503
A counting method for measuring the volumes of tissue
components in microscopical sections
By A. DOUGLAS HALLY, M.D.
(From the Department of Anatomy, University of Glasgow, Scotland. Present address:
Department of Anatomy, University of Newcastle upon Tyne, England)
Summary
Several methods are available for estimating the relative volume of a tissue component from a study of tissue sections. These methods are all based on the fact that
the mean relative area of a component in a series of random sections through a tissue
is a consistent estimate of its relative volume in the whole tissue. Thus the problem
is basically one of measuring area in a section, which can be done by the following
simple counting method.
The method consists of placing a regular pattern of points in the form of a square
lattice upon the section image, and counting the number of points over the section
N, and over the component n.
Relative area of component = njN
The method also measures absolute area, and where d is the distance between
adjacent points,
, .
,.
r
absolute area of component == nd2.
This capacity to measure absolute area means that the method is particularly
suitable for determining a component which has a low relative volume.
The accuracy of the method is influenced by several factors including the size of
the grid mesh, and the relative area, shape, and spatial arrangement of the component. With reasonable care the error will not be larger than that of a truly random
system of points, as expressed by the following:
relative standard error = V(* ~P)hfn,
where p is the relative area of the component, and the relative standard error (R.S.E.)
is S.E./relative area of component.
The method is equally applicable to either light or electron microscopy. A series
of measurements on electron micrographs of rat cardiac muscle revealed a close
agreement between the counting method and planimetry.
The method is rapid, simple and accurate, and requires no complex apparatus.
Introduction
A PROBLEM which arises frequently in biology is that of estimating the
relative volume of tissue components from a study of tissue sections.
The available methods include planimetry, paper weighing, lineal analysis 1 ' 12 ' 13 , and random point analysis2, and are all based on the reasonable
assumption that the mean relative area of a component in a series of random
sections will be a consistent estimate of its relative volume throughout the entire
tissue. Thus the problem reduces to that of measuring area in a section, which
can be done by the following simple counting method, using a non-random
pattern of points.
[Quart. J. micr. Sci., Vol. 105, pt. 4, pp. 503-17, 1964.]
Hally—Measurement of objects in sections
5°4
The method is basically that of Glagolev6'7, but has been further developed
by Chayes3>4's, and is now widely used by geologists. Initially Haug8 and
later Hennig and his co-worker9'I0 applied it to biological problems.
This paper emphasizes the value of the method in both electron and light
microscopy, considers certain factors which may affect its accuracy, and shows
that it is particularly suitable for the measurement of tissue components with
a low relative volume.
Method
A regular pattern of points, in the form of a square lattice, is superimposed
on the section image (fig. i), and the number of points over the various components is counted. Before each additional count the grid or lattice is lifted
and replaced at random over the section.
A
B
• t
FIG. I.
A, diagram of section containing an irregular component, x
B, estimation of relative area
A section with a superimposed lattice of points. N, the number of points over the" entire
section = 30, and n the number over component x = 12.
Relative area of x = njN = 12/30 = 2/5.
c, estimation of absolute area
Lattice superimposed on component x divides area into hypothetical squares, each rf2 in
area, where d is the distance between adjacent lattice points. The interrupted line indicates
the estimated absolute area of x.
Absolute area of x = nd2.
In fig. I, B,
the relative area of x = xjaiea. of section == njN
(i)
where n is the number of points or 'hits' on x, and N is the number over the
entire section.
It is important to appreciate that this method, because it adopts a regular
pattern of points, is capable of measuring absolute as well as relative areas. In
fig. 1, c the absolute area of the irregular shape x is estimated in terms of an
integral number of hypothetical squares. Each square is centred on a lattice
point and is d2 in area, where d is the distance between adjacent lattice points.
Hally—Measurement of objects in sections
505
Absolute area of x = nd2
(ii)
and for repeated counts,
absolute area of x == ndz,
where n is the average number of hits on x per count.
