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Scholarly Review of Geometry and Compliance: Biomechanical Perspectives on Vascular Injury and Healing

Ho, Paul C.; Melbin, Julius*; Nesto, Richard W.

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The contribution of physical forces to cardiovascular disease has long been a focus of study for engineers, mathematicians, and physicists. The concept of physical factors playing an important role in cardiovascular tissue behavior dates back to the 15th Century when Leonardo Da Vinci 1 postulated that blood flow through heart valves was affected by the geometry of surrounding tissues. Similar notions were shared centuries later by Virchow 2 and Rokitansky 3 who postulated that the nonuniformity of atherosclerotic plaques was a consequence of hemodynamic forces. In recent decades, these early perceptions were confirmed through experimental observations aimed at associating arterial disease and hemodynamic factors. 4–9 Specifically, locations of atherosclerosis were matched to regions of disturbed blood flow, suggesting involvement of local physical factors in the injury of endothelial cells. 10,11 Mechanical cellular injuries such as disruption of the plasma membrane, were observed 12,13 that further allowed for influx of various macromolecules into the cytoplasm 14 and release of secretory products. 15–18 Such initial observations of physical perturbations and vascular tissue response inspired the “reaction to injury” hypothesis of atherogenesis. 10,11

The wealth of information concerning biological aspects of the mechanical-vascular connection has provided great insight at the molecular and genetic levels. In the endothelium, physical stimuli are realized to be capable of inducing conformational changes in the cellular membrane or cytoskeleton, 19,20 which can activate intracellular signal processes. Examples of molecules involved in signal activation include the membrane K+ channel, G proteins, intracellular Ca2+, cAMP, cGMP, inositol trisphosphate, protein kinase C, mitogen-activated protein kinases, GTPases, and protein tyrosine kinases. 21–30 Such signal activation further induces gene expression of various “response” moieties such as upregulation of nitric oxide (a vasodilator), 31,32 cyclooxygenase-2 (an antithrombotic enzyme), 33 thrombomodulin (an anticoagulant activator), 34 tissue plasminogen activator (a thrombolytic agent), 35 Mn superoxide dismutase (an antioxidant), 33 platelet derived growth factor and basic fibroblast growth factor, 36,37c-fos (a proto-oncogene), 38 and downregulation of endothelin-1 (a vasoconstrictor), 39,40 VCAM-1 (an adhesion molecule), 41,42 MCP-1 (a chemoattractant), 43 and others. At the vascular smooth muscle level, an array of molecular and genetic expressions have also been associated with physical stimuli. 44 In addition, cardiovascular calcification has also been linked to mechanical factors. 45–48 Therefore, in the presence of mechanical stress, the vascular response to the perturbed physical environment results in an altered balance of the vessel’s vasoactive, thrombotic, fibrinolytic, coagulatory, inflammatory, and mitogenic states. The culmination is a protective, biological counteraction to the imposed physical stress. In pathologic conditions, this mechanical-vascular balance may be inadequate, leading to disease initiation and progression. 49–51 Implications of these and similar molecular data to clinical disease processes, however, remain largely speculative.

The physical aspect of the mechanical-vascular coupling, unlike the biological, has been less represented in the cardiovascular literature. Mechanical forces alone, irrespective of triggered biological responses, are capable of causing material failure in the circulatory system. Examples are found in prosthetic implant failure involving strut fractures of aortic prosthetic valves, 52 mechanical dilatations of synthetic peripheral bypass grafts, and suture breakage. 53,54 Other natural examples include aneurysm rupture and aortic dissection. Mechanical forces in the arteries, therefore, may act by either triggering biological processes or directly causing material failure. The forces perceived by vascular cells may themselves alter due to changes in the artery’s architecture (compliance and geometry) as a consequence of conditions such as aging, plaque burden, surgical/percutaneous interventions, and changes in cardiac output. Even with stable mechanical conditions, the vessel’s biological responses may vary as a function of age related molecular changes, 55,56 location specific vascular biology, 57 or genetic predisposition. 58 This balance between varying mechanical conditions and vascular biology determines the health or disease state of the vasculature (Figure 1). The necessity of a multidisciplinary approach to cardiovascular diseases necessitates that cardiovascular specialists be familiar with the biomechanical aspects of vascular pathophysiology. In this article, fundamental concepts, clinical relevance, and the potential for manipulating these biomechanical forces to optimize vascular outcomes are considered.

