- Research
- Open Access
Effect of guidewire on contribution of loss due to momentum change and viscous loss to the translesional pressure drop across coronary artery stenosis: An analytical approach
- Ehsan Rajabi-Jaghargh†^{1},
- Kranthi K Kolli†^{1},
- Lloyd H Back^{2} and
- Rupak K Banerjee^{1}Email author
https://doi.org/10.1186/1475-925X-10-51
© Rajabi-Jaghargh et al; licensee BioMed Central Ltd. 2011
- Received: 16 March 2011
- Accepted: 10 June 2011
- Published: 10 June 2011
Abstract
Background
Guidewire (GW) size and stenosis dimensions are the two major factors affecting the translesional pressure drop. Studying the combined effect of these parameters on the mean pressure drop (Δp) across the stenosis is of high practical importance.
Methods
In this study, time averaged mass and momentum conservation equations are solved analytically to obtain pressure drop-flow, Δp-Q, curves for three different percentage area blockages corresponding to moderate (64%), intermediate (80%), and severe (90%) stenoses. Stenosis is considered to be axisymmetric consisting of three different sections namely converging, throat, and diverging regions. Analytical expressions for pressure drop are obtained for each of these regions separately. Using this approach, effects of lesion length and GW insertion on the mean translesional pressure drop and its component (loss due to momentum change and viscous loss) are analyzed.
Results and Conclusion
It is observed that for a given percent area stenosis (AS), increase in the throat length only increases the viscous loss. However, increase in the severity of stenosis and GW insertion increase both loss due to momentum change and viscous loss. GW insertion has greater contribution to the rise in viscous loss (increase by 2.14 and 2.72 times for 64% and 90% AS, respectively) than loss due to momentum change (1.34% increase for 64% AS and 25% decrease for 90% AS). It also alters the hyperemic pressure drop in moderate (48% increase) to intermediate (30% increase) stenoses significantly. However, in severe stenoses GW insertion has a negligible effect (0.5% increase) on hyperemic translesional pressure drop. It is also observed that pressure drop in a severe stenosis is less sensitive to lesion length variation (4% and 14% increase in Δp for without and with GW, respectively) as compared to intermediate (10% and 30% increase in Δp for without and with GW, respectively) and moderate stenoses (22% and 48% increase in Δp for without and with GW, respectively). Based on the contribution of pressure drop components to the total translesional pressure drop, it is found that viscous losses are dominant in moderate stenoses, while in severe stenoses losses due to momentum changes are significant. It is also shown that this simple analytical solution can provide valuable information regarding interpretation of coronary diagnostic parameters such as fractional flow reserve (FFR).
Keywords
- Throat length
- throat diameter
- guidewire
- hyperemic flow
- basal flow
- viscous loss
- mo-mentum change
- diagnostic parameter
- stenosis
1. Background
Formation of stenosis in coronary arteries is the leading cause of myocardial infarction and death in United States [1], and therefore, accurate assessment of the stenosis severity is crucial to the interventional cardiologists. In interventional cardiology fractional flow reserve (FFR; the ratio of average pressure distal [p _{ d } ] and proximal to stenosis [p _{ a } ] measured at maximal flow (hyperemia)) and coronary flow reserve (CFR; the ratio of blood flow rates at hyperemic to basal condition) are measured to find the functional severity of coronary stenosis [2]. It can be noted that these diagnostic parameters (FFR and CFR) are either ratio of pressure drop or blood flow rate. However, recent studies [3, 4] have proposed that the combined use of translesional pressure drop and blood flow rate can improve the functional assessment of stenosis severity. Accordingly, in the newly proposed diagnostic parameters translesional pressure drop is scaled either linearly or quadratically with flow rate. It may be noted that an appropriate choice of scaling factor can result in non-dimensional diagnostic parameters by including fluid properties (viscosity and density) and geometric information (diameter). Therefore, there is a need to analytically determine the appropriate scaling approach that can be applied to different ranges of stenoses severity and flow rates. An appropriate scaling approach can be determined and put into practice by exploring the pressure drop and its components (viscous losses [linear relation with flow rate] and losses due to momentum changes [quadratic relation with flow rate]) along the stenosis for different flow rates and stenoses severity.
