J. Biomedical Science and Engineering, 2011, 4, 34-45
doi:10.4236/jbise.2011.41005 Published Online January 2011 (http://www.SciRP.org/journal/jbise/
JBiSE
).
Published Online January 2011 in SciRes. http://www.scirp.org/journal/JBiSE
Patients’ specific simulations of coronary fluxes in case of
three-vessel disease
Mahmoud Maasrani1, Issam Abouliatim2,3, Majid Harmouche2,3, Jean-Philippe Verhoye2,3,
Hervé Corbineau2,3, Agnès Drochon4
1Faculty of Sciences, Lebanese University, Tripoli, Lebanon;
2Department of Thoracic and Cardiovascular Surgery, Rennes Hospital Center, Rennes, France;
3Research Unit INSERM U642, Rennes, France;
4University of Technology of Compiègne, Compiègne, France.
Received 2 November 2010; revised 4 November 2010; accepted 8 November 2010.
ABSTRACT
In this work, we propose a model of the coronary
circulation based on hydraulic/electric analogy. This
model aims to provide quantitative estimations of the
distribution of flows and pressures across the coro-
nary network for patients with stenoses of the left
main coronary artery (LMCA), left anterior de-
scending artery (LAD) and left circumflex branch
(LCx), and chronic occlusion of the right coronary
artery (RCA), undergoing off-pump coronary sur-
gery. The results of the simulations are presented for
10 patients with various stenoses grades and collat-
eral supply. For each patient, the four revasculariza-
tion situations (no graft operating, pathological situa-
tion (0G); right graft only (1G), left grafts only (2G),
complete revascularization (3G)) are considered. It is
shown that: 1) the complete revascularization is fully
justified for these patients because neither the right
graft alone, nor the left grafts alone can ensure a suf-
ficient perfusion improvement for the heart; 2) the
capillary and collateral resistances (and the propor-
tion between them) have a major impact on the flows
and pressures everywhere in the network; 3) in the
presence of the left grafts, the flows in the native
stenosed arteries become low and this could promote
the development of the native disease in these
branches.
Keywords: Coronary Disease; Off-Pump Surgery;
Patients’ Specific Simulations; Lumped Parameter Model
1. INTRODUCTION
Bypass grafting is commonly performed to obtain myo-
cardial reperfusion distal to critical coronary stenoses or
thromboses. An accurate model of the coronary circula-
tion would be helpful for predicting the effects of
pathophysiological and pharmacological interventions,
especially bypass grafts. Developing such a model is a
complicated task due to the complexity of coronary
hemodynamics, especially in pathological situations.
Coronary artery diseases may induce the development of
a coronary collateral circulation, which is generally es-
timated from a patient’s angiogram. However, well de-
veloped collaterals are a risk factor for restenosis after
bypass grafting due to competing hemodynamic forces.
In this work, we study the case of severe coronary dis-
eases: patients have stenoses of the left main coronary
artery (LMCA), left anterior descending artery (LAD)
and left circumflex branch (LCx), and chronic occlusion
of the right coronary artery (RCA). In this clinical situa-
tion, the collateral circulation to the occluded artery is
difficult to ascertain from preoperative measurements
and it is debatable whether it is necessary to revascular-
ize the right artery.
In a previously published paper [1], we proposed a
model based on hydraulic/electric analogy that describes
the coronary artery system mathematically. This model
has been modified in order to include possible stenosis
of the LMCA and to take into account variable stenosis
coefficients on all left branches. In the present paper, we
provide the detailed hemodynamic results for 10 patients
with different stenoses severities. The simulations allow
to know the pressures and flow rates in the stenosed na-
tive arteries, the collateral branches, the capillary areas,
depending on the revascularization status (no grafts,
right graft only, left grafts only, complete revasculariza-
tion).
It is hoped that these computations will augment the
surgeons professional experience in decision making
process.

