The explanation proposed in this article for the fact that in bridged coronary arteries atherosclerosis develops mainly at the bridge entrance is based on the concept that axial wall stress becomes cyclically excessive at this site, and that this abnormal stress induces wall damages. The underlying postulate that cyclic axial overloads are less well tolerated than the cyclic increases in circumferential stress generated by the normal pressure pulses is based on the fact that the structure of arterial walls is mainly circumferential, and that the axial sections of the vessel are less coupled. The effect of axial wall stress is therefore of much more interest for wall damages than circumferential stress, which is present during the cardiac cycle.
In vessel modeling one has to simplify many things and to make assumptions. The numerical results presented in the preceding section can therefore not be accurate to a few percents. Nevertheless, they clearly indicate that axial wall stress may considerably augment in the segment immediately proximal to the bridge entrance in the course of each cardiac cycle. Due to the variability of the many parameters involved (pressures, flow, bridge morphology, compression strength, etc), the cyclic stress increase exhibits most probably a considerable inter-individual variability; in some bridges, it will perhaps be greater than 100% while in other ones it will be much less. Furthermore, the magnitude of axial wall stress, and of increases of this stress, along a vessel segment also depends on the action of the tethering forces exerted by the surrounding tissues. But the increase in axial stress will be maximal at the bridge entrance because, further upstream, the cyclic axial wall force F is progressively absorbed by the surrounding tissues.
According to Fig. 4, axial wall stress becomes clearly excessive only at high DS values. As a consequence, the proximal segment of bridged arteries in which the lumen of the tunneled segment is not strongly reduced during systole should exhibit less atherosclerosis than the proximal segment of bridged arteries in which the tunneled segment undergoes a strong compression.
The proposed concept of excessive axial stress does not apply to the tunneled segment itself. For this segment, one can make the following considerations. If the tunneled segment is firmly attached to the myocardium, axial forces of hemodynamic origin cannot induce appreciable cyclic variations of its length. If the axial forces generated by the deformation of the myocardium in the region of the bridge are, moreover, also negligible, then axial wall stress remains constant, irrespectively of the actual wall thickness of the segment. Thus, atherosclerotic modifications of the wall, if any, should not be due in this case to excessive variations of axial wall stress. This is independent of the actual value of axial stress, which may be lower, equal, or higher than in the proximal and distal epicardial segments. The actual stress value cannot be predicted; one can presume, at most, that it is comparable to the "normal" axial wall stress of the proximal and distal segments.
If the tethering forces acting axially on the tunneled segment are, on the contrary, negligible, as it is possibly the case when the segment is embedded in a thick layer of fat and thus not firmly attached to the myocardium, then the axial wall stress of the tunneled segment will be greater than in the proximal and distal epicardial segments if the diastolic lumen area of the tunneled segment is equal to (or smaller than) the lumen area of the proximal and distal segments, and the wall thinner (Thereby, constant length of the tunneled segment and wall incompressibility are assumed). But this difference in axial stress will be permanent because it is due to the smaller lumen and/or the smaller wall thickness of the tunneled segment. Since the SMC inside the wall of the tunneled segment are in this case not submitted to axial wall elongations induced by cyclic increases of axial stress, this permanent stress difference has (presumably) no deleterious effects Pathologic modifications that are exceptionally found well inside the tunneled segment [5, 7] may therefore be due to excessive shear stress inside this segment during systole or early diastole, or to a cyclic elongation of this segment due to morphologic changes in the bridge region.
If there is a lumen reduction at the bridge exit during diastole (which does not mean that such cases really exist), and if the tunneled segment is embedded in a thick layer of fat, then a cyclic elongation of the arterial wall of the tunneled segment is possible, particularly just proximal to the bridge exit. This prediction is consistent with the fact that pathologic wall modifications are more frequent when the fat layer around the tunneled vessel segment is thick .
