This section provides an overview of the most important methods that have been used to acquire data from which to extract or calculate arterial tree morphometry. Such methods fall into two major categories: the older destructive techniques, including histology and vascular casting, and the increasingly useful nondestructive imaging methods that are the subject of this chapter.
Historically, most anatomical data, particularly "microana-tomical" data on the remodeling of small arteries, was derived from histological sections. In fact, the words "morphometry" and "morphometric" are probably used more often in the histological context, where they are synonymous with ultrastructure and ultrastructural, than they are in the context in which we use them here. This dual usage of the terminology can become particularly confounding, since many of the changes in gross tree structure in which we are interested, including luminal narrowing and decreases in the number of parallel vessels in the arterial network, are preceded chronologically and etiologically by changes, usually in arterial wall structure, that are best observed and quantified by histological studies. Serial section reconstructions in histology are obtained by microtomy and microscopy of tissue blocks followed by stacking of the digitized photomicrographs in software . Although histological observation of stained sections of vessels is an excellent way to observe the cellular and subcellular changes implicated in conditions and diseases that affect arterial tree structure and function, this method does not lend itself to appreciation of the intact tree structure as a whole. It is difficult, if not impossible, to precisely maintain the position and orientation of the thousands of sections it would take to "reconstruct" any significant portion of the tree, and therefore tedious to arrange them in software in a visually meaningful and accurate way. Thus, serial section reconstruction methods have been applied to most advantage to the peripheral zones of the circulatory system, including the capillaries [19-21].
Corrosion casting refers to a class of methods in which a polymeric material (such as silicon rubber or Batsons No. 17, a modified methyl methacrylate whose viscosity can be adjusted over a wide range to govern, in conjunction with the injection pressure, the microvascular level to which the tree is filled) is injected into the arterial or venous tree [9,22-29]. After filling the vessels, the cast hardens and the tissue of the surrounding organ is corroded away, typically with a potassium hydroxide solution. The remaining plastic structure can represent an intricate and beautiful positive cast of the vascular lumen, though dimensional accuracy may vary with experimental conditions. These casts can then be conductively coated and viewed and photomicrographed in the scanning electron microscope (SEM). Morphometry has then been carried out by many groups, by meticulously breaking apart the tree and measuring the segments' and branching angles' geometric parameters. Some researchers have used morphometric parameters obtained in this manner as inputs to mathematical hemodynamic models [6,30]. Although highly valuable, casting methods are tedious to carry out and do not lend themselves to studies involving significant numbers of specimens or animals. In fact, until recently, most of the available data on the geometry of the pulmonary arterial tree, for example, were obtained from measurements on only a few (one rat , one cat , two dog [6,32], and three human ) plastic corrosion casts. Tedious measurements of individual vessel segment numbers, lengths and diameters have been made and several methods used to summarize these data as discussed later. One of the promises of newer 3D imaging technologies is to provide similar structural data in a nondestructive manner and with the potential for much higher throughput.
Microsphere embolization has been used in a number of laboratories as an indirect method to quantify the structure or, perhaps more accurately, the delivery capacity of the arterial tree [34-36]. For example, Seiler et al. found a linear relationship between myocardial mass supplied and the sum of branch segment lengths distal to any point in the coronary arterial tree and a two-thirds power relationship between luminal cross-sectional area and myocardial mass supplied . Again, though valuable, these techniques are acute and, used in isolation, do not retain information about the 3D branching structure of the tree or, for example, about focal obstructions in large vessels, both of which are sought in imaging studies.
We are primarily concerned with imaging and image processing methods for quantifying arterial tree morphometry. As mentioned previously, for many decades after 1895 planar X-rays recorded on film were the predominant method available to image the vasculature or any other type of structure. Today, the inherently planar methods available to clinicians and researchers for studying vascular structure and disorders fall into the two broad categories of radiography (including mobile units in the clinical setting) and fluoroscopy. Fluoroscopic methods include the highly specialized and sophisticated variants employed in angiography suites and cardiac catheterization laboratories. Whereas radiographic methods are static in nature, fluoroscopic methods permit dynamic image acquisition (15 to 60 frames per second) and are therefore useful for freezing the motion of structures such as the beating heart in the interest of extracting accurate quantitative measurements. Arteriography is a term which has been variously applied to both static  and dynamic (fluoroscopic) [39-41] imaging methods.
