships represent the classical
ratio estimators of stereology
Geometrical probes
The probability that a particular probe hits (or
intersects) a feature is dependent on both the type
of probe and feature.
• A point thrown at random within a reference space
hits features within the reference space with a
probability proportional to the volume of the
features.
• A line of isotropic random orientation thrown into a
reference space will hit features with a probability
proportional to the surface area of the features.
In practice:
• It is difficult to microscopically probe a 3D object
directly with a set of point or line probes.
• Therefore, the material is sectioned first and a suitable
test system of points, lines, sample frames etc. is
randomly superimposed on the planar sections.
This probing of a section is equivalent to direct probing
of 3D space and is the essence of the stereological
approach.
7
Calculation method for determining the chances
(frequency) of obtaining different-sized transections
from transecting a sphere in size class k of diameter
2R with many parallel planes (R=4r/ π).
Bias in stereological measurements - Holmes effect
Finite section thickness introduces two opposite
sources of bias into conventional stereological
measurements.
• These have been recognized for a long time
[Holmes, A. (1927) "Petrographic methods and
calculations" Murby, London].
• Various correction procedures have been derived
(Cruz-Orive, 1983).
Bias in stereological measurements Holmes effect
Correction procedures are not used widely.
• Because they require knowing the section thickness.
• Making some assumptions about feature shape and
uniformity (also isotropy of orientation).
• and/or performing more measurement work to obtain
results.
• In practice, sections with no more than 3µm
thick are useful to most of stereological
procedures (an exception to the use of thin
sections is the disector's method, where
thick section over 20µm could be used).
8
Section Thicknes
Small method
Unbiased estimation of the Area of a 2D object
The area (A) of a
2D object can be
estimated without
bias by randomly
translating a point
grid over the object
in both the x and y
direction.
A=P(loln)/Mag2
The 2D nucleator
Area = π li2
The 2D nucleator estimates
the area of any object
irrespective of its size,
shape, and orientation by
measuring the distance
between a "central point"
in the object and the
intersections between the
object boundary and the
four radiating test lines.
9
The 2D nucleator
Area = π li2
Calculation of Linear Magnification
One key piece of information
required in stereological work
is the magnification (Mag) of
the images used for
measurements.
The best method for
determining the final linear
magnification of an image is
to capture an image of a
calibrated graticule at the
same magnification used for
the images.
1mm
Unbiased estimation of 2D Boundary length in the plane
(The method is a result of the famous Buffons needle problem.)
Boundary length is
estimated by randomly
translating and rotating
a grid of parallel lines,
of grid spacing D, over
the object of interest.
The number of
intersections between
the object boundary
and the set of test lines
is counted (I) and the
boundary length is then
estimated from
B = (π / 2) I D
10
Buffons needle problem
LA = LINEAR ELEMENTS IN A PLANE
LA = LINEAR ELEMENTS IN A PLANE
The average number of intersections formed by an array
of test lines which are uniformly distributed in both
position and orientation is:
2π d λ
dφ
dλ
π /2
dλ
d N L (φ) = ∫
cos( φ)
=
[
cos(φ ) d φ] =
0 L2
2π 2π L2 ∫0
π L2
2dλ
2λ
=
NL=∫
λ π L 2 π L2
LA = (π/2) NL = (π/2) I D
11
B =(π/2) (A/L) I
Unbiased estimation of root length in the plane
B = (π / 2)( I / 2 ) D
= (π / 4) I D
Mag = 2.35
I = 41
D = 6cm / 2.35 = 2.56
B = 82.4 cm
Mag = 25
I = 71
D = 4mm / 25 = 160µm
B = (π / 2) I D
12
12
14
20
12
13
B = 17.8 mm
Unbiased estimation of 2D Objects per Unit Area (Na)
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Conversions
Area
Conversion of mm 2 to m2
1 mm 2 = (1 × 10 -3 m)2 = (10 -3 m)2 = 10 -6 m 2
Conversion of µm 2 to m 2
1 mm 2 = (1 × 10 -6 m)2 = (10 -6 m)2 = 10 -12 m 2
Volume
Conversion of µm 3 to m 3
1 µm 3 = (1 × 10 -6 m)3 = 10 -18 m 3
Reciprocal Length (L 2/L 3=L -1 e.g. surface density)
Conversion of µm -1 to m -1
1 µm -1 = (1 × 10 -6 m)-1 = (10 -6 m)-1 = 1/(10-6 m) = 10 6 m -1
13
Design-based and
model-based stereology
• Model-based - assumptions of
randomness are made about the
structure (probes are fixed, structure
random).
