Analytical modeling of the instantaneous maximal transvalvular pressure gradient in aortic stenosis

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1 RTICLE IN PRESS Journal of Biomechanics 39 (26) nalytical modeling of the instantaneous maximal transvalvular ressure gradient in aortic stenosis Damien Garcia a,, Lyes Kadem a, David Savéry b, Philie Pibarot c, Louis-Gilles Durand a a Biomedical Engineering Laboratory, Clinical Research Institute of Montreal, IRCM, Pine West venue, Montreal, QC, Canada H2W R7 b Philis Research US, Healthcare Systems and Information Technology, Briarcliff Manor, NY, US c Research Center of Laval Hosital-Quebec Heart Institute, Laval University, Sainte-Foy, QC, Canada cceted 2 October 25 bstract In resence of aortic stenosis, a jet is roduced downstream of the aortic valve annulus during systole. The vena contracta corresonds to the location where the cross-sectional area of the flow jet is minimal. The maximal transvalvular ressure gradient (TPG max ) is the difference between the static ressure in the left ventricle and that in the vena contracta. TPG max is highly timedeendent over systole and is known to deend uon the transvalvular flow rate, the effective orifice area (EO) of the aortic valve and the cross-sectional area of the left ventricular outflow tract. However, it is still unclear how these arameters modify the TPG max waveform. We thus derived an exlicit analytical model to describe the instantaneous TPG max across the aortic valve during systole. This theoretical model was validated with in vivo exeriments obtained in 9 igs with suravalvular aortic stenosis. Instantaneous TPG max was measured by catheter and its waveform was comared with the one determined from the derived equation. Our results showed a very good concordance between the measured and redicted instantaneous TPG max. Total relative error and mean absolute error were on average % and 2.7. mmhg, resectively. The analytical model roosed and validated in this study rovides new insight into the behaviour of the TPG max and thus of the aortic ressure at the level of vena contracta. Because the static ressure at the coronary inlet is similar to that at the vena contracta, the roosed equation will ermit to further examine the imact of aortic stenosis on coronary blood flow. r 25 Elsevier Ltd. ll rights reserved. Keywords: ortic stenosis; Heart valve; Pressure gradient; nalytical modeling; nimal data; Hemodynamics. Introduction ortic stenosis (S) is the most frequent cause of valvular relacement in develoed countries (Tornos, 2). s the blood flow asses through an S, it forms a jet, which contracts to a minimum cross-sectional area (so-called effective orifice area, EO) at the level of the vena contracta (location 2, Fig. ). The difference between the left ventricular (LV) static ressure (P LV, location, Fig. ) and the static ressure at the vena contracta (P VC, location 2) is the maximal transvalvular Corresonding author. Tel.: ; fax: address: damien.garcia@ircm.qc.ca (D. Garcia). ressure gradient (TPG max ) whereas the difference between P LV and the recovered aortic ressure (P, location 3) is the net transvalvular ressure gradient (TPG net ). Using the exression of instantaneous TPG net, we have develoed a mathematical model (V 3 model) describing the ventricular valvular vascular interaction under various athohysiologic conditions (Garcia et al., 25a). LV failure occurs when demand of LV myocardial oxygen exceeds suly, the latter being tightly couled to the left coronary blood flow (Braunwald et al., 958). The left coronary circulation is essentially determined by LV ressure and by the left coronary inlet ressure (Hoffman and Saan, 99). The vena contracta and the left coronary inlet are located at about the same distance 2-929/$ - see front matter r 25 Elsevier Ltd. ll rights reserved. doi:.6/j.jbiomech.25..3