M. Maasrani et al. / J. Biomedical Science and Engineering 4 (2011) 34-45
35
2. MATERIALS AND METHODS
2.1. Clinical Measurements for Each Patient
Informed, signed consent was obtained from the patients
before participating in the study.
The reductions in diameter and area of the stenosed
arteries were estimated from angiographic observations,
before surgery.
The off-pump coronary surgical procedure has been
described previously [2]. The RCA is first revascularized
via a saphenous vein graft. Two series of measurements
are performed: Pao (aortic pressure), Pv (central venous
pressure), Pw (pressure distal to the RCA occlusion),
with the right graft clamped (0G); and Pao, Pv, QRCAg
(flow rate in the RCA graft) with the right graft opened
(1G). The left coronary arteries are then revascularized
via the internal thoracic arteries. Two additional series of
measurements are performed: Pao, Pv, Pw, QLADg and
QLCxg (flow rates in the LAD and LCx grafts) with the
right graft clamped (2G); and Pao, Pv, QLADg, QLCxg and
QRCAg with the right graft opened (3G). Flow rates are
measured with an ultrasonic transit time flowmeter
(Butterfly Flowmeter 2001; Medi-Stim, Oslo, Norway),
after hemodynamic stabilization.
2.2. Biomechanical Model
A lumped biomechanical model of this coronary network
is proposed in Figure 1. In this model, the capillaries are
represented by their hydraulic resistances (RLADc, RLCXc,
RRCAc are the resistances of the capillaries vascularized
by the LAD, LCx and RCA arteries, respectively). The
blood flow rates across the LAD, LCx and RCA capil-
laries are denoted by QLADc, QLCXc, QRCAc respectively.
Qcol1 and Qcol4 are the collateral flow rates from LAD
towards RCA (before and after LAD stenoses, respec-
tively), Qcol2 and Qcol5 are the collateral flow rates from
LCx (before and after LCx stenoses, respectively) and
Qcol3 is the collateral flow rate from the aorta towards the
RCA.
Full resolution of the fluid mechanics equations in
such a network is complicated. As suggested by several
authors, prediction of flows and pressures can be facili-
tated by the use of an analog electrical model [3].
2.3. The Electrical Model
The electrical analog model corresponding to the coro-
nary network of Figure 1 is shown in Figure 2. Each
segment of the coronary artery can be simulated by an
Figure 1. Simplified schematics of the coronary network in the case of patients with chronic
occlusion of the right coronary artery (RCA), and stenoses on the left arteries (LMCA, LAD
and LCx). The bypasses on the left arteries are performed with the internal mammary arteries
(IMAG). The RCA is revascularized by saphenous vein graft (SVG).

M. Maasrani et al. / J. Biomedical Science and Engineering 4 (2011) 34-45
36
Figure 2. Analog electrical model for the network shown in Figure 1. The grafts are repre-
sented with dotted lines.
equivalent analog circuit model with resistance R (hy-
draulic resistance of the vessel), capacitance C (compli-
ance of the vessel) and inductance L (inertia of the
flowing blood). The LMCA is represented by RLMCA,
CLMCA and LLMCA. The LAD is represented by RLAD,
CLAD and LLAD. The LCx is represented by RLCX, CLCX
and LLCX. The RCA is represented by RRCA, CRCA and
LRCA. Our coronary artery system was modeled in the
presence of bypasses. Pietrabissa et al. suggested some
representation of internal mammary artery grafts (IMAG,
used for left coronary artery bypasses) and saphenous
vein grafts (SVG, used for the RCA) that are supported
by physiological observations [4]. To take into account
tapering, the IMAG is artificially divided into two seg-
ments of equal length (70 mm) but of different diameters
(2.8 mm and 2 mm, respectively); consequently, it is
modeled by five elements: two resistances RIMAG1, RI-
MAG2; two coils LIMAG1, LIMAG2; and a capacitor CIMAG.
SVG is modeled by two elements: a resistance RSVG and
a coil LSVG. The SVG model does not include an electric
capacitance as experimental data confirm that when a
vein is exposed to arterial pressure it loses its high com-
pliance characteristics. The myocardial capillaries fed by
the left and right coronary arteries are represented only
by their resistances RLADc, RLCXc and RRCAc. This ap-
proximation is convenient since the resistive effects are
preponderant for small diameter vessels like capillaries
[3]. For the same reason, the collateral vessels are also
represented only by their resistances Rcoli, i = 1-5.
In this hydraulic/electric analogy, pressure and flow
rate correspond to electrical voltage and current, respec-
tively.
According to the network presented in Figure 2, the
flow rate in the LAD artery, QLAD, is defined as the sum
of the flow rate in the LAD graft, QLADg, (if it exists),
and of the flow rate in the stenosed native artery, QLAD1.
The same notations are used for the LCx branch.
The sum of the right and left coronary flows is de-
noted Qt:
tLADLCxRC
QQQ Q
A

