Single Object Stereology (Part 3)

Posted: 22 Jan 03This article describes the vertical spatial grid and a method for estimating curve length in 3D from total vertical projections.

Surface Area - Vertical Spatial Grid

Using exhaustive vertical sections, the surface area, S, of a bounded object, M, can be estimated from two series of vertical sections (n = 2), separated by a distance T, whose orientations are f₀ Î UR[0, p/2) and f₁ = f₀ + p/2. Let I_i be the number of intersections between the boundary of M on the sections of series i and the rolling cycloids. The surface area of M is given by

(26)

The vertical spatial grid (VSG), devised by Cruz-Orive and Howard (1995), is a test probe comprising two systems of rolling cycloids that allows the intersection counts I₀ and I₁ to be observed from a single series of vertical sections through M. The rolling cycloids are contained in two mutually perpendicular sets of planes parallel to the sections in series 0 and series 1. Let series 0 be the observation series. The VSG, shown in Figure 16, consists of rolling cycloids parallel to the observation series that ride upon rolling cycloids normal to the observation series.

Figure 16: The vertical spatial grid and its fundamental tile, J₀. The vertical spatial grid is a test probe comprising two systems of rolling cycloids that are contained in mutually orthogonal sets of planes (xy and yz). These systems of rolling cycloids sweep out the grey surface shown here.

Suppose a single (n = 1) observation series of m parallel vertical sections (T₀, …, T_m-1) is constructed that cuts through M and whose orientation is f₀ Î UR[0, p/2). As shown in Figure 17, let the distance between successive vertical sections be T. On the first vertical section, T₀, overlay a UR test system of points and rolling cycloids (see Figure 14(b)). Let I_xy,₀ be the number of intersections between the boundary of M on T₀ and the rolling cycloids. Let P₀ be the number of test points hitting the two dimensional cross-section of M on T₀. Test points that contribute to P₀ are considered "switched on" or highlighted.

Figure 17: An observation series of m parallel vertical sections. The object of interest M, through which the series cuts, is omitted for clarity.

On the second section, T₁, of the observation series, overlay the same test system that appeared on T₀, but shifted vertically by the signed distance D₀. The calculation of the vertical shift, D₀, is described below. In the same way as described in the previous paragraph, count I_xy,1 and P₁. On T₁ (and subsequent sections) make an additional count P_0,1 of highlighted points on T₀ that remain highlighted on T₁. Each of the three counts described is illustrated in Figure 18.

Figure 18: Point and intersection counting on the observation series. In this example, I_xy,i = 6, P_i = 8, I_xy,i+1 = 6, P_i+1 = 9 and P_i,i+1 = 5.

This process is repeated for all sections T₀, …, T_m-1 belonging to the observation series. The total number of intersections, I₀, between the boundary of M on the observation series sections and the test systems of rolling cycloids is

Consider a second constructed series of parallel vertical sections (shown in Figure 19), perpendicular to the observation series and horizontal plane, whose orientation is f₁ = f₀ + p/2. Let the distance between successive vertical sections be pr. Position the constructed series so that test points on the observation series also lie on the new series. The vertical shifts Di are calculated so that test points on the constructed series trace out constructed rolling cyloids that are normal to the observation series.

Figure 19: A constructed series of vertical sections perpendicular to the observation series. In this example, the distance between successive vertical sections on both series is pr.

The vertical shifts D_i are calculated as follows. Let Z₀ Î UR[0, T) and Z_i = Z₀ + (i×T) (i = 0, …, m - 1) be systematic abscissae along a rolling cycloid, radius r, whose corresponding ordinates are Y_i. As shown in Figure 20, the vertical shifts D_i (i = 0, …, m - 2) are given by

However, given a cycloid abscissa Z_i, the corresponding ordinate Y_i is not available explicitly. (23) must be solved numerically for q and this value of q substituted into (24). For each i = 0, …, m - 1 let

The solutions q_i are taken to be q_i,N as soon as | q_i,N - q_{i,N - 1} | < e or N > N₀, where e is a small positive number and N₀ is the maximum number of iterations. Cruz-Orive and Howard (1995) suggest e = 0.001 and N₀ = 100. Greater accuracy is readily achieved (by decreasing e and increasing N₀) when modern computers are employed. Finally the solutions q_i are substituted into (24) to obtain (i = 0, …, m - 2)

Figure 20: A constructed vertical section through M. Test points on the observation series T₀,…,T_m-1 trace constructed rolling cycloids. Test points within M are represented by rectangles. Test points highlighted for more than one section (moving in the direction T₀ to T_m-1) are circled. The number of intersections between the cross-section of M and the constructed rolling cycloids is I₁ = 2×(4 - 2) = 4.

If a test point is highlighted on T₀ and switched off on T₁, then a constructed rolling cycloid has traversed the boundary of M (moving from the inside to the outside of M). For every inside-outside intersection, there must be a corresponding outside-inside intersection. Therefore, the total number of intersections, I₁, between the boundary of M on the new series of sections and the constructed rolling cycloids is

Equation (26) cannot be used directly to calculate the surface area of M when the VSG is employed. The extra constraints imposed on the positioning of the rolling cycloid test systems mean a new equation must be formulated. This is achieved by referring back to (21). If the boundary of M is intersected with an unbounded VSG then

L/V is the mean length of rolling cycloids per unit volume of space. As shown in Figure 17, the VSG has as its fundamental tile, J₀, a rectangular box with volume V = 64pr³. The mean length of rolling cycloid, EL, that falls within the volume is made up of two components.