• Design-based - no assumptions are
made, the sampling uses carefully
randomised probes (structure fixed,
probes random).
Design-based and
model-based stereology
• Material science problems are often
cast in a model-based setting.
• Biological problems have
increasingly been seen as ideal for
design-based methods.
• Neither of these approaches relies
upon unrealistic model assumptions of
sphericity or uses ill conditioned
unfolding methods.
DELESSE'S PRINCIPLE
Unbiased estimation of Volume Fraction – Vv
• The volume fraction of a phase within a 3D
reference volume can be estimated without bias by
randomly translating a point grid over a uniform
random cross-sectional image in both the x and y
direction.
• The number of points hitting the reference space
(Pref) and the number of points hitting the phase of
interest Y , (PY) are counted and volume fraction of
Y is estimated from
Vv (Y,ref) =
14
Σ AY / ΣAref = ΣLY / ΣLref = ΣPY / Σ Pref
The Delesse's principle is
based on homogenous
structures.
In stereological viewpoint in "isotropic and
uniform random, IUR, sections".
Vv = Volume Fraction
f(z) is the probability of
finding a test plane at a
position between z and z+dz
Vv = Volume Fraction
If the test planes are uniformally
distributed in position, so that the
probability of finding a test plane in
any interval is the same as that for
any interval, then f(z)dz = dz / L
L
dz
L A(z) dz V
= 3 =V
3
v
L
L
A A = ∫0 A A(z ) L = ∫0
15
Unbiased estimation of Volume Fraction - Vv
The number of points
hitting the reference space
(light and dark grey) is 15
and the number hitting Y
(light grey phase only ) is 6
(5+2x0.5) , thus the volume
fraction of Y in the reference
volume is estimated to be,
1
2
4
3
5
Vv = 6 / 15 = 0.40
A test system of cycloids
Reference Volume
Many of the methods used in stereology provide
estimates of the amount of a feature per unit
reference volume i.e. densities .
For example:
Volume density, Vv, the volume proportion of the
phase of interest within a reference volume.
Surface density, Sv, the area of an interface within
a unit reference volume.
Length density, Lv, the length of a linear feature
within a unit reference volume.
Numerical density, Nv, the number of discrete
objects in a unit reference volume.
16
'Reference Trap'
Final data should be presented per animal
and not as density
e.g. Hypertrophy vs Hyperplasia.
Sv = 6 (2r)2 / (2r)3 = 3 / r
Sv = 4πr2 / (4πr 3/3) = 3 / r
Sv = Surface Density = Surface to volume ratio
SV = Surface Density = Surface to volume ratio
π 2π dS
sin(Φ )dΦ dΘ
dS
d NL = ∫ ∫
cos(Θ )
=
0 0 L3
4π
2L3
The number of intersections
dS
S
=
formed all surface area in the
N L = ∫∫S
3
2L
2L3
structure is the sum:
The total extent of surface area per unit volume = SV = 2 NL
17
SV = Surface density is directly related to the
number of interactions (Ii) formed with test
lines of length Lt :
SV= 2 Ii / Lt
COHERENT SQUARE GRID
SV = ΣIi / d ΣP i
COHERENT MULTIPURPOSE GRID
SV = 4 ΣIi / z ΣP i
Intercept length and grain size
• Surfaces within real specimens
can have very large amounts of
area occupying a relatively
small volume.
• The mean linear intercept λ of
a structure is often a useful
measure of the scale of that
structure, and is related to the
surface-to-volume ratio of the
features, since
λ = 4VV / SV= 4V / S
Chandreshakar (1943) showed that for a
random distribution of stars in space
•
•
18
The mean nearest neighbor distance
is L = 0.554•Nv-1/3 where Nv is the
number of points (stars) per unit
volume.