 (1)
2.4. Parameter Determination
2.4.1. Vessel Resistance, Inductance and Compliance
As suggested by Wang et al. [5] and Pietrabissa et al. [4],
R, L and C can be calculated for each vessel segment as

M. Maasrani et al. / J. Biomedical Science and Engineering 4 (2011) 34-45
37
follows:
4
128 l
RD


 (2a)
2
4l
LD


 (2b)
3
4
Dl
CEh

 (2c)
where  = 4  10-3 kg.m-1.s-1 is the blood viscosity;  =
103 kg.m-3 is the blood density; E = 2  105 Pa is the
Young modulus of the vessel; l (m) is the vessel length;
D (m) is the vessel diameter and h (m) is the vessel wall
thickness (estimated as: h = 0.08D).
Table 1 shows the values of R, L and C for the left
and right coronary arteries and grafts. These values were
provided by Pietrabissa et al. [4] except for the data for
the RCA. In our study, the useful length of the RCA
corresponds to the part distal to the thrombosis. This has
been estimated as half of the LAD length and R, C and L
for the RCA were deduced from those provided by
Pietrabissa et al. for the LAD [4].
Left coronary stenoses were considered by varying the
parameters of specific segments of the net as follows [5]:
2
0
RR


 (3a)
32
0
CC

 (3b)
1
0
LL


 (3c)
where  = 1 – p, p is the percentage of area reduction of
the stenosed vessel. R0, C0, and L0 are the values when p =
0.
2.4.2. Capillary Resistances
These resistances (RLADc, RLCXc and RRCAc) are patient–
specific. Their determination is performed using the ex-
perimental data of the case (3G) for each patient. The
detailed calculations are given in [6].
Ta b le 1 . Values of resistance R, inductance L and capacitance
C for the vessels represented in the model.
Vessel type Resistance
(mmHg.s/ml) Inductance (mmHg.s2/ml)
Capacitance
(ml/mmHg)
LMCA 0.1 0.02 0.002
LAD 0.5 0.03 0.0015
LCx 0.3 0.02 0.0011
RCA 0.3 0.02 0.0008
IMAGI 1.4 0.08 0.0054
IMAGII 5.3 0.17
SVG 0.2 0.04
2.4.3. Collater al Resi st ances
Due to the difficulty of determining the exact character-
istics of the collateral pathways, it was assumed that all
the collateral resistances are the same [1]:
12345colcol colcol colcol
RRRRRR

