S 47

Quantitative computed tomography: From linear absorption coefficients to bone mass J. Cordey Laboratory for experimental surgery, Davos, Switzerland

1. Problem Computed axial tomography was introduced in the early 70's and awakened increasing interest in the world of orthopaedics because of its ability to display cross-sectional views of limbs, a goal impossible to achieve using conventional radiography. A significant limitation was (and still is) that metallic implants jeopardize accurate picture reconstruction. The main two clinical interests in this method are the determination of rotational deformities of long bones (and fragment recognition~ and, the display ~f complex pelvic fractures after extensive PICture analysis , The first application is fairly simple; th~ sec~nd, h,owe~e:' requires elaborate computer. work and, in s~lte of Its cli~­ cal relevance, is not yet available to all hospitals, The third application is bone densitometry. This method was introduced more than 15 years ago for quantitative assessment of osteoporosis but is the object of renewed interest now due to the availability of powerful microcomputers. We have been using this method since 1977 in orthopaedic research. In this communication, we restrict our atten tion to the different problems and their solution from the point of view of the researcher. 2. What is a computed tomograph? A computed tomograph is an instrument which allows the display of a cross-sectional picture of a limb. The principle of this instrument is based upon the scanning beam of the X-ray (the y-ray) passing through matter and on the continuous measurement of the absorption values. An image of the cross-section is subsequently reconstructed from comp uted calculation (e.g. using back projections) which displays in a visual form values of the local linear absorption coefficient Such an image differs from that of a conventional radiograph in that it is not a shadow including the superimposition of all the X-ray absorption along the beam but it displays local values of absorption. What is a tomogram? A tomogram displays a table (matrix) of cross-sectional voxels, which are values of the absorption coefficient

Problems related to the quality of the tomograms The quality of the tomograms is limited by the following facts: 1) Reduced contrast, particularly for soft tissue visual evaluation. 2) Reduced resolution due to the size of the beam. 3) Beam hardening artefact 4) Metal implant artefact

Picture contrast It is obvious that pictures are generated for visual examination. The question is, how good are they for quantitative data analysis? When conventional computed tomographs are used, it must be remembered that techniques are used in order to improve the "visual quality", for example, contrast enhancement which leads to values improper for data analysis. Appropriate machines and programmes, however, allow for quantitative data analysis of bone density.

Resolution The size of the radiation beam is of the order of magnitude of 1 rom2. For reasons of speed and reduction of irradiation, the scanning rate must be fast and the irradiation limited. These limitations result in a picture with a spatial resolution far below that of a fine X-ray. The order of magnitude of this resolution is usually of the order of the mm'2 (the Densiscan being a "formula 1" machine with an extreme resolution of 0.1 mm4. The individual values of the matrix of data are related in pixels (picture elements). The cr generates X or y-ray beams of rectangular cross-section with a size transverse to the limb of about 1 mm and a size longitudinally to the limb of about 1 mm. The tomogram is, therefore, a picture of a thin slice, with a thickness of 1 - 3 mm. Better than calling the picture elements "pixels" the name voxels (volume elements) should be preferred. It must be remembered that we are speaking about matter and that matter requires a certain amount of volume! The error of density of voxels used to assess a sharp edge (like the outside of a cortical bone) is called the partial volume error. This partial volume error may affect the analysis of density.

S 48

Beam hardening artefact An imperfection in accurate density data is due to the fact that not all the X-rays are absorbed in the same quantity within the material; low energy X-rays are more easily absorbed than high energy ones. Due to this fact, the X-rays which penetrate the bone are richer in low energy than those which are in the medullary cavity. This physical effect is named beam hardening. When this fact is not taken into account by appropriate reconstruction programmes, then it seems that the cortical bone next to the periosteum is more dense than that next to the endosteum. This is the beam hardening artefact Due to the beam hardening, the cortical bone next to the periosteum seems to be denser than that next to the endosteum. This has been demonstrated by scanning homogeneous fantorns [3], whereby the apparant density results in a "cuvette effect" (Figure 1); for bones, this effect might be erroneously interpretated as an endosteal porosis.

Dens ity

Fig. 1: Beam hardening effect, resulting in a "cuvette effect". This effect might be erroneously interpretated as an endosteal porosis. Left: uncorrected, right: corrected. The metallic implant artefact

Tomographs are limited by the fact that X-rays are easily absorbed by metallic implants. This makes it almost impossible to reconstruct a valid picture. The reconstructed picture is affected by apparent high density shadows, the so-called metallic artefacts. For purposes of bone assessment, these artefacts are just at the place which would be of interest Ruegsegger [1, 2, 3] and co-workers made programmes which were able to achieve adequate reconstruction for intramedullary implants (e.g. hip prostheses). To my knowledge, these programmes never became available for conventional instruments. Limitations of quality of the tomogram: The quality of the tomogram is affected by: 1. the required contrast enhancement, 2 the limited resolution, 3. the beam hardening artefact, 4. the ugly artefacts generated by metallic implants which may disturb the picture entirely.