For small features on a 2D plane the
similar relationship is L = 0.5•NA-1/2
where NA is the number per unit area.
Normalized count:
PA= 13 counts / 1600 µm2 =
0.0081 counts/µm2
Geometric property
LV = 2 P A
2 • 0.0081 =
0.016 ( µm/µm3 ) =
16 ( km/m3 )
Disector
NA = NV (D + t)
t = slice width
A = Lookup plane area
NA2 – NA1 = NV (t 2 – t1)
(NA 2 – NA 1) A = Nv (t 2 – t1) A
NV = N / V (slice)
19
Increased Islet Volume but Unchanged Islet Number
in ob/ob Mice.
Bock T, Pakkenberg B, Buschard K. Diabetes. 52:1716-22, 2003.
The figure illustrates the sampling of histological sections. The
pancreas was exhaustively sectioned, and both the sections marked
as primary and the reference sections were used in the stereological
investigations.
Each sampled primary section
was systematically investigated
as illustrated in the top of the
figure. This was performed in
two sessions during which
either the counting frame
(lower left in the figure) was
attached to the table to count
the islets as described in the
text, or the point-counting grid
(lower right in the figure) was
attached to the table to count
the total volume of islets, the
total volume of pancreas, and
the volume-weighted mean islet
volume
In the primary section, islets
denoted B, C, and D are defined as
within the counting frame because
they do not touch the (full)
exclusion lines, and they are either
completely within the frame or
touch the (dashed) inclusion lines.
The islets denoted A, E, and F are
defined as being outside the frame
because they are either completely
outside the frame or touch the
exclusion lines. In the reference
section shown at the bottom of the
figure, islets B and D are still
present, and therefore they are not
counted, whereas islet C is not
present, and therefore it is counted
20
Disector - NV of convex features in three dimensional space.
This count: N = 6
Relationship: NV = N / V0
Geometric property: NV = 6 / 32 µm3 = 0.19 µm
–3
= 1.9x1011 per cm 3
Why Start with 100-200 Counts
and 10-20 Sections?
• The 100-200 counts stems from the fact that the
sampling with disector probes by itself must be
viewed as independent random sampling.
• At the moment, there is no mathematical basis
for calculating the variance of a two or three
dimensional systematic random sample and one
must resort to the formula for independent
sampling.
Thus, the basis for suggesting that 10-20 sections
be used is primarily empirical.
Bonaventura Francesco Cavalieri
Born: 1598 in Milan, Duchy of Milan, Habsburg Empire (now Italy)
Died: 30 Nov 1647 in Bologna, Papal States (now Italy)
• Bonaventura Cavalieri joined the
religious order Jesuati in Milan in 1615
while he was still a boy.
• In 1616 he was transferred to the Jesuit
monastery in Pisa. His interest in
mathematics was stimulated by
Euclid's works and after meeting
Galileo, considered himself a disciple
of the astronomer.
• The meeting with Galileo was set up
by Cardinal Federico Borromeo who
saw clearly the genius in Cavalieri
while he was at the monastery in
Milan.
21
• In Pisa, Cavalieri was taught mathematics by Benedetto Castelli, a
lecturer in mathematics at the University of Pisa. He taught Cavalieri
geometry and he showed such promise that Cavalieri sometimes took
over Castelli's lectures at the university.
• Cavalieri applied for the chair of mathematics in Bologna in 1619
but was not successful since he was considered too young for a
position of this seniority. He also failed to get the chair of
mathematics at Pisa when Castelli left for Rome.
• In 1621 Cavalieri became a deacon and assistant to Cardinal
Federico Borromeo at the monastery in Milan. He taught theology
there until 1623 when he became prior of St Peter's at Lodi. Aft er
three years at Lodi he went to the Jesuit monastery in Parma, where
he was to spend another three years.
• In 1629 Cavalieri was appointed to the chair of mathematics at
Bologna but by this time he had already developed a method of
indivisibles which became a factor in the development of the integral
calculus.
