QJM vol. 97 no. 11 © Association of Physicians 2004; all rights reserved.
Mathematical estimation of the potential effect of vascular remodelling/dilatation on B-mode ultrasound intima-medial thickness
From the Departments of 1Epidemiology and 2Biostatistics, University of Arkansas for Medical Sciences College of Public Health, 3AP Calculus, Little Rock Christian Academy, 4 Department of Biostatistics, University of North Carolina at Chapel Hill, 5 Department of Pathology, Louisiana State University Health Science Center, New Orleans, and 6Departments of Internal Medicine, Physiology, and Biophysics, University of Arkansas for Medical Sciences, USA
Received 2 February 2004 and in revised form 30 June 2004
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Summary |
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Background: Arterial diameter changes are known to impact wall thickness, but the clinical relevance of the changes is unclear.
Aim: To use known mathematical relationships to estimate anticipated changes in arterial wall thicknesses occurring with enlargement of atherosclerotic regions.
Design: Mathematical relationships between a cylinder's diameter and its wall thickness were used to calculate the theoretical effect of diameter enlargement on the thickness of an atherosclerotic wall.
Methods: Equating the wall areas of two cylinders, one of smaller diameter than the other, allowed estimation of the degree of thickening that would be needed to maintain intima-medial thickness (IMT) after arterial remodelling. The difference in cylinder diameters was based on arterial diameter enlargement reported with atherosclerosis progression. Thus, the calculated wall changes estimate arterial changes which could go undetected if only IMT is measured by ultrasound.
Results: The expected IMT change for diameter enlargement is not a linear function of the diameter change, but varies depending upon initial size (diameter and IMT). Thus a 0.6 mm arterial diameter enlargement would be expected to cause a 0.039–0.235 mm change in IMT, depending on artery size. The estimated IMT change is similar to that associated with major atherosclerotic risk factors.
Discussion: The level of vascular remodelling reported with atherosclerosis could have a measurable impact on IMT, suggesting that indicators incorporating both diameter and IMT may be better disease indicators than IMT alone. Arterial diameters, as well as IMT, should be obtained in ultrasound studies of atherosclerosis.
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Introduction |
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The need to identify people at risk of heart attacks and strokes prior to the development of clinical disease is well recognized.1–6 Since atherosclerosis, a major cause of coronary artery disease and stroke, is a diffuse process, estimation of carotid artery intima-medial thickness (IMT) on B-mode ultrasound is often used to indicate increasing atherosclerosis severity and clinical disease risk.1–3,5–22 IMT is generally preferred over the identification of plaques because it can be more easily standardized.23 A recent review suggested that mean maximum carotid IMT may be preferable over the mean IMT, but issues remain. Whether early intimal thickening is an adaptive physiological response 24 or an early stage of atherosclerosis has not been resolved.25,26 Another issue is that the intima and media cannot be separated reliably using B-mode ultrasound, so that medial hypertrophy could also contribute to wall thickening.23
In addition, arterial diameter may change with mechanical stress,27,28 which can have an impact on wall thickness.29,30 The term ‘arterial remodelling’ can mean either an increase (outward, compensatory, positive, or Glagovian remodelling) or a decrease (inward, paradoxical shrinkage, negative, constrictive, or antiglagovian remodelling) in the arterial diameter compared to normal.31,32 Atherosclerosis of the common carotid artery is often associated with outward remodelling.13,33 Outward remodelling has been associated with medial injury/atrophy in the area of the plaque in humans34,35 and with focal lamellar degeneration in the area of plaques in apoE-deficient mice.36 The resulting IMT would be thinner relative to the IMT prior to remodelling, as shown schematically in Figure 1A, and could result in an elliptical wall shape. Diffuse diameter enlargement (eutrophic or hypotrophic outward remodelling) may occur as a physiologic response to increased blood flow37,38 or as a change in ageing, and could result in reduction in wall thickness from stretching of the arterial wall or from diffuse damage to elastic lamellae30 (Figures 1B, 1C). This is similar to the thinning of the wall of a balloon as it is blown up. Studies of increased aortic blood flow in rats resulted in enlarged aortic diameters,37,38 and increased wall area,38 but decreased wall thicknesses.37,38 However, diffuse diameter enlargement, such as occurs with ageing, is often gradual and occurs concurrently with small increases in wall thickness because of mechanical stresses.28 Thus, even though the wall is increasing in area, the counter-balancing effect of diameter enlargement on IMT could mask the increase in the wall that is taking place, just as it did in the animal studies. Since hypertension33 and many other risk factors13 are associated with a larger arterial diameter, not accounting for the diameter increase could result in underestimation of the severity of vascular disease.