Figure 1
Figure 1:
Schematic diagram demonstrates the effects of structural parameters,i.e., geometry and compliance, on local vascular mechanics. The changes in local mechanical forces, in turn, shape the local vascular response. Influence on vascular mechanics as a result of the interplay between geometry and compliance is highlighted.

Arterial Biomechanical Forces

Mechanical forces in blood vessels are commonly expressed in conventional three-dimensional orthogonal planes, i.e., in radial, axial, and circumferential directions. Fundamental hemodynamic factors such as transmural pressure, local vasoactive tone, fluid shear as related to blood flow, and arterial pulsations interact in the development of a variety of mechanical stresses in the tethered vascular wall. A schematic of biomechanical forces in an arterial wall segment is represented in Figure 2. These hemodynamic factors fashion unique patterns of forces, and their effects on vascular biology and clinical syndromes have been observed.

Figure 2
Figure 2:
Schematic illustration summarizes all of the fundamental mechanical stresses in a representative segment of blood vessel: ςR, radial stress; ςA, axial stress; ςC, circumferential stress; ςS, shear stress; ςp, pulsatile stress; P, intraluminal pressure or transmural pressure, if the extraluminal pressure is assumed to be small; R, radius of the artery; Th, thickness of the artery wall.

Transmural Pressure and Mechanical Stress Modulation.

The transmural pressure (intraluminal minus extravascular) establishes the baseline, global physical environment in the arterial system. The arterial wall must possess sufficient tissue “strength or integrity” to contain the luminal pressure. When luminal conditions exceed the vessel’s ability to compensate within the realm of normal physiology, pathologic vascular behavior leading to clinical syndromes can occur.

Transmural pressure induces stress in the arterial wall according to the Law of Laplace. This indicates an idealized relationship of the intraluminal pressure to wall stress, i.e., (Wall Stress) = (Transmural Pressure) · (Arterial Radius) / 2 · (Wall Thickness). Note that in most cases the extravascular pressure is small and constant; the intraluminal pressure may approximate the transmural pressure. As intraluminal or transmural pressure rises, mechanical stresses are induced in all directions within the arterial wall. Radial compressive stress may occur and is enhanced when the less compliant adventitia prevents the media and intima from expanding, resulting in further compression. 59 Stresses in the axial and circumferential directions, on the other hand, are generally tensile in nature, leading to tissue lengthening. 60 For example, an increase of transmural pressure from 0 to 125 mm Hg was found to cause a passive increase in the axial and circumferential lengths of isolated uterine arteries by 50% and 100%, respectively. 60

The abnormal vascular mechanics associated with elevated transmural pressure or hypertension may result in numerous pathologic manifestations. Clinical syndromes may include atherosclerosis (usually configured uniformly unless augmentation of local forces occurs) 61;smooth muscle hypertrophy in peripheral arteries and arterioles, leading to increase in peripheral vascular resistance 62;degenerative medial changes in the aorta and large elastic arteries attributable to tissue fatigue; and material degradation, resulting from chronic and persistently elevated cyclic stress, 63 in most extreme cases leading to dissecting aortic aneurysms. 64

Arterial mechanics related to elevated transmural pressure provides insights into certain clinical scenarios. In patients with hypertensive diseases and arterial stiffening from other causes as well, there exists a tendency for augmentation of the arterial pressure and enhancement of pulsewave reflections. 62,65 These findings are commonly associated with a higher likelihood of peripheral vascular disease 66 and left ventricular diastolic dysfunction. 67 Arterial branch points are major sites of pulsewave reflections; enhancement of pulsewave reflections infers a mechanism for the observed localization of atherosclerotic disease at branch points and other major sites of reflections, especially in the hypertensive population (see below: Arterial Pulsatility and Dynamic Stress section).

Vasoactive Tone and Mechanical Stress Modulation.