Cavalieri’s estimator
The honorary application of Cavalieri’s name to the estimator has
led to some confusion, at least linguistically, with regard to his
contribution to the stereologic method, as manifest in usages
such as Cavalieri’s method and Cavalieri’s principle.
The theorems of Cavalieri deal with ratios of volumes of solids
based on all the planes, the indivisibles, through the solid, not a
finite number of sections of known thickness (divisibles).
The use of slices to estimate the volumes of regular solids
was in use long before Cavalieri’s work.
The major virtue of Cavalieri’s analysis, as Sterio points out,
was the generalization of the use of indivisibles to the
ratios of volumes of any two solids .
Thus it seems best to designate the stereologic method as the
Cavalieri estimator, thereby honoring him without incorrectly
attributing the method or the principle to him.
This estimator of volume,
named in honor of
Cavalieri, is the product of
the sum of the areas of
parallel sections cut
through a particle and the
thickness of the individual
sections, ( ΣAi )*∆x, where
Ai is the profile area of the
i-th section and ∆x is the
mean section thickness.
22
Threedimensional
surface
reconstruction
of a spheroid
test object.
Duerstock et al. J. of
Microscopy 210 , 13
8-148, 2003.
A computer generated plot showing the number of particles required to be
sectioned to reach a 95% confidence interval of ±5% of the mean volume for
varying dimensions of oblate ellipsoids, varying standard deviations of the
volumes in the parent populations and varying ratios of the section thickness
to the major axis.
200
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H B F F
P
J
B
F
HJ
J
H
F
J
P
B
H
0.5
J
J
YES
REPEAT ?
NO
4 0
GOTO STATISTICS AND
GRAPHICS SUBRO UTINES
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Section thickness / Major axis
SERIAL
S E C T I O N I N G (e.g. MRI, CONFOCAL)
ALONG EITHER MAJOR AXIS OR MINOR AXIS
Ellipsoid, axial ratio 1 : 3 : 3
minor axis = 0.596 unit ength
l
major axis = 1.79 unit length
0.5
0.5
MIN OR A XIS
INTERVAL
CONFIDENCE
DO
Section thicknes = 0.2-0.4
Along the major Axis
0.4
0.3
0.3
0.2
95%
NOT RECOMMENDED
Many serial sections
MAJOR AXIS
0.4
.
± .
.
± .
.
± .
.
± .
0.2
0.1
0.1
0
0
0
50
100
150
200
0
50
100
150
200
100
150
200
Ellipsoid, axial ratio 1 : 1 : 3
minor axis = 0.860 unit ength
l
major axis = 2.58 unit length
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
50
100
150
200
NUMBER OF
23
SECTI ON TH C
I K N ES S
0
50
SAMPLED PARTICLES
Nucleator and Star Volume
vn* = 4π/3 li3
Correlation
between
mitochondria
volume fraction
and star volume
during
pancreatic
acinar cell
regranulation
24
Surface-weighted star volume
Gittes, F. (1990) Estimating mean particle volume and
number from random sections by sampling profile
boundaries. J. Microsc . 158: 1-18.
v s* = 2π/3 li3
CV = σ/ vN = (vV / vN –1)1/2
Evaluation of the measure of disperssion (CV = σ/vN) demonstrate
that even with large number of measurements
the precision of CV estimation is poor.
Number of sampled profiles
AUTORADIOGRAPHY
3-5 minutes
25
15-30 minutes
2 hours
Defective cytoplasmic granule formation. II. Differences in patterns
of radiolabeling of secretory granules in beige versus normal mouse
pancreatic acinar cells after [3H]-glycine administration in vivo.
Hammel I, Dvorak AM, Fox P, Shimoni E and Galli SJ. Cell Tissue
Res. 1998; 293:445-452.
Data can be presented as Silver Grain Density or
Percent of Labeling
Correlation of size with “age”
26
Brain map
Densitometry
Autoradiography
We are now at the end of our journey,
and the two ideas worth repeating are:
1. $200 is a lot cheaper than $100,000.
A little education can save a lot.
2. I am very proud to be part of that
rapidly growing branch of microscopy
known as “quantitative microscopy".
It is the future of structural biology.
27