Luis M. Cruz-Orive

Estimation of surface area from vertical sections

‘Vertical’ sections are plane sections longitudinal to a fixed (but arbitrary) axial direction. Examples are sections of a cylinder parallel to the central axis; and sections of a flat slab normal to the plane of the slab. Vertical sections of any object can be generated by placing the object on a table and taking sections perpendicular to the plane of the table.

Application of the Cavalieri principle and vertical sections method to lung: estimation of volume and pleural surface area

A practical methodology is proposed for the stereological analysis of lung and other organs using recently developed unbiased procedures. This study concentrates on the unbiased estimation of lung volume using Cavalieris principle compared with the fluid displacement method and measurement of pleural surface area using vertical sections. Furthermore, the proposed design, in addition to the sampling of extensive slices for the initial steps, also allows sampling of vertical sections for light and electron‐microscopical stereology.

Sampling designs for stereology

The purpose of this paper is to propose the necessary sampling techniques for estimating a global parameter defined in a solid opaque specimen (e.g. the total volume of mitochondria in a given liver, the total capillary surface area in a given lung, etc.). The geometry of the specimen often suggests a multi‐level or cascade sampling design at different magnifications, whereby the object phase at one level becomes the reference phase in the next level. The final parameter is then estimated as the product of the intermediate ratios with the volume of the specimen, which is estimated independently. Each level can be regarded as an independent sampling design; a given stereological project may be planned in terms of one or more of these designs.

Quantitation of chondrocyte performance in growth-plate cartilage during longitudinal bone growth.

The longitudinal growth of bone depends on the activities of individual chondrocytes of the growth plate. Each chondrocyte remains in a fixed location throughout its life, and there accomplishes all of its functions. Although a cell may perform several or all of its activities simultaneously, one of these will usually predominate during a particular phase of its life. The two most prominent stages are those of cellular proliferation and hypertrophy (including the mineralization of matrix) before the resorption of tissue during vascular invasion. By applying recently developed stereological procedures and improved methods for the fixation of cartilage, we compared cellular shape modulation, various ultrastructural parameters (surface areas or volumes of endoplasmic reticulum, Golgi membranes, and mitochondria), the production of matrix, and cellular turnover for proliferating and hypertrophic chondrocytes within the proximal tibial growth plate of the rat. By the late hypertrophic stage, fourfold and tenfold increases in the mean cellular height and volume, respectively, and a threefold increase in the mean volume of the matrix per cell were achieved. The high metabolic activity of hypertrophic cells was reflected by a twofold to fivefold increase in the mean cellular surface area of rough endoplasmic reticulum, the Golgi membranes, and the mean cellular mitochondrial volume. Rates of longitudinal growth were determined by fluorochrome labeling and incident-light fluorescence microscopy. Using these values and the stereological estimators describing cellular height, the rates of cellular turnover were calculated. The rapid progression of the vascular invasion front was found to eliminate, for each column of cells, one chondrocyte every three hours; that is, eight cells a day. The maintenance of a steady-state structure for growth-plate cartilage in rats in a steady state of growth thus necessitates efficient compensation for these losses, which is achieved by a high rate of cellular proliferation and rapid hypertrophy.

Measuring error and sampling variation in stereology: comparison of the efficiency of various methods for planar image analysis

An evaluation is made of the relative efficiency (precision of the final estimate per unit time of measurement on a given set of sections) of different methods for planar analysis aimed at estimating aggregate, overall stereological parameters (such as Vv, Sv). The methods tested are point‐counting with different densities of test points (4 ≤ PT ≤ 900 per picture), semiautomatic computer image analysis with MOP and automatic image analysis with Quantimet, for obtaining Vv and Sv estimates. One biological sample as well as three synthetic model structures with known coefficients of variation between sections are used. The standard error of an estimate is mainly determined by the coefficient of variation between sampling units (= sections in the present paper) so that measuring each sample unit with a very high precision is not necessary. Automatic image analysis and point‐counting with a 100‐point grid were the most efficient methods for reducing the relative standard errors of the Vv and Sv estimates to equivalent levels in the synthetic models. Using a 64‐point grid was as precise, and about 11 times faster than using a tracing device for obtaining the estimate of Vv in the biological sample.

