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BioMedical Engineering OnLine 2006, 5:42http://www.biomedical-engineering-online.com/content/5/1/42 Mean Figure fractional 13 flow reserve in dependence on degree of deformation (left) and segment length (right)Mean fractional flow reserve in dependence on degree of deformation (left) and segment length (right). Themean pressure derived fractional flow reserve of the baseline environment shown in Figure 8 is plotted as a function of deformation(left) defined in equation (3) and versus the segment length (right). The computations via the boundary layer theory(BLT) are compared to Hagen-Poiseuille (HP) flow profile (γ = 2). It is seen that the mean pressure derived fractional flowreserve is overestimated under the assumption of fully developed flow. Generally the mean FFR is larger in the dynamic case(dashed lines), because the pressure recovers during the relaxation phase. The segment length of three dynamic lesions at adeformation of ζ 0 = 0.7, 0.5 and 0.3 was varied in a physiological range between 5 mm and 40 mm.In Figure 13 we have shown the coronary pressure-derivedfractional flow reserve for the baseline case as a measureof coronary stenosis severity (left) and in dependence ondeformation length (right). In both plots we have usedbaseline inflow conditions with either fixed (solid lines)and dynamic walls (dashed lines). Initially we found thatin developing flow conditions the FFR is overestimated bythe assumption of Hagen-Poiseuille flow (HP), and, asexpected, that the mean FFR in dynamic lesions is muchlarger than in fixed stenotic environment, because the distalpressure recovers during relaxation phase. The graph atthe right shows that the mean FFR depends essentially linearlyon deformation length for different severities, ζ 0 =0.7, 0.5, 0.3, suggesting that the losses are mainly viscousand that the non-linear term plays a minor role forobserved deformations.The values indicate that the FFR depends on the degreeand length of the stenosis. As mentioned earlier the lossesin fixed environments are more pronounced. Further, wepoint out that series stenoses separated by more than thelength of the stenosis drop below the cut-off value of 0.75markedly earlier than single stenosis with the same degreeand extent.Pressure-flow relationThe pressure-flow relations for different stenotic environmentsin the coronary arteries are shown in Figure 14. Wehave applied a stationary flow rate q LMCA at the entrance ofthe LMCA in the range between 4.5 cm 3 /s and 9 cm 3 /s. Theresultant mean flow in the myocardial bridge, q MB wasbetween 1.57 cm 3 /s and 3.13 cm 3 /s. The pressure-flow relationsin this range are essentially linear and extrapolate tothe origin at q LMCA = 0 and Δp = 0. This is consistent withthe assumption of long waves in short tubes, where thepressure-drop is linearily dependent on the flow. The singledeformation with length 12 mm is denoted by (S 12)and with length 24 mm by (S 24). We further show a doubledeformation with length 12 mm (D 12). In general itis seen that the pressure drop increases with severity andlength of the deformation, however the differencebetween S 24 and D 12, which have essentially the samedeformation length, result from inlet and outlet effects ofthe double environment.Influence of wall velocityIn contrast to fixed stenoses, the velocity of the wall ν w ≠ 0in a dynamic environment, i.e. positive during compressionand negative while the vessel is relaxing. At the bottomof Figure 15 we have shown the thickness of theboundary layer during inward (left) and outward motion(right) of the wall compared to fixed deformations. Thesituation depicts two different states where the influenceis close to its maximum. Compared to fixed deformations(ν w = 0) the boundary layer thickness in systole isincreased in the entrance region of the deformation, whilePage 20 of 25(page number not for citation purposes)
BioMedical Engineering OnLine 2006, 5:42http://www.biomedical-engineering-online.com/content/5/1/42 Pressure Figure 14 drop vs. volume flowPressure drop vs. volume flow. The pressure-drop flow characteristics of stenoses with different degree and extent whereobtained under stationary flow conditions. We compare two single dynamic stenosis of length 12 mm (S12) and 24 mm (S24)with a double stenosis with length of 12 mm each (D12). Further D12 is compared for three severities ζ 0 = 0.25, 0.5 and 0.75.The relations are essentially linear, extrapolation to lower inflow values leads to the origin.it is decreased in the outlet region, and the situation is viceversa in diastole. In fact this is due to an additional pressuregradient, which causes fluid acceleration or decelerationdepending on axial position in the constriction andwhether the wall moves inwards or outwards. For exampleduring inwards motion of the wall the fluid is deceleratedat the entrance and accelerated at the outlet of the deformation,while the situation is opposite if the vessel wallmoves outwards. Due to the symmetry of the indentationthere is no additional acceleration or deceleration in thecentre of the deformation. The influence of the wall velocityonto the boundary layer properties is more pronouncedif V W /V is reasonably large, i.e. if thedeformation that the axial flow experiences during passageof the constriction is comparable to the radius of thetube. The influence of wall velocity on viscous friction andwall shear stress is small and thus the difference in pressureloss over the deformation with ν w ≠ 0 compared ν w =0 is small (δp ≈ 1% of Δp).DiscussionThe results have demonstrated that the formation anddevelopment of flow separation and reattachment in thepost-stenotic region of a time dependent constriction arePage 21 of 25(page number not for citation purposes)
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