Brain volume estimation from serial section measurements - CiteSeerX

Journal of Neuroscience Methods, 35 (1990) 115-124

115

Elsevier NSM 01156

Brain volume estimation from serial section measurements: a comparison of methodologies G l e n n D. R o s e n 1 a n d J a s o n D. H a r r y 2 1 Dyslexia Neuroanatomical Research Laboratory and the Charles A. Dana Research Institute, and Department of Neurology, Beth Israel Hospital, Boston, MA 02215 (U.S.A.)and Harvard Medical School, Boston, MA 02215 (U.S.A.), and 2 Division of Engineering, Box D, Brown University, Providence, R I 02912 (U.S.A.)

(Received 8 January 1990) (Revised version received 11 June 1990) (Accepted 18 July 1990)

K e y words: Cavalieri's rule; S i m p s o n ' s rule; Trapezoid; Brain volume; Serial sections

Estimation of brain volume from serial sections typically involves using a rectangular, Cavalieri's, parabolic (Simpson's), or a trapezoidal rule to integrate numerically a curve of cross-sectional area measurements plotted against section number. We practically compare the efficacy of each of these methods using mathematical simulations of regularly- and irregularly-shaped "brain volumes" as well as actual morphometric measures from brain regions. There are no meaningful differences between the various estimates when many sections are used - - with fewer sections, Cavalieri's estimator is most accurate. This confirms previous theoretical reports demonstrating the efficiency and accuracy of the Cavalieri estimator of volume, particularly when few sections are analyzed. While the Cavalieri approach provides a better approximation of volume under some circumstances, it requires equally spaced sections. We therefore describe methods for the estimation of brain volume from unequally spaced sections, including an estimator based on the fitting of piece-wise parabolic curves to the data. We outline a series of guidelines for the use of these mathematical rules in the estimation of brain volume from serial sections.

Introduction T h e accurate m o r p h o m e t r i c c o m p u t a t i o n of the v o l u m e of b r a i n regions is a n issue of some concern to n e u r o a n a t o m i s t s from a wide variety of fields (cf. G o t t l i e b et al., 1985; H o f m a n , 1988; R o s e n et al., 1989; Williams a n d Rakic, 1988a). F o r example, i n order to best estimate n e u r o n a l or glial n u m b e r s from m e a s u r e m e n t s of p a c k i n g density within a given architectonic region, one m u s t have a n accurate estimate of the volume of that region ( H o f m a n et al., 1988; Williams a n d Rakic,

Correspondence." Glenn D. Rosen, Ph.D., Neurological Unit, Beth Israel Hospital, 330 Brookline Ave., Boston, MA 02215, U.S.A.

1988a). D e t e r m i n a t i o n of d e v e l o p m e n t a l changes of particular n e u r a l structures requires c o n s i s t e n t measures (Gottlieb et al., 1985), a n d research o n a s y m m e t r y of cortical a n d subcortical regions (Rosen et al., 1989; v a n E d e n et al., 1984), or of n e u r o n a l c o u n t s in the two hemispheres (Williams a n d Rakic, 1988a), also requires a n accurate estimate of b r a i n volume. I n the c u r r e n t report, the histological p r o b l e m s associated with the accurate m e a s u r e m e n t of b r a i n v o l u m e (e.g., accurate d e t e r m i n a t i o n of section thickness, a n a t o m i c parcellation, etc.) are n o t discussed. T h e reader is referred to U y l i n g s et al. (1986) for a complete discussion of these issues. W e focus i n s t e a d o n c o m p a r i n g the available c o m p u t a t i o n a l techniques. Currently, e s t i m a t i n g b r a i n v o l u m e b y analyz-

0165-0270/90/$03.50 © 1990 Elsevier Science Publishers B.V. (Biomedical Division)