2 RTICLE IN PRESS D. Garcia et al. / Journal of Biomechanics 39 (26) Fig.. Schema of the flow across an aortic stenosis during systole and corresonding ressure field along the flow axis. Locations, 2 and 3 corresond to the detachment of the flow from the left ventricular outflow tract, the vena contracta and the location where aortic ressure is totally recovered. TPG is the transvalvular ressure gradient and EO is the valvular effective orifice area. P LV ¼ left ventricular ressure, P VC ¼ ressure in the vena contracta, P ¼ aortic ressure, LV ¼ cross-sectional area of the left ventricular outflow tract, ¼ aortic cross-sectional area. downstream of the valve annulus, and it has been shown by Sung et al. (997) that the static ressure does not change significantly from the flow axis to the wall. Thus, static ressure in the left coronary inlet is similar to that in the vena contracta (P VC ). Knowing the exressions of instantaneous TPG max and LV ressure might thus rovide the instantaneous coronary inlet ressure. This would allow us to combine the V 3 model with a lumed model of the left coronary circulation such as the one roosed by Judd and Mates (99) and Mates and Judd (993). Such a comrehensive model will allow studying the imact of S on left coronary blood flow. Therefore, with this forthcoming urose in view, the rimary objective of the resent study was to roose an analytical model of the instantaneous TPG max. In the following sections, TPG max will refer to instantaneous TPG max, unless mentioned otherwise. 2. Methods: mathematical derivation of TPG max 2.. Hyotheses Systole is defined as the eriod where the transvalvular flow rate Q4 (LV ejection eriod). The flow attern across an aortic stenosis is mainly characterized by a contraction as far as the vena contracta, followed by an abrut exansion (Fig. ). Within the exansion, some dynamic ressure is converted to static ressure. This rocess is unstable and generates turbulence, which means that a art of the initial fluid energy is irreversibly lost. On the contrary, no significant turbulence occurs ustream from the vena contracta since the conversion of static to dynamic ressure is stable and is a lowenergy dissiation rocess (Ward-Smith, 98). Thus, we first suose that the fluid is ideal (i.e. incomressible and non viscous) ustream from the vena contracta. Second, we assume that the aortic valve oens and closes instantaneously and that its EO remains constant during systole. Furthermore, we assume that the flow velocity rofile is flat throughout the region of interest (shaded zone located between location and location 3, Fig. ). Finally the cross-sectional area of the LV outflow tract (LVOT, location ) and that of the ascending aorta (location 3) are suosed invariant throughout systole. For simlicity, they are suosed similar and noted Derivation of the TPG max During systole, the blood accelerates between the LVOT (location, Fig. ) and the vena contracta (location 2). No significant energy loss occurs during this convective acceleration (Miller, 996). Neglecting the effects of gravitation, the generalized Bernoulli equation used along a streamline linking location with location 2 yields TPG max : TPG max ¼ P P 2 ¼ Z 2 2 rðv 2 2 V 2 Þþr qv dl, () qt where P, V, and r are the static ressure, the velocity and the density of the fluid, resectively. The coordinate l is the curvilinear coordinate along the streamline. Using the conservation of mass, the transvalvular flow rate Q can be written as Q ¼ V ¼ EO V 2 ¼ ðlþ V ðlþ, where ðlþ is the cross-sectional area occuied by the through-flow at the location l. If we assume that ðlþ is time indeendent, Eq. () becomes: TPG max ¼ 2 rq2 EO 2 2 þ r qq Z 2 qt ðlþ dl. (2) Time integration of Eq. (2) over systole eliminates the time derivative of Q, which leads to the mean TPG max : TPG max ¼ 2 rq2 EO 2 2, (3) where the overline denotes the systolic temoral mean. We now define the arameter l *, which is homogeneous to a length, as follows: Z 2 l ¼ ðlþ dl. (4)