 (4)
This resistance is also specific to each patient.
In the case of RCA occlusion and three vessel disease,
the value of Rcol is strongly related to the value of pres-
sure Pw. Thus, the Pw value measured in case (2G) is
used as a convergence criterion to numerically determine
the convenient value of Rcol for the patient. The numeri-
cal simulations are performed using the Matlab Simulink
program. The value of Rcol is changed until the calcu-
lated Pw value converges towards the clinically measured
one.
2.5. Flow Rate and Pressure Simulations
2.5.1. Aortic Pressur e
The input of the model is the aortic pressure wave, Pao(t),
measured for each patient and each situation (0G, 1G, 2G,
3G).
2.5.2. M at lab Simulations
Once the model parameters are determined, flow and
pressure predictions can be performed in any branch of
the model and for all surgical cases. The calculated
flows and pressures are time-dependent, but we focus on
average cardiac cycle values. This is consistent with the
fact that the collected clinical data are also average car-
diac cycle values.
The influence of ventricular contraction upon coro-
nary vascular bed resistance and compliance is not taken
into account in our simulations. The impact of this as-
sumption is probably less important in the case of our
study than it would be for healthy patients. Wang et al.
[5] have indeed shown that when severe stenoses exist,
the significance of the collapse effect of intramural ar-
teries due to myocardial contraction is reduced.
3. RESULTS
3.1. Stenoses Severity
The percentages of area reduction of the stenosed vessels
for each patient are given in Table 2. For all patients, the
RCA is totally occluded. Patient 7 has no stenosis on
LAD; Patient 9 and 10 have no stenosis on LCx. Patients
1, 4 and 5 have moderate lesions on LMCA. Patient 1
has a very severe lesion on LAD.
3.2. Capillary and Collateral Resistances
The values of the capillary and collateral resistances for
each patient are presented in Table 3. Despite the fact
that the description of the coronary network presented in

M. Maasrani et al. / J. Biomedical Science and Engineering 4 (2011) 34-45
38
Tab le 2. Percentage of area reduction of the stenosed vessels
for each patient.
Patient % area LMCA % area LAD % area LCx
1 26 99 90
2 46 89 95
3 91.6 84.8 95.6
4 19 86 97
5 20 88 92
6 85 94 82
7 80 0 85
8 87 70 90
9 83 78 0
10 75 93 0
Table 3. Values of the capillary resistances, RLADc , RLCx c ,
RRCAc, and collateral resistances, Rcol, for the patients consid-
ered in the study. All these values are given in mmHg.s/ml.
Patient RLADc R
LCxc R RCAc R col
1 83.3 207.9 54.1 160
2 174.6 210.9 96.9 430
3 213 94.2 62.8 350
4 47.5 119.1 147.2 565
5 175.3 68.7 56.1 205
6 240.4 135.5 117.6 1055
7 50.2 118.4 76 650
8 77.6 196 347.6 970
9 374.8 33.7 80.7 420
10 155.9 62.1 213.8 405
Mean ± σ 159.3 ± 96.6 124.6 ± 59.9 125.3 ± 87.6 521 ± 282.3
this paper is more precise than that presented in the pre-
vious paper [1], the mean values obtained for these 10
patients are in the same range than those previously pub-
lished. As discussed in [1], these values are in good
agreement with literature data. Important values of Rcol