scribe bone in terms of numeric values. That is the scope of quantitative computed tomography (QCI'). !he largest field of interest of QCf is osteoporosis. Here it IS used to assess the amount of bone within the section (bone mass) and its relative distribution (bone density). A further field of interest for us is bone healing in relation to callus production, osteopenia related to bone plates and osteogenesis under callus distraction, e.g. lengthening. Quantitative computed tomography aims to assess two

things. The morphology (geometry) of the cross-section and its mineralization (bone mass and bone density). Morphological analysis of the tomogram The visual information provided by the tomogram aims to answer the following question: Where is bone and how dense is this bone 7 Radiologists prefer black and white pictures because they are used to looking at them! Visualization and discrimination of density is, however, better displayed by a discrete series of colours! which allows for comparison of bone of disimilar density at a distance (the human eye is not an accurate intensity instrument). The information which is also directly available from the tomogram is the cortical thickness which can be measured on the picture at any selected position. This cortical thickness, together with the presence or absence of less dense endosteal bone, is the first estimation of the extent of osteoporosis of the bone. Geometrical values may be directly calculated from the tomogram (and/or from the matrix of densities). For instance, the bone area may be calculated either by the number of pixels of densities higher than a given thresh?ld, i.e.. bone density, mutiplied by the pixel area, or by integration from contour recognition algorithms (this solution is also appropriate for measuring the area of the medullary cavity). Corresponding areas are then calculated by addition of the corresponding number of pixels. The bone area A is A = Rx,y A(x,y)[D(x,y) > Dol whereby A(x,y) [x,y = 1, ... to n] is the area of a pixel and the density D(x,y) is larger than a bone threshold density value Do A value which is of interest in evaluation of osteoporosis is the cortical index which is the ratio of the cortical bone area to the total cross-sectional area (including the less dense bone and the medullary cavity). Assessment of further geometrical properties is feasible, for example, the centre of mass of the section (centroid) and the moments of inertia (second moment of area). Experience shows that the beam hardening artefact does not significantly affect the geometrical measurements; the partial volume artefacts have to be taken into account particularly when bones cover only a small part of the picture area.

3. Quantitative computed tomogmphy There are instances in which computed tomography is used less for its ability to provide pictures but more to de-

1 Few, i.e. less than six, grey levels might also be used and accurately discriminated.

S 49

Cordey: Quantitative computed tomography

Geometry Quantitative computed tomography provides the following data: Bone and medullary cavity cross-sectional area, moments of inertia, cortical index (as ratio cortical bone area / total bone + medullary cavity area). Densitometri.cal analysis of the tomognun Densitometry is the method of measuring the mineralization of the bone. The local density measurements allow calculation of the global mineral content within the thick cross-section. A computed tomograph used for QCT does not use contrast enhancement in its programme. Furthermore, it is mandatory to take beam hardening into account. y-rays may be used for reducing the beam hardening artefact, for example, in the Isotom which was developed by Elsasser and Ruegsegger [1, 2, 3] in the mid-70's in order to achieve a nearly monochromatic beam. Another method is to use X-rays to filtrate the beam hardening artefact and to use an appropriate picture recon-struction software which takes this effect into account, e.g. in the more recent Densiscan developed by Ruegsegger and Naeff. How do the voxels data relate to the bone density 7 The linear absorption coefficient is a physical value. In the conventional machines, it is expressed using the Housfield's numbers which are values compared with the absorption of water (representative of soft tissues). For densitometric machines, the units may be different. The lsotom on-line analysis provides values in cm-1 (which are the real physical values for the energy of 27 kV, due to the y-ray generated by an 1251 source; the matrix of data provided is, however, not transformed into these values (it is used for generating colours in the display unit and such a transformation is not necessary). The quantitative computed tomographs provide values in terms (if not in units) of mineral density.

How much bone (or mineral) do we have in a tomogram, Le, within the cross-sectional slice which relates to this tomogram 7 Bone mass: A mass is defined as a quantity of matter. Therefore, we defined as bone mass the quantity of mineralized bone which is present within the tomogram (remember that it is an - 3 mm thick slice of bone!). How can it be measured ? This is simple. After exclusion of the pixels (or voxels) which have a density lower than a given threshold, i.e, soft tissues, we multiply the remaining pixels which represent bone of varying densities by their density, and we add them. The result of this truly weighted addition is the bone mass. Summation on the pixels of area x density. The true unit should then be in kglm or glmm (This is because the mass is calculated for a

slice of a length of resrctively 0.003 m or 3 mm along the long axis of the bone) . The bone mass is given by the summation of the pixel area and the pixel density BM

=

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A(x,y) . D(x,y)

whereby A(x,y) [x,y = 1, ... to n] is the area of a pixel and D(x,y) its mineral density. Bone density The average bone density is the opposite of the bone mass: D = A/BM With exceptions, this value is generally not of interest. More interesting is the distribution of bone densities. Several facts have been established from the measurements of hundreds of human femora and tibiae as well as sheep tibiae: The average cortical bone density is fairly constant within the midshaft, slightly higher for sheep tibiae and slightly lower for human femora. Looking at a histogram of density (or local bone mass), these values vary according to a normal law by about 15,000 vax units. Less dense bone is found either in callus formation or within the distracted callus generated by leg lengthening, for example. The behaviour of new bone was particularly interesting in a segment transfer experiment (Brunner et al., 1992 [4]), whereby new bone did appear as a new normal distribution peak at a lower value. It was therefore possible to approximate the bone mass distribution around increasing values of densities as the addition of three components (Figure 2):

1.