It is often convenient to choose a grid where d = 1 cm because then
absolute area of x = n cm2.
(iii)
Subsequently it will become clear that this ability to determine absolute
area is of considerable value in certain problems.
Accuracy of method
Like the other methods, the present one is extremely accurate provided a
sufficiently large number of points is counted, and fig. 2 shows the decrease in
L
73-
o
72-
-+2%
-+1%
T
T
o
S"
o_
c_
a
- 0% 5
o
- 1 %
"2%
71-
70-
0
1000
no- of hits, n
2000
FIG. Z. Graph of measurements of large circle
A circle of radius 4/81 was counted 16 times with a grid of 1 cm2 squares. The graph shows
the estimations and their relative standard errors. The 'true' area was calculated from the
measured radius.
As n, the total number of hits, becomes larger, the relative standard error diminishes, and
there is an increasingly good agreement between the estimated area and the 'true' area.
standard error which occurs as n increases and the close agreement between
the estimated area of a circle obtained by counting, and the 'true' area calculated from the measured radius.
506
Hally—Measurement of objects in sections
However, the efficiency of the method, that is to say the accuracy attainable
with a given number of points, depends on a number of factors, including the
size of the grid squares relative to the component, and also the relative area,
shape, spacing, and orientation of the component.
Relative area or volume of component
Glagolev6*7 assumed that the arrangement of points was effectively random,
and from Gaussian probability deduced that the equation for the standard
error was
standard error (S.E.) = J{p(i-p)}HN,
(iv)
where p is the relative area of the component, and N is the total number of
points counted over the entire section. But this method can measure absolute
area, which makes possible certain short cuts in which JV need not be counted,
and therefore it is often more convenient to use the following derivation of
Glagolev's equation:
Relative standard error (R.S.E.) = ^j{i—p)j^n,
(v)
where n is the number of hits on the component, and the relative standard
error is the S.E./relative area of component. The use of the R.S.E. is probably
more logical than the S.E., because in practice the degree of accuracy required
will depend on the relative area of the component. For example, suppose two
components x and y have relative volumes of 50% and 1%. If an S.E. of 5%
is chosen then the results will be x == 50% ± S.E. of 5%, and y = i%±S.E.
of 5%. Clearly the result for x is reasonable, but that for y is very inaccurate.
On the other hand, if an R.S.E. of 10% is chosen, then more reasonable values
will be obtained: * = 5O%±S.E. of 5%, andy = i%±S.E. of o-i%.
Equation (v) is derived in the following way:
Standard error = <j{p{i—p)IN}.
But if N is large, then
p == njN.
(iv)
.'. N === njp.
Substituting for N, in equation (iv),
standard error == •\/{p(i—p)l(nlp)}
But relative standard error = standard error/relative area, p.
_ , .
, ,
pJ{i—p)Nn
Relative standard error = *-^
——
P
== J(i-p)Hn.
(v)
From equation (v) the R.S.E. decreases as^>, the relative area, increases, and
conversely it reaches a maximum as p —*• o, when
R.S.E. = i/V«.
(vi)
Hally—Measurement of objects in sections
507
Grid size
Suppose component x is a single circle. With a large mesh grid, where d
(the distance between adjacent points) is greater than the diameter of the
circle x, none of the grid squares will be filled completely, and applying
equation (v),
R.S.E. = V(i-*>)/V«,
where p is the relative area of x. (As absolute area is being measured here, it
is illogical to define p as a relative area, and in actual fact p is the probability
of hitting x with a single count, which can however be shown to equal its
relative area.)
From equation (v), as the grid becomes coarser and the circle relatively
smaller, then
.
DOT?//
maximum K.b.ii. == i/Vre.
Thus for a coarse grid
R.S.E. ^ I/VM.