Pressure drop across a stenosis is a function of blood flow rate and lesion anatomy (lesion dimensions and stenosis severity [AS]). In current clinical practice geometric (or anatomic) information can be obtained using bi-planar quantitative coronary angiography (QCA). Furthermore, the functional (hemodynamic) endpoints can be assessed using Doppler flow guidewire (GW) and/or piezoelectric pressure wires [4–6]. This study proposes a quick and inexpensive analytical approach that can potentially utilize the information from QCA and GW to evaluate the translesional pressure drop and thus, diagnostic parameters during the cardiac catheterization procedure.
In the catheterization lab FFR is the current clinical diagnostic gold standard for detecting the severity of stenosis. If FFR falls below 0.75, then it is clinically considered as an ischemic condition and the patient may be treated by coronary intervention (e.g. angioplasty). Brosh et al. [7] studied 63 patients suffering from coronary artery disease and found out that lesion length and in particular stenosis severity have significant impact on the FFR values of intermediate coronary stenoses. The effects of GW and vessel diameter along with percent area stenosis (AS) were also discussed in an in vitro study by De Bruyne et al. [8], where the lesion was modeled as an orifice. Numerical validation of pressure drop measured in in vivo experiments and GW flow obstruction effect was quantified by Banerjee et al. [9, 10].
Pressure drop-flow, Δp-Q, relation in the stenosis region has been studied by many researchers for a wide range of geometries [8, 11–17] and flow rates [18–20]. However, there is not much analytical work, that can be used in clinical practice, to assess the combined effect of throat geometry and GW on transstenotic pressure drop. Thus, the goal of this study is to find the effect of the throat length, AS, and influence of GW on pressure drop and its components (viscous losses and losses due to momentum changes) across the stenosis. Studying the components of pressure drop would allow us to determine their contribution to the total pressure drop. Thus, in turn, will allow better scaling of diagnostic parameters and possibly improved quantification of coronary artery impairment in the cardiac catheterization lab.
2. Method
In this study, mean pressure drop is obtained analytically for moderate (64%), intermediate (80%), and severe (90%) stenoses using the approach proposed by Back et al. [21]. Effects of lesion length, GW insertion, and plaque severity on translesional pressure drop are analyzed. Details on stenosis configuration and mathematical formulation are presented below.
2.1 Geometry
Geometry of stenosis (All dimensions are in mm)
Geometry | d _{ e } = d _{ r } | l _{ c } | d _{ m } | l _{ m } | l _{ m } /d _{ m } | l _{ r } |
---|---|---|---|---|---|---|
3.0 | 6.0 | 1.80 | 1.50 | 0.83 | 1.5 | |
64% | 3.0 | 6.0 | 1.80 | 3.00 | 1.67 | 1.5 |
3.0 | 6.0 | 1.80 | 4.50 | 2.50 | 1.5 | |
3.0 | 6.0 | 1.35 | 0.75 | 0.57 | 1.5 | |
80% | 3.0 | 6.0 | 1.35 | 1.50 | 1.14 | 1.5 |
3.0 | 6.0 | 1.35 | 2.25 | 1.70 | 1.5 | |
3.0 | 6.0 | 0.95 | 0.25 | 0.26 | 1.5 | |
90% | 3.0 | 6.0 | 0.95 | 0.50 | 0.52 | 1.5 |
3.0 | 6.0 | 0.95 | 0.75 | 0.79 | 1.5 |
A coronary artery diameter of 3 mm and a constant converging length (l _{ c } ) of 6 mm is assumed for all AS [20, 21]. Previous studies have shown that pressure drop along the stenosis can be somewhat affected by the stenosis exit angle [23, 24]. However, previous studies by Lipscomb et al. [13] have shown that pressure drop across the stenosis is not affected by varying the stenosis exit angle from 10° to 90°. This analytical approach doesn't consider the effect of exit angle which may have some effect on Δp and thus needs to be assessed in future studies. Constant diverging length (l _{ r } ) of 1.5 mm is assumed for all AS. For a particular AS with throat diameter of d _{ m } , throat lengths (l _{ m } ) are chosen such that the distal pressure remains within the physiological range (> 55 mmHg) [25]. Therefore, to satisfy this criterion, throat lengths for 90%, 80%, and 64% area stenoses are chosen in the ranges of 0.25 to 0.75 mm, 0.75 to 2.25 mm, and 1.5 to 4.5 mm, respectively. Moreover, proximal and distal diameters are assumed to be identical (d _{ e } = d _{ r } ). Assuming a constant l _{ c } and l _{ r } , the total lesion length (L = l _{ c } + l _{ m } + l _{ r } ) varies only with l _{ m } . Thus, it is possible to assess the effect of throat length on translesional pressure drop, with and without GW. Previous studies have used GW and catheters with large diameters (0.66 mm [18, 26] to 1.4 mm [21]), however, in this study a GW diameter (d _{ i } ) of 0.35 mm is considered, which is the most commonly used GW under current clinical practice. Pressure drop is calculated at each section (converging, throat, and diverging) separately. The analytical formulation is summarized in the following section.
