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BioMedical Engineering OnLine 2006, 5:42http://www.biomedical-engineering-online.com/content/5/1/42 Figure Influence 15of the wall velocity on boundary layer thickness, friction coefficient and wall shear stressInfluence of the wall velocity on boundary layer thickness, friction coefficient and wall shear stress. The bottomfigure shows the normalised thickness of the boundary layer during in- (left) and outward motion (right) of the wall in thedynamic configuration presented in the third Additional file. Compared to zero wall velocity the boundary layer thickness insystole (ν w = 5.1 mm/s) is increased in the entrance region of the deformation while it is decreased in the outlet region, in diastole(ν w = -6.8 mm/s) the situation is vice versa. The influence on viscous friction and wall shear stress is small and thus the differencein pressure loss over the deformation compared to ν w = 0 is small (δp ≈ 1% of Δp).very complicated, especially secondary fluid motion inthe systolic deceleration phase can cause situations ofreverse flow. The post-stenotic flow is influenced by anumber of factors, including the degree of stenosis, theflow and deformation waveform, and the geometry of theconstriction. The above calculations suggest is that percentdiameter stenosis alone does not adequately characterisethe flow through myocardial bridges, and that geometricand physiological features such as the curvature, extent,and asymmetry of the stenosis, and the shape of the pulsatilewaveform have substantial effects on the haemodynamicconditions.However, we found that the total perfusion to the myocardiumis strongly dependent on the severity and length ofthe muscle bridge. The mean FFR in fixed environment isgenerally smaller than in the dynamic case because thelosses are not persistent during periods of small deformation,so that the pressure distal the bridge recovers duringthis time span. Consequently the pressure drop and flowPage 22 of 25(page number not for citation purposes)
BioMedical Engineering OnLine 2006, 5:42http://www.biomedical-engineering-online.com/content/5/1/42reduction across fixed stenoses are more pronounced thanin dynamic environments.As previously noted [8] found that the pressure proximalto the myocardial bridge was higher than the aortic pressure,and concluded that the disturbance of blood flowand high wall stress proximal to the myocardial bridgewas the main contributor to the development of atherosclerosisin the proximal segment. The observed wall shearstress distributions indicate that the proximal segment ismore susceptible to the development of atherosclerosis,firstly because the pressure is increased there and secondlyfor reasons that the wall shear stress and their oscillationsare maximum in the entrance region of the deformation.In contrast bridged segments are relatively spared becausethe wall shear stress fades towards the end of the deformation.In a series of myocardial bridges it is likely that theintima between the deformations and distal to the myocardialbridge are protected from atherosclerosis becausethe wall shear stress is very low and negative in separatedflow regions.ConclusionWe have presented a method for simulation of unsteadyblood flow in a time dependent vessel geometry using anintegral boundary layer method. The strong interaction ofthe viscous boundary layer and the inviscid core flow proposedby Veldman [58] models the pressure and theextent of the separation region by assuming Falkner-Skanflow profiles. The equations were modified to the flow situationunder consideration. Numerical simulations wereperformed for idealised stenosis geometries with a timedependent, smooth wall contour, but with a physiologicallyrealistic coronary artery flow waveform. The predictedvalues of fractional flow reserve in dynamic lesionsagree well with the clinical findings in [38], however, furtherquantification in more defined geometries isrequired.Regarding the wall shear stresses and the development ofatherosclerosis the findings are consistent with [1], wherethe intima beneath the bridge is protected from atherosclerosis,and the proximal segment is more susceptible tothe development of atherosclerotic lesions.Besides the advantage of computational time taken for thesimulation, the choice of parameters, such as location,length and severity of the lesion are easily determined bycoronary angiography. Due to the assumptions made inthe boundary layer model, the approximation fails for theprediction of reverse flow and flow where the boundarylayers merge, i.e. fully developed flow. Under theseaspects severe deformations cannot be calculated, becausethe boundary layers merge in the deformation. Further,the length of the computational domain is restricted bythe entrance length, which depends on Reynolds number.And finally the pulsatile frequency and the frequency ofwall motion has to be sufficiently low (a few Hz), so thatthe approximation of quasi-stationary evolution of theboundary layers is satisfied.We believe that the parameters and equations in this articleare detailed enough to describe the physiological consequencesalso in a clinical setting, however, this remainsto be confirmed by in vivo studies. The functional consequence,especially for severe systolic compression, is consistentwith clinical findings published in the literature [1-3,6,8,38,51], where myocardial bridging is found to beresponsible for myocardial ischaemia. The comparison ofour findings with the published data from patient studiesin [38,51] supports a potential clinical relevance of oursimulation.Authors' contributionsSB developed the boundary layer model, carried out thesimulations and drafted the manuscript. SM drafted thesection on the clinical situation. AT participated in modeldevelopment in its design and coordination. All authorsread and approved the final manuscript.Additional materialAdditional file 1Animation of the left descending coronary artery assuming Hagen-Poiseuille flow. The animation shows the pressure in the main segmentsof the left coronary arterial tree during one pulsatile cycle. The unit of thecolour bar is kPa, the playback speed is slower by a factor of three thanrealtime.Click here for file[http://www.biomedcentral.com/content/supplementary/1475-925X-5-42-S1.mov]Additional file 2Animation of a double myocardial bridge in the left descending coronaryartery assuming Hagen-Poiseuille flow. As in the first animationthe pressure in the main segments of the left coronary arterial tree areshown over one heart cycle. A series of two myocardial bridges with length8 mm and deformation ζ 0 = 0.25 are located in the mid LAD. The pressuredrop is underestimated by the assumption of Hagen-Poiseuille flow,consequently perfusion to the myocardium remains nearly unaffected. Theunit of the colour bar is kPa, the playback speed is slower by a factor ofthree than realtime.Click here for file[http://www.biomedcentral.com/content/supplementary/1475-925X-5-42-S2.mov]Page 23 of 25(page number not for citation purposes)
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