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Even with its limitations, IMT has been useful in epidemiological and clinical studies,23 but less so than had been anticipated,39–42 and alternative measures have been considered.29,36,43–46 Cross-sectional arterial wall area for round arteries has been estimated mathematically (Figure 2).47 Recently, plaque area has been determined from manual tracing of carotid plaques on B-mode ultrasound and was associated with increased risk, even after adjusting for other risk factors.48
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This study explored the potential magnitude of effect of diameter enlargement on IMT using known mathematical relationship between a cylinder's diameter and its wall thickness.47 This has not been previously reported to our knowledge. Secondly, using known trigonometric relationships, we described the mathematical steps that can be used to calculate arterial wall areas of round or elliptically shaped arteries using routine B-mode ultrasound measurements, thus taking into consideration both IMT and diameter.
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The mathematical formula relating radius to the area of a circle was used to calculate arterial wall area (e.g. a cylinder wall area). If the radius of the circle defined by the media-adventitial interface is r, then the radius of the arterial lumen is r–IMT (Figure 2). The arterial wall cross sectional area (A) would encompass both the intima and media, and is represented mathematically as the area encompassed by the outer circle (media-adventitia interface) minus the area within the inner circle (intima–luminal interface).47 This relationship is described by the mathematical formula:
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r is the amount of artery radius enlargement (i.e. one-half diameter change) and A is the cross-sectional area of the arterial wall. For different levels of diameter enlargement, the change in IMT (IMT) was calculated by subtracting the original IMT from the new value of IMT (IMTN):
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The amount of arterial diameter change that our study evaluated was the maximum (1.2 mm) and half of the maximum (0.6 mm) change found in a prospective clinical study of 32 patients with plaque progression.49 Since arterial diameter and IMT can vary, and may impact the estimated changes in IMT, several different initial arterial diameters and IMT levels were tested (initial arterial diameters of 6 mm, 7 mm, and 8 mm diameter and initial IMTs of 0.5 mm, 1 mm, and 1.5 mm). In the discussion, the magnitude of the estimated IMT reduction from remodelling is compared to change in IMT that has been associated with several known risk factors.
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Results |
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Figures 3 and 4 display the differences in the thickness of cylinders of differing initial diameters and wall thicknesses as an estimate of the change in IMT that would occur theoretically with diameter enlargement. The mathematical relationship indicates that arteries ranging from 6 mm to 8 mm in initial diameter would need to have additional tissue deposition, or there would be a decrease in IMT as the arterial diameter enlarges (Figures 3 and 4). In other words, to maintain the original IMT, there would need to be an increase in arterial wall area. Without an increase in wall area, the reduction in IMT that would be predicted to occur with a 0.6 mm diameter increase ranged from 0.039 mm to a maximum of 0.235 mm depending upon the initial artery diameter and IMT. For a diameter increase of 1.2 mm, the predicted IMT reduction ranged from 0.073 mm up to 0.392 mm.
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Figures 3 and 4 indicate how initial arterial diameter and wall thickness influence the magnitude of IMT change predicted to occur with arterial enlargement. In Figure 3, the influence of diameter can be seen by comparing the bottom line in each of the three panels that represents IMT change in arteries with an initial wall thickness of 1.5 mm. Moving from left to right panel (small to larger diameter arteries), the change in IMT becomes less as the initial diameter becomes larger. Similar differences are seen for arteries with thinner initial walls (Figure 3). This is can also be seen by examining the horizontal slope in Figure 4, especially the middle and right panels. Smaller arteries had a larger predicted decrease in IMT for any given initial wall thickness. For arteries of 6, 7, and 8 mm initial diameters and initial IMTs of 0.5 mm, a 0.6 mm arterial diameter enlargement would be associated with a predicted decrease in IMT of 0.053 mm, 0.045 mm, and 0.039 mm, respectively.
If diameters increased less in small arteries compared to large arteries, the impact of initial artery diameter would be less important. However, data suggests that diameter increase may not be proportional to size. A small study of 72 patients and 8 controls found larger percent diameter increase for smaller arteries.50 The known mathematical relationships indicate that a small artery would need to experience a greater diameter increase than a large artery to achieve a given increase in lumen area.