2.2 Mathematical Model
The conservation of mass and momentum equations in their integral form are applied to find the relation between pressure drop, flow, and lesion geometry.
Where ū is the average axial velocity during the cycle and is the energy correction factor based on the linear flow theory. The symbol (~) represents the time average of the corresponding parameters which will be dropped from the rest of equations in the paper. Therefore, as described above mean pressure drop along the stenosis is obtained under steady, laminar flow, and Newtonian fluid assumptions.
2.2.1 Pressure Drop Calculations
Stenosis geometry in general is comprised of converging, throat, and diverging sections. In the converging section mean blood flow velocity increases and flow accelerates. Therefore, in this region pressure drops due to both momentum change and viscous loss. As the flow advances towards the throat area, flow momentum varies due to entrance effects, however, in the current analytical approach the induced loss due to momentum change is considered to be negligible when compared to viscous loss. Further as the flow enters the diverging section due to the adverse pressure gradients, flow separates from the wall forming a recirculation zone near to the wall along with a high momentum stenotic jet in the center. The pressure recovery in this section is of the order of the throat's dynamic pressure and is estimated using a pressure recovery coefficient. Inserting the GW shifts the flow maximum velocity pocket towards the GW surface inducing high shear forces. Due to GW blockage effect loss due to momentum change also increases. To sum up, losses in different regions of a stenosis are dominated with either viscous or momentum change or both of them. Therefore, it is of interest to determine the contribution of loss due to momentum change and viscous loss to translesional pressure drop for different area stenoses.
where is shear force integral with H defined as
In these equations r _{ o } varies with axial length (x) as r _{ o } = r _{ e } - x tan(λ), where λ= tan^{-1}((r _{ e } -r _{ m } )/l _{ c } ) is the slope of converging section and x is calculated from the stenosis leading edge as shown in Figure 1. Flow rate is calculated using the following equation: Q = ū _{ e } A _{ E } =ū _{ m } A _{ M } , where ū _{ e } and ū _{ m } are, respectively, the average velocities at proximal and at the throat section of stenosis; while A _{ E } = (A _{ e } - A _{ i } ) and A _{ M } = (A _{ m } - A _{ i } ) represent the arterial cross-sections at corresponding regions, respectively.
where is the increase in flow resistance due to the presence of catheter, and depends only on the ratio of catheter to vessel radius (or diameter). In the absence of a catheter r _{ i } → 0 and F → 1, thus, reducing equation (7) to Poiseuille flow relation.
Equations (9) to (11) can be solved to obtain each component of pressure drop. Integral I _{ s } is solved numerically along the entire converging length using trapezoidal integration method. Pressure drops are calculated for different AS considering three different throat lengths for each stenosis in the absence and presence of GW. The Δp-Q characteristic curves are obtained for each case and the contribution of loss due to momentum change and viscous loss to the total pressure drop are evaluated.