(vii)
With a grid of finer mesh, where the circle or irregular area x fills certain of
the squares completely, there is a substantial gain in accuracy. This improvement is most easily explained graphically (fig. 3), because although it has been
proved mathematically for circles and ovals11'14, no such proof exists for
irregular areas. In fig. 3 the fine mesh grid is superimposed on x, and as before
effectively divides the area into a number of squares each centred on a point.
Each point 'represents' the square around it, an area of d2. The central portion
of x fills a number of squares completely, and is therefore exactly 'measured'
by the points within them. But at the periphery of x there is a strip of partly
filled squares containing points which may or may not be counted as hits
according to whether they lie inside or outside x. This peripheral portion is
inaccurately measured, because the area of * within each square will either be
over- or under-estimated, and will have an R.S.E. similar to a series of small
R.S.E. of peripheral strip == i/Vwp,
(viii)
where np is the number of hits on the partly filled squares. Consequently the
total error is reduced by choosing a fine mesh grid, because of this 'central
effect' whereby the central part of a component is exactly measured. This
conclusion is supported by the data in table 1 (see appendix, p. 516) and by
fig. 4, which shows that for a given value of n, the R.S.E. becomes smaller as
the grid becomes finer.
Thus, provided a fine mesh grid is adopted, this method is more efficient
than one based on a random series of points.
Shape of component and grid orientation
The method is demonstrably accurate for measuring circles, but they have
an ideal shape, because the ratio of peripheral strip to central portion is at
a minimum, and also the orientation of the grid has no effect on the error.
Consider therefore a different component shape which is less satisfactory.
5o8
Hatty—Measurement of objects in sections
accurately measured central portion[~~J]
/ overestimated areap"-\]
peripheral portion/
underestimated area ] .-"NJ
FIG. 3. Graphic explanation of accuracy of method
The area of component x is estimated in terms of hypothetical squares, each d* in area,
where d is the distance between adjacent points. The area of each square is 'represented' by
the point at its centre. The interrupted line indicates the estimated absolute area.
The central portion of x fills certain squares completely, and is therefore exactly 'measured'
by the points within them. But the peripheral portion of x only partly fills individual squares,
and consequently is either over- or under-estimated, which results in a random error. The
finer the grid, the greater the proportion of completely filled squares, and the more accurate
the method.
The R.S.E. of estimations on a narrow component x, which is less than d
in width, is similar to that of a small circle, provided the grid is orientated so
that the long axis of x is inclined to a grid axis of symmetry (fig. 5, A). There
is a marked rise in R.S.E. when these axes nearly coincide (table 2, p. 517),
and therefore these axes should be inclined to each other.
Spatial distribution of component
We have seen that with care the R.S.E. of measuring a single component
will be less than i/Vw, but as this method adopts a regular pattern of points,
Hally—Measurement of objects in sections
100
200
300
400
total number of hits, n
5°9
S00
FIG. 4. The relative standard errors of measuring circles
Some data from table i, relating the R.S.E. of estimations of the area of a circle to n, the total
number of hits, d is the distance between lattice points, and zr the diameter of the circle being
measured. The ratio djzr indicates the relative size of the grid, and diminishes as the grid
becomes finer.
The graph shows that R.S.E. < i/Vn. Moreover, for a given value of n, the R.S.E. diminishes as the grid becomes finer.
there is a possibility that a component consisting of a number of regularly
arranged separate items might be distributed in such a way that it was in
register with the points (fig. 6, A), which would result in a very large R.S.E.
Fortunately in such cases the error can be reduced to reasonable dimensions
by inclining the grid to the rows of items (fig. 6, B). The remedy is therefore
similar to that used in estimating narrow components, but observe that the
angle of inclination in both cases should be about 200 to 300 and not 450,
because a square grid has an axis of symmetry every 45°. However, such
regular arrangements will rarely occur in biology, and when present would
probably be easily recognized. Moreover, casual inspection of a few counts
on a single section would reveal whether or not the R.S.E. was unusually large,
510
Hally—Measurement of objects in sections
and if it was in fact unsatisfactory then the total number of hits could be
increased to reduce it to the desired level.