indicate that the collateral network is not very developed
or not functional. The values of resistances presented in
Ta b le 3 are affected by a great variability. This finding
supports the motivation of our work: the necessity to
provide patients’ specific simulations.
3.3. Flows in LAD Branch and in LCx Branch
In the following, all the quantities are calculated values,
except when indicated with the notation “Cli” (used to
denote measured clinical values).
The values of the flow rates in the native stenosed
LAD (QLAD1), in the LAD graft (QLADg) and the total
flow rate in the LAD branch (QLAD) are given in Table
4.
The same type of values, for the LCx branch, are
given in Table 5.
One can notice that the flow rates in the native
stenosed artery, QLAD1 or QLCx1, is only slightly affected
by the presence of the right graft (mean value of QLAD1
in the case (1G) = 25.3 ± 24.2 and in the case (0G) =
28.4 ± 24.1; mean value of QLCx1 in the case (1G) = 28.0 ±
23.4 and in the case (0G) = 31.3 ±2 3.5). This slight de-
crease is due to the fact that, in the presence of the right
graft, the pressure drop across the LAD or LCx stenosis
is slightly modified (because the pressure gradients
across the collaterals are modified).
On the contrary, there is an important decrease of
QLAD1 and QLCx1 in the presence of the left grafts (mean
value of QLAD1 in the case (2G) = 7.9 ± 17.2, in the case
(3G) = 7.4 ± 15.7 and in the case (0G) = 28.4 ± 24.1;
mean value of QLCx1 in the case (2G) = 11.4 ± 18.8, in
the case (3G) = 12.5 ± 22.3 and in the case (0G) = 31.3 ±
23.5). This decrease is less important for Patient 7 (no
stenosis on LAD) and for Patient 9 and 10 (no stenosis
on LCx). When the LAD graft or LCx graft are present,
the pressure drops across the LAD or LCx stenosis are
reduced and since the hydraulic resistances of these
stenosed arteries remain the same, the flow rate drops.
Some sort of flow compensation appears between the
graft and the native artery, especially if the native artery
is not too severely obstructed (for example, this remark
does not apply for the Patient 1, whose LAD branch is
almost totally obstructed). Overall, the perfusion of the
LAD territory (QLAD) or of the LCx territory (QLCx) is
improved in the presence of the left grafts, but this im-
provement remains moderate (lower than 10ml/min).
This finding is important because it means that the graft
could thus promote progression of native disease. It is
hoped that these results will bring some new arguments
to the controversy that exists regarding the competitive
flows between the native stenosed artery and the graft
[7-11].
For Patient 9, a negative QLAD1 flow is obtained in the
presence of the left grafts (cases (2G) and (3G)).
For this patient, the LAD capillary resistance, RLADc,
is quite high when compared to RLCxc, and is of the same
order as Rcol. Besides, the LAD stenosis is not too severe;
so that the blood brought by the LAD graft will flow
preferentially in the native LAD branch rather than
through RLADc or Rcol4. The same explanation prevails for
Patient 3, but this patient has more severe obstructions of

