IOP PUBLISHING
`
`Phys. Med. Biol. 53 (2008) 3995–4011
`
`PHYSICS IN MEDICINE AND BIOLOGY
`
`doi:10.1088/0031-9155/53/14/017
`
`Determination of fractional flow reserve (FFR) based
`on scaling laws: a simulation study
`
`Jerry T Wong and Sabee Molloi
`
`Department of Radiological Sciences, University of California, Irvine-92697, CA, USA
`
`E-mail: symolloi@uci.edu
`
`Received 8 April 2008, in final form 8 May 2008
`Published 3 July 2008
`Online at stacks.iop.org/PMB/53/3995
`
`Abstract
`Fractional flow reserve (FFR) provides an objective physiological evaluation of
`stenosis severity. A technique that can measure FFR using only angiographic
`images would be a valuable tool in the cardiac catheterization laboratory.
`To perform this,
`the diseased blood flow can be measured with a first
`pass distribution analysis and the theoretical normal blood flow can be
`estimated from the total coronary arterial volume based on scaling laws.
`A computer simulation of the coronary arterial network was used to gain
`a better understanding of how hemodynamic conditions and coronary artery
`disease can affect blood flow, arterial volume and FFR estimation. Changes
`in coronary arterial flow and volume due to coronary stenosis, aortic pressure
`and venous pressure were examined to evaluate the potential use of flow and
`volume for FFR determination. This study showed that FFR can be estimated
`using arterial volume and a scaling coefficient corrected for aortic pressure.
`However, variations in venous pressure were found to introduce some error
`in FFR estimation. A relative form of FFR was introduced and was found to
`cancel out the influence of pressure on coronary flow, arterial volume and FFR
`estimation. The use of coronary flow and arterial volume for FFR determination
`appears promising.
`
`(Some figures in this article are in colour only in the electronic version)
`
`Glossary
`
`AM
`AN
`AS
`CE
`
`CV
`
`minimum cross-sectional area in an arterial stenosis (cm2)
`cross-sectional area of a normal arterial segment (cm2)
`cross-sectional area of a stenosed arterial segment (cm2)
`
`
`−7 to mmHg/(ml min−1)2)
`(conversion factor needed from g cm
`
`
`−3 to mmHg/(ml min−1))
`(conversion factor needed from poise cm
`
`−1)2)coefficient of pressure loss due to flow separation (mmHg/(ml min
`−1))coefficient of pressure loss due to viscous friction (mmHg/(ml/min
`
`0031-9155/08/143995+17$30.00 © 2008 Institute of Physics and Engineering in Medicine Printed in the UK
`
`3995
`
`CATHWORKS EXHIBIT 1006
`Page 3995 of 4011
`
`