FIG. 5.
A, the effect of orientation of square grid on estimation of narrow component
2
The true area of component x = ijW . The only possible counts on x, when the major
grid axis coincides with that of the component (position Xi), are either 3 or o. However, inclination of the axes to each other reduces the range of counts, thereby reducing the standard
errors of the estimates.
B, the effect of orientation of rectangular grid on estimation of narrow component
Diagram similar to 5, A, except that the grid is rectangular instead of square. In position X\
the counts are either 6 or o, which gives a standard error greater than that obtained with the
square grid. Moreover, inclining the axes does not reduce the error to the same extent.
Type of grid
Glagolev7 used a single fixed point which traced out a square lattice as the
section was moved stepwise by a mechanical stage. Any method which involves movement introduces a small additional error due to irregularities in
the mechanism itself,3 but this is avoided in the present method by the
adoption of a stationary grid. Chayes4 chose a method similar to that of Glagolev, except that the single point traced out a rectangular grid in which the
point frequencies along the two main axes of symmetry were unequal. This
type of grid is less accurate for measuring narrow components (fig. 5, B and
table 2), and although this factor was unimportant in Chayes's petrographic
analysis, it cannot be ignored in biology, especially in electron microscopy
where narrow membranous components are common.
Hally—Measurement of objects in sections
511
A triangular grid consisting of points situated at the angles of equilateral
triangles has 6 axes of symmetry, and might prove more accurate than the
square lattice, but for routine use the square grid has the advantage of being
much simpler to construct.
•D-D-D-D-D-D-D-D•D-D-D-D-
D '•. • D
D -D Q D
DP'D D
• b.ri D
B
FIG. 6. Effect of grid orientation on estimation of a regular pattern of items
As this method employs a non-random system of points, a component consisting of regularly
spaced items may be in register with the grid points. For example, in fig. A the grid is orientated so that only counts of either o or 16 are possible, resulting in a large standard error.
However, if the major grid axis is inclined to the rows of items, as in fig. B, then the R.S.E.
drops to < l/Vn.
To summarize this assessment of the accuracy of the method, it appears
that although several factors affect the efficiency of the method, provided
reasonable care is taken in the measurement of narrow, highly orientated, or
regularly arranged objects, the relative standard error is unlikely to exceed
i/Vn, and will probably approximate closely to the expected error of a truly
random system of points, where
R.S.E. = V(i-/0/Vn.
(v)
Si2
Hally—Measurement of objects in sections
Moreover, with a fine grid the central effect will further reduce the error
below that of a random point system.
In practice there is little difference between the expressions i/Vre and
*J(I~P)Hn if p is < 0-25, but as p increases the theoretical error diverges
from I/VK, and the value of ^(i—p)j^!n is only o-5/V» when p — 0-75, and
drops to O-2/VK when ^> = 0-95. As a working rule, therefore, the error can be
taken as
R.S.E. = i/Vn
(vii)
if p ^ 0-25, but if p > 0-25 then the more accurate expression
R.S.E. = V(I-/>)/VM
(V)
should be used to calculate the number of hits required to reach the desired
degree of accuracy. Suppose a component is to be measured to give an R.S.E.
of 2-5% on a sample consisting of 8 sections. Inspection reveals that the relative area of the component is much less than 25%, that is^> < 0-25, so equation
(vii) is applicable and substituting,
2*5/100 = i/Vra,
therefore
,
V« = 40,
giving
n = 1,600.
The total of i,6oo points can be equally divided over the 8 sections, so that
200 points are counted on each section.
The advantage of measuring absolute area in the measurement of components with
a low percentage volume
The capacity of the present method to measure absolute areas is advantageous when estimating components with a small percentage volume, because
in such cases the direct determination of relative areas is very tedious. If the
relative area of a component x in a series of sections is about 1% then from
equation (i),
rektive ^
Q{ x = J / I O O = njN^
therefore
N = IOOM.
