Trabecular bone (TB) (also known as cancellous bone), which occurs in most of the axial skeleton and at locations toward the ends of the long bones, consists of a lattice of interconnected plates and rods that confer mechanical strength to the skeleton at minimum weight. In addition to the volume fraction of the trabecular bone (often quantified in terms of bone density), the three-dimensional (3D) arrangement of the trabecular network is a major determinant of elastic modulus (Odgaard et al., .J Biomechan. 30:487-495 (1997)) and ultimate strength.
In general, characterization of the strength of trabecular lattices from three-dimensional (3D) images can be divided into three major categories: material, scale and topology (DeHoff et al., J. Microscopy 95:69-91 (1972)). ‘Material properties’ describe the bone material; ‘scale properties’ describe the size and thickness (local volume properties) of the trabecular elements; and the ‘topological properties’ describe the spatial arrangement of the bone material in the network. These parameters change characteristically with subject age. It is also widely accepted that the mechanical competence of trabecular bone (i.e., its resistance to fracturing) is a function of both mass density and architecture and that disease processes such as osteoporosis entail both loss in net bone mass and architectural deterioration (World Health Organization [WHO] Technical Report Series No. 843, 1994).
A common diagnostic screening method for osteoporosis is based on ‘dual-energy X-ray absorptiometry’ (DEXA) (Wahner et al., The Evaluation of Osteoporosis: Dual Energy X-Ray Absorptiometry in Clinical Practice, Cambridge: University Press, 1994) to measure integral bone mineral density (BMD). This method, however, does not distinguish between trabecular and cortical bone and ignores the role of structure as a contributor to mechanical competence.
It has been shown that the 3D arrangement (Mosekilde, Bone 9:247-250 (1988)) and the nature of the trabeculae (e.g., plate-like versus rod-like) (Morita et al., Ann. Biomed. Eng. 22:532-539 (1994)) can help explain the variation in elastic moduli and ultimate strength of trabecular bone networks that is unaccounted for by density alone (Ciarelli et al., J. Orthopaedic Res. 9:674-682 (1991); Goulet et al., J. Biomechanics 27:375-389 (1994)). In models of trabecular bone, Jensen et al. (Bone 11:417-423 (1990)) demonstrated that the apparent stiffness can vary by a factor of 5-10 when the arrangement of the network goes from a perfect cubic lattice to maximum irregularity, even though trabecular bone volume remains almost constant. The independent role of architecture in conferring strength to the trabecular network is supported by a large number of experimental studies (Gordon et al. (Can. Assoc. Radiol. J. 49:390-397 (1998); Oden et al. (Calcif. Tissue Int. 63:67-73 (1998); Siffert et al., Bone 18:197-206 (1996); Hwang et al., Med. Phys. 24:1255-1261 (1997)). It is generally agreed that 50%-60% of the mechanical competence of the bone can be explained by variations in the apparent density (bone mass/tissue volume). The clinical evidence for an independent contribution from trabecular architecture is equally compelling.
In the past, studies concerned with the quantitative description of trabecular bone have used histomorphometry from sections in conjunction with stereology to reconstruct the third dimension (Parfitt, In Bone Histomorphometry: Techniques and Interpretation, Boca Raton, Fla.: CRC Press, 1981, pp. 53-87 (1981)). Several studies involving histomorphometry in patients with and without vertebral fractures, matched for gender and BMD, found the two groups of subjects to differ in histomorphometric indices. Kleerekoper et al. (Calcif Tissue Int. 37:594-597 (1985)) first demonstrated that women with osteoporosis and vertebral compression deformities had a significantly lower mean trabecular plate density than did women without deformities, matched for age and BMD. Similarly, Recker (Calcif Tissue Int. 53 Suppl. 1:S139-142 (1993)) showed that a subset of patients with vertebral crush fractures, matched to an equal number of controls for trabecular bone volume, had considerably decreased trabecular plate density and increased marrow star volume (Vesterby et al., Bone 12:219-224 (1989)). Legrand et al. (J. Bone Miner. Res. 15:13-19 (2000)) studied 108 men with osteoporosis, of whom 62 had at least one vertebral fracture, and determined that the patients with fractures did not differ in age, bone mineral density of the spine or hip from those without fractures. However, trabecular number was lower and trabecular separation greater in the fracture group. Other distinguishing histomorphometric parameters were found to relate to network connectivity.
As pointed out, most measurements of trabecular architecture are based on two-dimensional (2D) images (Gordon et al., Physics in Medicine and Biology 41:495-508 (1996); Parfitt, 1981; Parfitt et al., J. Clin. Invest. 72:1396-1409 (1983); Vesterby, Bone 11:149-155 (1990); Hahn et al., Bone 13:327-330 (1992); Garrahan et al., J. Microscopy. 142:341-349 (1986)). Yet, it is well known that connectivity analysis of 2D sections does not accurately define the 3D structures found in trabecular bone networks (Odgaard et al., Bone 14:173-182 (1993). For example, what appears to be a rod in a 2D section, may actually be a cross-section of a plate-like structure in the 3D network, or even a junction between two plates. A 3D analysis is thus essential to unambiguously establish the topology of the trabecular architecture (Wessels et al., Medical Physics 24:1409-20 (1997)).