The mathematical relationships also indicated that the initial arterial wall thickness (IMT) would have an even greater impact on the magnitude of change in IMT for a given degree of arterial diameter enlargement (Figures 3 and 4). Considering the left panel in Figure 3, the lines indicating the IMT decrease have much steeper declines for larger initial wall thicknesses (bottom line) than for smaller IMTs (middle and top lines). The associations hold true for larger initial diameters as well (Figure 3, middle and right panels). In Figure 4 the absolute decrease in IMT is minimal for arteries of initial IMT of 0.5 mm (left panel), but is substantial for arteries of initial IMT of 1.0 (middle panel) and is largest for arteries having an initial wall thickness of 1.5 mm (right panel). Thus, for arteries of the same initial diameter, those with thicker initial IMT would have a larger change in IMT for the same degree of arterial diameter enlargement. The mathematical relationships indicate that the reduction in IMT occurring with acute diameter increase is not a linear function of the diameter change, but is larger for arteries with thicker walls and smaller diameters.
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Discussion |
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Traditional risk factors do not account for all cardiovascular disease in patients.51 While studies indicate carotid IMT is strongly associated with various cardiovascular risk factors12,13,16,52,53 and cardiovascular diseases,5,7,8,14,54–57 IMT improves predictive models only marginally over traditional risk factors.39–42 These limitations could be for physiological and/or technical reasons.39,58 For IMT to be useful in clinical and epidemiological studies, models with IMT should offer improvement over traditional risk factor models, or provide an explanation of how traditional risk factors contribute to clinical disease. The mathematical relationships explored in the current study indicate that the degree of vascular diameter enlargement that has been reported clinically could have a measurable and biologically relevant impact on wall thickness, which would have immediate relevance in studies of plaque progression or regression.
The B-mode ultrasound equipment used in some major epidemiological studies has a resolution of approximately 0.067 mm,21 and so most of the changes found in this study should be detectable. In fact, the mathematical model indicates that the impact of arterial diameter enlargement on IMT is of similar magnitude to that reported for many known risk factors. Cross-sectional data from the Atherosclerosis Risk in Communities (ARIC) Study found the following differences in common carotid IMT associated with the indicated risk factors after adjusting for age, height, race and centre: presence of diabetes (men, 0.067 mm; women, 0.074 mm), hypertension (men, 0.049 mm; women, 0.048 mm), 40 mg/dl increase in LDL-C (men, 0.027 mm; women, 0.025 mm), for lifetime smoking (men, 0.023 mm; women, 0.020 mm), and increase in BMI of 5 kg /m2 (men, 0.018 mm; women, 0.010 mm).13 So, the magnitude of IMT change estimated to occur with diameter enlargement was similar (and in many instances greater) than the increase in IMT that has been associated with several risk factors. The IMT progression in various studies summarized by Aminbakhsh indicate the relevance as well, since the progression rate of only 0.01 mm per year was found in volunteers 59–71 years of age and an increase of far-wall common carotid of 0.034 mm per year increased the risk of events significantly.59
The idea that remodelling or diameter change could impact other arterial measures such as lumen area60 and IMT29 is not new. It is known that IMT can vary even during the cardiac cycle.13 The new finding in the current study is that the degree of clinically occurring arterial diameter enlargement with atherosclerosis is sufficient to theoretically impact IMT in a measurable and meaningful way. Both early intimal fibroplasia40 and plaques in atherosclerosis61 are associated with diameter enlargement, which our study indicates could counterbalance intimal thickening. While the arterial wall is unlikely to remain static after vascular remodelling or after the diameter enlargement that accompanies ageing, an IMT in an artery whose diameter has enlarged is unlikely to indicate the same degree of injury as the same IMT value in an artery that has not enlarged. If vascular diameter enlargement occurred uniformly with age and with atherosclerosis, arterial diameter enlargement's impact on IMT would not be relevant. However, previous studies indicate that enlargement does not occur uniformly for all risk factors,13,61 with different underlying mechanisms of atherogenesis,62 in different arterial segments,63 or even in individuals.46 Since IMT is used as measure of vascular injury,54,59,64 the impact of arterial enlargement on IMT is important and could result in misclassification of vascular damage. In a study of borderline hypertension, a measure of intima-medial area that took diameter into consideration was a more potent discriminator between borderline hypertensives and controls than IMT alone.29 Our results suggest reasons that area measures provided a better discriminatory function than IMT in some relationships.
