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`
` on November 14, 2014
`
`J. R. Soc. Interface (2012) 9, 1325–1338
`doi:10.1098/rsif.2011.0605
`Published online 23 November 2011
`
`A validated predictive model of
`coronary fractional flow reserve
`
`Yunlong Huo1, Mark Svendsen2, Jenny Susana Choy1, Z.-D. Zhang1
`and Ghassan S. Kassab1,*
`
`1Department of Biomedical Engineering, Surgery, and Cellular and Integrative Physiology,
`Indiana University Purdue University Indianapolis (IUPUI ), Indianapolis, IN 46202, USA
`2Weldon School of Biomedical Engineering, Purdue University, West Lafayette,
`IN 47907, USA
`
`is
`Myocardial fractional flow reserve (FFR), an important index of coronary stenosis,
`measured by a pressure sensor guidewire. The determination of FFR, only based on the
`dimensions (lumen diameters and length) of stenosis and hyperaemic coronary flow with
`no other ad hoc parameters, is currently not possible. We propose an analytical model derived
`from conservation of energy, which considers various energy losses along the length of a
`stenosis, i.e. convective and diffusive energy losses as well as energy loss due to sudden
`constriction and expansion in lumen area. In vitro (constrictions were created in isolated
`arteries using symmetric and asymmetric tubes as well as an inflatable occluder cuff) and
`in vivo (constrictions were induced in coronary arteries of eight swine by an occluder cuff)
`experiments were used to validate the proposed analytical model. The proposed model
`agreed well with the experimental measurements. A least-squares fit showed a linear relation
`as (Dp or FFR)experiment ¼ a(Dp or FFR)theory þ b, where a and b were 1.08 and 21.15 mmHg
`(r2 ¼ 0.99) for in vitro Dp, 0.96 and 1.79 mmHg (r2 ¼ 0.75) for in vivo Dp, and 0.85 and 0.1
`(r2 ¼ 0.7) for FFR. Flow pulsatility and stenosis shape (e.g. eccentricity, exit angle
`divergence, etc.) had a negligible effect on myocardial FFR, while the entrance effect in a
`coronary stenosis was found to contribute significantly to the pressure drop. We present a
`physics-based experimentally validated analytical model of coronary stenosis, which allows
`prediction of FFR based on stenosis dimensions and hyperaemic coronary flow with no
`empirical parameters.
`
`Keywords: fractional flow reserve; lesion; coronary artery disease;
`Bernoulli’s equation; model
`
`1. INTRODUCTION
`
`Myocardial fractional flow reserve (FFR), the ratio of
`distal to proximal pressure of a lesion under hyperae-
`mic conditions [1], is an important index of coronary
`stenosis because it has lower variability and higher
`reproducibility than coronary flow reserve (CFR) and
`hyperaemic stenosis resistance (HSR) [2,3]. A recent
`landmark study showed a clear benefit of FFR in guid-
`ing percutaneous coronary intervention for better
`clinical outcome [4]. The current method for the
`measurement of FFR requires the use of a pressure
`wire inserted through the stenosis [2,3]. Although
`recent advancements in sensor guidewire technology
`allow simultaneous measurement of distal pressure
`and flow velocity, there are still high variability and
`instability of flow velocity, and occasional signal shift
`for pressure and guidewire obstruction of flow [2,3].
`The placement of a pressure wire near a stenosis can
`also lead to overestimation of FFR. To avoid these pro-
`cedural shortcomings and the expense of pressure wire,
`
`*Author for correspondence (gkassab@iupui.edu).
`
`Received 7 September 2011
`Accepted 3 November 2011
`
`a non-invasive predictive validated model of FFR would
`be very valuable.
`Although angiographically based methods for the
`measurement of coronary flow [5,6] and lesion dimen-
`sion [7,8] have been well established, there is still a
`lack of fundamental theory that can determine the
`pressure drop (Dp) and hence FFR, based on the dimen-
`sion of lesion (i.e. the cross-sectional area-CSA along
`the lesion and the length of lesion) and hyperaemic cor-
`onary flow with no empirical parameters. The objective
`of this study is to introduce such a predictive analytical
`model and to validate it using in vitro and in vivo
`experiments and finite-element (FE) method.
`In vessel segments without a stenosis, the pressure–
`flow curve is nearly linear in the physiological pressure
`range during maximal vasodilation [9]. The linear
`pressure–flow relation is altered when stenosis is pre-
`sent [10]. Young and co-workers [11–14] showed a
`quadratic relation between pressure gradient and flow
`rate as Dp ¼ AQ þ BQ2, where A and B were empirical
`parameters determined through a curve fit of exper-
`imental data. Although the quadratic relation was
`experimentally validated for coronary stenosis [10],
`
`1325
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`1326 A validated FFR model Y. Huo et al.