M. Maasrani et al. / J. Biomedical Science and Engineering 4 (2011) 34-45
39
Table 4. Values of the flow rates (ml/min) in the native ste-
nosed LAD (QLAD1), in the LAD graft (QLADg), and total flow
rate in the LAD branch (QLAD). These values are given for each
patient, in the four different revascularization situations (0G,
1G, 2G, 3G).
Patient QLAD1 (0G) QLAD1 (1G)QLAD1 (2G) QLAD1 (3G)
1 0.1 0.1 0 0
2 23.6 19.7 3.6 3.3
3 20.7 15.6 -1.7 2.5
4 56.2 56.2 15 15.7
5 25.9 18 4.3 3.5
6 11.6 8.3 0.3 0.3
7 77.5 73.2 52.1 46.1
8 45.7 44.6 14.9 14.5
9 10.4 6.5 -10.2 -12.2
10 12.1 10.7 0.5 0.5
Mean ± σ 28.4 ± 24.1 25.3 ± 24.27.9 ± 17.2 7.4 ± 15.7
Patient QLADgCli (2G) QLADg (2G) QLADgCli (3G) QLADg (3G)
1 34 39.6 40 38.9
2 23 24 21 21.1
3 22 28.5 19 19
4 59 54.3 57 56.8
5 24 22.3 18 18.3
6 11 17.9 14 14.3
7 28 36.3 28 28.2
8 38 31.8 28 28.1
9 24 23.1 23 22.9
10 20 21.7 18 18
Mean ± σ 28.3 ± 13.1 30.0 ± 11.026.6 ± 13.0 26.6 ± 12.8
Patient QLAD (0G) QLAD (1G)QLAD (2G) QLAD (3G)
1 0.1 0.1 39.3 38.6
2 23.6 19.7 27.1 24
3 20.7 15.6 26.1 21.1
4 56.2 56.2 69.1 72.2
5 25.9 18 25.8 21.1
6 11.6 8.3 17.8 14.3
7 77.5 73.2 87.8 73.8
8 45.7 44.6 46.5 42.3
9 10.4 6.5 12.5 10.3
10 12.1 10.7 21.9 18.2
Mean ± σ 28.4 ± 24.1 25.3 ± 24.237.4 ± 24.1 33.6 ± 23.0
Table 5. Values of the flow rates (ml/min) in the native
stenosed LCx (QLCx 1 ), in the LCx graft (QLCxg ), and total flow
rate in the LCx branch (QLCx ). These values are given for each
patient, in the four different revascularization situations.
Patient QLCx1 (0G)QLCx1 (1G) QLCx1 (2G) QLCx1 (3G)
1 19.9 13.7 3.5 2.8
2 15.4 12.4 1.3 1.2
3 15.1 13.9 0.7 1.5
4 8 7.4 0.6 0.5
5 37.5 31.8 6.3 6.2
6 29.8 22.6 7.2 7.3
7 31.2 27.5 2.6 2.3
8 17.3 15.9 0 0.2
9 87.1 84.8 54.7 69.2
10 51.9 49.5 37.1 33.9
Mean ± σ31.3 ± 23.528.0 ± 23.4 11.4 ± 18.8 12.5 ± 22.3
Patient QLCxgCli (2G)QLCxg (2G) QLCxgCli (3G) QLCxg (3G)
1 27 17.6 14 13.9
2 32 22.4 19 19.2
3 48 49.2 45 44.7
4 40 30.2 30 30.1
5 56 46.6 46 45.6
6 12 22.6 18 18.1
7 43 37.5 29 29.2
8 16 19.9 17 17.2
9 60 41.1 45 44.6
10 7 15.9 13 13
Mean ± σ34.1 ± 18.430.3 ± 12.4 27.6 ± 13.4 27.6 ± 13.3
Patient QLCx (0G)QLCx (1G) QLCx (2G) QLCx (3G)
1 19.9 13.7 20.9 16.5
2 15.4 12.4 23.2 19.9
3 15.1 13.9 49.1 45.8
4 8 7.4 30.5 30.2
5 37.5 31.8 52.2 51
6 29.8 22.6 29.5 25.1
7 31.2 27.5 39.5 31.1
8 17.3 15.9 19.6 17.1
9 87.1 84.8 95.4 113.4
10 51.9 49.5 52.7 46.7
Mean ± σ31.3 ± 23.528.0 ± 23.4 41.3 ± 22.9 39.7 ± 28.8