3 338 RTICLE IN PRESS D. Garcia et al. / Journal of Biomechanics 39 (26) To obtain the comlete exression of TPG max, one has to calculate =l. It can be noted that when EO (i.e. ðl ¼ 2Þ), converges towards zero, this term tends towards +N. On the contrary, when the stenosis becomes less and less severe, converging towards a nostenotic case, location 2 tends towards location. Hence, if EO ¼ (no stenosis), =l is zero and TPG max ¼. The ustream flow geometry deends mainly uon EO and (Gurevich, 965; Ward-Smith, 98) so that a dimensional analysis gives the following relation linking =l with EO/: ffiffiffiffi l ¼ f. EO (5) simle tye of functions f defined on [, +N[ that meets the two aforementioned boundary conditions (i.e. f ðxþ!þ when x!þ and f ðþ ¼) is the following: ffiffiffiffi " # b g l ¼ a, (6) EO where a, b and g are three strictly ositive constants to be solved. Consequently, from Eqs. (2), (4) and (6), the exression of TPG max becomes: TPG max ¼ 2 rq2 EO 2 2 þ ra ffiffiffiffi qq qt " # b g. ð7þ EO The constants resent in Eq. (7) were determined analytically as described below Determination of the three constants We note L 23 the recovery length that is the distance searating the vena contracta from the location beyond where static ressure is totally recovered. To determine a, b and g, one will study the behaviour of L 23. ccording to our revious study (see Eq. () in (Garcia et al., 25b)), L 23 is related to =l as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L 23 ¼ 2 EO l (8) and more recisely, it follows from Eq. (6) that: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # L 23 ffiffiffiffi ¼ 2 EO b g a. (9) EO It is straightforward to verify that L 23 is zero if EO ¼. Because L 23 may not be infinite, it is meaningful to ostulate that (L 23 = ) must converge L 23 / 2π π.5 EO / Fig. 2. Normalized recovery length as a function of normalized EO. is the aortic cross-sectional area. towards a strictly ositive constant when /EO tends towards +N, i.e. when the stenosis becomes more and more severe (EO-). This essential ostulate leads to the unique following solution (see endix ): a ¼ 2; b ¼ =2 and g ¼. () Finally, the recovery length can be exressed as L 23 ¼ 2 ffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi! EO þ. () EO Fig. 2 illustrates the dimensionless recovery length (L 23 = ) as a function of the dimensionless EO (EO/). Using Eqs. (7) and (), one finally obtains: TPG max ¼ 2 rq2 EO 2 2 þ 2r qq ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffi. ð2þ qt EO For comarison, TPG net is written as (see (Garcia et al., 25b) for details): TPG net ¼ 2 rq2 EO 2 þ 2r qq rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qt EO. ð3þ

4 The exression of TPG max given by Eq. (2) has been validated in vivo as described hereafter. RTICLE IN PRESS D. Garcia et al. / Journal of Biomechanics 39 (26) Methods: in vivo validation of the TPG max equation 3.. In vivo exeriments The animal rotocol has been reviously described in detail (Garcia et al., 23; Kadem et al., 25). suravalvular S was created in 9 igs using umbilical tae tightened around the aorta 2 cm downstream from the aortic valve annulus. The ressure measurements were erformed using a Millar catheter (customized model, Millar Instruments) with a distal (P ¼ P LV ), intermediary (P 2 ¼ P VC ), and roximal (P 3 ¼ P ) sensor. The P sensor was located cm ustream from the stenosis. The P 2 was ositioned at the level of the vena contracta. The P 3 sensor, located at 4 cm of the intermediary sensor (P 2 ), was used to measure the aortic ressure after recovery. Cardiac outut was measured using an ultrasonic flowmeter (T26, Transonic Systems), with the robe ositioned around the main ulmonary artery. The three ressure signals were simultaneously recorded using a samling frequency of 4 Hz. The systolic trans-stenotic ressure gradients were calculated as follows: TPG max ¼ P 2P 2 ( ¼ P LV P VC ) and TPG net ¼ P 2P 3 ( ¼ P LV P ). Doler echocardiograhic measurements were erformed with a Sonos 55. Doler EO was calculated using the standard continuity equation. The diameter of the ascending aorta was measured at 2 cm downstream of the stenosis by bi-dimensional echocardiograhy. The cross-sectional area was calculated assuming a circular shae. These measurements were obtained