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ing parallel, systematically sampled serial sections through the material is the most common technique (Hofman et al., 1988; Rosen et al., 1989; van Eden et al., 1984; Williams and Rakic, 1988a). A calculation of volume involves essentially the numerical integration of a curve that plots area against the appropriate x, y, or z coordinate (i.e., the coordinate orthogonal to the plane of section). Numerical integration is typically performed using either Cavalieri's rule (Uylings et al., 1986), a trapezoidal (Rosen et al., 1989; van Eden et al., 1984) or parabolic (Simpson's or Durand's) rule (Gottlieb et al., 1985; Williams and Rakic, 1988a). From a mathematical perspective, Cavalieri's rule is an unbiased estimator based on the rectangular approximation of the area under a curve (CruzOrive, 1985; Gundersen and Jensen, 1987), the trapezoidal approximation further enhances this rectangular approximation, and the parabolic approach is an improvement over the trapezoidal, with Simpson's rule being the preferred implementation (Burington, 1973). Some workers have argued strongly for the parabolic approximation (Williams and Rakic, 1988b) while others have supported the use of Cavalieri's rule, claiming its efficiency for volume estimation, especially for irregular brain regions (Uylings et al., 1986). Given that these calculations are now almost always performed on a computer, the differences in computational effort among Cavalieri's, the trapezoidal, and parabolic estimations of volume are insignificant. It seems reasonable, therefore, to opt always for the most accurate of the three methods. There are, however, practical differences between the methods. Simpson's rule requires an odd number of cross-sectional area measurements at equally spaced intervals through the brain (Burington, 1973), and while "Simpson's threeeighths" or Durand's rule will work on odd or even numbers of sections, it also requires equal spacing (Burington, 1973; James et al., 1985). Cavalieri's rule also requires equal spacing of points (Cruz-Orive, 1985; Gundersen and Jensen, 1987). These are conditions not always possible to meet in brain histology as some sections from a series are not well suited (because of reasons of cutting or staining artifact, for example) for morphometric measurement. This begs the question of

whether the various computational approaches are equivalent in the more demanding setting of real brain measurements of volume. Our goal, them was to make a real-world comparison of these techniques as they apply to brain morphometry in an effort to establish practical and useful guidelines for their use. Using both idealized (i.e., mathematically contrived) data and morphometric measures from actual brain material, these methods were directly tested against one another. In addition, we describe a more general mathematical treatment of piece-wise parabolic integration which relaxes the equi-spaced requirement of the traditional parabolic rule. Such a formulation permits greater flexibility in sectioning protocols, but has its own idiosyncratic problems which cannot be ignored.

Methods In order to test these rules, we p e r f o i : ~ the following tests: (1) We compared t h e accuracy of results computed from systematic sampling of two mathematically simulated brain regions o f known volume using the parabolic, trapezoidal. Cavatieri. and rectangular estimators. (2) We measured one regularly-shaped, and one irregularly-shaped simulated brain region. Regularly shaped brain regions are defined as those regions with cross-sectional areas symmetric about the midline of the axis orthogonal to the plane of section, and irregularly-shaped regions as those with cross-sectional areas a s y m n ~ r i c about the mid!ine of this axis. In addition, we examined the effect of increase of sampling distance on the accuracy of these estimations. (3) We randomly "lost" sections from the samples and determined which of the rectaagular, trapezoidal, and piece-wise parabolk estimators were best on unequally spaced sections. (4) FinaUy, we estimated volume of the visual cortex (a regularly-shaped region) and thec~u~t~3~..~utamen complex (an irregularly-shaped r e ~ t ) of the rat. After a brief discussion of sampling protocols, we catalog the equations that can be used to compute volume from serial cross-section area measurements. Most of these can be found readily in other sources and are included here for corn-

117 pleteness. Some additional mathematical detail is provided for the description of the parabolic calculations that permit unequally spaced sections. Finally, we describe how these various methods were applied to the mathematically simulated brain regions.

Sampling protocols In the estimation of brain volume from serial sections, one basic presumption is that not every section is to be analyzed; usually every fifth or tenth section is used. There are basically two possibilities for selecting sections for use in the analysis - - "non-systematic" and "systematic" sampling - - and the choice between these is critical in determining the proper volume estimator. A non-systematic sample is obtained by looking through all the sections to find the very first one in which the structure of interest appears. That section, and all subsequent sections that are the desired distance apart, is analyzed until the structure of interest no longer appears. The systematic sampling paradigm, in practice, involves the analysis of an equally spaced series of sections beginning with the first section in which the region appears in a randomly chosen series, and continuing for each section of the series that contains the structure. Thus, in the systematic approach the first section analyzed may or may not be the first section on which the region of interest appears. In this paper, we focus on analysis using systematic sampling.

Rectangular estimation of morphometric volume The rectangular estimate of morphometric volume for equally spaced sections (VRequa,) is determined by the following formula: n--]

VR~qua,=d E (y,)

For unequal spacing, the rectangular estimation of morphometric v o l u m e (VRunequal) is defined in Eqn. 2: n--1 VRunequaI =

E ( X i + I -i=1

Xi)(Yi)

(2)

where x, = the distance orthogonal to the plane of section of the i-th section (usually section number multiplied by section thickness). The rectangular approach, as others have pointed out (Uylings et al., 1986), is appropriate only for non-systematic sampling. For systematic sampling, the rectangular method is incorrect and Cavalieri's estimate or the "basic volume estimator" (see below) should be used. We include the rectangular approach in our analysis as an empirical illustration of the potential problems of using this approach with the systematic sampling procedure.