Soft tissues as decreasing exponential value,

2.

less dense bone as Gauss distribution with a density smaller than the cortical bone,

3.

cortical bone, as Gauss distribution about a density value of 15,000.

Both the average density of less dense bone and the amount of less dense bone, i.e. bone mass, are parameters which can be taken into account when assessing the development of the callus.

2 For our comparative purpose, we did not transform the density given in vax units into the physical unit of density (gIcm3). We arbitrarily take A(x,y) to 1 and D(x,y) to its value in vax units.

550

ISOTOM ANALYSIS

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Low density (so ft ti ssues ) Less dense bone Cortical bone

Fig. 2: Histogram of bone mass in relation to the density of the voxels. The bone mass distribution in relationship to the density may be approximated by superimposing

sity, average "trabecular"4 density and cortical bone density. Further variables of interest were calculated, like the percentage of bone within the cross-section (percentage of total bone area related to the cross-sectional area). Where should the bones be measured ? In a preliminary series where tomograms were obtained every centimetre along the long axis of the bones, it was established that the appropriate location for the measurements was in the distal third of the tibiae and the proximal third for the femora [8]: At this location, a small variation in position produces minimal changes (Figure 3)~ this location is that of the smallest cross-sectional area and that of the largest average cross-sectional density. Many variables are provided by QCf; how can the bones be sorted? Using a method derived from multivariate statistics, namely, exploratory principle component analysis, we found that only two variables are required to describe and subsequently to select these bones. The first one relates to the extent of osteoporosis (for instance, the percentage of bone within the cross-section) and the second one to the size of the bones (for instance, the crosssectional area). De nS I

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Normal long bones When we carry out mechanical experiments on long bones, it is prerequisite that they are sorted according to morphology. This has been done for a number of projects [5, 6, 7] using quantitative computed tomography.

Fig. 3: Densitometrical and geometrical measurements along the long axis of an human tibia (typical example). One third of the bone is then chosen as a standard position for measuring a large series of bones.

Analysis of normal and osteoporotic' human bones (tibiae and femora): The following variables have been assessed: Total cross-sectional area (area of bone + medullary cavity), total bone area (cortical + less dense bone), cortical bone area for the geometry; average cross-sectional den-

The dramatic variation in bone size between individuals is displayed in Figure 4: The cross-sectional area ranges between 250 and 475 mm 2 and the bone area between 50 and more than 300 mm 2. It is therefore obvious that a small and thin bone (with a cross-sectional area about

paper to discuss the concept and the definition of osteoporosis. We use this word because it is the main reason for the osteopenia observed particularly in female bones, and which results in mainly endosteal bone loss.

4 The average "trabecular" density is the average density of the remaining area which amounts to 50% of the crosssectional area, obtained using an erosion proced ure on the tomogram. This method is relevant for appendicular QCT [1,2].

3 It is beyond the scope of this

Cordey: Quantitative computed tomography

300 mm 2 and a bone area about 100 mm 2) will be far less stiff and will fail earlier than a strong one (with a crosssectional area of about 450 mm.2 and a bone area about 300 mm 2); the load required to produce the same effect will be 2.5 - 3 times larger for the large bone.

551

Average Bone Density (em-I) 3.

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Figures 5 and 6 display the variation of the average bone density and the percentage of bone in relationsliip to the cross-sectional area. For both these variables, lower values are observed for smaller bones. The cortical bone density is fairly constant (Figure 7), about 3.45 cm'l with a coefficient of variation of about 6 7%. Few values are however significantly skewed below the mean; this might be explained either by the partial volume effect on thin cortices or by a truly porotic bone. However, these values demonstrate that cortical bone is fairly homogeneous. A very high correlation (Figure 8) is seen between the percentage of bone within the section and the average cross-sectional density. This suggests that bone, as material, is homogeneous (cortical bone) and that the linear absorption coefficient (which is our estimate of density) ranges between that of soft tissues (about 0.5 cm'l for a percentage of bone of 0%) and that of cortical bone (about 3.5 cm- l for a percentage of bone of 100%, see Fig. 7). Neither of the variables displayed in Figure 8 depend upon the size of the bones (but the fact that the smaller bones of women are more prone to be osteoporotic); therefore, either the percentage of bone or the average crosssectional density are good estimates for describing the ex, tent of osteoporosis. Cortical Bone Area (mm2)

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