`

`3996
`
`d0
`danat
`dmax
`dM
`dN
`dS
`FFRP
`FFRQ
`FFRV
`FFRVD
`FFRVN
`FFRVR
`k
`kVD
`kVN
`KE
`KV
`L
`(cid:1)
`(cid:1)S
`µ
` P
`P0.5
`P
`Pa
`Pc
`PV
`Q
`QD
`QN
`QVD
`QVN
`ρ
`r
`R
`Rd
`RS
`RT
`SEE
`V3/4
`VVD
`VVN
`
`J T Wong and S Molloi
`
`normalized diameter at zero pressure (unitless)
`normal diameter at 100 mmHg (cm)
`maximum normalized diameter (unitless)
`minimum diameter in a stenosis (cm)
`diameter of a normal arterial segment (cm)
`diameter of a stenosed arterial segment (cm)
`pressure-based fractional flow reserve (unitless)
`flow-based fractional flow reserve (unitless)
`volume-based fractional flow reserve (unitless)
`volume-based fractional flow reserve of diseased artery (unitless)
`volume-based fractional flow reserve of reference artery (unitless)
`relative volume-based fractional flow reserve (unitless)
`−1)
`scaling coefficient relating QN and V3/4 (ml min
`−1)scaling coefficient of a diseased artery (ml min
`
`−1)
`scaling coefficient of a reference artery (ml min
`dimensionless coefficient used in CE (unitless)
`dimensionless coefficient used in CV (unitless)
`cumulative arterial branch length (cm)
`arterial segment length (cm)
`stenosis length (cm)
`absolute viscosity (poise)
`pressure drop (mmHg)
`pressure when diameter is midway between d0 and dmax (mmHg)
`pressure (mmHg)
`aortic pressure (mmHg)
`pre-capillary pressure (mmHg)
`venous pressure (mmHg)
`−1)
`coronary blood flow (ml min
`−1)
`diseased hyperemic blood flow (ml min
`−1)normal hyperemic blood flow (ml min
`
`−1)hyperemic coronary blood flow in a diseased artery (ml min
`
`−1)hyperemic coronary blood flow in a reference artery (ml min
`
`−3)
`blood density (g cm
`correlation coefficient (unitless)
`−1))
`resistance (mmHg/(ml min
`−1))effective resistance of distal tree to a segment (mmHg/(ml min
`
`−1))flow dependent resistance due to a stenosis (mmHg/(ml min
`
`−1))total effective resistance of a subtree (mmHg/(ml min
`
`−1)
`standard error of estimate (ml min
`cumulative arterial volume raised to an exponent of 3/4 (unitless)
`cumulative arterial volume of a diseased artery (unitless)
`cumulative arterial volume of a reference artery (unitless)
`
`1. Introduction
`
`Limitations in the visual assessment of intermediate severity stenoses by coronary angiography
`are known to suffer from intra- and inter-observer variability as well as discordance with their
`true physiological importance (DeRouen et al 1977, Detre et al 1975, Robbins et al 1966,
`Vlodaver et al 1973, White et al 1984, Zir et al 1976). Fractional flow reserve (FFR)
`
`CATHWORKS EXHIBIT 1006
`Page 3996 of 4011
`
`

`

`Determination of fractional flow reserve (FFR) based on scaling laws
`
`3997
`
`was introduced to provide a physiological measure of coronary stenosis by quantifying the
`reduction in maximum coronary blood flow from a theoretical maximum normal flow in
`the presence of a lesion. Currently, FFR is approximated by dividing the pressure distal to
`the stenosis by the aortic pressure. The distal pressure is measured using a pressure-sensing
`wire that has passed across the stenosis, and the aortic pressure is measured simultaneously at
`the catheter tip with a pressure transducer. Pressure-based fractional flow reserve (FFRP) has
`proven to aid the cardiologist in evaluating the flow-limiting potential of stenoses as well as the
`therapeutic gain of angioplasties (Kern et al 2006, Pijls et al 1996). However, an alternative
`technique that can measure FFR using only angiographic images would be a valuable tool in
`the cardiac catheterization laboratory because the acquired images used for visual assessment
`of stenosis severity can also be used to quantify physiological alterations imposed by the
`stenosis. In other words, FFR could potentially be measured using only image data without
`the need to pass a pressure wire across a stenosis.
`FFR is defined as the hyperemic blood flow in an artery with a stenosis divided by
`the hypothetical hyperemic blood flow in the same artery without the stenosis. Thus the
`two parameters that need to be measured are the real flow and the hypothetical normal flow.
`Previous reports indicate that a first pass distribution analysis technique can be used to measure
`absolute coronary blood flow by analyzing the propagation of a contrast material signal in
`the coronary system (Hangiandreou et al 1991, Marinus et al 1990, Molloi et al 1998, 1996).
`The challenge is to determine the maximum normal flow in the absence of a stenosis. This
`maximum normal flow can be estimated by using a measurable parameter that correlates well
`with it. For instance, allometry between various biological parameters and body mass has
`been studied extensively in comparative physiology. A key example of allometry that is
`generally accepted and widely used is the 3/4 scaling of the basal metabolic rate with body
`mass (Kleiber 1932), although unequivocal experimental and theoretical explanation for it
`remains ambiguous. With coronary blood flow shown to correlate linearly with the oxygen
`consumption (or metabolic need) of the heart (Alella et al 1955), an allometric relationship
`between maximum normal flow and myocardial mass is also expected. However, myocardial
`mass cannot be measured accurately with angiography in vivo.
`There have been previous reports of an angiographic method to obtain the size of the
`dependent regional myocardial mass by measuring the cumulative arterial branch lengths that
`supply it (Seiler et al 1992, 1993). These studies found a systematic correlation between the
`cumulative arterial branch length and the dependent regional myocardial mass with sufficiently
`small variability to be clinically useful. Similarly, recent studies have shown that cumulative
`arterial branch length is correlated with lumen volume through a power-law relation (Kassab
`2005, Zhou et al 1999, 2002). Recent simulation studies also have shown that flow through
`any point in the epicardial coronary arterial tree is related to the sum of its distal coronary
`arterial lumen volume (Molloi and Wong 2007). This is consistent with previous observations
`in comparative physiology that blood flow scales with myocardial mass and that the total
`mammalian blood volume scales linearly with body mass (Prothero 1980). Therefore, total
`arterial blood volume appears to be a suitable surrogate measure of normal blood flow. Arterial
`blood volume has also been shown to be accurately measurable with angiography (Molloi
`et al 2001). Therefore, the combination of volumetric coronary blood flow and arterial
`volume measurements with angiography can be used to determine FFR.
`The purpose of this study is to gain a better understanding of how hemodynamic
`conditions and coronary artery disease can affect arterial blood flow and volume used for FFR
`determination. The response of arterial volume to different hemodynamic conditions can be
`different than that of normal blood flow. Moreover, the presence of a stenosis can alter arterial
`volume. Arterial volume changes due to variations in pressures and disease can potentially
`
`CATHWORKS EXHIBIT 1006
`Page 3997 of 4011
`
`