If an R.S.E. of 5% is required, then from equation (vii) n must be about
400, and therefore, substituting,
i V = 100x400
= 40,000.
Fortunately the counting of such a huge number of points can be avoided
in the present method by estimating the absolute areas of x and the section
independently, and then relating them. The same reasoning applies to lineal
analysis, provided D, the interval between scan lines, is kept constant. If s is
the total intercept length on component x,
absolute area of * = sD.
Hatty—Measurement of objects in sections
513
The area of the section is estimated by a few counts a, until N (the total
number of points) reaches 400, and then further counts b are made in which
only n the hits on x are recorded until n reaches 400. Then from equation (ii)
absolute area of section = Ndzja,
absolute area of *
== ndzjb,
therefore relative area of * == najNb, and only about 800 points have been
counted.
The measurement of absolute area also makes it possible to use grids of
various sizes on a single section. Suppose an electron micrograph 20 cm X 20
cm in size contains two components * and y which occupy approximately 25%
and 1% of the area. On a single count with a grid of 1 cm2 squares, the counts
will be approximately as follows: on whole section, 400; on x, 100; and on;y
only about 4. If an R.S.E. of 5% is required on this section, then from
equation (vii) the hits on each component must be at least 400, and therefore
while a few counts are adequate for the section and x, another 100 are necessary
for the estimation of y. It is therefore simpler, and probably more accurate,
because of the central effect, to select a finer grid for y and count it separately.
With a grid of squares 0-25 cm2 in area, for example, only 25 counts would be
required, and from equation (ii),
absolute area of y == fid2.
As d2 = 0-25 cm2,
absolute area of y == 0-25^ cm2.
This area can easily be related to the absolute areas of * and the section.
To extend this flexibility of the present method further, there is no reason
why certain components should not be measured by counting, and others in the
same section estimated by any other more convenient method. For example,
the total area of an electron micrograph can be measured accurately with a
ruler much more readily than by counting or planimetry.
Scope of the method, and practical procedure
The method is equally applicable to both light and electron microscopy.
Light or electron micrographs may be measured by a superimposed lattice,
consisting of 1 cm2 squares ruled with a pencil on a transparent plastic sheet.
A suitable transparent matt film (Drafton 253-080) is obtainable from Ozalid,
Division of General Aniline and Film Corp., Johnson City, N.Y. Alternatively,
light microscopic sections can be examined either through a lattice inserted
into the eyepiece, or by projecting the microscopic image on to a white sheet
of cardboard ruled with a lattice.
To illustrate its application in electron microscopy, the volume of mitochondria relative to cytoplasmic volume, and the volume of mitochondrial
cristae relative to mitochondrial volume, were determined in rat cardiac
muscle.
514
Hatty—Measurement of objects in sections
Estimation of mitochondrial volume/total cytoplasmic volume
Seven electron micrographs of rat cardiac muscle, at magnifications of
about 24,000, were each counted 3 times, with a randomly placed grid of
1 cm2 squares. The hits on the mitochondria and cytoplasm were recorded
separately, and the relative volume of mitochondria determined from the ratio
of the hits on mitochondria to those on total cytoplasm. The relative standard
IS
\
4
\
13
\
12
0
\
\
\
9
\
8
7
6
S
4
4
"*"--.
3
2
*
•
*
1
0
100
200
300
400
total number of hits, n
S00
FIG. 7. Relative standard errors of measurements on electron micrographs of rat cardiac
muscle
On each micrograph both the mitochondrial volume/cytoplasmic volume (•) and the
volume of mitochondrial cristae/cytoplasmic volume ( + ) were determined by the counting
method.
The relative standard errors of estimations are graphed against n, the total number of hits.
