1

Stereological Estimation of Brain Volume and Surface Area from MR Images

Niyazi Acer1 and Mehmet Turgut2

1 Department of Anatomy, Erciyes University School of Medicine, Kayseri, Turkey

2 Department of Neurosurgery, Adnan Menderes University School of Medicine, Aydın, Turkey

Background

Stereology combines mathematical and statistical approaches to estimate three-dimensional (3D) parameters of biological objects based on two-dimensional (2D) observations obtained from sections through arbitrary-shaped objects (for reviews of design-based stereology, see Howard and Reed, 1998; Mouton, 2002, 2011; Evans et al., 2004). Among the first-order parameters quantified using unbiased stereology are length using plane or sphere probes, surface area using lines, volume using points, and number using the 3D disector probe. These approaches estimate stereology parameters with known precision for any object regardless of its shape.

These criteria for stereological estimation of volume and surface area are met by standard magnetic resonance imaging (MRI) and computed tomography (CT) scans, as well as tissue sections separated by a known distance with systematic random sampling, that is, taking a random first section followed by systematic sampling through the entire reference space (Gundersen and Jensen, 1987; Regeur and Pakkenberg, 1989; Roberts et al., 2000; Mouton, 2002, 2011; García-Fiñana et al., 2003; Acer et al., 2008, 2010). Numerous studies have been reported using MRI to estimate brain and related volumes by stereologic and segmentation methods in adults (Gur et al., 2002; Allen et al., 2003; Acer et al., 2007, 2008; Jovicich et al., 2009), children (Knickmeyer et al., 2008), and newborns (Anbeek et al., 2008; Weisenfeld and Warfield, 2009; Nisari et al., 2012).

The Cavalieri Principle

Named after the Italian mathematician Bonaventura Cavalieri (1598–1647), the Cavalieri principle estimates the first-order parameter volume (V) from an equidistant and parallel set of 2D slices through the 3D object. As detailed later, the approach uses the area on the cut surfaces of sections through the reference space (region of interest) to estimate size (volume) of whole organs and subregions of interest. The point counting technique for area estimation uses a point-grid system superimposed with random placement onto each section through the reference space (Gundersen and Jensen, 1987). The number of points falling within the reference area is counted for each section (Figure 1.1). Total V of a 3D object, x, is estimated by Equation 1.1:

(1.1)

where A(x) is the area of the section of the object passing through the point x ε (a, b), and b is the caliper diameter of the object perpendicular to section planes. The function A(x) is bounded and integratable in a bounded domain (a, b), which represents the orthogonal linear projection of the object on the sampling axis (García-Fiñana et al., 2003; Kubínová et al., 2005).

Figure 1.1 Illustration of point counting grid overlaid on one brain section.

The Cavalieri estimator of volume is constructed from a sample of equidistant observations of f, with a distance T apart, as follows (Eq. 1.2):

(1.2)

where x0 is a uniform random variable in the interval (0,T) and {f1, f2, … , fn} is the set of equidistant observations of f at the sampling points which lie in (a, b). In many applications, Q represents the volume of a structure, and f(x) is the area of the intersection between the structure and a plane that is perpendicular to a given sampling axis at the point of abscissa x (García-Fiñana et al., 2003; García-Fiñana, 2006; García-Fiñana et al., 2009).

Cavalieri Principle with Point Counting

Unbiased and efficient volume estimates with known precision (Roberts et al., 2000; García-Fiñana et al., 2003) can be obtained from a set of parallel slices separated by a known distance (T), and sampled in a systematic random manner. These criteria are easily obtained from standard sets of MRI and CT scans (Roberts et al., 2000; Acer et al., 2008, 2010).

To apply the point-counting method, a square grid system is superimposed with random placement onto each Cavalieri section or slice and the number of points falling within the reference area (area of interest) counted on each section (Figure 1.1). Finally, an unbiased estimate of volume is calculated from Equation 1.3:

(1.3)

where n is the number of sections, P1, P2, … , Pn show point counts, a/p represents the area associated with each test point, and T is the sectioning interval.