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`da;
`
`1 a
`
`pmLstenosis
`4rQ
`
`
`
`2
`
`)
`
`1 a
`
`
`
`(
`
`
`ð
`
`0
`
`and
`
`expansion ¼ rQ2
`
`DPa0:05
`2
`
`þ 2
`
`
`
`stenosis, is well-bound and follows the streamlines, the
`energy loss due to a sudden constriction is relatively
`small (loss coefficient  0.1 generally) and negligible
`such that DPconstriction ¼ 0.
`Although DPdiffusive is generally caused by the vis-
`cosity in the fully developed region (i.e. viscous energy
`loss in this case), the pressure drop serves both to accel-
`erate the flow and to overcome viscous drag in the
`entrance region of a stenosis, which contributes to the
`diffusive energy loss. For the entrance region of stenosis,
`we define a dimensionless radius of inviscid core (a), in
`which the flow velocity is uniform such as a¼ r at the
`inlet, 0 , a, r from the inlet to the fully developed
`region and a¼ 0 at the fully developed region, as
`ð
`shown in figure 6a–c, respectively. The dimensionless
`radius of inviscid core (a) is calculated from
`ð1 aÞð6 þ aÞð1 þ 4aþ 9a2 þ 4a3Þ
`¼ 1
`5að3 þ 2aÞð3 þ 2aþ a2Þ2
`4
`where Lstenosis is the length of stenosis [17]. Lvessel is the
`length of vessel, which is composed of both normal
`vessel and stenosis. If a 0.05 (most common for cor-
`ð
`onary artery dimensions and lesion lengths), DPdiffusive
`and DPexpansion are expressed as
`ð1 þ 4aþ 9a2 þ 4a3Þ
`rQ2
`96
`diffusive ¼
`DPa0:05
`að3 þ 2aÞð3 þ 2aþ a2Þ2 da
`2CSA2
`5
`stenosis
`LvesselLstenosis
`8pm
`þ
`CSA2 Q dx
`
`1
`CSAstenosis
`
`
`
`1
`CSAdistal
`
`
`
`#
`
`
`
`1
`CSAdistal
`1
`CSAdistal
`2
`
`
`
`1
`CSAstenosis
`1
` 1
`CSAstenosis
`3
`
`
`
`1
`CSAstenosis
`
`
`
`1
`CSAdistal
`
`ð1 aÞ2
`
`:
`
`If a, 0.05, the entire stenosis is divided into entrance
`and fully developed regions, and the entrance length
`Lentrance is obtained from
`ð1aÞð6þaÞð1þ 4aþ 9a2 þ 4a3Þ
`¼ 1
`5að3þ 2aÞð3þ 2aþa2Þ2
`4
`ð
`
`da;
`
`1 0
`
`:05
`
`ð
`
`ð1þ4aþ9a2þ4a3Þ
`að3þ2aÞð3þ2aþa2Þ2 da
`
`8pm
`CSA2 Q dx
`
`1
`
`CSAstenosis
`1
`1
`CSAstenosis
`3
`
`1 0
`
`:05
`
`rQ2
`96
`2CSA2
`5
`stenosis
`LvesselLentrance
`
`ð
`
`þ
`
`0
`
`
`
`
`expansion ¼rQ2
`and DPa,0:05
`
`
`
`1
`CSAdistal
`1
`CSAdistal
`
`:
`
`pmLentrance
`4rQ
`
`such that
`diffusive ¼
`DPa,0:05
`
`the empirical parameters (A and B) are not known a
`priori [15]. Hence, there is a need for a model of Dp or
`FFR that does not contain any empirical parameters,
`but only depends on measurable quantities of coronary
`artery lesion.
`Here, we present an analytical model derived from
`the
`general Bernoulli
`equation (conservation of
`energy), which considers various energy losses along
`the length of a lesion. The input model variables are
`lesion lumen CSA (the proximal, distal and minimal
`CSA along the lesion), and the length of lesion and
`hyperaemic volumetric flow rate through the lesion.
`There are no empirical parameters in the analytical
`model unlike previous models [11–14]. The in vitro
`and in vivo experiments (swine) as well as a Galerkin
`FE model were used to validate the proposed analytical
`model. The significance, limitations and implications of
`the validated model are contemplated.
`
`2. METHODS
`
`2.1. Analytical model
`
`;
`
`ð2:2Þ
`
`Myocardial FFR is a functional parameter of stenosis
`severity. FFR during hyperaemic flow is expressed as
`FFR ¼ Pdistal Pv
`ð2:1Þ
`Pa Pv
`;
`where Pa is the mean aortic pressure (Pa  Pproximal
`assuming a proximal lesion or no significant diffuse cor-
`onary artery disease in a distal lesion); Pv is the central
`venous pressure; Pproximal and Pdistal are the hyperaemic
`coronary pressure proximal and distal to stenosis,
`respectively [1].