The concept of cyclically excessive axial stress appears to be also consistent with results published by different authors. For instance, Ishikawa et al. studied 108 rabbits fed with a cholesterol diet (ChoR) and 29 control rabbits (ConR) . In the rabbits they used, a part of the LAD is always tunneled. Groups of ChoR were sacrificed at 1 week intervals up to the 20th week, and groups of ConR were sacrificed after 1, 8, and 20 weeks. The last 3 mm segment immediately proximal to the tunneled LAD (called EpiLAD) and the first 3 mm of the tunneled segment (MyoLAD) were examined. The tunneled segment appeared to be still normal in the ChoR and the ConR. The EpiLAD of the ConR were also normal but 1A4 (alpha smooth muscle actin, Dakopatts, Denmark) was found in the cytoplasm of smooth muscle cells of the media. In the EpiLAD of the ChoR, raised lesions grew very rapidly after the 10th week. If one considers that cholesterol played in that study the role of a "marker" of favorable conditions for atherosclerosis, then the results show that such conditions are totally absent in tunneled segments but fulfilled in the EpiLAD of the ChoR, and probably in the EpiLAD of the ConR, too. Since the endothelial cells had different shapes in the MyoLAD and the EpiLAD, Ishigawa attributed the different behavior of MyoLAD and EpiLAD to shear stress differences. However, one can as well come to the conclusion that the observed differences were due to excessive axial wall stress in the arterial segment immediately proximal to the tunneled segment. One can, of course, not exclude that also circumferential stress increased too much during early systole. Distal segments were not examined. Boucek and co-authors found that the metabolism of glycoaminoglycan (GAG) is much higher in the segment immediately proximal to the bridge than in the tunneled segment . This is also an important observation because it shows that increased GAG metabolism is indeed found there where increased axial stresses can be expected. Of note is, moreover, that they attributed this phenomenon to axial stress, which is quite unusual in the literature about atherosclerosis. Their findings are thus in agreement with our concept. The same applies to the results of Polacek who found intima thickening in the segment immediately proximal to the bridge .
A further observation that supports the concept of excessive axial stress is that atheroma at the bridge entrance is more severe when the fat layer between myocardium and tunneled segment is thick . This observation is easily explainable by the fact that in this case the force Ftissues (see Appendix) is weaker.
Like non bridged coronary arteries, bridged ones can be angiographically normal during diastole . This does not prove, however, that they are free of atherosclerosis because uniform intima thickening is seldom detectable angiographically. Inversely, our concept does not exclude that some bridged arteries may be non diseased. This might be the case for instance when there is no great diameter differences between epicardial and tunneled segments during diastole.
As previously mentioned, cyclic increases of axial wall stress may also be due to morphologic changes in the bridge region. The tortuosities observed by Klues et al. and Channer et al. [29, 35] at bridge entries or exits during diastole may be a consequence of such a cyclic axial pulling at the tunneled segment.
The mechanical model used in the present contribution was originally developed for conventional stenoses affecting conductance or distribution arteries . It was shown later to be consistent with published observations about radioactive stents, catheter-based brachytherapy, and conventional stents [36, 37]. The explanation of atherosclerosis in bridged coronary arteries proposed in this article is quite different from the one proposed by Ge and coauthors  who suggested increased circumferential and WSS as probable reasons. It is also different from the ones of Klues and coauthors  and of Bernhard and coauthors  who also incriminated WSS. It must be underlined, however, that our concept cannot invalidate these different explanations (and inversely). In fact, it is quite compatible with these explanations. It is also possible that excessive axial wall stress and WSS have a combined causal action. On the other hand, the concept of excessive axial wall stress provides also an explanation for the fact that the intensity of atherosclerotic developments in the proximal LAD segment is greater when the fat layer between tunneled segment and myocardium is thick  or when the bridge is situated on the upper segment of the LAD [4, 11, 30]. This fact may not be easily explainable by excessive shear or circumferential stresses [15, 29].