`

`3998
`
`J T Wong and S Molloi
`
`hinder its usefulness for predicting normal blood flow and, thus, estimating FFR. In this study,
`a computer simulation model was used to evaluate the impact various parameters such as
`pressure, stenosis length and percent diameter stenosis have on FFR determination using an
`arterial volume-based approach. The computer simulation model used a fully reconstructed
`porcine coronary arterial system with maximally vasodilated arterial segments responsive
`to perfusion pressures to study pressure-flow distributions and their effect on total arterial
`volume during normal and diseased states. Then the volume-based determination of FFR was
`evaluated against the true FFR quantification using a flow ratio and the clinically implemented
`FFR measurement technique using a pressure ratio. The concept of estimating FFR using
`arterial volume is new and the effects of hemodynamic conditions and arterial stenosis on the
`quantification of normal flow and FFR are not currently known.
`
`2. Methods
`
`2.1. Theoretical considerations
`
`Zhou et al, by applying the principle of minimum energy to the design of the coronary arterial
`system, predicted a linear relationship between blood flow (Q) through a proximal segment
`and the cumulative arterial length of all its distal branches (L) (Zhou et al 1999, 2002). Zhou
`et al also predicted a power law relation between the cumulative lumen volume (V) and
`the cumulative branch length of the arterial branches. These relationships are expressed in
`equations (1) and (2), respectively:
`Q ∝ L,
`L ∝ V 3/4,
`Q ∝ V 3/4.
`
`(1)
`
`(2)
`
`(3)
`
`The combination of equations (1) and (2) yields the relationship between Q and V as
`expressed in equation (3).
`In comparison with Kleiber’s allometry (Kleiber 1932), which
`relates the metabolic rate to body mass raised to an exponent of 3/4, equation (3) is consistent
`with this 3/4 allometry law after considering that metabolic rate scales with blood flow (Alella
`et al 1955) and that mass scales with total blood volume (Choy and Kassab 2008, Le et al
`2008, Prothero 1980).
`Assuming that blood flow correlates well with volume as expressed in equation (3), FFR
`can be estimated from measuring the diseased blood flow and estimating the normal blood
`flow using the measured arterial volume. The hyperemic blood flow (QN) in a normal artery
`tree can be determined from its total arterial lumen volume using equation (3) derived from
`an analysis of normal arterial trees:
`QN = kV 3/4,
`(4)
`where k is the scaling coefficient that is determined from a linear regression of hyperemic
`normal flow versus normalized blood volume raised to an exponent of 3/4. Blood
`volumes were normalized by dividing them by a reference volume of 1.0 ml. As a
`−1.
`From the definition of
`result,
`the proportionality constant k has units of ml min
`FFR = QD/QN where QD is the hyperemic flow through an artery with a stenosis and QN is
`the normal hyperemic flow through the same artery without disease, an alternative expression
`(equation (5)) for FFR can be obtained in terms of the measured blood flow and total arterial
`volume from angiographic images after substituting the right-hand term in equation (4) for
`QN:
`
`CATHWORKS EXHIBIT 1006
`Page 3998 of 4011
`
`