`If the central venous pressure is
`assumed to be negligible, equation (2.1) is generally
`approximated as
`¼ Pa Dp
`FFR ¼ Pdistal
`Pa
`Pa
`where Dp is the pressure gradient along the axis of vessel
`segment from proximal to distal position of stenosis.
`Here, we propose a model to determine Dp speci-
`fically for
`the
`coronary arteries
`(see details
`in
`appendix A). Briefly, since gravity is negligible in the
`coronary circulation [16], the general Bernoulli equation
`can be written as
`DP ¼ DPconvective þ DPconstriction þ DPdiffusive
`ð2:3Þ
`þ DPexpansion;
`DPconvective, DPconstriction, DPdiffusive and DPexpansion are
`energy losses due to flow convection, sudden constric-
`tion in CSA from proximal normal vessel to stenosis,
`flow diffusion and sudden expansion in CSA from steno-
`sis to distal normal vessel, respectively.
`DPconvective ¼ rQ2
`2
`
`
`
`!
`
`1
`CSA2
`outlet
`
`
`
`1
`CSA2
`inlet
`
`;
`
`where CSAinlet and CSAoutlet are the inlet and outlet
`cross-sectional areas, respectively; Q is the hyperaemic
`flow rate in a vessel segment; r is the density of blood.
`If the flow transition, from proximal normal vessel to
`
`J. R. Soc. Interface (2012)
`
`CATHWORKS EXHIBIT 1011
`Page 1326 of 1338
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`

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`
` on November 14, 2014
`
`A validated predictive model Y. Huo et al. 1327
`
`Table 1. Geometrical parameters and flow rates in blood vessel and stenosis of in vitro experiments.
`
`stenosis set-ups
`
`symmetric long tubing
`symmetric tubings
`asymmetric tubings
`occluder cuff
`
`stenotic
`diameters (mm)
`
`2.1
`0.85, 1.2 or 1.7
`0.85 or 1.3
`1.25, 1.65 or 2
`
`stenotic
`lengths (mm)
`
`normal vessel
`diameters (mm)
`
`flow rates
`(ml s – 1)
`
`110
`10–30
`8–22
`5
`
`3.7
`3.5–5
`
`1.3, 1.7 or 2.1
`1.1, 1.7, 2.3 or 2.7
`
`Here, the entrance effect plus the viscosity (Poiseuille
`formula in the fully developed region) leads to the diffu-
`sive energy loss. We also account for the energy loss due
`to a sudden expansion in CSA, based on the outlet flow
`pattern that represents the growth of the boundary
`layer from the inlet of stenosis to the outlet. Equation
`(2.3) was combined with equation (2.2) to determine
`FFR from the stenosis geometry and hyperaemic flow.
`We also compared the present model (equation (2.3))
`with other models that do not consider the entrance
`effect [13,14,18]:
`¼ rQ2
`1
`1
`
`CSA2
`CSA2
`2
`inlet
`outlet
`diffusive þ DPno entranceexpansion :
`þ DPno entrance
`ð
`If the entrance effect is omitted,
`diffusive ¼
`DPno entrance
`
`
`
`
`!
`
`DPno entrance
`theory
`
`
`
`ð2:4Þ
`
`Lvessel
`
`2
`
`
`
`1
`CSAdistal
`
`
`
`1
`CSAstenosis
`(uniform outlet velocity);
`expansion ¼ 1:52rQ2
`1
`1
`
`DPnoentrance
`CSAdistal
`CSAstenosis
`2
`ðthe product of a constant and uniform outlet velocityÞ
`
`
`½13Š or
`
`expansion ¼ rQ2
`DPnoentrance
`2
`1
` 1
`CSAstenosis
`3
`
`respectively. The viscosity (m) and density (r) of the
`solution were selected as 4.5 cp and 1.06 g cm – 3,
`respectively, to mimic blood flow with a haematocrit
`of about 45 per cent in medium size arteries. Various
`dimensions of stenosis (area stenosis varying from 20
`to 90% and length of stenosis from 0.5 to 2 cm) and
`flow velocities (5–50 cm s – 1) were used to calculate
`the pressure drop.
`In vitro and in vivo experiments were used to vali-
`date the analytical model of pressure drop (equation
`(2.3)) and FFR (equation (2.2)). The dimensions of
`the stenosis and flow rates for in vitro and in vivo exper-
`iments are listed in tables 1 and 2, respectively. The
`prediction of other models (DPno entrance
`in equation
`diffusive
`(2.4)) was also compared with the measurements.
`Studies were performed on eight domestic swine
`weighing 60–70 kg.