The first component, EL₀, is the length of rolling cycloid contributed by the observation series. Each observation section that cuts through J₀ contains one full period of rolling cycloid within J₀. The expected number of observation sections that intersect J₀ is 4pr/T - i.e., the length of J₀ in the constructed series direction divided by the distance between observation series. A proof of this fact will be written up in the future. Since one period of rolling cycloid has length 16r, EL₀ = 16r×4pr/T = 64pr²/T.

The second component, EL₁, is the length of constructed rolling cycloid contributed by the constructed series. Four constructed sections, a distance pr apart, cut through J₀ and each section contains one full period of rolling cycloid within J₀. Therefore EL₁ = 4×16r = 64r, so that

Equation 27 (27)

The rolling cycloids that comprise the VSG have test area per test point, a/p = pr×4r = 4pr² and test length per test point, l/p = 4r. When these values are substituted into (27), Cruz-Orive and Howard's result for estimating surface area using the VSG is obtained:

(28)

Implementing the VSG is not straightforward. Some guidelines are stated here. Care should be taken to avoid periodicity between the step length T and the horizontal width of a cycloid, pr, as this causes unwanted repetitions of the vertical shifts D_i and decreases precision. Large values of T are also discouraged as this causes constructed cycloids (shown in Figure 20) to be poorly approximated and thus intersections contributing to I₁ may be omitted.

Another issue is oversampling on either the observation series or the constructed series. This will occur if either VSG component (observed or constructed) has significantly greater length per unit volume (L/V) than the other. Figure 19 illustrates a VSG that has L/V identical for both components. The observation and constructed series intersect each other in a square lattice. Unfortunately, this configuration implies T = pr with unwanted repetitions of the vertical shifts D_i resulting. These conflicting guidelines suggest the following VSG implementation:

Set T considerably smaller than pr, e.g. T < pr/4, (but not pr divided by some positive integer). Count intersections, I₁, contributed by the constructed cycloids as described in the text. Intersections, I₀, on the observation series should only be counted on every [pr/T]'th section (where [x] means the integer part of x). The distance between successive sections comprising the observation series is now T₂ = [pr/T]×T, and this value should be used instead of T in (28).

When the VSG is implemented using tools that allow vertical scanning, such as a confocal scanning laser microscope, the intersection count I₁ need not be counted as described in the text. Instead virtual cycloid probes are imagined travelling normal to the observation series that pass through points on the observation cycloids (marked as vertical and horizontal lines in Figure 18).

Furthermore, 3D digital images can be reformatted using image analysis tools to produce 2D image sections that comprise observation and constructed series. For images sectioned in this way, true horizontal rolling cycloids can be overlain on both observation and constructed sections with intersection counts I₀ and I₁ recorded in the normal way.

Curve Length from Total Vertical Projections

Total vertical projections (TVPs) can be obtained by fixing a vertical axis with respect to the object under study. Then, with a random starting angle and continuing at uniform intervals, the whole object is projected onto a plane in a systematic set of directions between 0 and p radians. An estimator of curve length from TVPs can be obtained with the help of (19). The following derivation is illustrated in Figure 22.

Figure 22: Estimating curve length from total vertical projections. Redrawn from Cruz-Orive (1997).

Consider a curve in Â³, of length L, in a bounded reference space of volume V. Choose an arbitrary horizontal plane with vertical axis perpendicular. Next, sit a vertical slab of known thickness t and width w on the horizontal plane. The vertical slab must be positioned so that its rectangular base is IUR in the horizontal plane. Construct a test plane perpendicular to the slab. The test plane must be IUR in Â³. Therefore, the angle q between the normal to the plane and the vertical axis must have probability density function sinq. Furthermore, given q, the position of the test plane must be UR on any bounded interval along the normal to the plane. The intersection of the test plane with the vertical slab forms an IUR rectangle, R, whose area is t×l, where l = w/cosq. If there are Q transects between the curve and R, then from (19):

(29)

Project the curve and R onto an observation plane parallel to the vertical slab. R becomes a line segment of length l. The line segment is cosine-weighted because its normal is sine-weighted. If there are no overlapping problems the number of transects, Q, between the curve and R is equal to the number of intersections, I, between the projections of R and the curve. Equation (29) becomes

(30)

a result originally obtained by Gokhale (1990). This result has been applied by McMillan et al. (1994) and Batra et al. (1995) to estimate capillary length.

Instead of repeated sampling of cosine-weighted straight lines, it is convenient to use a test system of cycloids with the minor principal axis perpendicular to the vertical axis. Let t -> ¥. The projection is now a total vertical projection (TVP). Onto the TVP superimpose a uniform random test system of (cosine-weighted) cycloids, with known length per unit area l/a. From (30) Cruz-Orive and Howard (1991) obtained:

I is the total number of intersections between the TVP of the curve and the test system of cycloids on the projection plane. When m TVPs are obtained, curve length L is estimated by:

Equation 31 (31)

TVPs have been used by Roberts et al. (1991) with MRI to estimate the length of blood vessels, while Howard et al. (1992, 1993) and Roberts and Cruz-Orive (1993) have estimated neuron dendritic length.