`

`Determination of fractional flow reserve (FFR) based on scaling laws
`
`FFRV = QD
`kV 3/4 ,
`FFRVR = FFRVD
`FFRVN
`
`= kVN
`kVD
`
`(QVD/QVN)
`(VVD/VVN)3/4 .
`
`3999
`
`(5)
`
`(6)
`
`A relative form of volume-based FFR (FFRVR) is also introduced to offset flow and volume
`changes due to pressure variations. FFRVR, expressed in equation (6), is defined as the ratio
`of FFRV values from a diseased artery (FFRVD) and a reference normal artery (FFRVN), where
`the VD subscripts refer to the diseased artery under investigation and the VN subscripts refer
`to another artery used as the normal reference. The use of FFRVR requires the existence of a
`normal reference artery.
`
`2.2. Model overview
`
`The coronary arterial network was simulated using existing morphometric data of a porcine
`heart (Kassab et al 1993). These fully reconstructed morphometric data were used as the
`quintessential porcine coronary arterial network. Previous computer reconstructions using
`morphometric data from Kassab et al showed small variations in both morphometry and
`hemodynamics (Mittal et al 2005a, 2005b). More recently, ex vivo studies on a number of
`porcine hearts showed that all the hearts followed a similar allometric scaling law (Choy and
`Kassab 2008, Le et al 2008). Thus the effects of pressure and focal stenosis on coronary blood
`flow and total arterial volume were studied on this quintessential arterial network and assumed
`to be generally applicable. The arterial segments were modeled as rigid (non-collapsible), fully
`dilated tubes with dependence on distending pressure only. Segment diameters were dependent
`on local pressures following observations from previous studies. Both the myogenic response
`and flow-induced dilation of the segments were considered to be absent at full dilation, such
`as during the influence of high-concentration adenosine. Coronary blood flow through the
`reconstructed arterial tree was assumed laminar and steady. The pressure–flow distribution in
`the coronary arterial circulation was determined using an electrical circuit model consisting
`of resistances in series and parallel. A simulated stenosis was modeled as a flow-dependent
`resistor in series with distal segments. Collateral formation in diseased states was not modeled.
`This simulation of the coronary arterial network was not meant to account for complex
`physiological conditions such as microvascular dysfunction, diffuse disease or congestive
`heart failure.
`
`2.3. Description of the normal model
`
`Morphometric information on the proximal branches of the left anterior descending (LAD),
`left circumflex (LCX) and right coronary (RCA) arteries were obtained from previous
`polymer cast data (Kassab et al 1993). Moreover, a method for the reconstruction of a
`fully reconstructed coronary vascular tree from partial measurements has been developed
`recently (Mittal et al 2005b). The cast data of proximal arterial segments were used as
`seed points for growing the distal arterial subtrees. Growing started from arterial segments
`having diameters greater than 8 µm because these segments in the cast were assumed to be
`prematurely broken before reaching the capillaries. Using intact mother–daughter bifurcation
`units or arteriole subtree units, broken segments were replaced with intact units whose mother
`segment diameters resembled that of the broken segments. This growing process continued
`until all terminal arterioles had diameters of 8 µm because a previous report had observed that
`pre-capillary arterioles that bifurcate into capillaries have diameters in the range of 8–10 µm
`
`CATHWORKS EXHIBIT 1006
`Page 3999 of 4011
`
`