`Surgical anaesthesia was induced with TKX
`(Telazol 500 mg, ketamine 250 mg, xylazine 250 mg)
`and maintained with 2 per cent isoflurane [21]. The
`animal was intubated and ventilated with room air
`and oxygen by a respiratory pump. A side branch from
`the left jugular vein was dissected and cannulated with
`a 7 Fr sheath for administration of drugs (e.g. heparin,
`lidocaine,
`levophed and saline as needed). The right
`femoral artery was cannulated with a 7 Fr sheath and
`then a guide catheter was inserted to measure the aortic
`blood pressure using a transducer (Summit Disposable
`Pressure Transducer, Baxter Healthcare; error of +2%
`at full scale).
`For the in vitro experiments, carotid arteries were
`dissected and isolated, and small side branches were
`ligated by suture. Several tubings (concentric and
`eccentric) and an inflatable occluder cuff (OC4, In
`Vivo Metric) were used to create various stenoses
`(figure 1a,b). In one in vitro set-up, various sizes of
`concentric and eccentric tubings were inserted into
`carotid artery and ligated against the vessel wall to
`form symmetric and asymmetric stenoses, as shown in
`figure 1a. Table 1 shows the geometry and flow rate
`in carotid arteries and tubings. The stenosis eccen-
`tricity ranged from zero to 0.8 (defined as Daxis/
`Rproximal, where Daxis
`is the distance of centrelines
`between stenosis and proximal vessel segment and
`Rproximal is the radius of the proximal vessel segment
`to stenosis).
`In another in vitro set-up, an arterial occluder
`was mounted around the carotid artery to create ste-
`noses of different degrees (as shown in figure 1b). The
`occluder cuff has an inner diameter and length of
`4 mm and 5 mm, respectively, which can induce zero
`
`8pm
`CSA2 Q dx:
`0
`
`The energy loss due to a sudden expansion can be mod-
`elled as:
`expansion ¼ rQ2
`DPnoentrance
`2
`
`
`
`2
`
`
`
`
`
`1
`1
`CSAstenosis
`CSAdistal
`1
`ðparabolic outlet velocityÞ:
`CSAdistal
`
`
`
`2.2. Validation
`
`A Galerkin FE model was compared with the analytical
`model of equation (2.3). The Navier–Stokes equations
`were solved for velocity and pressure subject to a non-
`slip wall boundary condition and a stress-free outlet
`boundary condition [19,20]. The axisymmetric control
`volume was a rectangle of vessel radius  vessel length
`(0.15  6 cm). After a mesh-independent test, the con-
`trol volume was divided into 20  200 quadrilateral
`and hybrid FEs for healthy and stenotic vessels,
`
`J. R. Soc. Interface (2012)
`
`CATHWORKS EXHIBIT 1011
`Page 1327 of 1338
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`1328 A validated FFR model Y. Huo et al.
`
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`
`Table 2. Geometrical parameters and flow rates in blood vessel and stenosis of in vivo experiments. The stenotic length
`is 5 mm.
`
`coronary stenoses
`
`LAD stenosis 1
`LAD stenosis 2
`LAD stenosis 3
`LAD stenosis 4
`LAD stenosis 5
`LAD stenosis 6
`LAD stenosis 7
`LAD stenosis 8
`LAD stenosis 9
`RCA stenosis 1
`RCA stenosis 2
`RCA stenosis 3
`RCA stenosis 4
`RCA stenosis 5
`RCA stenosis 6
`RCA stenosis 7
`RCA stenosis 8
`RCA stenosis 9
`RCA stenosis 10
`RCA stenosis 11
`
`stenotic
`diameters (mm)
`
`normal vessel
`diameters (mm)
`
`flow rates
`(ml s – 1)
`
`aortic pressure
`(mmHg)
`
`2.1
`2.3
`1.4
`1.6
`1.4
`1.3
`1.4
`1.7
`1.2
`1.2
`1.2
`1.1
`1.4
`1.3
`1.3
`1.4
`1.4
`1.4
`1.3
`1.2
`
`3.8
`3.8
`4.1
`4.3
`3.3
`3.3
`4.5
`4.0
`3.5
`3.7
`3.7
`3.9
`4.6
`4.6
`4.5
`4.5
`4.7
`4.7
`4.7
`3.5
`
`3.27
`3.52
`2.43
`2.88
`2.31
`1.73
`1.57
`3.05
`1.59
`1.6
`1.1
`0.97
`1.41
`1.39
`1.57
`1.9
`2.17
`2.36
`1.79
`1.39
`
`69.1
`71.6
`67.3
`76
`62.7
`63.6
`66
`70
`66.6
`72.6
`66
`62.4
`58.3
`62
`60
`59.5
`62.1
`76
`64.5
`60.5
`
`(no stenosis) to unity (full stenosis) area stenoses. The
`volumetric flow rate (Q) was measured by a perivascu-
`lar flow probe (Transonic Systems Inc.; a relative error
`of +2% at full scale). The arteries were cannulated to
`T-junctions at both ends. The pressure transducers
`were connected to the T-junctions to measure the prox-
`(Pproximal and Pdistal,
`imal and distal pressures
`respectively) of the stenosis in order to determine the
`pressure gradient (Dp ¼ Pproximal 2 Pdistal). Pulsatile
`pressure and flow were continuously recorded using a
`Biopac MP 150 data acquisition system (Biopac Sys-
`tems,
`Inc., Goleta, CA). A cast was made at
`100 mmHg after the stenotic vessel was fixed with
`6.25 per cent glutaraldehyde solution in 0.1 sodium
`cacodylate
`buffer
`(osmolarity
`of
`fixative was
`1100 mosM) [22]. Photographs of small rings sectioned
`from the vessel and stenosis casts were then taken
`(figure 1b). The CSA measurements were made using
`the NIS-ELEMENTS imaging software for the cast.