We used software that allowed the user to automatically sum the area of each slice and determine brain volumes by the Cavalieri principle. An unbiased estimate of volume was obtained as the sum of the estimated areas of the structure transects on consecutive systematic sections multiplied by the distance between sections, that is, V = ΣA • T. The program allowed the user to determine contrast, select true threshold value to estimate the point count automatically (Denby et al., 2009).

Coefficient of Error (CE) and Confidence Interval (CI) for Volume Estimation

The precision of volume estimation by the Cavalieri method was estimated by CE. Based on the original work of Matheron, the CE was adapted to the Cavalieri volume estimator by Gundersen and Jensen (1987) and more recently simplied by a number of stereologists (Gundersen et al., 1999; García-Fiñana et al., 2003; Cruz-Orive, 2006; Ertekin et al., 2010; Hall and Ziegel, 2011). The CE is useful for estimating the contribution of sampling error to the overall (total) variation for stereological estimates. A pilot study of the mean CE estimate allows the user to optimize sampling parameters, for example, mean CE less than one-half of total variation; to select the appropriate number of MRI sections through the reference space; and to set the optimal density of the point or cycloid grid.

García-Fiñana (2006) pointed out that the asymptotic distribution of the parameter volume as its variance is strongly connected with the smoothness properties of the measurement function. Using the Cavalieri method, we constructed both CE and a CI value for estimation of brain volumes. The first calculation involved the estimation of volume, variance of the volume estimate, and bounded intervals for the volume by Eq. (1.3).

Second, Var (QT) was estimated via Eq. (1.4) according to Kiêu (1997), which first requires calculation of α(q), C0, C1, C2, and C4 (Table 1.1):

(1.4)

Table 1.1 Calculation of the constants C0, C1, C2, and C4 for brain volume

Eq. (1.5) leads to

(1.5)

The quantities C0, C1, and C2 can be computed from the systematic data sample (García-Fiñana et al., 2003).

The smoothness constant (q) is then estimated from Eq. (1.6) as given in the following:

(1.6)

The coefficient α(q) has the following expression:

(1.7)

where Γ and ζ denote the gamma function and the Riemann zeta function, respectively. For fairly regular, quasi-ellipsoidal objects, q approaches 1, and for irregular objects, q approaches 0. Under these circumstances, α(0) = 0.83 and α(1) = 0.0041 (García-Fiñana et al., 2003).

The bounded interval for the cerebral volume was obtained by Eq. (1.8):

(1.8)

Note that Eq. (1.8) gives the approximate lower and upper bounds for V2 − V1.

Example for Cerebral Volume, CE, and CI

Examples are provided for estimation of cerebral volume with upper and lower CI values and CE. To estimate brain volume, we used the total data set of 158 images with slice thickness 1 mm split into 15 Cavalieri planes, that is, every 10th magnetic resonance (MR) image with a different random starting point. Thus, each Cavalieri sample represents the area of cerebral cortex of a set of MR images at distance T = 10 • 1 mm = 10 mm apart (Table 1.1).

We illustrate the calculation steps involved in the estimation of a lower and upper bound for the true cerebral cortex volume by applying Eq. (1.8) to one set of Cavalieri planes (i.e., 26, 66, 98, 132, 156, 163, 158, 150, 115, 85, 22). This data sample represents the area of cerebral cortex in square centimeters on n = 11 MR sections a distance T = 1 cm apart. The Cavalieri volume for cerebrum was obtained using Eq. (1.3) as follows:

(1.9)

(1.10)

The smoothness constant (q) is estimated from Eq. (1.6) as follows:

(1.11)

Applying Eq. (1.7) with q = 0.815 leads to α(q) as follows:

(1.12)

Therefore, the estimate of ...