In contrast to passive structural components, vascular smooth muscle cells determine the active tension in the vasculature. The function of vascular smooth muscle is affected by multiple factors, including autonomic nervous inputs, circulating humeral factors, local hemodynamic conditions, and the disease burden of the vessel. Because of the circumferential orientation of smooth muscle cells, cumulative contractions lead to a decrease in the vessel’s luminal diameter while increasing wall thickness and axial length. 68 The result is modification of the wall’s geometry and mechanics in all three of the conventional, spatial planes. From the structural standpoint, reduction in luminal diameter and increase in wall thickness from an active contraction may lead to reduction in local wall stress. The independent contribution to the overall wall stress from the actively contracting myogenic tone may have to be considered.

Vascular smooth muscle can contract solely in response to direct mechanical stimulation. Several early animal models revealed a relationship between transmural loading and vasoactivity. 69,70 In a later experiment with isolated cat mesenteric arterioles, 71 sustained contraction was demonstrated with elevated static intraluminal pressure in the absence of flow. Such vascular behavior is thought to be a function of altered wall stress sensed by the smooth muscle apparatus 71–73 with subsequent myogenic adjustment as entirely a mechanical process independent of flow, neurogenic, and humeral inputs. Veins, as well as arteries, have also been shown to exhibit such behavior. 74 This physiologic feature may explain, in part, the tendency for atherosclerotic disease to spare muscular arteries such as the radial and internal mammary arteries. 75

An important perspective here is that the vasculature embodies the intrinsic capability to counteract an increase in wall stress by altering its geometry. In circumstances where active myogenic tone may be impaired or artificially abolished such as in patients with atherosclerosis, hypertension, or diabetes; in patients status post percutaneous arterial interventions; or at surgical anastomotic sites, the lack of capacity for local stress modification leads to vessel vulnerability and, perhaps, predisposes vascular diseases. This natural geometric reduction of stress may be instructive when planning for future treatment strategies (see below: Geometry Versus Compliance section).

Blood Flow and Shear Stress Modulation.

Unlike orthogonal forces such as those induced by transmural pressure, fluid shear generated by friction or viscous drag due to blood flow acts tangentially upon the vascular wall and endothelium (Figure 2). The measure of fluid shear is dependent upon factors such as local geometry of the vessel, local velocity of blood flow, and the viscosity of blood as largely related to the hematocrit. 76–78 Clinically important shear stresses tend to occur in regions of abrupt changes in the vessel architecture, i.e., geometry and compliance. 79,80 Conversely, vascular architecture and endothelial morphology may themselves be modified by shearing forces. 80,81

Clinically important shear tends to occur at arterial branch points, tortuousities, bends, or acute dilatations where eddy currents or secondary flows can form. 79,80 At locations such as the flow divider of a branch point, the “inner” wall just distal to a bifurcation and at the outer curvature of a bend such as the aortic arch, large amplitude shear appears to promote a local conditioning effect against injury. At these locations, endothelial cells tend to elongate and orient in the direction of shear and atherosclerotic changes are rarely observed. 75,80,82–85 Predilection for disease correlates rather with areas of low, variable, or fluctuating shear, shear gradients, and relative stasis. 79,80 Examples of such locations are the inner curvature of a bend, the outer wall distal to a bifurcation, and the carotid bulb. 75,83–85

The luminal diameter has been noted to enlarge in response to increased flow rates in arteries. In contrast to the myogenic response, however, the sensed parameter is flow, not pressure, and occurs at the endothelial level. In experimental models where the endothelium is removed, fluid shear mediated geometric alterations are abolished. 86 Vascular smooth muscle cells, on the other hand, are physically insulated from fluid shear by the intima (endothelium and elastic lamina), but some of the frictional forces from blood flow may be transmitted to the media by means of focal contacts or myoendothelial junctions, including gap junctions. 87,88 Data on the direct effect of shear on the vascular smooth muscle is limited. In atherectomy related injuries where the intimal lining is removed, the paucity of geometric adjustment and the exposure of vascular smooth muscle to shear become relevant.

Turbulence should be considered as discrete from fluid shear. Turbulence refers to the break-up of simple, organized, or laminar flow at certain physical thresholds as a function of local vessel geometry, flow velocity, and blood viscosity: commonly represented as a dimensionless variable, the Reynolds number (NR), 89 NR= D · v · ρ/μ, where D is the internal diameter of the artery, v is the mean flow velocity, ρ is the fluid density, and μ is the viscosity coefficient. If local conditions allow for the Reynolds number to rise above a certain value, 90 turbulence may arise leading to formation of local eddy currents or secondary flow. A gamut of high, low, or variable shear stresses may result locally. At predisposed sites, such as atherosclerotic stenoses, the combination of a narrowed lumen, accelerated flow (by means of Bernoulli’s effect), and perhaps concomitant anemia may further enhance turbulence provoking disease progression. 91 Other clinical findings generally related to turbulence include cardiac murmurs, 92 vascular bruits, poststenotic dilatations, and aneurysm formation. 93

Arterial Pulsatility and Dynamic Stress Modulation.