`For the in vivo experiments, the analytical model
`was validated in coronary arteries. A sheath was intro-
`duced through the femoral artery to access the right
`coronary artery (RCA), left anterior descending artery
`(LAD artery) and left circumflex artery (LCx artery).
`After a midline sternotomy, the main trunk of these
`arteries was dissected free from the surrounding tissue
`in preparation for the placement of a flow probe and
`an inflatable occluder with no apparent major branches
`in between them. The coronary artery was gradually
`occluded by an inflatable occluder cuff to create differ-
`ent degrees of stenoses. The hyperaemic volumetric
`flow rate (intracoronary injection of adenosine: 120 mg
`for both left and right coronary arteries) was deter-
`mined by a flow probe (Transonic Systems Inc.; a
`relative error of +2% at full scale). The distal pressure
`to coronary stenosis (Pdistal) was measured by a Vol-
`cano ComboWire (Volcano Corp., San Diego, CA,
`
`J. R. Soc. Interface (2012)
`
`USA), which was inserted into the coronary artery
`through a sheath. The proximal, distal and minimal
`CSAs were obtained from coronary angiograms, using
`previous method [7,8].
`
`2.3. Data analysis
`The proximal, distal and minimal CSA and stenosis
`length as well as hyperaemic flow rate were used to calcu-
`late the pressure drop (equation (2.3)), which was
`compared with the measurement obtained from in
`vitro and in vivo experiments. The relation of the
`pressure drop between the analytical (or theoretical)
`model and experimental measurements was expressed
`as DPexperiment ¼ aDPtheory þ b. Myocardial FFR was
`calculated from the theoretical model (a combination
`of equations (2.2) and (2.3)) in comparison with the
`in vivo coronary measurements. The empirical constants,
`a and b, were determined by a linear least-squares fit
`with corresponding correlation coefficients (r2). In a
`Bland–Altman diagram, the difference of pressure
`drop and myocardial FFR between the theoretical
`model and experimental measurements was plotted
`against their means. In the scatter diagram, the precision
`and bias of the analytical model were quantified. We also
`determined the root mean square error (r.m.s.e.) to
`further assess the accuracy of the theoretical model.
`
`2.4. Sensitivity analysis
`To determine the sensitivity of the model to various
`inputs (e.g. CSA and length of the lesion, hyperaemic
`flow), we varied these parameters over a range of values
`and determined the effect on pressure drop. The normal-
`ized pressure drop ((DPperturbed 2 DPactual)/(DPactual))
`was
`determined
`as
`a
`function
`of
`parameter
`X((Xperturbed 2 Xactual)/(Xactual)), which refers to distal
`
`CATHWORKS EXHIBIT 1011
`Page 1328 of 1338
`
`

`

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`
`A validated predictive model Y. Huo et al. 1329
`
`(a)
`
`T-junction connected with pressure transducer
`
`sutured against
`vessel wall
`
`tubing inside
`of vessel
`
`adapter to control pressure
`
`cross-sectional area
`of symmetric stenosis
`
`cross-sectional area
`of asymmetric stenosis
`
`carotid artery
`
`(b)
`
`T-junction connected with pressure transducer
`
`flow direction
`
`pump
`
`tubing inside of vessel
`
`adapter to control pressure
`
`occluder
`
`pump
`
`flow direction
`
`carotid artery
`
`Figure 1. (a–b) Schematic of in vitro stenosis set-ups: (a) insertion of known sizes of concentric and eccentric tubings into carotid
`artery to mimic various stenoses; (b) an inflatable arterial occluder to create various stenoses.
`
`CSAstenosis
`
`CSAproximal
`
`CSA, stenosis CSA, stenosis length and flow rate in a
`vessel (actual or reference values of (p4.52/4) mm2,
`(p1.72/4) mm2, 10 mm and 111 ml min – 1). The proximal
`CSA was not considered, as it has a negligible effect on
`pressure drop. The actual pressure drop (DPactual)
`equalled to 9.4 mmHg when the dynamic viscosity of
`blood is
`4.5 cp. The perturbed pressure drop
`(DPperturbed) was calculated by equation (2.3) when
`Xactual was changed to Xperturbed.