under different grades of stenosis severity ( 3 grades er ig). In total, 33 comlete series of acquisitions were obtained. It should be noted that the normal native aortic valve does not affect the flow through the suravalvular stenosis since the valve is widely oen during systole Determination of the instantaneous transvalvular flow rate The instantaneous transvalvular flow rate (Q) was necessary to redict TPG max (see Eq. (2)). However, only the ulmonary flow waveform was measured. The transvalvular flow waveform was therefore determined from the continuous-wave Doler sectrum measured in the trans-stenotic flow jet: the resective enveloes of three sectra were semi-automatically extracted using a Matlab (MathWorks, Inc.) rogramme and their average was used as a substitute of the Q waveform (Fig. 3). This aroach is valid because the geometric area of the stenosis did not significantly change during CW Doler Cardiac outut measured by electromagnetic flowmeter systole. Then, the amlitude of the reconstructed flow signal was adjusted to match the time-averaged ulmonary flow ( ¼ cardiac outut) Comarison between measured and redicted TPG max s shown by Eq. (2), TPG max is deendent uon qq=qt. However, it is difficult to obtain an accurate qq=qt waveform from Doler signal, esecially at the ejection onset. We therefore chose to redict TPG max from Q 2 and TPG net, rather than from Q 2 and qq=qt (as in Eq. (2)). Indeed, by means of Eqs. (2) and (3), TPG max can be rewritten as follows: TPG max ¼ 2 r a b 2 a 2 b Q 2 þ b TPG net b 2 b 8 2 b ¼ = ffiffiffiffiffiffiffiffiffiffiffi EO = ffiffiffiffi >< b 2 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi =EO = where a ¼ =EO 2 = 2 >: a 2 ¼ð=EO =Þ 2 : ð4þ EO was determined by Doler echocardiograhy (see Section 3.) and also by catheter with the use of the mean systolic exression of Eq. (3): s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! EO ¼ þ 2 TPG net. (5) r Q nalysis of the results 36 Time (ms) TPG max was redicted from Eq. (4) with the use of either Doler or catheter EO (Eq. (5)). Total relative errors between the measured and redicted instantaneous TPG max were calculated as: jtpg measured TPG estimated jc jtpg measured j, where jj reresents the vectorial Flow rate (ml/s) 2 Fig. 3. Determination of the transvalvular flow waveform from the enveloe of the continuous-wave (CW) Doler sectrum and cardiac outut measured by electromagnetic flowmeter. See Section 3.2 for details.

5 34 RTICLE IN PRESS D. Garcia et al. / Journal of Biomechanics 39 (26) norm. Mean absolute errors were calculated as: jtpg measured TPG estimated j/n, where N reresents the vector length. Doler and catheter EOs were comared using a linear regression and the Bland ltman method (Bland and ltman, 986). 4. Results Doler EO (cm 2 ) Doler versus catheter EOs The Doler EO ranged from.24 to.6 cm 2 and the catheter EO from.3 to.8 cm 2. The equation of the regression line linking Doler and catheter EOs was y ¼ :97x þ :6 with a coefficient of determination of r 2 ¼ :83 (Fig. 4). The mean (7SD) of the difference (Doler EO catheter EO) was.47. cm 2.No correlation was found between the differences and catheter EO (r 2 o:). Our results thus suggest that there was a very good concordance between Doler and catheter EOs and that their differences behaved randomly Predicted versus measured TPG max Catheter EO (cm 2 ) Fig. 4. Doler EO versus catheter EO (n ¼ 33). The dotted line is the identity line. The solid line is the regression line. Overall there was a very good concordance between measured and redicted TPG max when using catheter EO (Fig. 5), which validates Eq. (4). Total relative error ranged between 3.6% and 26% (Fig. 6) and was on average % (median ¼ 8.6%). Mean absolute 4 EO =.3 cm 2 4 EO =.48 cm 2 TPGmax (mmhg) EO =.85 cm 2 4 EO =.67 cm 2 TPGmax (mmhg) Fig. 5. Predicted (dotted curves) versus measured (solid curves) TPG max. These examles corresond to the four median results in terms of total relative error (white dots in left anel, Fig. 6). Here, total relative error is around 8.5%.

