Nonlinear Wave Propagation in Viscoelastic Tubes: Application to Aortic Rupture*

Abstract
Recent statistical surveys into the causes of automobile Fatalities have shown that traumatic rupture of the aorta followed by immediate exsanguination is responsible for a significant percentage of traffic deaths in the United States. The object of this investigation is to understand a possible mechanism for this failure. A mathematical analysis is presented of the motion of blood in a distensible viscoelastic segment of aorta subjected to a deceleratlve force field. Calculations of axial wall strain and strain-rate indicate that wave propagation resulting in abrupt shock-like transitions along the aortic wall may well account for the transverse ruptures observed. when compared with the limited amount of rupture data presently available. The analytic method and numerical solution by a two-step Lax-Wendroff differencing scheme are sufficientlv general to describe a wide variety of initial and boundary conditions related to blunt impact to the thorax.

Blunt impact to the thorax often results in traumatic rupture of the aorta, leading to immediate exsanguination. Current interest in the mechanisms of this failure is great (Roberts and Beckman, 1970), particularly with regard to vehicular fatilities in which passengers are subjected to high levels of deceleration. An estimated 61-83 per cent of all such aortic ruptures occur in automobile collisions. Mechanical forces acting on the wall of the aorta may derive: from the intra-aortic pressure field which changes significantly during impact. from sudden local stretching of the wall at points of relative fixation, or from the effects of nonlinear wave propagation along the aorta. The dominant mechanism(s) responsible for rupture still constitute a subject of considerable controversy in the medical literature (see references). In the next section, a review is presented of the current hypotheses concerning the mechanisms of aortic rupture. followed by a theoretical analysis of the coupled problem of fluid and wall motions for a straight aortic segment subjected to various levels of acceleration.

In the present work, a simple experimental scheme is described for loading dynamically a fresh aortic segment so as to achieve very nearly a uniaxial state of stress. The results indicate a stress-strain relation of the same exponential character as observed for other soft �tissues such as muscles by Hill (1953), Fung (1970) and numerous other researchers. In particular, it appears that the constitutive relations are definitely dependent on the magnitude of the strain-rate.

Rindfleisch (1893) was among the first to observe pathological rupture of the aorta. proposing that its location is in all cases determined by the degree of attachment of the aortic arch and the pulmonary artery. Often a split occurs between the media and adventitia which results in complete isolation of the aorta in an aneurysmic �sack�. This may then burst outwards, leading to immediate exsanguination. Greendyke (1966) found, in a statistical study. that the common site of rupture was the aortic isthmus. (just distal to the insertion of the ligamentum arteriosum) comprising more than half the cases examined. He affirmed the traditional explanation for the high incidence of ruptures at this location as that due to inertial forces developed during the deceleration. These forces pull on the vessel at its points of fixation, particularly at the arch, where the greatest strain results. Ruptures may also occur in the ascending aorta, proximal to its zone of fixation near the arch. The attachment of the proximal end of the ascending aorta in the heart does not possess the same degree of rigidity. due to the mobility of the heart. Relative mobility of the aorta also exists at the descending thoracic aorta at the diaphragmatic hiatus, and the abdominal aorta. proximal to its bifurcation. The cases of aortic rupture examined by Greendyke in which trauma to the chest was absent caused him to confirm the idea that violent horizontal deceleration alone is adequate to cause rupture. It is noted that rupture was twice as common in occupants who were ejected from the automobile during collision. March and Moore (1957) quoted by Greendyke. estimate that although a vehicle stopping from 30 m.p.h. in a distance of two feet is subjected to a deceleration of 15 p; the passengers. who may stop in inches (relative to the vehicle) may be subjected to I50 g. Head-on collisions may at higher speeds. increase these figures by a factor of X. In some cases. this effect alone is sufficient to rupture the aorta.

Rutherford (1951) reported on four cases of aortic rupture. all occurring in the isthmus region. Since in two of thecases, thisconstituted theonly injury, with no evidence of crushing or flexing of the chest or spine, the mechanism is believed due to a pulling away of the aorta from the well-anchored arch. This produces a tear or rupture immediately distal to the arch. Fidler (1949) observed that spontaneous rupture (due to medio-necrosis) often occurs in the ascending portion of the aorta, and only rarely in the descending portion. Traumatic rupture, on the other hand, may be directed to certain parts of the aorta by virtue of its anatomical attachments. In the age group 2341, the aorta of adult males is of fairly uniform strength. The great majority of traumatic ruptures occur nonetheless at the aortic insertion of the ligamentum fibrosum. The inherent weakness of the aorta at this point does not appear to be a factor of primary importance.

It is not clear therefore, that rupture is due to high internal pressure in all cases. Certainly in aortic rupture due to falls from great heights, the internal pressure in the aortic arch would be lower than normal if the body strikes the ground on the caudal (tail) end. In this case, it is more reasonable to attribute rupture to mechanical strain.