`

`4000
`
`J T Wong and S Molloi
`
`(Kassab et al 1993). Morphometric data (such as diameter and length) along with the node-
`to-node connectivity for every segment were stored in an efficient array system for easy
`recall. Initial simulations revealed higher arteriolar resistances than expected in maximally
`dilated coronary arterial trees based on our in vivo studies as well as previously reported
`pressure–flow relation (Pantely et al 1984) and microvascular resistance (Fearon et al 2004)
`for a swine LAD. Thus, arterioles (diameters (cid:1)400 µm) of the LCX, LAD and RCA were
`uniformly dilated by a factor of 1.6 in order to achieve a reasonable agreement with previously
`reported pressure–flow relation and microvascular resistance. The final reconstructed normal
`coronary networks of the LCX, LAD and RCA had effective resistances of 0.726, 0.350 and
`−1), respectively, and inlet flows of 116, 242 and 214 ml min
`−1,
`0.397 mmHg/(ml min
`respectively, when aortic pressure was 100 mmHg and venous pressure was 5 mmHg.
`
`2.4. Description of the diseased model
`
`(cid:2)
`
`(cid:1)
`A stenosis of a given severity was implemented by reducing the diameters of discrete segments
`by an appropriate amount: percent diameter stenosis = 100
`1 − dSd
`−1
`/d0, where dS and dN
`N
`are the diameters of the constricted and normal segments, respectively. The converging and
`diverging ends of real stenoses were not simulated. A focal stenosis, one that does not span
`across bifurcation nodes, was implemented in the first segment of the reconstructed LCX tree.
`The first segment in the reconstructed LCX tree was artificially lengthened from 5 mm to
`20 mm in order to accommodate the range of stenosis lengths that were studied. The
`first segment of the LAD and RCA were also lengthened to 20 mm for consistency. This
`introduction of a longer input segment did not affect the hemodynamics of the distal tree.
`Stenosis resistance was calculated from the following equations (Kirkeeide 1991):
`RS = CV + CEQ,
`CV = KVµ
`(AN − AS)2
`CE = KEρ
`2A2
`A2
`N
`S
`
`,
`
`dNAN
`
`.
`
`(7)
`
`(8)
`
`(9)
`
`In equation (7), RS is the flow-dependent resistance due to the stenosis, CV is the coefficient
`of pressure loss due to viscous friction, and CE is the coefficient of pressure loss downstream
`from the stenosis due to flow separation, turbulence, or both. In equations (8) and (9), AN is
`the cross-sectional area of the normal segment, AS is the cross-sectional area of the stenosed
`part, µ is the absolute blood viscosity, ρ is blood density and KV and KE are dimensionless
`coefficients that take stenosis diameter, length and entrance effects into consideration. KV and
`(cid:3)
`(cid:4)
`KE are defined as follows:
`KV = 32
`
`0.45
`
`AN
`AM
`
`d2
`N
`d2
`M
`
`+ 0.86
`
`AN(cid:1)S
`ASdN
`
`d2
`N
`d2
`S
`
`(10)
`
`KE = 1.21 + 0.08
`
`(cid:1)S
`dN
`
`.
`
`(11)
`
`In equations (10) and (11), (cid:1)S is the length of the stenosis, and dM and AM are the minimum
`diameter and minimum cross-sectional area found in the stenosis. In the current simulation,
`dM = dS and AM = AS. The above expressions for predicting flow-dependent resistances from
`geometrical dimensions have been validated in an extensive series of in vitro experiments
`(Kirkeeide 1991).
`
`CATHWORKS EXHIBIT 1006
`Page 4000 of 4011
`
`

`

`Determination of fractional flow reserve (FFR) based on scaling laws
`
`4001
`
`1.2
`
`u
`
`_A
`
` ‘
`0.8
`
`1
`
`
`
`Normalizeddiameter
`
`0
`
`40
`
`80
`
`1 20
`
`1 60
`
`Pressure (mmHg)
`
`Figure 1. Passive pressure—diameter relation using the dataset ofconducting arteries with diameters
`ranging from 0.74 to 2.8 mm. Diameters were normalized to the diameter at 100 mmHg. The solid
`line represents the best—fit curve based on equation (12) with do = 0.77, d.” = 1.15 and P05 =
`63.8 man.
`
`2.5. Modeling pressure—diameter relations
`
`The compliance of resistance and conducting arteries has been previously reported
`(Comelissen et al 2000, Kassab and Molloi 2001). Comelissen et al showed that arterioles
`normalized by their diameters at a pressure of 100 mmHg demonstrated a similar pressure—
`diarneter curve. Arteriole diameters distended at 100 mmHg were designated as the anatomical
`diameters (dam). The following equation was derived for the curve that best fitted existing
`experimental data (Comelissen et a! 2000, Liao and Kuo 1997):
`d
`P P
`—=do+(dmax—do)
`’°"
`danat
`l + P/ P05 I
`
`(12)
`
`The constants dnm, do and PM have values of 1.07, 0.70 and 22.7 mml-lg, respectively,
`based on the dataset of resistance vessels (diameters $400 um) (Comelissen et al 2000). The
`constants do and dm are unitless since normalized diameters (d/dml) are used. Similarly,
`figure 1 shows the best-fit curve using equation (12) for the dataset of conducting arteries
`(diameters > 400 um) (Kassab and Molloi 2001), where dnm, do and P05 were determined to
`be 1. 15, 0.77 and 63.8 mmHg, respectively. All reconstructed segment diameters were assigned
`pressures of 80 mmHg because arterial casting was performed at this pressure under zero flow
`(Kassab et al 1993). The anatomical diameters of all segments dam were then calculated
`from these pressure—diameter associations using equation (12). Subsequent responses to local
`pressures were determined with dam and equation ( 12).
`
`2.6. Modeling pressure—flow distributions
`
`The coronary arterial network was modeled as an electrical circuit with each segment acting
`as a resistor. The resistance of each segment was determined with Poiseuille’s equation:
`
`_ AP _ 128,1!
`Q W
`
`"3’
`
`CATHWORKS EXHIBIT 1006
`
`Page 4001 of 4011
`
`CATHWORKS EXHIBIT 1006
`Page 4001 of 4011
`
`

