Bone elongation in children occurs through the process of endochondral ossification in cartilaginous growth plates at the ends of long bones. Clonal expansion of stem cells results in columns of chondrocytes whose spatial position within the growth plate mirrors their differentiation stage: cellular proliferation, cellular enlargement (hypertrophy), and cellular apoptotic death followed by replacement of bone on the previously calcified cartilaginous matrix (Figure 1). The extent of bone elongation achieved depends on the kinetics of chondrocytic activity at each stage of differentiation, and on the rate of regulated transitions between stages. A complex interplay of genetic and epigenetic factors (e.g., endocrine, paracrine, autocrine, nutritional, biomechanical) influences postnatal longitudinal bone growth, acting primarily at the cellular level through differential effects at specific phases of chondrocytic development and maturation. For recent reviews, see Farnum and Wilsman, 2001 and 2002.
Observed originally in rabbit kidney cells (Zimmerman, 1898), the primary cilium has been suggested to constitute a regular structural feature of virtually all eukaryotic cells within both vertebrates and invertebrates, most characteristically at the incidence of one per cell (for a website on the primary cilium, see: http://members.global2000.net/bowser/cilialist.html). The axonemal structure of primary cilia is characterized by nine doublet microtubules that extend through the axonemal length (Singla and Reiter, 2006). The monocilia of the nodal cells in the embryo show dynein arms which are hypothesized to generate a characteristic propeller-like movement (Tabin, 2006; Hirokawa, 2006). In general, the primary cilia, that are not nodal, lack dynein arms and are considered to be non-motile in the sense that they lack the ability to generate either a propeller-like movement as for the nodal cilia or a waveform characteristic of the motile cilia whose axonemal core consists of two central microtubules (Bisgrove and Yost, 2006), such as those found in cells of the airway epithelium.. Primary cilia have been observed in the cells of multiple connective tissues including osteoblasts (Tonna and Lampen, 1972); osteocytes (Federman and Nichols, 1974), odontoblasts (Garant et al., 1968), ligament fibroblasts (Bray et al., 2005), meniscal fibroblasts (Le Gaverand et al., 2001), periodontal cells (Beersten et al., 1975), adipocytes (Geerts et al.,1990), and in chondrocytes of articular (Wilsman, 1978) and elastic (Cox and Peacock, 1977) cartilage.
Recent papers recognize primary cilia as sensory organelles for detection and transmission of signals from the extracellular environment to the cell, essential for tissue homeostasis and function (Pazour and Witman, 2003; Whitfield, 2003; Davenport and Yoder, 2005; Schneider et al., 2005; Olsen, 2005). In connective tissues the cilium projects into the extracellular matrix (ECM) and is closely associated with the Golgi apparatus of the cell. Given the highly anisotropic organization of most connective tissues, it has been suggested that the primary cilium may act as a mechanosensor to the local biomechanical environment, and may be significant in the establishment of cellular orientation and directed secretion of ECM components from the Golgi apparatus (Quarmby and Parker, 2005). Poole et al. (1997, 2001) demonstrated that the degree to which the primary cilium extends into the ECM, and whether its axoneme is straight or bent, is variable in articular chondrocytes, and that the configuration of the cilium relative to the chondrocyte changes as fluid flow in the environment changes. Poole has hypothesized that the chondrocytic primary cilium acts as a probe of the ECM and, because of its close association with the Golgi and the microtubule organizing center of the cell, is a key player in establishing cellular shape (Poole et al., 1985, 1997, 2001; Badano et al., 2005). A similar hypothesis has been proposed for the primary cilium in osteoblasts (Wheatley et al., 1996; Quarles, 2005). A hypothesis that the primary cilium is the osteocyte's strain-rate sensing flowmeter unites mechanical and fluid-flow sensory functions (Whitfield, 2003). Attractive as these hypotheses are, they are very difficult to test in the living animal.
If the primary cilium of connective tissue cells is a sensory organelle involved with receiving biomechanical signals that result in directed secretion of the surrounding ECM, one could hypothesize that the orientation of the cilium in three dimensional space should be consistent with the orientation of the cell itself (i.e. the long axis of the cell on longitudinal sections), or of the orientation of the cells within the tissue (i.e. the long axis of the macroscopic bone). The growth plate is a particularly appropriate connective tissue to investigate this hypothesis since cellular profiles and their orientation have been studied in growth plate cartilage using stereologically based approaches, and it is clear that the long axis of the cell relative to the long axis of the bone changes as chondrocytes progress from proliferation through their terminal differentiation characterized by cellular enlargement during hypertrophy (Farnum et al., 1990; Breur et al., 1991; Hunziker et al., 1987; Buckwalter et al., 1986; Hunziker and Schenk, 1989; Wilsman et al., 1993; Cruz-Orive and Hunziker, 1986).
The purpose of the current study was to develop an experimental technique for analysis of the orientation of the cilium in the growth plate through a new application of mathematical concepts originally developed by Euler to describe the orientation of a segment in a three dimensional space (Euler L., 1755). Adaptation of the classical concepts to the specifics of the biological context is a necessary, important and not obvious step, requiring knowledge of both the mathematical concepts and the biological specifications of the tissue of interest.
The mathematical methodology described in this paper is part of the rapidly growing field of mathematical methods that are developed on images of a given specimen for the purpose of modeling the specimen through the two-step process of data collection from images and consequent computational algorithms on collected data. Angenent et al. (2006) have recently emphasized how this major impetus for new algorithms in signal and image processing has stemmed from the last decade's advent of a variety of faster, more accurate and less invasive imaging devices. All such mathematical algorithms lead to interactive procedures. That is, in each case there is a human user in the loop who is the ultimate judge of the tuning of parameters. There is a major need for further mathematical techniques and modifications of previous approaches that lead to more automatic, and easier-to-interpret, imaging.
The mathematical application described here assesses cellular orientation with respect to the direction of bone elongation and ciliary orientation with respect to cellular orientation in a three dimensional environment. Rapid cold precipitation-based fixation is used to minimize possible artifactual post-mortem alterations of ciliary orientation with the ultimate goal of assessing ciliary orientation in situ. The mathematical method is applied to serial optical sections of growth plate chondrocytes captured by multiphoton microscopy (MPM), after immunohistochemistry to demonstrate acetylated-α-tubulin in the ciliary axoneme. The combined application of the experimental and mathematical methods yields results that support the high anisotropy among the distinct zones of the growth plate that are identified by the organization of chondrocytes in relationship to the long axis of the bone. Such anisotropy is well characterized by the orientation of the ciliary axoneme with respect to the orientation of the chondrocyte. Having developed this methodology, we are now in a position to analyze ciliary orientation in three dimensional space using versatile coordinate systems applicable to a wide variety of connective tissues. The methodology developed is applicable to achieve three-dimensional reconstruction from z-stacks obtained by any microscopical modality (confocal microscopy, transmission electron microscopy), in addition to MPM.