3. Results
Flow is assumed to be steady and laminar. Blood is treated as Newtonian fluid with the viscosity of 3.5 cP and density of 1050 kg/m^{3}. Pressure drop variation with flow rate is obtained for different throat lengths and stenosis severity, considering the effect of GW. Throat lengths for severe (90%), intermediate (80%), and moderate (64%) stenoses vary from 0.25 to 0.75 mm, 0.75 to 2.25 mm, and 1.5 to 4.5 mm, respectively. Flow rate varies from basal to hyperemic in all the figures. Basal flow for all the cases is 50 ml/min, while the hyperemic flow rate is different for each of the cases considered in this study. The physiological cut off value of hyperemic flow rate has an inverse correlation with the flow resistance. That is, severe stenosis with the highest flow resistance, among other plaques, has the lowest hyperemic flow rate. The cut off values for hyperemic flow rates before insertion of GW for 64%, 80%, and 90% area stenoses are 180, 165, and 115 ml/min, respectively. However, in the presence of the GW these values reduce to 173, 150, and 85 ml/min, respectively. The cut off values for hyperemic flow rates in the absence of GW are chosen based on the pre- and post- angioplasty data of Wilson et al. [20], while the hyperemic flow rates in the presence of GW are adopted from the study of Roy et al. [28]. It should be noted that, all percentages presented for the comparison of pressure drops between with and without GW cases are obtained from pressure drops at their corresponding hyperemic flow rates.
3.1 Variation of Pressure Drop (Δp) with Flow Rate (Q)
At basal flow and before insertion of GW for 64% area stenosis (moderate) as shown in Figure 2 pressure drop increases from 0.59 to 0.86 mmHg, a 46% (= [{ (0.86-0.59)}/0.59] × 100) increase as l _{ m } increases from 1.5 to 4.5 mm. In hyperemic condition (180 ml/min), Δp increases from 4.42 to 5.39 mmHg (22% increment) for the same l _{ m } range. In the presence of GW and for same l _{ m } range, pressure drop increases from 1.03 to 1.66 mmHg (61% increase) at basal flow and from 5.92 to 8.08 mmHg (36% increase) at hyperemic flow (173 ml/min). Moreover, at maximum throat length (l _{ m } = 4.5 mm) hyperemic Δp increases by 48% (= [{(8.08-5.45)}/5.45] × 100) due to GW insertion only. This confirms the obstruction effect of GW which has also been shown by Roy et al. [28].
The Δp-Q characteristic for 80% area stenosis (intermediate) where l _{ m } ranges from 0.75 to 2.25 mm is shown in Figure 3. Without GW at basal flow (50 ml/min), Δp increases from 1.76 to 2.20 mmHg (25% increase) as l _{ m } increases from 0.75 to 2.25 mm, while for hyperemic flow (165 ml/min) and same range of l _{ m } , Δp increases from 14.30 to 15.72 mmHg (10% rise). As the GW is inserted at basal flow and hyperemic flow, for the same range of throat lengths, Δp increases from 2.95 to 4.17 mmHg (41% rise) and 16.73 to 20.39 mmHg (22% rise), respectively. In the intermediate stenosis, the difference between Δp for same flow rate but different l _{m} is diminished in both without and with GW insertion when compared to moderate stenosis. Although, there is a relative increase in overall Δp because of GW insertion, its dependency on l _{m} is reduced as compared to 64% AS. As the throat length increases for 64% AS at hyperemic flow rates with and without GW, Δp increases from 22% to 36%, respectively, while these percentages reduce to 10% and 22% for intermediate stenosis. Moreover, percentage pressure drop increase at hyperemic condition due to GW insertion (30%) is less than that for 64% AS (48%). This indicates a decreasing trend in sensitivity of Δp to GW insertion in stenoses with higher severity.