`

`4002
`
`J T Wong and S Molloi
`
`Figure 2. A circuit model of a bifurcating arterial network. Definitions for the symbols are given
`in the glossary.
`
`where R is the resistance, P is the pressure drop, Q is the blood flow, µ is the absolute blood
`viscosity, (cid:1) is the arterial segment length and d is the arterial lumen diameter. The viscosity
`term is diameter dependent because of the Fåhræus–Lindqvist effect (Fåhræus and Lindqvist
`1931). The expression for calculating blood viscosity was given by Pries et al (1994). The
`pressure–flow distribution in the network was calculated based on known resistances, mass
`balance and Poiseuille’s equation. The current method of determining the pressure–flow
`distribution is different from solving a large set of linear equations as done previously in
`our lab (Zhou et al 1999). The current method was utilized because it was computationally
`efficient and did not require working with matrices to solve a system of equations. The specific
`details for determining the pressure–flow distribution, using figure 2 as an example, are as
`follows.
`
`(1) Anatomical diameters danat were first determined based on reconstructed diameters and a
`casting pressure of 80 mmHg.
`(2) Segment resistances were calculated based on equation (13) and segment diameters.
`(3) Starting from the most distal segments, the effective resistance of the coronary network
`and constituent subtrees was determined by adding segment resistances in series, parallel
`or both. For example, the total effective resistance (RT) is calculated using the following
`expression:
`
`.
`
`(14)
`
`(cid:2)(cid:2)
`)(R3 + R
`(R2 + R
`d )
`(cid:2)(cid:2)
`(R2 + R
`) + (R3 + R
`d )
`
`(cid:2)d
`
`(cid:2)d
`
`RT = R1 +
`
`(4) For known aortic and venous pressures, flow through each segment (Q1) was determined
`from the pressure drop from the inlet of the segment (P0) to the outlet of pre-capillary
`
`CATHWORKS EXHIBIT 1006
`Page 4002 of 4011
`
`