The average R.S.E. < i/Vn and approximates closely to -J(i —p)j«Jn. (p = 0-4 in both sets
of data, so the single line applies to both.)
error of the triple estimations for each micrograph was calculated and found
on average to be less than i/Vra (table 3, p. 517, and fig. 7). In fact the average
error was better than ^/(i—p)l*Jn, the expected error for a truly random
system, and this discrepancy was probably due to the central effect, because
Hatty—Measurement of objects in sections
515
the mitochondrial profiles were usually large enough to fill several squares
completely. Planimetric measurement of the relative volume in each of the
micrographs confirmed the accuracy of the counting method, as the final
values for the relative mitochondrial volume obtained by the two methods
agreed very closely (table 3). The relative mitochondrial volume by counting
was 34-8%.
Estimation of volume of mitochondrial cristae\mitochondrial volume
The proportion of mitochondrial volume occupied by cristae in rat cardiac
muscle was measured in 6 electron micrographs at magnifications of about
100,000. Only mitochondria in which the cristae lay parallel to the electron
beam were selected. Each mitochondrion was counted several times until the
number of hits on the cristae reached about 200, and the grid was placed at
random over the micrograph so that the long axes of the cristae and the grid
axis of symmetry were at an angle of 30°. The relative standard error on
each micrograph approximated to V(i—/>)/V« (fig. 7). This is satisfactory,
because the cristae in cardiac muscle form a highly orientated system of
narrow components, which might have resulted in a high R.S.E. if care had
not been taken in orientating the grid.
On this small sample the proportion of mitochondrial volume occupied
by cristae was 427% (S.E.±i-6%).
Discussion
This method of regular point analysis is extremely simple and rapid. It
requires little apparatus, although Chayes4 recommends a Clay Adams blood
cell counter to ease the counting. Possibly the main advantage of the counting
method over other methods such as planimetry, paper-weighing, and lineal
analysis, is that it is almost as quick to count a component composed of a large
number of separate items, as to count one consisting of a single item.
The reason why a regular pattern of points functions as though it were
random is partly because it effectively divides the area into squares within
which the portions of the components tend to lie at random. Moreover, where
sections contain regularly arranged items, the randomness is introduced
deliberately by the analyst, first by orientating the grid so that it is out of
register with the components, and second by counting the section several
times, the grid being replaced at random before each count. Consequently,
although the efficiency of the method has been shown to depend on a number
of factors, with reasonable care it will correspond closely to that of a random
system, except where a fine mesh grid is used, when the central effect can be
expected to reduce the error further.
Finally, the fact that regular point analysis, unlike random point analysis,
is capable of determining absolute area, is advantageous in the measurement of
low-volume components, and increases the flexibility with which it can be
combined with other methods of sectional analysis.
5i6
Hatty—Measurement of objects in sections
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Carpenter, A. M., and Lazarow, A., 1962. J. Histochem. Cytochem., 10, 329.
Chalkley, H. W., 1943-44. J- Nat. Cane. Inst., 4, 47.
Chayes, F., 1949. J. Geol., 62, 92.
1949- Amer. Mineral., 34, 1.
and Fairbairn, H. W., 1954. Ibid., 36, 704.
Glagolev., A. A., 1933. Trans. Inst. Econ. Min. (Moscow), 59, 5. (In Russian: English
abstract.)
1934. Engr. Min. J., 135, 399.
Haug, H., 1955. Zeit. Anat. Ent.Gesch., 118, 302.
Hennig, A., 1957. Mikroscopie, 12, 174.
and Meyer-Arendt, J. R., 1963. J. lab. Invest., iz, 460.
Kendall, D. G., 1948. Q. J. Math., 19, 1.
Lazarow, A., and Carpenter, A. M., 1962. J. Histochem. Cytochem., 10, 324.
Loud, A. V., 1962. J. Cell Biol., 15, 481.
Sawyer, D. B., 1953. Q. J. Math., 4, 284.