`
`3. RESULTS
`
`A comparison of pressure drop between the theoretical
`model (equation (2.3)) and FE simulation shows a
`relation as DPFE model ¼ 0.98DPtheory 2 0.14
`linear
`
`J. R. Soc. Interface (2012)
`
`(r2 ¼ 1). Figure 1 shows the in vitro stenosis set-up in
`the carotid artery using concentric and eccentric tub-
`ings as well as an inflatable occluder cuff,
`the
`dimensions of which are shown in table 1. The flow
`rates were varied in the range of 65–170 ml min – 1.
`Figure 2a shows a comparison of pressure gradient
`between the present theoretical model (i.e. equation
`(2.3)) and in vitro experiments, which has the linear
`relation DPexperiment ¼ 1.08DPtheory 2 1.15 (r2 ¼ 0.99)
`(table 3). Moreover, the difference of pressure gradients
`(DPtheory 2 DPexperiment) was plotted against the mean
`((DPtheory þ DPexperiment)/2),
`as
`shown
`in
`value
`figure 2b. The mean systematic error (or bias) of the
`difference of pressure drops (20.59 mmHg) was nearly
`zero, which suggests consistency of the theoretical
`model and experimental measurements. Therefore, the
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`1330 A validated FFR model Y. Huo et al.
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`mean + 2s.d.
`
`7 5 3 1
`
`–1
`
`0
`
`10
`
`20
`
`30
`
`40
`
`50
`
`mean
`
`60
`
`–3
`
`–5
`
`–7
`
`mean – 2s.d.
`
` DPtheory + DPexperiment (mmHg)
`
`2
`
`(b)
`
` DPtheory – DPexperiment (mmHg)
`
`60
`
`50
`
`40
`
`30
`
`20
`
`10
`
`(a)
`
`DPexperiment (mmHg)
`
`0
`
`10
`
`30
`20
`40
` DPtheory (mmHg)
`
`50
`
`60
`
`Figure 2. (a) A comparison of pressure gradient between theoretical model (i.e. equation (2.3)) and in vitro carotid experiments
`(DPtheory versus DPexperiment) using symmetric (squares) and asymmetric (triangles) tubings as well as an inflatable occluder cuff
`(asterisks). A least-squares fit shows a relation: DPexperiment ¼ 1.08  DPtheory 2 1.15 (r2 ¼ 0.99). (b) Bland–Altman plots for the
`pairwise comparisons of pressure gradient between theoretical model and in vitro experiments, where the mean + s.d. of pressure
`gradient difference (DPtheory 2 DPexperiment) are 20.59 and 2.61 mmHg, which is not significantly different from zero ( p  0.05).
`The r.m.s.e. of pressure gradient difference between theoretical model and in vitro experiments is 2.66 mmHg. The dynamic vis-
`cosity of saline solution is 1.0 cp.
`
`Table 3. A comparison of the present model (equation (2.3)) with other models that do not consider the entrance effect
`(equation (2.4)), and in vitro experimental measurements DPtheory 2 DPexperiment. A least-squares fit was used to determine
`parameters for DPexperiment ¼ aDPtheory þ b and DPexperiment ¼ aDPno entrance
`þ b.
`theory
`
`model
`
`present model (equation (2.3))
`
`other models (equation (2.4)) with
`different outlet flow patterns
`uniform outlet velocity
`a constant (1.52) times
`uniform outlet velocity
`parabolic outlet velocity
`
`r2
`a
`b
`DPexperiment ¼ aDPtheory þ b
`1.08
`21.15
`0.99
`DPexperiment ¼ aDPno entrance
`þ b
`
`theory
`
`1.98
`1.45
`
`1.21
`
`20.72
`20.77
`
`21.28
`
`0.99
`0.98
`
`0.95
`
`mean systematic error (or bias)
`
`r.m.s.e.
`
`DPtheory 2 DPexperiment (mmHg)
`
`20.59
`DPno entrance
`theory
`
` DPexperiment ðmmHgÞ
`
`210.0
`26.04
`
`21.82
`
`2.66
`
`15.3
`9.79
`
`5.37
`
`1 s.d. value (2.61 mmHg) was similar to the r.m.s.e.
`(2.66 mmHg) for the pressure difference. Figure 2
`shows a good correlation between DPtheory and
`DPexperiment. It should be noted that the pressure drop
`was accurately predicted by the model in comparison
`with experiments for various stenotic segments (con-
`centric, eccentric, cuff and various lengths).
`A comparison of pressure drop between other
`analytical models (i.e. equation (2.4)) and in vitro
`experiments is shown in table 3. The experimental
`results were more in agreement with the proposed
`model (equation (2.3)) when both entrance effects at
`the inlet of stenosis and flow velocity profiles at the
`outlet of stenosis were considered and hence all sub-
`sequent
`calculations accounted for
`those
`factors.