In addition to steady forces such as those associated with mean transmural pressure, the arterial wall also experiences cyclic stretches as a result of the pulsatile nature of arterial blood flow (Figure 2). The mechanical effects of pulsatility appears to be ubiquitous in the entire arterial system, from the aorta to the arterioles. During a cardiac cycle, for example, epicardial coronary arteries have been shown to distend 10 to 22%94; carotid, brachial, and femoral arteries, and the aorta approximately 6 to 10%;95–98 and smaller arteries and arterioles < 5%. 44 Another study discovered low frequency components of pulses in capillary beds. 99

Cyclic motion in the cardiovascular system is considered in the development of atherosclerotic disease. For example, deformation of the coronary arteries caused by contraction of the heart, especially at bifurcations, was associated with local atherosclerosis. 100–102 At the molecular level, cyclic stretching of vascular smooth muscle cells of the rabbit aorta was found to stimulate production of matrix substances such as collagen, 103 and to cause morphologic preservation of cytoplasmic integrity and myofilaments. 104 Others considered cyclic motion associated with tethering at a suture line to be a significant contributor to stress at vascular anastomoses. 105,106 Furthermore, cyclic shear has been shown to affect endothelial signaling such as the modulation of platelet derived growth factor production and proto-oncogene expression. 36,38

Even in the absence of bulk vessel wall motion, arterial pulsatility can affect disease processes in a more subtle manner by means of reflection of the pulses. Arterial pulsewave reflections normally occur at sites of arterial geometric and/or mechanical property alterations. 107 These locations include arterial branch points, vessel taper, vasoactive sites, and areas of varying vessel compliance. Physiologic significance of pulse reflections is demonstrated by their directional disparity 108 and diastolic pressure augmentation 109: Reflection of antegrade (from the heart) pulses is less than that of retrograde pulses suggesting a preference for antegrade transmission of pulse energy; pulses propagating retrograde (toward the heart) develop significantly larger reflections and are again mostly reflected peripherally, thus supporting forward blood flow 108 and uncoupling the heart from excessive retrograde returning pulse energy. Furthermore, of the reflected pulsewaves that do return to the ascending aorta (mostly low frequencies), they are timed to occur during diastole such that augmentation of diastolic pressure and coronary flow are achieved without unnecessary elevation of the systolic left ventricular afterload. 110

Unnatural reflections, however, may be deleterious to the health of the heart and arteries. Aberrant reflections may occur in circumstances where the natural arterial architecture is altered as a result of disease processes and/or surgical/percutaneous interventions. 48 Such locations may include arterial stenoses, aneurysms, dissections, calcification, other arterial stiffening processes such as from hypertension and aging, arterial bypass grafting, percutaneous angioplasty, and stent implantation. Both geometry and compliance are commonly altered simultaneously. Local geometric and compliance mismatch can induce untoward mechanical stresses solely derived from pressure pulsations and aberrant reflections. 48 Experimentally, reflection related stresses and gradients were correlated with passive, acellular calcification in vitro, and dystrophic calcification of bovine-pericardial cardiac patches in vivo. 46–48,111 Furthermore, in global arterial stiffening such as that related to hypertension and aging (see below: Arterial Compliance section), pulse velocity may be enhanced and reflected waves to the ascending aorta increased during systole, thus increasing the left ventricular afterload. 61 Vasodilators have been shown to attenuate and delay peripheral wave reflections, and perhaps this process represents a mechanism for their afterload reduction effect. 112,113

Physical Substrates

The architecture or physical parameters of the vasculature serve as “substrates” for static and dynamic biomechanical stresses to act upon, resulting in a new, revised, or altered biomechanical environment that can ultimately be expressed as vascular response (see Figure 1). The degree of change in the physical parameters (geometry and compliance) will gauge the extent of alteration in biomechanical forces. Although perhaps counterintuitive, the interplay between geometry and compliance can also contribute greatly to the final determination of the biomechanical profile of the artery.