M. Maasrani et al. / J. Biomedical Science and Engineering 4 (2011) 34-45
40
LAD and LCx, so that retrograde flow is lower. Some
situations of retrograde flow have been reported previ-
ously in the literature: for example, in arterial conduits
grafted to coronary arteries with lower grade stenosis [9,
11,12]. However, the situation studied in this paper is
somewhat different because of the LMCA stenosis.
This observation about Patient 9 is consistent with the
high flow rates predicted in its LCx branch (this patient
has no stenosis on LCx and a low RLCxc value). Patient
10 has also no stenosis on LCx, but he does not present
such a disproportion between RLCxc and RLADc.
This demonstrates that the values of the capillary and
collateral resistances have a major impact on all the
pressures and flow rates, including the flow rates in the
grafts. For example, an elevated value of QLADg is ob-
tained for Patient 4 because RLADc is specially low for
this patient (on the contrary, Patient 4 has severe stenosis
on LCx; consequently, low QLCx flow rates are predicted
for this patient in the case 0G and 1G). Patients who
have rather high values of RLCxc have rather low flow
rates in the LCx graft (for example, Patients 1, 2, 6, 8).
Conversely, patients who have rather low values of RLCxc
have rather large flow rates in the LCx graft (for exam-
ple, Patients 3, 5, 9). Such a correlation between graft
flow and grafted perfusion area has been demonstrated
previously by Hirotani et al. [13].
As we previously found in [14], a small decrease (a
few ml/min) of the flow rates in the left grafts can be
noticed in the case (3G) compared to the case (2G).
3.4. Flows and Pressures in the RCA Branch
The values of the pressures distal to the thrombosis on
RCA (Pw), of the flow rates in the RCA graft (QRCAg),
and of the flow rates in the right capillary area (QRCAc)
are presented in Table 6.
As we previously found in [2], no significant change
in the mean value of Pw in the presence of the left grafts
(case (2G) compared to case (0G)) can be demonstrated.
Moreover, no evident correlation appears between the
values of Pw and the corresponding flow rates in the
right capillary area, QRCAc .
The flow rates QRCAg (1G) are slightly higher than
QRCAg (3G) (mean value of QRCAg in the case (1G) = 51.5
± 27.6 and in the case (3G) = 44.7 ± 23.2) because of the
negative collateral flows that exist in the case (1G) (For
example, for Patient1, because of the severe obstruction
of the LAD artery, Qcol4 is important (= -15ml/min, see
Table 7)).
For all patients, the perfusion of the right territory,
QRCAc, is improved in the presence of the right graft
(mean value of QRCAc in the case (1G) = 44.7 ± 23.5, and
in the case (3G) = 43.6 ± 22.8; mean value of QRCAc in
the case (0G) = 19.3 ± 10.3, and in the case (2G) = 20.5 ±
Table 6. Values of the pressures distal to the thrombosis on
RCA (Pw, in mmHg), of the flow rates (ml/min) in the RCA
graft (QRCAg), and in the right capillary area (QRCAc).
Patient PwCli (0G)Pw (0G) PwCli (2G) Pw (2G)
1 35 31.6 31 31.3
2 49 44.5 49 49
3 40 33.1 40 40.1
4 43 38.3 42 42.3
5 53 41.4 36 35.7
6 35 28.2 28 28.4
7 29 37 40 40.2
8 46 45.1 43 44.5
9 37 37.9 40 40
10 47 44.9 48 48.2
Mean ± σ41.4 ± 7.538.2 ± 5.9 39.7 ± 6.6 40.0 ± 6.7
Patient QRCAg (1G)QRCAgCli (3G) QRCAg (3G)
1 88.2 66 67.6
2 52.4 45 45.4
3 86.6 74 75.3
4 35.1 26 27
5 85.4 69 70.5
6 31.9 30 30.3
7 55.9 51 52
8 14.6 10 10.5
9 45.6 51 53.2
10 18.9 14 14.8
Mean ± σ51.5 ± 27.643.6 ± 22.8 44.7 ± 23.2
Patient QRCAc (0G)QRCAc (1G) QRC Ac (2G) QRCAc (3G)
1 31.5 70.3 34.6 66
2 21.9 47.4 22.3 45
3 25.8 73.9 31.6 74
4 11.9 28 13.2 26
5 38.8 75.3 34.9 68.9
6 11.3 30 11.4 30
7 18.1 53.4 20.7 51
8 6.7 12.1 6.6 10
9 17.7 41.4 19.3 51
10 9.8 15.1 10.7 14
Mean ± σ19.3 ± 10.344.7 ± 23.5 20.5 ± 10.3 43.6 ± 22.8