As mentioned before from diagnostic viewpoint, intermediate stenosis is a clinically challenging case. GW diagnostic is widely used to assess lesion severity by measuring FFR under hyperemic condition. Insertion of GW adds extra resistance to flow which results in sharp rise in Δp and consequently a reduction in FFR value. At the maximum l _{m} of 2.25 mm, Δp increases by 30% (= [{20.39-15.72}/15.72] × 100) as the result of GW insertion, which underestimates FFR value. In 80% AS with proximal pressure (p _{a}) of 86 mmHg [28], distal pressure (p _{d}) reduces from 70.5 to 66 mmHg as the GW is inserted. This would change FFR from 0.85 (= 70.5/86) to 0.77 (= 66/86). Thus, in the presence of GW, FFR shows values around the limiting condition of 0.75 which may lead to misdiagnosis of lesion severity. With this regard, the Δp-Q curves can be helpful in interpreting Δp values and consequently the FFR results.
Similar to Figures 2 and 3, Δp-Q curve is also obtained for the severe stenosis (90%) case with throat lengths ranging from 0.25 to 0.75 mm (Figure 4). In the absence of GW with increase in lesion length, Δp shows 11% increase (from 6.90 to 7.48 mmHg) at basal flow (50 ml/min) and 4% rise (from 32.48 to 33.81 mmHg) at hyperemic condition (115 ml/min). In the presence of GW and for the same range of throat lengths pressure drop increases by 20% (from 12.21 to 14.70 mmHg) at basal flow (50 ml/min) and 14% (29.75 to 33.99 mmHg) at hyperemic flow (85 ml/min). Moreover, for the maximum l _{m} of 0.75 mm, hyperemic Δp increases only by 0.5% (= [{33.99-33.81}/33.81] × 100) due to GW insertion. Hence Δp variation in severely stenosed arteries has a weak dependency on GW insertion when compared to that of moderate and intermediate stenoses.
Slope and constant as per Δp = k × Q^{ n }
Area stenosis | Without GW | With GW | Without GW | With GW | Without GW | With GW | |
---|---|---|---|---|---|---|---|
l _{ m } | 1.5 | 3.00 | 4.50 | ||||
64% | n | 1.6191 | 1.4145 | 1.5263 | 1.3168 | 1.4530 | 1.2523 |
k | 0.0001 | 0.0039 | 0.0017 | 0.0076 | 0.0028 | 0.0123 | |
l _{ m } | 0.75 | 1.5 | 2.25 | ||||
80% | n | 1.7276 | 1.5130 | 1.6592 | 1.4206 | 1.5997 | 1.3538 |
k | 0.0020 | 0.0081 | 0.0030 | 0.0142 | 0.0042 | 0.0217 | |
l _{ m } | 0.25 | 0.50 | 0.75 | ||||
90% | n | 1.7653 | 1.5292 | 1.7244 | 1.4478 | 1.6862 | 1.3926 |
k | 0.0070 | 0.0314 | 0.0086 | 0.0477 | 0.0103 | 0.0647 |
3.2 Loss Due to Momentum change and Viscous Loss
It should be noted that inserting GW in all cases increases the viscous loss relatively more than the loss due to momentum change, however, based on the stenosis severity, one of these losses becomes dominant. For example in the moderate stenoses the viscous loss tends to dominate the loss due to momentum change, while this trend is reversed in the severe stenoses. Therefore, insertion of GW decreases n in Δp = kQ^{ n } relationship (see Table 2). Furthermore, it is noteworthy that regardless of stenosis severity, changes in lesion length only affect the viscous loss.
The transitional percent area blockage is defined as the point where loss due to momentum change overcomes viscous loss. For basal flow before insertion of GW and for l _{ m } /d _{m} = 0.25 and 4.0 as shown in Figure 9A, transitional percent area blockages are 64% and 90%, respectively. As the flow rate increases to hyperemic condition these values reduce to 52% and 80%, respectively (Figure 9B). This reduction is due to higher dependency of loss due to momentum change to the second power of flow rate, while viscous loss is linearly related to flow rate. Thus transitional percent area blockages are reduced as flow is increased. Moreover, for l _{ m } /d _{m} = 0.25 and before insertion of GW loss due to momentum change is dominant for moderate to severe stenoses, while for the l _{ m } /d _{m} = 4.0 viscous loss is dominant even for severe stenoses. Furthermore, by insertion of GW (Figures 9C and 9D) viscous loss is dominant at both basal and hyperemic flow rates in the entire range of area stenoses for l _{ m } /d _{m} = 4.0 However, for l _{ m } /d _{m} = 0.25 the transitional percent area blockage are 79% and 67% at basal and hyperemic flow rates, respectively. These values are higher than the corresponding transitional area blockages for l _{ m } /d _{m} = 0.25 without GW. This again shows that GW insertion contributes more to the rise in viscous loss rather than loss due to momentum change.