Shennan (1928) affirmed that the normal aorta can withstand any increased blood pressure due exclusively to a strongly acting left ventricle. Failure of the wall most often occurs by degeneration of various elements of the media.

Wilson and Roome (1933) cited by McDonald and Campbell (1945) suggest that injury is most likely at the start of diastole when the aorta is fully distended with blood, whereas Warfield (1933) (quoted by McDonald and Campbell) contends that the important factor is the condition of full inspiration, when the heart is caught between the sternum and the fully inflated lungs. McDonald and Campbell support the theory of Rindfleisch concerning the importance of fixation of the aorta in localizing the rupture area. McKnight et al. (1964) (quoted by Pate et al., 1968) state that it is the aortic arch that is mobile, the descending aorta being attached to the left anterolateral border of the vertebral column. They agree with Rindfleisch only in that points of fixation in general determine the zone of concentrated stresses, but differ on the ways in which fixation is produced.

Strassman (1947) presented the findings of 72 cases of aortic rupture examined in New York City during 1936-1942 from a total of approximately 7000 autopsies. The age distribution was from under 10 to over 80 yr of age. In all cases of spontaneous rupture, the tear started within the media. In all cases of traumatic rupture, however, the aorta was completely severed, making it difficult to determine in which layer the tear had begun, although in some survivors the adventitia remained intact. It would appear that in some cases of traumatic rupture, the tear begins in the intima. Strassman found that the majority of traumatic ruptures occurred at the isthmus where the aorta is narrower and relatively fixed by the ligamentum arteriosum. He concluded that the most likely explanation of aortic rupture in a few cases in which there was no evidence of bone fracture or external injury was the sudden increase in intra-arterial pressure caused by the blunt compression of the aorta against the vertebral column.

Gable and Townsend (1963) found in a study of 459 cases of fatal injuries of the cardiovascular system resulting from accelerative forces, that the aorta and its branches were the most commonly involved of all the major blood vessels. In fact, the aorta is far more susceptible to injury than are the other major blood vessels. They also affirmed the importance of accelerative force in causing cardiovascular injury but were not able to choose definitively between that and the hypothesis of Rindfleisch (rupture due to intravascular pressure). They noted, however, the remarkable concurrence of findings among many researchers that the highest incidence of injury was just distal to the left subclavian artery; i.e. in the region of the ductus, for cases of isolated aortic ruptures. When heart lesions were present, the incidence of rupture above the aortic valve was double that of the ductus region.

Taylor (1962) has demonstrated on pigs that, during acceleration, an emptying of the distal half of the thoracic aorta occurs with engorgement of the upper half and of the arch. This retrograde flow may increase the pressure sufficiently in the region of the arch to cause rupture here.

Lundevall (1964) has suggested that geometric distortion of the aorta in the sagittal plane during deceleration will cause local longitudinal stretching of the aortic wall at the two points of fixation; i.e. at the base of the heart and at the isthmus.

In traffic accidents, the contact of the lower portion of the steering wheel with the abdomen may push the abdominal viscera upwards. The left lung may press upward against the aortic arch, causing increased bending or even kinking of the arch. A transverse rupture results near the isthmus. In addition, the cervical vessels may stretch during head motion, exerting longitudinal forces on the aortic wall. Internal pressure in the aortic arch may rise suddenly, due both to compression of the heart and forward inertial motion of the blood already in the arch.

The equations of motion for the blood and the aortic wall are formulated in the next section. Their numerical solution in the following section then allows one to evaluate the dependence of wall stresses (in terms of strain and strain rate) on the magnitude of acceleration, wall viscoelasticity. and on the geometry of the aortic segment as characterized by its length, taper, and wail thickness. Calculations are made up to acceleration levels of 150 g in order to evaluate some of the current hypotheses described above.

4.1 Mathematical Formulation
One adopts a quasi one-dimensional model (Olsen and Shapiro. 1967; Rudinger, 1970; Lambert, 1958) for the flow of an incompressible fluid in a distensible tube. based on the assumptions that (a) the wave length is long compared to the tube diameter and (b) that the tube is constrained from longitudinal motion. The wall material is assumed to be viscoelastic. but fluid viscosity is neglected since its effect on the flow in the larger arteries is insignificant. Under these conditions the governing equations of motion are . . . Continue to article and footnotes (PDF)

Dr. Richard Collins is the Former University Professor of Biomedical and Human Factors Engineering, Professor of Medicine, Professor of Mechanical and Aerospace Engineering. Dr. Collins has Demonstrated Accuracy in Research and Scientific Analysis in Mechanical and Biomechanical Engineering and Regulatory Assessments. His work has been Extensively Published in National and International peer-reviewed Literature. He is an Internationally recognized engineering and medical expert, having earned his Ph.D. from the University of California, Berkeley. Dr. Collins is qualified as an expert witness in civil and criminal courts.

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