Appendix
TABLE I
The influence of n, the total number of hits, on the accuracy of the method
Circle
radius
(cm)
0-445
No. of
counts
c
64
128
256
512
819
072
238
37
77
165
315
SIS
Estimated
area
n\c
163
54
64
107
167
210
1 64
410
161
2
34
70
139
279
16
32
2
4
8
16
32
554
147
287
S73
1156
2314
nr*
0622
128
256
4
'True'
calculated
area
0-578
0601
0645
0614
0629
168
32
8
4-81
Total
no. of
points
n
170
- 7 1
- 3 4
+ 37
-i-3
+ i'i
17-72
17-5
17-38
17'44
I7'3i
73-5O
7*75
7163
72-25
7239
% Relative
% Absolute
S.E.
error
S.E./esti{nlc—irr*)lirr- mated area
72-47
+ 3'i
+ 2-5
+ o-6
.
1 0 0
6-9
48
3'5
2-7
Si
36
-0-9
26
19
-3-9
5-8
— IT
-i-8
5-4
27
- 1 4
2 2
— 22
I'S
+ i-4
0-7
18
— 10
— 12
—03
— 0-2
1 0
07
04
Single circles of differing radii were counted repeatedly with a randomly placed
lattice. The estimated area = ndz, but d2 the area of a lattice square was 1 cm2, and
therefore the estimated area was simply the mean number of hits per count. The
'true' area was calculated from the measured radius.
Clearly, for any circle the errors decrease as n the total number of hits increase;
and for a given value of n, the method becomes more accurate as the radius of the
circle increases.
Hally—Measurement of objects in sections
TABLE 2
The relationship between grid orientation and the R.S.E. of
measurements on a narrow component
0°
Square grid
% R.S.E.
n-6
Angle between grid
and component axes
S°
7"2
10°
5'5
iS°
30°
3-2
30
45°
59
Rectangular grid
% R.S.E.
14/6
109
96
3'i
3 i
The components and grids are those in figs. 5, A, B. At each angle component x
was counted repeatedly until n = 200, so that if R.S.E. = i/Vn, then R.S.E. would
be about 7%. The actual R.S.E. is greater than 7% when the long axis of * and an
axis of grid symmetry coincide, but as the angle between the axes increases the error
falls. The most suitable angle is about 20° to 35°, which should be chosen when
narrow objects are measured.
Observe that the rectangular grid is less accurate than the square one for this
shape of component.
TABLE 3
Estimation of mitochondrial volumejcytoplasmic volume in rat cardiac muscle
by both the counting method and planimetry
Cou nting me hod
Electron micrograph no.
H I
H 2
H
3
H
4
Planimetry
Total no. of
Single estihits on mitomates of mitochondr al
M an MIC
volume (MIC)
chondria
x
x
«
44-2
362
43 4
360
39-8
3°'5
41-7
41-6
42-9
29-4
% Relative
standard
error
R.S.E./x
Milochondrial
vol/cytoplasmic
volume
xp
% Absolute
error
(x-xp)lxp
41-3
392
± 7'l
395
+ 4'6
37-4
401
± 4'1
362
+ 3-3
42-1
433
± 1-9
41-6
4- 1-2
293
252
± 035
290
4- 1 0
271
280
± 2-1
266
4- 17
280
229
± 33
271
4- 33
39'"
— 1-8
294
H
5
29-1
278
360
2
H 6
Hj
7'5
29-7
26-8
a
7'4
40'o
377
37-5
38-4
To tal
243 -6
Gr and
me an 34-8
336
± 2'I
Total 239-1
Grand
34-2
mean
Mean
error
+ i-8
Each electron micrograph was counted 3 times, with a randomly placed grid of
1 cm2 squares. The standard errors approximate to V(i - £)/Vra (see fig. 7), and there
is a very good agreement between the values obtained by counting and those
measured by planimetry.