Quantitative measurements are important for the analysis of medical images and the diagnosis of diseases. Performing the measurement directly on the image data provides a noninvasive way to get the physical and physiological properties of the anatomic structures. Sometimes image measurement is the primary goal of a clinical trial.

Measurement is usually performed after image segmentation to determine the properties of the segmented region. Quantitative measurements can be used to compare an individual patient to other patients or reference values. Measurements derived from a single patient may be compared over time, which is useful in tracking disease progression and monitoring treatment. Measurements of anatomic structures include physical properties such as size, shape, and density, and physiological attributes such as tissue perfusion and vascular permeability.

Size of a structure is the most intuitive measurement in many cases. Size of an object often tells the significance of a pathological region and is an important indication of a disease. Examples of size measurement include the longest diameter, which is the longest distance between any two points on the boundary; the effective diameter, which is the diameter of a circle having the same area as the region of interest; and the area/volume of a region.

Shape of an anatomic structure can also be quantified. The compactness of a shape can measure how closely a structure is related to a circle or a sphere. Compactness in 2D can be written as C = P2/4pA, where P is the perimeter and A is the area of the shape. In 3D space, compactness can be written as C = A3/36p V2, where A is the surface area and V is the volume of the shape. In either case, the compactness of circle and sphere is one and that of other shapes is larger than one. Compactness has been used for quantifying mammographic calcification and breast tumors. Boundary curvatures such as Gaussian curvature and mean curvature can be used to quantify the curvedness and smoothness of a shape (11). Other shape properties such as shape index and spatial moments have been proposed to quantify anatomic shapes (12). Angular measurements often reflect the deformity of an object and the relationship between two objects. Angle measurement on bony structures is especially important for patho-anatomical analysis, diagnosis, and therapeutic planning in orthopedic diseases (13).

The distribution and statistics of pixel intensity within a region can reveal the smoothness, contrast, regularity, or homogeneity of tissues. Texture analysis such as statistical moments, and co-occurrence matrix provides ways to describe the tissue appearance (12). The statistical moments are computed based on the intensity histogram. The second moment of the histogram measures the intensity variance within the region, which correlates with the roughness perception. The third and fourth moments, skewness and kurtosis, reflect the asymmetry and uniformity of the intensity distribution. The co-occurrence matrix is also known as spatial gray level dependence matrix in the sense that it combines spatial information and intensity statistics (14). The inertia of the co-occurrence matrix characterizes the texture contrast of a region. The entropy of the matrix quantifies the level of randomness in the region. The angular second moment of the co-occurrence matrix can be used to describe the homogeneity of a region. Co-occurrence matrix was used in the analysis of prostate tumor and breast calcification (14,15).

Two examples of physical measurement application are bone mineral densito-metry (BMD) and mammographic density measurement. BMD is a method to quantify the bone mass in the body (16). Osteoporosis is a common bone disease, which makes bone fragile and easy to fracture. Future risk of fracture can be predicted through a BMD measurement. Mammographic density measurement is a tool to measure the regions of brightness associated with fibroglandular tissues in the mammography, which is directly linked with the breast cancer risk. Mammographic density can be computed from the mammography using histogram analysis, or fractal analysis (17,18).

In addition to the physical and anatomical measurements, functional and physiological measurements can be obtained from functional imaging modalities such as PET, functional MRI (fMRI), and DCE MRI. The functional quantifications are computed by applying pharmacokinetic or functional models to the original data, and generating parameter maps. The parameter maps are usually color-encoded and overlaid with the original images.

As an example of functional images, DCE MRI is a method to reveal the physiology of the microcirculation. DCE MRI measurement correlates well with tumor angio-genesis, which is the formation of new blood vessels that allow tumors to grow. After intravenous administration of Gd-DTPA, DCE MRI yields a description of the measured signal-time curves in terms of pharmacokinetic parameters. The signal-time curves are analyzed pixel-by-pixel to preserve the spatial information of MR images. A two-compartment pharmacokinetic model had been proposed to analyze DCE MRI data (19). In the model, two characteristic parameters, transfer rate k21 and amplitude A, are fit to characterize the tissue physiological properties. Transfer rate k21 is the rate for transfer of Gd-DTPA from the extracellular space to the plasma, which characterizes perfusion and vascular permeability of the lesion. The amplitude A reflects the degree of signal enhancement in a region. A tumor region is expected to have high k21 and A values. The two-compartment model has been successfully applied in analyzing breast tumors, brain tumors, and osteosarcoma. Another model characterizing the uptake slope in enhancing phase of Gd-DTPA in the cells has been used in the diagnosis of prostate tumors (20). Figure 5 shows color-encoded A parameter map (Fig. 5A) and k21 parameter map (Fig. 5B) of DCE MRI breast study, and take-off slope map (Fig. 5C) of a prostate study. The parameter maps are superimposed on the original images.

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