4. Discussion
The purpose of this study is to acquire a more fundamental understanding regarding the effect of GW insertion, lesion length, and stenosis severity on the total pressure drop across a stenosed coronary artery. Mean pressure drop along the stenosis is obtained under steady flow and Newtonian fluid assumptions. Contribution of the loss due to momentum change and viscous loss to the overall pressure drop along the lesion is also investigated.
Numerical [28], analytical, and corrected analytical pressure drops along with the corresponding FFR values for different stenosis severity at hyperemic flow rate (Q)
AS | l _{ m }(mm) | Q (ml/min) | p _{ a }(mmHg) | Δp_{numerical} | Δp_{analytical} | Δp_{corrected} = 1.16 × Δp_{analytical} | FFR_{numerical} | FFR_{analytical} | FFR_{corrected} |
---|---|---|---|---|---|---|---|---|---|
64% | 3 | 173 | 84 | 9.9 | 7 | 8.13 | 0.88 | 0.92 | 0.9 |
80% | 0.75 | 150 | 86 | 18.9 | 16.74 | 19.42 | 0.78 | 0.81 | 0.77 |
90% | 0.75 | 85 | 89 | 43 | 34.02 | 39.46 | 0.52 | 0.62 | 0.56 |
In addition to steady state assumption, considering blood as a Newtonian fluid is another limitation of this study. Blood viscosity may affect the total pressure drop in the viscous dominated regions. From the results of this work it can be concluded that GW insertion tends to increase the viscous loss more than the loss due to momentum change (see Figures 7 and 8). However, due to high shear rate in the converging and throat sections, non-Newtonian behavior of blood is of minor importance. The adverse pressure gradient results in recirculation zone distal to the plaque where the non-Newtonian behavior of blood becomes important. However, previous studies have shown that Newtonian assumption has lesser influence on flow field in medium to large sized arteries such as coronary artery [31].
In previous studies [10, 32, 33] the occurrence of shear layer instabilities has been observed for intermediate to severe stenoses. Shear layer instability is considered as low Reynolds number turbulence flow phenomenon which can be observed experimentally, but are not easy to detect by numerical computations (may need a refined mesh and higher order numerical schemes). It should be mentioned that in this study the throat Reynolds number (Re _{ m } ) is limited to 734 (where and, ) for the limiting case (i.e. 90% area stenosis without GW at hyperemic flow condition). Therefore, the authors are assuming the flow regime to be laminar. However, although laminar assumption is valid for the cases considered in this study, there are still chances of occurrence of shear layer instabilities in physiological flows and experimental studies due to possible disturbances in the cardiac pulse and irregularities in plaque anatomy. It should be noted that the current analytical approach is based on laminar flow assumption, and thus, cannot account for the shear layer instabilities which is one of the limitations of this work.
5. Conclusion
Translesional pressure drops in stenosed coronary artery with different area blockages with and without GW presence are studied. Pressure drop-flow rate characteristics are obtained analytically for different area blockages (64%, 80%, and 90%) with different throat lengths. Variations in lesion length primarily affect the viscous loss. However, this effect diminishes as the stenosis severity increases from moderate to intermediate stenoses. In the severe stenoses, effect of lesion length is almost negligible.
Similar to lesion length effect, insertion of GW increases the viscous loss significantly. In moderate stenoses, viscous effects in the presence of GW can surpass the loss due to momentum change in the entire range of flow rates. In contrast, for the severe stenoses although the GW increases the viscous effects, the loss due to momentum change is completely dominant in the entire range of flow rates. It is noteworthy that insertion of GW, as compared to without GW case, increases the hyperemic pressure drop in the moderate to intermediate stenoses significantly. However, for the severe stenosis GW insertion has a negligible effect on the hyperemic translesional pressure drop. This finding which is in agreement with the previous study of Verberne et al. [34] might be due to appreciable reduction in flow rate as compared to without GW case. Moreover, insertion of GW increases the dominancy of viscous losses regardless of ste-nosis severity which can be observed with the reducing values of n (comparing without and with GW cases) in Table 2.