6 RTICLE IN PRESS D. Garcia et al. / Journal of Biomechanics 39 (26) Relative error (%) Mean absolute error (mmhg) Mean TPGmax (mmhg) Mean TPGmax (mmhg) Fig. 6. Total relative errors (left anel) and mean absolute errors (right anel), between redicted and measured TPG max, as a function of the mean systolic measured TPG max. White dots in the left anel corresond to the curves of Fig. 5. error ranged between.7 and 5.9 mmhg (Fig. 6) and was on average 2.7. mmhg (median ¼.9 mmhg). When the mean TPG max was less than 2 mmhg, relative error was greater than %. Nevertheless, in this articular range, the mean absolute error remained around.5 mmhg only (Fig. 6). When TPG max was redicted using Doler EO, relative and absolute errors were 27% (median ¼ 8%) and (median ¼ 4.) mmhg, resectively. Hence, although there was a very good concordance between Doler and catheter EOs, TPG max estimated when using Doler EO did not redict measured TPG max as well as with catheter EO. This suggests that the roosed equations are sensitive to error measurements in EO. 5. Discussion The mean systolic values of TPGs are commonly utilized during clinical examination to evaluate the severity of S, and they can be determined either by cardiac catheterization or by Doler echocardiograhy. Since Doler echocardiograhy uses the velocity measured in the vena contracta, it rovides the mean TPG max as given by the simlified Bernoulli equation resulting from Eq. (3). In contrast, catheterization measures the recovered ressure at some distance from the vena contracta and thus gives the mean TPG net which may be exressed from Eq. (3) as: TPG net ¼ 2 rq2 EO 2. (6) Because the amount of overestimation of Doler gradients comared to catheter gradients may be clinically significant in atients with moderate or severe stenosis and a relatively small aortic cross-sectional area, mean systolic TPG max and TPG net have been widely studied (Baumgartner et al., 999; Voelker et al., 992). However, few investigations were carried out on instantaneous TPG net and, to our knowledge, none on instantaneous TPG max. Recently, we have roosed an exlicit analytic exression for the instantaneous TPG net (Garcia et al., 25b). To ursue this investigation of the transvalvular hemodynamics, we now derived the analytic equation of instantaneous TPG max. Some imortant features and otential clinical imlications of such a model are discussed below. 5.. Generalization of TPG net and TPG max In the resent study, we assumed that LVOT and aortic cross-sectional areas were similar. However, in atients with S, these areas are significantly different: from a database of 26 atients who underwent an echocardiograhic evaluation at the Quebec Heart Institute, LVOT area was on average cm 2 whereas aortic area measured at the sinotubular junction was cm 2. Nonetheless, we showed that the contribution of the LVOT area in TPG net is less than % in most of the atients (Garcia et al., 25b). Moreover, it is well established that the dimension and geometry of the flow downstream of the vena contracta has a minimal influence on the flow characteristics ustream from the vena contracta (Garcia et al., 24; Gurevich, 965; Ward-Smith, 98). Hence TPG max and TPG net can be generalized as follows: TPG max ¼ 2 rq2 EO 2 2 LV þ 2r qq qt ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi, ð7þ EO LV TPG net ¼ 2 rq2 EO 2 þ 2r qq rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qt EO, ð8þ