Cavalieri's estimator of morphometric volume Cavalieri's estimator of volume (Vc) is a statistically unbiased form of the rectangular approach which requires systematic sampling (Eqn. 3):

where YMAX is the maximum value of y, and t = section thickness. Their product is subtracted from the basic equation as a correction for overprojection. Correction for overprojection is required when section thickness is not negligible, but the volume of the object is large. Otherwise, the basic volume estimator d Y'.'/_I(Y,.) can be used (Uylings et al., 1986). Notice that the only difference between the basic volume estimator and the rectangular method is the ending index of the summation (Gundersen and Jensen, 1987).

(1)

i=1

where d = distance between the sections that are being analyzed (not to be confused with the section thickness t), y~ = cross-sectional area of the i-th section through the morphometric region, and n = total number of sections (Gundersen and Jensen, 1987).

Trapezoidal estimation of morphometric volume The trapezoidal estimate of morphometric volume for equally spaced sections (VTe~ual) is determined by the following formula (Bunngton, 1973):

V.requa,=d[½(y,+y.l+y2+y3+...y . ,]

(4)

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For unequal spacing between sections, the trapezoidal estimate of morphometric volume (VT~q~aJ) becomes: VTunequa' =

n--a [ (yi + Yi+l) ] E (Xi+l -- Xi) 2 i=l

(5)

Parabolic estimation o f morphometric volume

The Simpson's rule formulation for the determination of area under a curve (Vs) found in numerical analysis texts is a dosed form (i.e., exact) simplification of the area under a series of parabolas (Burington, 1973): d Vs = 3 [(Yl + Y , ) + 4(y2 +Y4 + - - - +Y,-1) +2(Y3 +Y5 + - ' - +Y,-2)]

(6)

where the total number of sections (n) is required to be odd. One can estimate the area under a curve with an even number of sections using either Simpson's three-eighths rule (Vs3/s; James et al., 1985): d Vs3/8 = -~ [(Yl + Y,-3) + 4(Y2 + 3'4 + . - . + Y,-4)

ax 2 + bx 1 + c = v1

(10)

a x 2 + bx 2 + c =Y2

(11)

ax~ + bx 3 + c =Y3

(12)

This is a set of 3 simultaneous equations for the three unknowns a, b, and c. Writing the equations in matrix form,

x2

x2

X32

x3

=

Y2

(13)

Y3

or

The unknowns a, b, and C can be found by multiplying the matrix Y by the inverse o f the matrix X ( X- l):

+2(y3 + y5 + ... +y,_5)] + 3 [Yn_3 + 3(Y,_2) + 3(y,_1) + y,]

than one method is available to do this (Davis and Rabinowitz, 1975), but perhaps the easiest to follow is shown here. Treat a, b, and c as unknowns and write 3 equations that must be true in order for the parabola to pass through the 3 known points (x 1, Yl), (x2, Y2), (x3, Y3):

(7)

or Durand's (Vo) equation (Burington, 1973): VD = d [0"4(yl +Y,) + l'l(Y2 +Y,-1) +Y3 +Y4 + "-" +Y,-2]

(8)

Of these two rules, the former (Vs3/8) is more accurate. To relax the need for equal spacing requires a slightly more general approach to the problem of fitting parabolas in a piece-wise fashion to the data points. The equation of a parabola is: y = ax 2 +

bx + c

(9)

where a, b, and c are constants. From simple geometry, it is clear that there is exactly one parabola that passes through three points in a plane. Finding that parabola is not a curve fit in the usual least squares sense of that term. It is possible to compute exactly the a, b, and c of the parabola that will pass through the points. More

This operation is easily implemented on a corn, puter. It is important to see that no assmnption on the spacing of the data is made in this analysis, Once the equation of the parabola has been found for these three data points, the :area u n ~ r the parabola (which is equivalent to the brain volume in the current instance) is computed by integrating: Area under parabola = f~i3( ax 2 + bx + c ) d x (16) In the case of estimating b r ~ volume from cross-sectional area me~urements, the~e ~ almost certainly he more thaa ~ e e ~ts available. ~ ~teasion of t ~ ~ f i ~ s above to this situation is straightforward. Simply divide the

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