Stereology for anisotropic cells: Application to growth cartilage*

A number of either new or recently available stereological methods are described for estimating volume, surface area and number of anisotropic cells. The methods are illustrated with direct reference to the epiphyseal growth plate. Different estimates of a given quantity are obtained by applying alternative methods to the same set of sections, in order to compare the relative merits of the methods. For instance, the surface area of the cells is estimated via the Dimroth–Watson model (which gives a measure of the degree of anisotropy in addition to the surface area estimate) and from vertical sections using cycloid test systems. Cell number is estimated by traditional unfolding methods and by the new disector method. Also, volume‐weighted mean cell volume is estimated from vertical sections via point‐sampled intercepts using two different kinds of rulers to classify intercept lengths. Finally, nested design statistics is applied to a set of data from twelve animals in order to compare the relative impacts of biological and stereological (sampling) variations on the observed coefficient of error of a group mean estimate. The preferred methods are listed in the final section.

Unbiased estimation of human body composition by the Cavalieri method using magnetic resonance imaging.

The classical methods for estimating the volume of human body compartments in vivo (e.g. skin‐fold thickness for fat, radioisotope counting for different compartments, etc.) are generally indirect and rely on essentially empirical relationships — hence they are biased to unknown degrees. The advent of modern non‐invasive scanning techniques, such as X‐ray computed tomography (CT) and magnetic resonance imaging (MRI) is now widening the scope of volume quantification, especially in combination with stereological methods. Apart from its superior soft tissue contrast, MRI enjoys the distinct advantage of not using ionizing radiations.

Estimating length density and quantifying anisotropy in skeletal muscle capillaries

The accurate estimation of stereological parameters defined on anisotropic structures is a long‐standing problem. In this paper we seek to estimate the capillary length density Jv in skeletal muscle tissue. A well‐known model for directional anisotropy in space, namely the ‘spherical normal’ or ‘Fisher axial distribution’ model, is found to fit the relevant data satisfactorily. Based on this model, a short‐cut estimation method is proposed and illustrated with a numerical example. This method essentially consists in taking the ratio of mean capillary profile counts, as obtained from transversal and longitudinal sections of the muscle tissue, and making use of a table or a graph given in the paper to estimate Jv. The conditions under which the methods are applicable and practicable are discussed in detail. Apart from an accurate estimation of Jv, an important feature of our method is the possibility of quantifying the degree of anisotropy by a coefficient K (called the concentration parameter of the Fisher axial distribution), which enjoys both a biological significance and a sound statistical basis.

Hippocampal volume and neuronal number in Ts65Dn mice: a murine model of down syndrome

Ts65Dn mouse displays a partial triplication of chromosome 16 and is adopted as a model for Down syndrome (DS). It is known that Ts65Dn mice present memory deficiencies. In order to gain insight into the cause of these deficiencies, we studied the possibility of changes in volumes and neuronal numbers in different regions of the hippocampus (dentate gyrus, CA3, CA2 and CA1) in trisomic mice as compared to control littermates using stereological methods. The mean hippocampal volumes of Ts65Dn mice did not show significant differences as compared to controls, except in CA2 where there was a barely significant decrease. However, mean neuron number was significantly lower in Ts65Dn mice than in controls in dentate gyrus (43.7 x 10(4), CV 21%, n = 5, vs. 30.4 x 10(4), CV 18.1%, n = 4) and higher in CA3 (23.1 x 10(4), CV 18.9% vs. 33.3 x 10(4), CV 14.9%). These quantitative changes may account for the memory deficiencies observed in Ts65Dn mice.

Precision of Cavalieri sections and slices with local errors

Cavalieri sections — and more recently Cavalieri slices, especially in combination with non‐invasive scanning — are widely used to estimate volumes. Physical Cavalieri slices are also increasingly used to estimate neuron numbers via the optical fractionator. In either case, the prediction of the error variance is important to assess optimal sample sizes. The error variance consists of two components, one due to the variation among the true contents of sections or slices, and the other due to local or ‘nugget’ errors. The latter may arise for instance when estimating section areas by point counting, or when counting discrete particles in slices or disectors. In this paper, a fairly comprehensive set of prediction formulae is presented to separate both variance components.