7 342 RTICLE IN PRESS D. Garcia et al. / Journal of Biomechanics 39 (26) Mild S Severe S 4 4 Flow rate (ml/s) TPGmax TPG (mmhg) 5 TPGnet TPGmax 5 TPGnet Fig. 7. Simulated TPG net and TPG max using the V 3 model for mild (EO ¼.5 cm 2 ) and severe (EO ¼.7 cm 2 ) S. Stroke volume is 7 ml, and heart rate is 7 bm. Note that TPG max is lower than TPG net at the very beginning of ejection. where LV and are the resective cross-sectional areas of the LVOT and the ascending aorta at the sinotubular junction (Fig. ) TPG max is not always maximal! TPG max is called this way because, when averaging throughout systole, it is indeed the maximal transvalvular ressure gradient, as could be easily demonstrated from Eqs. (3) and (6). However, at the very beginning of LV ejection, Q 2 is negligible comared to qq=qt so that TPG max and TPG net are essentially governed by the local inertia. In that case, TPG net (t)4tpg max (t)(fig. 7). Hence, the ressure at the vena contracta (location 2, Fig. ) is therefore higher than the recovered aortic ressure (location 3, Fig. ) attheonsetoflvejection(fig. 8) Recovery length The recovery length (L 23, Fig. 2) can be exressed as a function of the aortic cross-sectional area and EO alone, as shown by Eq. (). In this animal study, aortic area was on average cm 2 and catheter EO ranged from.3 to.2 cm 2. ccording to Eq. (), L 23 therefore ranged from 5.5 to 8 cm aroximately, which is higher than the Pressure (mmhg) 5 5 P inlet.5.5 distance searating P 3 and P 2 sensors (4 cm, see methods) in the catheter used for the exeriments. But, downstream of vena contracta, static ressure increases raidly in the first 5cm(Sung et al., 997). It should also be noted that L 23 is not flow-deendent because we assumed that the flow attern was deendent uon EO and only. This is robably true for high flow rates but should be verified for low flow rates. P P LV Fig. 8. Pressure waveforms obtained with the V 3 model ugraded with Eq. (7) and (9). P LV ¼ left ventricular ressure, P inlet ¼ left coronary inlet ressure ( ¼ P VC ), P ¼ aortic ressure. In this examle, a normotensive condition was simulated and stroke volume ¼ 7 ml, EO ¼ cm 2, LV ¼ 3.5 and ¼ 7cm 2.

8 RTICLE IN PRESS D. Garcia et al. / Journal of Biomechanics 39 (26) Integration of TPG max in the V 3 model and future investigations The resent study comletes the comrehensive descrition of the aortic valve hemodynamics and this may have several imortant clinical imlications. Indeed we have recently develoed and validated a mathematical model (V 3 model) describing the ventricular valvular vascular interaction (Garcia et al., 25a) which allows simulating left heart and vascular hemodynamics under many athohysiologic conditions (hyertension and/or S and/or LV failure). Integrating Eq. (7) of TPG max in the V 3 model will now ermit simulating the static ressure within the vena contracta and consequently the left coronary inlet ressure. Thus, to summarize, left coronary inlet ressure may be written as P inlet ðtþ ¼P LV ðtþ TPG max ðtþ during systole; (9) P inlet ðtþ ¼P ðtþ during diastole: This imlies that TPG max may have a critical effect on the left coronary circulation as reviously observed in atients (Jin, 24; Omran et al., 996; Rajaan et al., 22; Tamborini et al., 996). Combining the V 3 model ugraded with the new TPG max equation (Fig. 8) with the lumed model of left coronary circulation develoed by Judd and Mates (99) and Mates and Judd (993) will ermit to further examine the imact of S on coronary blood flow (CBF) and, more secifically, on coronary flow reserve (CFR). ccording to the Judd s model, the mean CBF is essentially governed by (P inlet 2 P LV). s the severity of S increases, it aears a substantial increase of P LV in combination with a lower ressure zone at the coronary inlet. This tends to significantly diminish systolic CBF which reresents, under normal conditions, 2 3% of total CBF. To comensate for this loss of CBF, autoregulatory coronary