Trabecular bone morphometry by volume projection

David A. Reimann, David P. Fyhrie, N.L. Fazzalari, and Mitchell B. Schaffler

In Transactions of the 38th Annual Meeting, page 561, 1992 Orthopaedic Research Society Meeting, Washington D.C., 17-20 February, 1992.

{\bf INTRODUCTION} The mathematics of computed tomography (CT) describe the methods by which the internal structure of an object is determined using x-ray projections. This projection data contains all the information about the object's internal structure. Using concepts arising from the mathematics of CT reconstruction, we have defined an approach to the quantification of bone and marrow space anisotropy based on the projection of bone and marrow volumes. This approach is rigorously related to the standard stereological parameters: bone volume fraction and specific intercept count (inverse of the mean intercept length). Idealized models of trabecular bone having different mechanical properties, but which cannot be distinguished by standard stereology were examined. The method of volume projection easily distinguished between the models, indicating that this approach may be useful in the quantification of bone mechanical properties. Preliminary application of the method to digitized bone samples indicates that the method will work for real bone. {\bf BACKGROUND} In x-ray projections of objects, the amount of radiation measured at a point on the detector is directly related to the quantity and type of materials between the detector and the x-ray source. In standard CT, the projection converts a two-dimensional (2D) image into a one-dimensional (1D) shadow image. This shadow image is transformed to an attenuation profile, $a(s,\theta)$. In this 2D case, $s$ denotes the position within the 1D image and $\theta$ the aiming angle in 2D space of the x-ray beam. The totality of the attenuation profiles for all angles of the 2D object completely determines its internal structure - this is the fundamental fact upon which all CT techniques are based (1). The following developments for the morphometric understanding of internal structure by volume projection are also rooted in this observation. {\bf METHODS} The bone volume fraction ($BV/TV$) and the specific intercept distribution ($P_l$) can be measured directly from a 2D image using a microscope and a grid graticule (2). The first is calculated as the percentage of point hits on bone per total points counted and the second is the number of point hits of lines with the bone-marrow interface per length of line. For two phase structures where one phase is relatively radiolucent, these measures are directly related to the properties of the attenuation profile. The bone volume fraction is equal to the mean of the ratio of measured attenuation and expected attenuation if the radiolucent phase were not present. This relationship holds because this ratio is numerically equal to $L_l$ (length of bone per length of line passing though the specimen). $P_l$ is related to the number of projected bone-marrow transitions in the image, $T_l$, where $T_l$ is the integral of the absolute value of the derivative of the attenuation ratio, $L_l$. It can be shown that $$ P_l(\theta) \le T_l(\theta) = \int_W \left|{\partial L_l(s,\theta)\over \partial s}\right| \,\,\,ds, $$ where $W$ is the width of the shadow profile at $\theta$. As a measure of structural orientation, the angular probability density function of the attenuation profile, $\rho_\theta(x)$ is used. The calculation of this function is straight-forward, since it is simply the probability of occurrence for each level of attenuation as a function of x-ray projection angle. Alignment angles are those angles for which there is a high probability of high or low attenuations. Bone alignment axes occur when attenuated regions are at local maximum. Marrow alignment axes occur when attenuated regions are at local minimum. These definitions are mathematically rigorous, and also conform to the intuitive notions of alignment stemming from visual inspection of an image. That is, if the bone parts line up, there are many highly attenuated regions on the attenuation profile and vice versa when the holes line up. (Normal x-rays are negative images, so high attenuation appears bright and vice versa). {\bf RESULTS} To demonstrate these concepts three phantoms were simulated having three $19.05$mm diameter circular holes in a square area of $7310.25$mm$^2$, only one of which is shown (Fig. 1). These three were chosen because although they have identical $BV/TV$ and $P_l$ properties, they have different mechanical properties (3). The purpose is to show that the proposed method of volume projection can distinguish between these phantoms, where the standard stereological measurements cannot. The volume projection results showed that $BV/TV$ and $T_l$ could be computed for each phantom. The relationship $P_l \le T_l$ was verified. The predicted volume alignments agreed with the visual structural alignments (Fig. 2,3) and were different between the three phantoms. {\bf DISCUSSION} Analysis of the phantoms demonstrates that volume projection has the ability to distinguish differences in structure that are indistinguishable by standard stereology. Additionally, the orientation measurement indicates secondary alignments not at right angles to the primary, unlike the alignment predictions of the mean intercept length. Preliminary testing of the method for a real trabecular bone cross-section indicates that it works similarly for the prediction of orientation and intercept count. Some practical considerations stem from the fact that the method is defined relative to projected structural data, rather than to an actual image of the the structure. Since a typical CT reconstruction of a structure is not a trivial operation, typically $8$ cpu hours for a $200\times 200\times 200$ element volume from $200$ projections around $180$ degrees, this method should provide a means of determining significant morphological data directly from the raw data, bypassing the reconstruction step. In addition, since obtaining standard stereological measurements requires a cross-section through the object, the standard methods are inherently destructive and require invasive methods such as biopsy or sacrifice. The volume projection method, however, provides the core concept for noninvasive investigation of stereological variables {\it in situ} using high resolution planar x-ray techniques. {\bf REFERENCES} 1) Kak AC, Slaney M, Principles of Computerized Tomographic Imaging, IEEE, New York, (1988). 2) Weibel ER, Stereological methods Academic Press, London, (1979). 3) Fyhrie DP, et al, 1992 Meeting of the ASB, submitted.

Copyright © 1992, David A. Reimann. All rights reserved.