Since the dose in X-ray imaging is always limited, the tradeoff between speed and image quality is an ever-present problem: The more time available for imaging and the more dose permissible, the greater the number of photons contributing to image formation and the lower the noise level in the image. Most of the physical processes involved in imaging, including X-ray generation, penetration, and detection, follow Poisson statistics, which means that, for large numbers of X-rays, the X-ray quantum noise is equal to the square root of the number of photons contributing to the image or image element. Thus, if 1000 X-rays contribute to the information available in an image pixel, the maximum possible signal-to-noise ratio (SNR) obtainable, limited by dose alone, would be 32. If 1,000,000 X-ray quanta contributed, the SNR could be 1000. So while it may appear as if there are diminishing returns, at the low fluences permissible in the clinic quantum noise is a significant problem, and many of the challenges faced in the processing of vascular imagery, particularly segmentation or measurement of the smallest vessels, are caused by the high noise levels. Additional sources of noise include the background fog always present in radiographic film, which limits its dynamic range to several hundred at best. The stochastic nature of light production and light scattering in radiographic intensifying screens used in conjunction with film exacerbates the noise problem. For image processing of radiographs to be possible, the film must first be digitized using a laser scanner or other mechanism, which not only reduces the resolution of the film but also increases the noise level relative to the signal.
Most X-ray imaging techniques for arterial morphometry do not employ film as the detector, but use a high-gain image intensifier to transform a very dim spatial pattern of X-ray intensities into a bright visible-light image at the output of the intensifier. This visible light image is then digitized, usually with a high-resolution, low-noise tube-based (Vidicon, Plumbicon, etc.)  or CCD (charge-coupled device) camera . The combination of the image intensifier, the lenses coupling the light from the output window to the camera sensor, and the camera chip itself is referred to as the "imaging chain" and is the source of additional noise. Besides additional statistical noise arising from the multistage conversion processes within the device (X-rays to light in a scintillator to electrons in a photocathode and back to light again — after dramatic acceleration of the electrons through several tens of keV — in an output phosphor), a background haze called veiling glare  is superimposed upon the image, and the illumination at the image periphery falls off relative to the center of the field, a phenomenon called vignetting. An added problem for quantitative imaging applications is the spatial distortion caused in the image by even the highest quality fluoroscopic imaging chains. These result primarily from the fact that the image formation process requires the focusing of high-energy electrons by electrostatic focusing plates, a process that is imperfect and variable, partly due to the earth's magnetic field. Image processing methods to correct the spatial distortions and to allow for robust performance in the measurement of vessels in the presence of high noise levels are discussed later.
Biplane angiography allows for the simultaneous acquisition of two near-orthogonal views of the contrast-enhanced vasculature in real time. This acquisition method has provided the input data to a large number of algorithms designed to reconstruct the arterial tree from as few as two projections . One requirement of all such few-view reconstruction approaches is that the imaging system geometry, that is, the relative positions of the two sources, the two detectors, and any fiducial markers placed on the patient, be very precisely known. Some methods have been developed to calibrate or calculate the system geometry from information available in the images [45-49].
The diffusion of computed tomographic imaging technology starting 25 years ago triggered the 3D revolution in diagnostic imaging. Of course, although the ability to reconstruct anatomical maps, based on attenuation properties (tissue or electron densities), of the interior of opaque objects immediately opened up the possibility of true 3D imaging and quantification, the perceptual, computational and algorithmic complexities of volumetric analysis have made real progress in this area quite slow. Starting in the 1980s, magnetic resonance imaging, with its exquisite sensitivity to soft tissue properties and ability to exploit a wide range of contrast mechanisms, added fuel to the fire of activity in this burgeoning field of research. MR techniques quickly gained in speed, with scan times being reduced from about an hour to less than 15 minutes for clinically useful volumes. Scan times continue to decrease, and 3D images of diagnostic quality covering several axial centimeters can now be acquired in minutes . Recently, CT research and diagnostic methods have been given new life and expanded capabilities by the extension, from single slice imaging, first to helical, and now to multislice helical scanning . Currently, four slices can be reconstructed per half-second revolution of the source and detector around the patient, making fast volumetric scanning clinically useful for imaging entire organs in a single breath hold. Recent developments in fast CTA [62-66] and MRA [60,67-71] are certain to increase the value of these techniques in clinical applications of quantitative 3D vascular morphometry. In the 1990s ultrahigh-frequency, high-resolution intravascular ultrasound (IVUS) methods were developed for imaging the constituents of vascular walls . All these improvements and new methods have given added impetus and possible value to the field of image processing for arterial tree morphometry.
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