Also, total translesional pressure drop can be written in the form of Δp = kQ^{ n } , in which n varies between 1 and 2, which are, respectively, the limits for viscous dominated and momentum dominated losses. Therefore, n can be used to assess the contribution of these two types of losses to the total pressure drop, and accordingly lesions can be cate-gorized into different groups of stenoses. Translesional pressure drop in the newly proposed diagnostic parameters is scaled either by viscous losses [4] or losses due to momentum changes [3]. Thus, evaluating n for specific flow rate and stenosis geometry can provide information on appropriate and accurate scaling approach for the diagnostic parameters. For the moderate stenosis with n values closer to 1, pressure drop may be scaled by viscous losses (or linear function of flow rate). However, for intermediate to severe stenosis with n values closer to 2, losses due to momentum changes (quadratic function of flow rate) can provide a better scaling for translesional pressure drop.
Moreover, pressure drop values obtained using this approach are comparable to the corresponding CFD results published in literature. Results of this approach can be further improved by modifying the current formulation to include a correction factor that can account for the pulsatile nature of coronary flows as well as the non-Newtonian behavior of blood. Also, with further improvements in clinical techniques such as QCA and Doppler flow catheters, this method has a potential to provide a quick evaluation of pressure drop and FFR values under bedside condition in the cardiac catheterization lab.
Nomenclature
A = flow cross-sectional area (m^{2}), , CFR = coronary flow reserve, d _{ e } = proximal vessel diameter (m), d _{ i } = catheter (or guidewire) diameter (m), d _{ m } = throat diameter (m), d _{ o } = mean vessel diameter (m), f= mean wall shear force = 2πrlτ _{ w } (N), F = flow resistance, FFR = fractional flow reserve, I_{s} = shear force integral, l _{ c } = length of converging region (m), l _{ m } = length of throat region (m), l _{ r } = length of diverging region (m), p = mean pressure (mmHg), p _{ a } = pressure proximal to the stenosis (mmHg), p _{ d } = pressure distal to the stenosis (mmHg), p _{ r } = Recovered pressure distal to stenosis (mmHg), Δp = overall mean pressure drop, Δp _{ viscous } = Overall pressure drop due to viscous loss (mmHg), Δp _{ momentum } = Overall pressure drop corresponding to loss due to momentum change (mmHg), Δp _{ cm } = pressure drop across constriction region (mmHg), Δp _{ m } = viscous pressure drop in throat region (mmHg), Δp _{ rm } = pressure recovery in divergent and distal region (mmHg), Q= volume flow rate (ml/min), r = radial distance (m), r _{ o } = mean vessel radius (m), t = time (s), ū = average axial velocity (m/s), x = axial distance (m), β = momentum coefficient, λ = conical half angle of constriction, μ = blood dynamic viscosity (Kg/m-s), υ = blood kinematic viscosity (μ/ρ) (cP), ρ = blood density (Kg/m^{3}), P _{ o } = vessel perimeter (m), τ _{ w } = wall shear stress (N/m^{2}), T = period of cardiac cycle (s), , ,
Subscripts:
e= proximal vessel, i = catheter, m = throat region, o = vessel, r = recovery point, w = wall condition
Superscripts:
(~) = time average over cardiac cycle, (-) = mean flow
Notes
Declarations
Acknowledgements
This work is supported by Grant-in-Aid of Great Rivers Affiliate, National-Scientific Development Grant of American Heart Association, and VA Merit Review Grant (Grant Reference #s: 0755236B, 0335270N, and I01CX000342-01, respectively). Authors would like to acknowledge the initial help of Koustubh Ashtekar who was unable to continue the study because of other commitments. The authors have corrected and significantly improved the initial results into its final form.
Authors’ Affiliations
References
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