`A comparison of in vitro pulsatile and steady-state
`flows shows a relative error of pressure drop less than
`+5% so that the time-averaged flow rate (over a cardiac
`cycle) is used in equation (2.3) for determination of
`pressure drop for the relatively small Womersley and
`Reynolds numbers in coronary arteries.
`
`Figure 3a shows a comparison of pressure drop between
`the theoretical model (equation (2.3)) and in vivo coronary
`experiments (DPtheory versus DPexperiment), whose geo-
`metrical and haemodynamic parameters are listed in
`table 2. A linear least-squares fit yielded a relation as
`DPexperiment ¼ 0.96DPtheory þ 1.79 (r2 ¼ 0.75). Figure 3b
`shows a Bland–Altman plot for the pairwise comparisons
`of pressure drop between the theoretical model and in vivo
`experiments, where the mean of pressure difference
`(DPtheory 2 DPexperiment) was2 1.01, which was not signifi-
`cantly different from zero ( p  0.05). The r.m.s.e. of
`pressure difference between the theoretical model and
`in vivo experiments was 3.65 mmHg.
`Figure 4a shows the relationship of FFR between the
`theoretical model (a combination of equations (2.2) and
`(2.3)) and in vivo coronary experiments (FFRtheory
`versus FFRexperiment), expressed as FFRexperiment ¼
`0.85FFRtheory þ 0.1 (r2 ¼ 0.7). Myocardial FFR was
`found to be less than 0.8 when the area stenosis was
`greater than 75 per cent (where CSAproximal is in the
`range of (p/4)3.32 2 (p/4)4.72 mm2). Similar to the
`
`J. R. Soc. Interface (2012)
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`A validated predictive model Y. Huo et al. 1331
`
`area stenosis > 75%£ 75%
`
`mean + 2s.d.
`
`0.5
`
`0.6
`
`0.7
`FFRtheory
`
`0.8
`
`0.9
`
`1.0
`
`0.5
`
`0.6
`
`0.7
`
`0.8
`
`0.9
`
`1.0
`
`mean
`
`mean – 2s.d.
`
`1.0
`
`0.9
`
`0.8
`
`0.7
`
`0.6
`
`0.5
`
`0.4
`0.4
`
`(a)
`
`FFRexperiment
`
`(b)
`
`0.15
`
`0.10
`
`0.05
`
`0
`0.4
`
`–0.05
`
`–0.10
`
`–0.15
`
`FFRtheory – FFRexperiment
`
`FFRtheory + FFRexperiment (mmHg)
`2
`
`Figure 4. (a) A comparison of myocardial FFR between theor-
`etical model (equations (2.2) and (2.3)) and in vivo coronary
`experiments (FFRtheory versus FFRexperiment). A least-squares
`fit shows a relation: FFRexperiment ¼ 0.85FFRtheory þ 0.1 (r2 ¼
`0.7). (b) Bland–Altman plots for the pairwise comparisons of
`myocardial FFR between theoretical model and in vivo coron-
`ary experiments, where the mean + s.d. of myocardial FFR
`difference (FFRtheory 2 FFRexperiment) are 0.01 and 0.06,
`which is not significantly different from zero ( p  0.05).
`The r.m.s.e. of myocardial FFR difference between theoretical
`model and in vivo coronary experiments is 0.06. The dynamic
`viscosity of blood flow is 4.5 cp.
`
`from stenosis dimensions and hyperaemic flow. The pro-
`posed analytical model (equation (2.3)) is derived from
`the general Bernoulli equation with diffusive energy
`loss and energy loss due to a sudden expansion post-
`stenosis. The only assumption of the present model is
`that the flow transition from proximal normal coronary
`artery to stenosis is well bounded and follows the
`streamlines so that the pressure loss due to a sudden
`constriction can be neglected (loss coefficient  0.1 if
`there is no plane of vena contracta for the incompres-
`sible, laminar coronary blood flow). This assumption
`is reasonable and consistent with previous models
`[11–14]. This flow transition results in an approxi-
`mately uniform distribution of streamlines at the inlet
`of stenosis so that the flow velocity has a uniform
`profile, and the entrance effect should be incorpora-
`ted into the diffusive energy loss, as described in
`appendix A (also see figure 6a–c).
`
`10
`
`30
`20
` DPtheory (mmHg)
`
`40
`
`mean + 2s.d.
`
`0
`
`10
`
`20
`
`30
`
`40
`
`mean
`
`mean – 2s.d.