`

`Determination of fractional flow reserve (FFR) based on scaling laws
`
`4003
`
`segments (Pc) divided by the effective resistance of the segment in series with the distal
`subtree:
`
`Q1 = P0 − Pc
`
`.
`
`(15)
`
`RT
`Pc was assumed to be 10 mmHg higher than the venous pressure based on previous
`hemodynamic studies on swine (Kassab et al 1999, Pantely et al 1984).
`(5) The outflow pressure of the segment (P1) was determined from the pressure drop calculated
`from the flow through the segment multiplied by the segment’s resistance:
`P1 = P0 − Q1R1.
`(16)
`(6) Starting from the most proximal segment, steps 4 and 5 were repeated until every segment
`in the arterial network was assigned a pressure and a flow.
`(7) Segment diameters were adjusted according to the local pressure.
`(8) Due to the interdependence of resistance, pressure and flow, these parameters were
`iteratively recalculated by repeating steps 2–7 until a steady state was reached.
`Steady state was reached when the maximum change in flow and diameter in every
`segment was less than 0.1%. This threshold was chosen to minimize computation time without
`compromising accuracy. For example, decreasing the threshold would require a greater number
`of iterations, but the parameter values for equation (12) are associated with larger errors than
`0.1%. Computation time substantially increased with the number of iterations because the
`number of segments in the simulated LAD, LCX and RCA were 745 999, 459 066 and
`703 866, respectively. Moreover, the mean number of segments in series connecting the
`most proximal inlet of the arterial tree to any outlet pre-capillary segments in the LAD, LCX
`and RCA were 38, 34 and 50, respectively. The corresponding standard deviations of the
`distributions were 11, 10 and 18 for the LAD, LCX and RCA, respectively.
`
`2.7. Study objectives
`
`The objectives of this study are (1) to determine the effects of pressure and focal stenosis
`on the relationship between maximum hyperemic blood flow and total arterial volume
`and (2) to predict the errors associated with estimating FFR with total arterial volume.
`A simulation program using fully reconstructed coronary arterial trees down to the pre-
`capillary arterioles was used to predict changes in flow–volume relations for a range of aortic
`(60–160 mmHg) and venous (0–20 mmHg) pressures.
`In addition, flow–volume relations
`for different combinations of stenosis diameters (0–90% diameter stenosis) and lengths
`(5–20 mm) were studied since both are important determinants of flow reduction and ischemia
`(Brosh et al 2005, Gould and Lipscomb 1974, Uren et al 1994). To better distinguish from
`diffuse coronary artery disease, stenosis lengths were limited to less than or equal to 20 mm.
`Total arterial volumes were calculated from the sum of all segments from a truncated dataset
`consisting of proximal segments with diameters greater than 0.5 mm. This truncation limit
`was chosen to approximately mimic the spatial resolution of an angiogram acquired using an
`image-intensifier x-ray system. Thus, references to total arterial volume in this study represent
`volume calculations based on a 0.5 mm resolution limit.
`The measured inlet flow and total arterial volume were used to determine FFRV of the
`diseased LCX system with equation (5). Since both blood flow and arterial volume were
`pressure dependent, the scaling coefficient k in equation (5) was made to be a function of
`aortic pressure, but not venous pressure, since aortic pressure is easily measured clinically.
`FFRVR was calculated from the ratio of FFRV of the diseased LCX to the normal LAD. An
`evaluation of FFRV and FFRVR to estimate true FFR was performed by comparing them to
`
`CATHWORKS EXHIBIT 1006
`Page 4003 of 4011
`
`

`

`4004
`
`Artery
`
`Z
`
`A
`−1.15
`0.000 0505
`−1.95
`0
`−2.43
`0.000 0793
`−2.73
`0
`−2.15
`0.000 0823
`−1.97
`0
`
`1.68
`0.001 04
`2.49
`2.49
`
`3.48
`0.001 76
`3.33
`3.33
`
`3.07
`0.001 95
`2.43
`2.43
`
`J T Wong and S Molloi
`Table 1. Parameter values for the general equation Z = A PV/mmHg + B Pa/mmHg + C. Both
`−1, but V3/4 is unitless.
`Q and k have units of ml min
`B
`C
`−44.4
`0.476
`−40.0
`−49.6
`−90.37
`0.705
`−47.4
`−60.8
`−79.2
`0.885
`−37.1
`−47.0
`
`LCX
`
`LAD
`
`RCA
`
`Q
`V3/4
`k
`k
`
`Q
`V3/4
`k
`k
`
`Q
`V3/4
`k
`k
`
`FFRQ. FFRQ was determined by dividing the diseased inlet flow (QD) by the inlet flow obtained
`when no disease was present (QN): FFRQ = QD/QN. FFRV and FFRVR were also compared
`to FFRP to assess their clinical applicability. FFRP was determined from the ratio of pressures
`distal (Pd) and proximal (Pa) to the stenosis: FFRP = Pd/Pa.
`
`3. Results
`
`Normal flow-volume studies were performed on the LCX, LAD and RCA systems and the
`results are summarized in table 1. In addition, a focal stenosis was implemented in the first
`segment of the reconstructed LCX tree. Thus FFRV measurements were made on the diseased
`model of the LCX tree, and FFRVR measurements were obtained using the normal LAD tree
`as reference.
`
`3.1. Normal model
`
`Figure 3 compares the simulated inlet flow versus aortic pressure with a previous study on zero-
`flow pressure (PZF) using maximally vasodilated LAD in swine (Pantely et al 1984). Figure 3
`shows a good agreement between in vivo and simulated data. In addition, the total resistance of
`−1) using an aortic pressure of 60 mmHg and PZF
`the simulated LAD was 0.40 mmHg/(ml min
`of 15, which agrees reasonably well with a previously reported mean microvascular resistance
`−1) under similar pressures (Fearon et al 2004). A linear pressure–
`of 0.43 mmHg/(ml min
`flow response was seen in the higher pressures, but the magnitude of pressure drop increased
`with decreasing flow at the lower pressures. This reflects the gradual increase in the arteriole
`resistance with decreasing local pressures. In figure 3, a linear fit of the pressure–flow curve
`gives an inverse slope (R = P/ Q) value of 0.367 mmHg/(ml min
`−1) and an extrapolated
`zero-flow pressure of 18.3 mmHg. The effects of aortic and venous pressures on the simulated
`inlet flow and V3/4 were also studied. Inlet flow was observed to have an inverse relationship
`with PV but a direct relationship with Pa. On the other hand, V3/4 increased slightly with
`increasing Pa, but remained approximately constant over the range of PV studied. Equations
`for planes that best fit simulation results and that describe inlet flow and V3/4 as a function of
`both PV and Pa are summarized in table 1. Similarly, the scaling coefficient was observed to
`
`CATHWORKS EXHIBIT 1006
`Page 4004 of 4011
`
`