The mathematical formula that we used to determine the impact of diameter enlargement on wall thickness has been previously used for calculating the cross-sectional arterial wall area for round arteries.47 Recently a method for estimating diseased cross-sectional areas was developed using technician tracings of B-mode plaques to estimate the plaque areas, which were found to predict vascular events.48 We propose calculations of arterial wall areas that are mathematically valid for both round and elliptically-shaped arteries (Figure 5). The measurements that are needed can be easily obtained from B-mode ultrasound, using methods previously described.65 The B-mode ultrasound arterial diameter and wall measurements that are taken on the longitudinal view of the common carotid artery (Figure 5, longitudinal view) can be projected onto a transverse (cross-sectional) view. If three measurements are taken (only one measurement is shown in the longitudinal view) for the same transverse plane but at different angles of interrogation, and if the angles of interrogation are known between the three measurements, the relationships shown in Figure 5A can be approximated. Calculations are relatively simple if one of the diameters is either the longest or shortest arterial diameter (major axis or minor axis, respectively) for the transected artery. The formula for the area of an ellipse is 
rarb, where ra is the major axis and rb is the minor axis. When ra (half the arterial diameter) is measured, trigonometric relationships can be used to find a point (x,y) that lies on the ellipse of which ra is a radius and where h (half a measured diagonal) is the hypotenuse of the right triangle created for the purpose of finding (x,y) and the angle opposite side x is known (or can be determined from the B-mode ultrasound angle of interrogation). In this example, the angle is 90° – 30° = 60°. The value for x is determined from the equation h(sin 60°) = x. Once x is known, the value for y can be determined using the Pythagorean Theorem. Finally, rb is determined from the equation for an ellipse, (x2/ra2) + (y2/rb2) = 1, where (x,y) is a point on the ellipse whose centre is (0,0) on a Cartesian plane, and ra and rb are radii. The lumen area, which can be calculated from the lumen radius, is then subtracted to determine the wall area. Because height, gender, and age are related to larger CCA diameters,13 the proposed measures should be normalized for those characteristics.
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Further refinement of the calculations can take several directions. Epidemiological studies have shown the feasibility of obtaining eleven B-mode ultrasound measures over the CCA of 1 cm length,65 and so even volume measurements of the CCA wall are possible by combining the estimated cross sectional areas for the eleven measurements using the distance between measures as the depth. Alternatively, an eccentricity measure comparing wall thicknesses or diameters could be used to indicate the presence of an eccentric plaque rather than merely intimal fibroplasias from ageing and to attempt to separate ageing components from atherosclerosis components.
A major limitation of the area calculation is that the maximum diameter and wall thickness may not occur in the same plane of B-mode interrogation within a person. Also, the plane containing the greatest wall dimensions may not be the same for all patients, thus making standardization more difficult, and suggesting that volume measures may be preferable to area measures. A limitation of using previously determined B-mode diameter measurements for the calculation of wall areas is that there may be insufficient information to determine whether the diameters that are available represent either the longest or shortest diameters. Also, the relationships between the B-mode measurements may not accurately reflect the relationship implied in Figure 5A; and so the calculated areas may be inaccurate. Accuracy of the mathematical estimations of the wall areas and the association with clinical disease need to be confirmed. Also, with inward remodelling, the lumen could be narrowed by a smaller increase in wall area than in an artery that has had outward remodelling and so increase risk. To eliminate some of these problems, measures relating IMT to the radius or diameter may need to be considered.
Even if the wall areas can be accurately calculated from B-mode measurements, associations with clinical events may not be strong. First, the proposed measure may not identify patients with vulnerable plaques, which are often the cause of clinical events.55–57 Since arterial segment changes are not uniform,12–14 even carotid artery wall area or volume measures may be a poor surrogate for cardiac and cerebral vessels. However, recent reports suggest that carotid plaque area measurements identified by other mechanisms were good indicators of risk of vascular events.48 The area and/or volume calculations we propose may help overcome some of the measurement errors and other technical limitations of carotid IMT,39 particularly since the formulae are valid for round or elliptical arteries.
In conclusion, known mathematical relationships suggest that clinically occurring arterial diameter enlargement could have an impact on IMT that would be biologically and clinically meaningful, and which should be detectable by B-mode ultrasound. Therefore, diameter measurement should be a routine part of the arterial B-mode ultrasound scans so that diameter and IMT can be considered in determining disease risk. Measures incorporating diameter (normalized for anthropomorphic characteristics) could provide simple and, potentially, better indicators of severity of atherosclerosis. The findings have immediate relevance to studies of plaque progression and regression, indicating the need for evaluation of these relationships in clinical and epidemiological studies.
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Acknowledgments |
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Support for this research was provided by National Heart, Lung, and Blood Institute, grant NHLBI 1R21 HL076833–01.
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Address correspondence to Dr M.L. Eigenbrodt, Department of Epidemiology, UAMS College of Public Health, 4301 W. Markham, Slot 820, Little Rock, Arkansas 72205, USA. e-mail: eigenbrodtmarshal{at}uams.edu
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