Arterial Compliance.

Compliance designates deformability of tissue to an applied force. This ease of tissue deformability or vessel distensibility is defined by volumetric alteration as a consequence of pressure. 76 In naturally occurring disease processes such as atherosclerosis and its sequelae, vascular calcification, or age related arterial changes, local vascular compliance can be significantly altered. Stiffening in the vessel mechanics can also be expected from surgical/percutaneous interventions such as in arterial bypass grafting or vascular stent implantation. Abrupt alterations in vessel compliance may restrict local wall motion from cyclic transmural pressure and arterial pulsatile conditions. The resultant local tethering and stretches stimulate local vascular response. In the presence of a vascular compliance mismatch, aberrant shear stresses and unnatural pulsewave reflections can also contribute to local pathophysiology. 48,111 When extensive alteration of mechanical properties is present, global arterial stiffening occurs with systemic consequences.

Age related arteriosclerosis, hypertension, diabetes, and Marfan’s syndrome have all been implicated in global stiffening of the arteries. Age related arteriosclerosis, which is usually confined to the aorta and central elastic arteries, 114 is thought to relate to a gradual breakdown of elastic (load-bearing) elements. As a result, the bearing of wall stress shifts to collagen, a stiffer wall component. 115 In hypertension and Marfan’s syndrome, the mechanism of arterial stiffness appears to be similar, reminiscent of an accelerated aging process in the arterial wall. 116 These changes tend to spare the peripheral muscular arteries, 114 perhaps ameliorated by vasoactive tone. Rather, accelerated intimal changes and atherosclerosis are observed, 117 perhaps related to greater flow disturbance and shear stress formation from vasoconstriction. In diabetes, accumulation of advance glycosylation end-products in collagen stiffens the arterial wall. 118 In contrast to aging or hypertension, atherosclerosis alone is generally considered not widespread enough to have global or systemic effects as related to left ventricular hypertrophy or widening of pulse pressure 119 (see above: Arterial Pulsatility and Dynamic Stress section). Atherosclerosis, however, causes significant local compliance mismatch, which can lead to plaque rupture. 120,121

The concept of local compliance mismatch has been considered primarily with respect to the problem of peripheral bypass graft failure. Vascular graft patency is found to have a direct relationship between the approximation of graft compliance and that of the host vessel. 122,123 Material designs and surgical techniques have been explored to improve compliance matching, especially at anastomotic junctions; however, under the best circumstance, the occlusion rates remain substantial. 124,125 It is becoming clear that the task of achieving complete compliance matching is formidable. For example, synthetic graft materials are isotropic and have fixed compliances, whereas arteries are anisotropic and become stiffer with increasing pressure (nonlinearity) such that the compliance of a synthetic graft can only match that of the host artery in the circumferential direction within a narrow pressure range. 126 Even suture materials can contribute to anastomotic stiffness. 127

Important flow related disturbances such as low fluid shear stress, variable oscillatory shear, and stagnation 128–130 are also associated with local compliance mismatch. As discussed, these phenomena are linked to platelet aggregation, thrombosis, endothelial injury, intimal hyperplasia, and other vascular responses (see above: Blood Flow and Shear Stress section). Additionally, compliance mismatch can generate unnatural pulse reflections and stress gradients that have been linked to cardiovascular calcific processes 48,111 (see above: Arterial Pulsatility and Dynamic Stress section). Lastly, degenerative forces induced by compliance mismatch have been implicated in suture line breakage, anastomotic rupture 131 and, in extreme cases, rupture of the donor ascending aorta after heart transplantation into a stiff, atherosclerotic, and calcified recipient aorta. 132

Arterial Geometry.

Geometry is also capable of modifying all biomechanical forces in the vasculature that affects wall stress, shear stress, pulse reflections, and cyclic stretches. Abrupt transitions affect local wall motion, local flow pattern, and pulse reflection behavior. As with compliance, arterial geometry can be altered by disease processes or surgical/percutaneous interventions.