vasodilation occurs to maintain an adequate myocardial erfusion to the detriment of the CFR. Thus, the imairment of the systolic CBF may greatly affect the CFR. This has been reviously reorted in atients with S by Hongo et al. (Hongo et al., 993) who found that CFR ositively correlates to the roortion of systolic CBF. Thus, as the severity of S increases, it is very likely that CFR decreases due to the increase of both TPG and P LV. Moreover, because the difference between TPG max and TPG net may be as high as 3 mmhg with severe S (Fig. 7), it is essential to use P inlet, yielded by Eq. (9), and not P when simulating the systolic coronary inlet ressure. 6. Limitations of the study The derivation of the analytical exression of TPG max necessitated the same essential hyotheses used reviously for TPG net (Garcia et al., 25b): () ideal flow ustream from the vena contracta, (2) flat velocity rofiles in the through-flow, (3) fixed flow geometry throughout systole and (4) equality of LVOT and ascending aorta cross-sectional areas. Hyotheses and 2 are commonly alied in some models of flowmetering devices even if some vortices may aear roximal to the vena contracta (Miller, 996; Munson et al., 994; Ward-Smith, 98). Moreover, because the transvalvular flow is not rectilinear nor axisymmetric, irregular velocity rofiles have been reorted both in LVOT and in ascending aorta (Haugen et al., 2; Segadal and Matre, 987). We also ostulated that EO is constant throughout ejection. Whereas this hyothesis is adequate in normal and mildly or moderately stenotic aortic valves, it has been reorted that EO may vary during ejection in atients with severely calcified valves (rsenault et al., 998; Beauchesne et al., 23). Hence, further studies should be necessary to evaluate the accuracy of the model in atients with various degrees of S and/or valve leaflet calcification. cknowledgements This work was suorted by an oerating grant from the Canadian Institutes of Health Research (MOP- 929). The authors thank Lynn tton, Guy Noe l, Justin Robillard, Guy Rossignol, Claudia Jones and Martine Lauzier for their technical assistance, and Christoher Hall for his suggestions. endix One seeks the values of the strictly ositive constants a, b and g so that: L 23 lim ffiffiffiffi ¼ constant4. (.) ð=eoþ!þ We define the variable X ¼ð=EO Þ. Thus, it follows from Eq. (9) that: L 23 = ffiffiffiffi ¼ 2X =2 a½x b ð þ X Þ b Š g. (.2) When X tends towards +N, one obtains: L 23 = ffiffiffiffi 2X =2 aðx b þ bx b Þ g 2X =2 ax bg ð þ bx X b Þ g 2X =2 ax bg ð þ bgx gx b Þ 2X =2 ax bg abgx bg þ agx bg b. ð:3þ Because ax bg þ abgx bg agx bg b ax bg when X tends towards +N, ostulate (.) is true if, and

9 RTICLE IN PRESS 344 D. Garcia et al. / Journal of Biomechanics 39 (26) only if: lim X!þ ð2x =2 ax bg Þ ¼ and lim ð abgx bg þ agx bg b Þ X!þ ¼ constant4 what necessarily yields: a ¼ 2; bg ¼ =2 and lim X!þ ð:4þ ð X =2 þ 2gX =2 =ð2gþ Þ¼constant4 ð:5þ and the unique condition for the limit in (.5) to converge towards a strictly ositive constant is g ¼, such that the corresonding constant is 2 and b ¼ 2. Finally, a ¼ 2; b ¼ =2 andg ¼. (.6) References rsenault, M., Masani, N., Magni, G., Yao, J., Deras, L., Pandian, N., 998. Variation of anatomic valve area during ejection in atients with valvular aortic stenosis evaluated by two-dimensional echocardiograhic lanimetry: comarison with traditional Doler data. Journal of the merican College of Cardiology 32, Baumgartner, H., Stefenelli, T., Niederberger, J., Schima, H., Maurer, G., 999. Overestimation of catheter gradients by Doler ultrasound in atients with aortic stenosis: a redictable manifestation of ressure recovery. 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