`

`Determination of fractional flow reserve (FFR) based on scaling laws
`
`4005
`
`PZFE
`PZF
`
` Pantely et al.
` Simulation
` Linear fit
`
`100
`75
`50
`25
`Aortic pressure (mmHg)
`
`125
`
`200
`
`150
`
`100
`
`50
`
`Inlet flow (ml/min)
`
`0
`
`0
`
`Figure 3. A plot of the normal pressure–flow relation for the LAD system. A linear fit through the
`−1) and an extrapolated zero-flow
`data points yields an arterial resistance of 0.367 mmHg/(ml min
`pressure (PZFE) of 18.3 mmHg at the limit when the inlet flow approaches zero. The venous (PV)
`and zero-flow (PZF) pressures were set at 5 and 15 mmHg, respectively.
`
`increase with increasing Pa, but decrease with increasing PV. Equations for the best-fit planes
`describing the scaling coefficient as a function of both PV and Pa are also summarized in
`table 1. Since PV is not readily available, a description of the scaling coefficient as a function
`of aortic pressure only is desirable. Thus, values of the scaling coefficient for the LCX system
`were plotted against aortic pressure alone in figure 4. A venous pressure of 5 mmHg was
`assumed based on a mean right atrial pressure of 3.7 mmHg from Pantely et al (1984). Similar
`results were observed in the LAD and RCA systems. Equations for best-fit lines correlating
`k with Pa and PV = 0 are summarized in table 1. The correlation coefficients (r) for the fits
`all equal 1.00 for all three arterial trees. The standard errors of estimate (SEE) for the fits are
`
`
`−1, 1.68 ml min−1 and 1.01 ml min−1 for the LCX, LAD and RCA, respectively.
`1.18 ml min
`A range of k values for venous pressures ranging from 0 to 10 mmHg is also provided. This
`range was selected based on the standard deviation of 2.5 mmHg in right atrial pressures
`in swine measured by Pantely et al (1984). The scatter in the data is due to the influence
`of venous pressure. For a given aortic pressure, larger k values correspond to lower venous
`pressures.
`
`3.2. Diseased model
`
`Figure 5 shows how inlet flow and V3/4 varied with different stenosis lengths and severities
`when aortic and venous pressures were held constant at 100 and 5 mmHg, respectively. The
`most severe stenosis (20 mm long 90% stenosis) decreased inlet flow by 98%, but decreased
`V3/4 by only 26%. However, an intermediate stenosis (10 mm long 70% stenosis) decreased
`inlet flow by only 35% and V3/4 by 6%. The rate of decrease in flow and volume increased
`substantially for stenoses greater than 60%. FFR values (FFRQ, FFRP, FFRV and FFRVR)
`were determined for different stenosis lengths and severities. Figure 6 compares the different
`FFR values for a 10 mm long stenosis with different percent diameter constrictions at Pa =
`100 mmHg and PV = 5 mmHg. A