The concept of “geometric risk factors” has been suggested as a predictor for atherosclerotic disease. Angiographic studies have demonstrated correlations between vessel tortuosity and aspects of branching (branch angle and area ratio) to subsequent disease development. 133–135 At branch points, geometric factors alone have been associated with local stress concentration. 136,137 At other sites of abrupt geometric changes such as at arterial stenoses and aneurysms, local disturbed flow patterns are frequently observed. 138–141 It is further speculated that these geometry related stress foci and flow patterns are primarily related to local atherogenesis and thrombosis. 141,142 In plaque rupture, structural analysis of atherosclerotic lesions indicated the importance of geometry of the fibrous cap and lipid pool. 121

In percutaneous arterial interventions, geometry has emerged as a dominant predictor of postprocedural restenosis. In the CAVEAT-I and STRESS trials, 143–145 an inverse relationship is found between postprocedural residual luminal diameter and the subsequent rate of restenosis. Along with the BOAT and OARS trials, 146,147 this relationship between the final vessel geometry and restenosis holds true regardless of the interventional device used, further confirming the importance of the geometry effect. Thus, “bigger is better” has become a maxim in interventional cardiology. 148–150

The “bigger is better” paradigm, however, may be inadequate to provide a complete explanation for improved outcomes. If the geometric effect simply provides a larger lumen, the restenotic process would still be expected to progress, but require a longer time before reaching critical renarrowing. Such delayed restenosis has not been observed. Finally, there may be a point of vessel expansion where bigger is no longer better. As discussed above, acute geometric alterations can have serious consequences related to wall stress, flow disturbance, cyclic stretch, and pulsewave reflections. This finding raises skepticism about excessive luminal enlargement such as is occasionally seen with coronary stent oversizing.

Geometry Versus Compliance.

Stress reduction as seen in active myogenic response demonstrates a physiologic solution to an aberrant mechanical environment. By altering its geometry, the vessel is capable of neutralizing increases in wall stress such as those caused by elevation in luminal pressure. This scenario highlights the possibility of manipulating vessel geometry to counteract untoward biomechanical conditions.

The study of vessels with elliptical cross-section, such as the pulmonary trunk, further inspired geometric solutions to counter untoward mechanical stress. As loading of a vessel occurs, the bending moment related to deformation of the elliptical cross-section to a circular shape affords additional mechanical compliance. 151 The concept of such reserved compliance has been used in constructing a theoretical vascular graft capable of reducing mechanical stress. The analytic model of the graft with an elliptical cross-section demonstrated a substantial anastomotic stress reduction compared with conventional circular grafts. 106 In theory, “geometric countermatching” can offset the untoward mechanical effects of significant material compliance mismatch at a vascular anastomotic site.

There is further indication that untoward mechanical effects from unnatural pulse reflections and compliance mismatch may also be countered by means of geometric matching. With respect to directional disparity of wave reflections in a vessel with geometric and compliance mismatch, pulse energy propagation is favored in the direction of smaller to larger internal diameter and stiffer to more compliant vessel wall. 48 At a stiffer, less compliant region of a vessel such as a stented segment of a coronary artery, unfavorable wave reflections arising from local compliance mismatch can theoretically be counterbalanced by appropriate geometric oversizing. This may be the explanation for the successful long-term outcome of a negative residual stenosis after stent deployment. Conversely, in this model, aberrant reflections can also result if excessive geometric overmatching occurs. Analytic models are being formulated to assess optimal geometric counterbalance for the local compliance disturbance imposed by a stent (Figure 3).

Figure 3
Figure 3:
Two-dimensional schematic illustrations represent compliance mismatch induced by intraarterial stent implantation leading to increased local vascular stress formation (A). After deliberate geometric mismatching or oversizing of the stented portion, reduced or compensated vascular stress is achieved (B). The exact amount of geometric oversizing to achieve optimal stress reduction is currently unknown.


Disturbances in the vascular architecture are inevitable sequelae of vascular disease processes. Alterations in the vessel’s local compliance and geometry will eventually induce unfavorable mechanical conditions that will result in vascular responses and pathology. Mechanical interventions can further exacerbate local compliance and/or geometric mismatch. Unless issues of compliance and geometric mismatch are addressed, unfavorable mechanical conditions can persist, despite aggressive drug therapy or repeat mechanical interventions. Optimal vascular biomechanical conditions may be achievable by introducing appropriate counterbalance matching of the vessel’s compliance and geometry.


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