In vivo assessment of trabecular bone architecture can be achieved by computed tomography (Gordon et al., 1996; Muller et al., J. Bone Mineral Res. 11:1745-1750 (1996); Laib et al., Bone 21(6):541-6 (1997); Laib et al., Tech. Health Care 6(5-6):329-337 (1998)), and magnetic resonance (MR) micro-imaging (Link et al., J. Bone Miner. Res. 13(7): 1175-1182 (1998); Majumdar et al., Bone 22(5):445-454 (1998); Wehrli et al., Techn. Health Care 6:307-320 (1998A)). Studies performed in patients are usually conducted at peripheral skeletal locations, such as the wrist (Majumdar et al., J. Bone Mineral Res. 12:111-118 (1997); Gordon et al., Med. Phys. 24:585-593 (1997); Wehrli et al., Radiology 206:347-357 (1998B)) and have been shown useful in deriving trabecular structural quantities. Recent advances have permitted analysis of the resulting images by probability-based image processing techniques (Hwang et al., Internat'l J. Imaging Systems and Technol. 10:186-198 (1999)).
One potentially powerful approach toward characterizing and quantifying trabecular network architecture is based on topological evaluation of the structure. Topology is the branch of mathematics concerned with the geometric properties of deformable objects (that are invariant in scale, rotation and translation) (Maunder, Algebraic Topology, Cambridge, UK: Cambridge Univ. Press, 1980). For example, topological criteria allow a determination as to whether a particular point in the network is part of a surface, curve, or junction. To illustrate the difference between topology and scale, one can consider a trabecular bone network that undergoes slight uniform thickening. Topologically, the network remains unaltered, but the scale properties have changed, which will result in changes to the mechanical properties. Conversely, given two networks with identical ‘bone volume fraction’ (BV/TV), one with more plate-like trabecular bone is assumed to be stronger than one that has trabeculae that are more rod-like. Here, the two networks differ in topology.
Feldkamp et al. (J. Bone Mineral Res. 4:3-11 (1989)) were among the first to use topological measures to describe trabecular lattices from 3D images obtained by micro-CT (μ-CT) images. The investigators reasoned that connectivity is impossible to derive from images of two-dimensional sections, hence they expressed connectivity in terms of a global network, referred to as the Euler characteristic or number. The Euler characteristic, derived from the Euler-Poincaré formula, assumes the number of bone objects to be one, and that marrow cavities do not exist in the network. These assumptions make the Euler characteristics equivalent to the first Betti number (Feldkamp et al., 1989), which is a measure of the number of loops in the network. This approach permitted characterization of osteoporotic changes in laboratory animals, demonstrating that the reduction in connectivity following bone loss and recovery parallels the reduction in Young's modulus for loading (Kinney et al., J. Bone Mineral Res. 13:839-845 (1998)).
Lane et al. (J. Bone Miner. Res. 14(2):206-214 (1999)) showed that ovariectomized rats, when treated with estrogen, could restore their pre-ovariectomy bone volume, but not the lost connections, thus demonstrating the disparate behavior of bone volume fraction and network topology. Pothuaud et al. (J. Microsc. 199:149-161 (2000)), extended the classical approach toward establishing topological quantities, such as the Euler-Poincaré number, with counts of branches and termini on the basis of skeleton graphs.
However, classical topology in terms of the Euler number ignores the existence of plates and rods, and fails to provide information on the spatial distribution of connectivity. Moreover, it may fail to detect the effect of osteoporotic erosion, as has been pointed out by Kinney et al., 1998. The first Betti number is inherently insensitive to trabecular erosion, which is known to result in perforation of trabecular bone plates and disconnection of rod-like trabecular bone (Amling et al., Arch. Orthop. Trauma Surg. 115:262-269 (1996); Parfitt, Bone 13:S41-47 (1992)). The First Betti number cannot distinguish between perforation of trabecularbone plates and disconnection of rods. The first Betti number will decrease from loss of rods, causing a reduction in the number of loops. However, it would increase as a result of perforation of plates, which increases the number of loops. Therefore, the first Betti number can not necessarily detect osteoporotic bone erosion.
Various approaches have been described to distinguish rod-like from plate-like architectures. Hahn et al., 1992, found the proportion of trabecular plates to rods to be reflected by the ratio of concave to convex surfaces, expressed in terms of the ‘trabecular bone pattern factor.’ An algorithm making use of the change in surface area for a differential change in radial expansion, led to the definition of a parameter denoted ‘structure-model index’ (SMI) Hildebrand et al., Computer Meth. Biomech. Biomed. Engin. 1:15-23 (1997). This metric was subsequently applied to the quantitative characterization of 3D μ-CT trabecular bone images from multiple anatomic locations known to be predominantly plate-like (SMI 0-1) or rod-like (SMI 2-3) (Hildebrand et al., J. Bone Miner. Res. 14(7):1167-1174 (1999)).
Nevertheless, none of the existing methods provide a reliable and efficient method for quantitatively characterizing the 3D architecture of cancellous bone networks, which are highly dependent on structural organization. Nor does the prior art provide methods for assessing bone strength or for predicting fracture risk in vivo in patients suffering from osteoporosis or metabolic bone disorders. It is an object of the present invention to meet these needs.
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