`
` DPtheory + DPexperiment
`2
`
`(mmHg)
`
`(a)
`
`40
`
`30
`
`20
`
`10
`
`0
`
`10
`8
`
`6 4 2 0
`
`–2
`–4
`–6
`–8
`–10
`
`DPexperiment (mmHg)
`
`(b)
`
` DPtheory – DPexperiment (mmHg)
`
`Figure 3. (a) A comparison of pressure gradient between
`theoretical model (i.e. equation (2.3)) and in vivo coronary
`experiments (DPtheory versus DPexperiment). A linear least-
`squares fit shows a relation: DPexperiment ¼ 0.96DPtheory þ
`1.79 (r2 ¼ 0.75). (b) Bland–Altman plots for the pairwise
`comparisons of pressure gradient between theoretical model
`and in vivo coronary experiments, where the mean + s.d. of
`pressure gradient difference (DPtheory 2 DPexperiment) are
`21.01 and 3.6 mmHg, which is not significantly different
`from zero ( p  0.05). The r.m.s.e. of pressure gradient differ-
`ence between theoretical model and in vivo experiments is
`3.65 mmHg. The dynamic viscosity of blood flow is 4.5 cp.
`
`comparison of pressure drop in figure 3b, figure 4b shows
`a Bland–Altman plot for the pairwise comparisons of
`FFR between the theoretical model and in vivo coron-
`ary experiments, where the mean + s.d. of myocardial
`FFR difference (FFRtheory 2 FFRexperiment) are 0.01
`and 0.06 as well as the r.m.s.e. is 0.06. There was a
`good agreement of FFR between the theoretical model
`and in vivo coronary experiments.
`Figure 5 shows a sensitivity analysis for the distal
`CSA, stenosis CSA, stenosis length and flow rate in a
`vessel. The pressure drop was strongly affected by ste-
`nosis CSA and flow rate, whereas proximal CSA (not
`shown), distal CSA and stenosis length had relatively
`small effects.
`
`4. DISCUSSION
`
`We validate a physics-based model of pressure drop and
`myocardial FFR for coronary stenosis derived strictly
`
`J. R. Soc. Interface (2012)
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`1332 A validated FFR model Y. Huo et al.
`
`
`
`(equation (2.3)) and previous models [11–14]. The
`in vitro experiments showed a good agreement with
`the present model predictions (equation (2.3)), as
`shown in figure 2 and table 3. We found that the
`pressure drop was significantly underestimated if the
`entrance effect of stenosis was not considered (e.g.
`equation (2.4)). Moreover, the last terms (DPno entrance
`expansion
`in equation (2.4)) cannot have a constant loss coeffi-
`cient (e.g. 1.52 in a previous study [13]) because it is
`strongly affected by an interaction of the outlet flow
`patterns and the degree of stenosis. Here, we used a
`second-order polynomial
`interpolation to determine
`the loss coefficient for the blunt velocity profile that is
`physically likely in most of coronary stenoses with a
`diameter of 0.8–2 mm and length of 1–30 mm. The
`second-order
`polynomial
`interpolation
`physically
`reflected the growth of flow boundary layer from the
`inlet to outlet of a coronary stenosis, where the flow
`velocity may vary from the uniform to parabolic profiles
`if the stenosis is sufficiently long.
`The predictions of equation (2.3) were consistent
`in vivo experiments, as
`with those of
`shown in
`figure 3. Moreover, figure 4 showed that the analytically
`computed FFR from equations (2.2) and (2.3) agreed
`reasonably well with those from the in vivo coronary
`experimental measurements using the Volcano Com-
`boWire given the potential error in the measurements.
`Similar to the previous studies [1,4,23], myocardial
`FFR computed from equations (2.2) and (2.3) was
`also found to be less than 0.8 when the area stenosis
`imposed to a normal coronary artery of swine heart
`was greater
`than 0.75. The consistency between
`theory and in vitro and in vivo experiments provides
`some validations of the initial assumptions of uniform
`and blunt/parabolic flow velocity profiles at the inlet
`and outlet of a coronary stenosis, respectively, and
`confirm the significant effects of stenotic entrance
`region and outlet flow velocity profiles on the pressure
`drop and FFR.
`
`4.2. Comparison with other studies
`
`Although the Bernoulli equation has many clinical
`applications [24],
`it has limited accuracy to predict
`the pressure drop across a stenosis. For example, the
`Bernoulli equation predicts a zero pressure drop across
`a stenosis if the proximal CSA is equal to the distal
`CSA. The models given by equation (2.4) incorporate
`the energy loss of viscosity and sudden CSA expansion
`into the Bernoulli equation, but have been shown to
`inadequately predict the pressure drop across a stenosis
`[15]. The present model includes the effects of both
`entrance region and various outlet flow velocity profiles
`in coronary stenoses when compared with previous
`studies that emphasized the energy loss of a sudden
`expansion at the outlet of stenosis or introduced
`empirical factors to fit the experimental measurements
`[11–14,25].
`The geometry of stenotic inflow region has been
`thought to affect the energy loss and hence there has
`been a significant focus on stenosis morphology [26]. If
`the change in lumen area is very sharp, which induces
`a vena contracta, equations (A 4)