M. Maasrani et al. / J. Biomedical Science and Engineering 4 (2011) 34-45
41
Ta b l e 7 . Values of the collateral flow rates (ml/min), for each
patient, in the four different revascularization situations.
Patient Qcol1 (0G) Qcol1 (1G) Qcol1 (2G) Qcol1 (3G)
1 10.6 0.1 7.4 0.1
2 5.6 0 4.6 0.1
3 6.2 -1.4 6.2 -0.1
4 3.9 0 2.8 0
5 10.4 0.1 7.4 0.1
6 2.5 -0.2 2.3 0
7 3.6 -0.5 4.1 -0.2
8 1.4 -0.4 1.3 -0.1
9 3.4 -1 3.7 -0.6
10 2.5 -0.3 2.1 -0.1
Mean ± σ 5.0 ± 3.2 -0.4 ± 0.5 4.2 ± 2.2 -0.1 ± 0.2
Patient Qcol3 (0G) Qcol3 (1G) Qcol3 (2G) Qcol3 (3G)
1 10.7 0.1 7.4 0.1
2 5.7 0.1 4.6 0.1
3 8.9 0.1 6.9 0.1
4 3.9 0.1 2.8 0
5 10.5 0.1 7.4 0.1
6 2.8 0 2.4 0
7 4.3 0.1 4.4 0.1
8 1.9 0 1.5 0
9 4.6 0.1 4.3 0.1
10 2.8 0 2.4 0
Mean ± σ 5.6 ± 3.3 0.1 ± 0.1 4.4 ± 2.2 0.1 ± 0.1
Patient Qcol4 (0G) Qcol4 (1G) Qcol4 (2G) Qcol4 (3G)
1 -7.0 -15.6 5.8 -1.4
2 3.3 -1.9 4.2 -0.3
3 4.9 -2.4 6.3 -0.3
4 1.5 -2.4 2.2 -0.7
5 6.2 -2.8 6.7 -0.4
6 1.1 -1.2 2.3 -0.1
7 3.6 -0.5 4.0 -0.3
8 1.2 -0.7 1.2 -0.2
9 3.1 -1.16 3.9 -0.3
10 -0.4 -2.8 2.0 -0.3
Mean ± σ 1.7 ± 3.6 -3.1 ± 4.4 3.9 ± 1.9 -0.4 ± 0.4
Patient Qcol5 (0G) Qcol5 (1G) Qcol5 (2G) Qcol5 (3G)
1 6.6 -2.6 6.7 -0.4
2 1.8 -3.1 4.3 -0.2
3 -0.4 -7.5 5.9 -0.8
4 -1.2 -4.7 2.5 -0.3
5 1.5 -7.4 5.9 -1.3
6 2.3 -0.4 2.2 -0.1
7 3.0 -1.0 4.0 -0.3
8 0.9 -0.9 1.3 -0.1
9 3.3 -1.0 3.7 -0.7
10 2.4 -0.3 2.1 -0.2
Mean ± σ2.0 ± 2.1 -2.9 ± 2.8 3.9 ± 1.8 -0.4 ± 0.4
10.3). However, for patients 8 and 10, this perfusion
appears to be critically low. This seems to be related to
their high values of the right capillary resistance RRCAc.
For patients 4 and 6, QRCAc is low in the case (0G) and
(2G), and this seems to be related to high values of the
collateral resistance Rcol.
For patients 7, 9 and 10, the important QLAD1 or QLCx1
flow rates due to the absence of stenoses on the LAD or
LCx branch are not associated with any particular im-
provement of the right territory perfusion.
More generally, it appears from the results of Table 6,
that, in the presence of the left grafts, the QRCAc flow
rates are not significantly modified (mean value of QRCAc
in the case (2G) = 20.5 ± 10.3, and mean value of QRCAc
in the case (0G) = 19.3 ± 10.3). Most of the blood
brought by the left grafts is used to perfuse the distal left
territories (QLADc and QLCxc). This observation is consis-
tent with our previous results [1,15].
3.5. Collateral Flows
The results of the collateral flows are presented in Table
7, for each patient and each revascularization situation.
Because of the assumption that the collateral resistances
are the same (Eq.4), we have PM – Pw= Rcol. Qcol1 = Rcol.
Qcol2 (see Figure 2); consequently, Qcol1 = Qcol2.
In the situation (0G) and (2G), the flow is delivered to
the right territory via the collaterals only. However, it
can be seen from Ta b le 7 that these collateral flows re-
main low in all cases. The presence of the left grafts
does not really improve the collateral flows. In the case
of Patient1, we obtain a negative Qcol4 value in the
pathological situation (0G). This indicates that blood
flows from the right artery to the LAD, which is almost
totally occluded. The same remark can be made for Pa-
tients 3 and 4, whose collateral flow rates Qcol5 (0G) are

M. Maasrani et al. / J. Biomedical Science and Engineering 4 (2011) 34-45
42
negative because severe stenoses are present on the LCx
artery.
Such reverse collateral flow also exists when the right
graft is present, especially in the case (1G): due to the
presence of the right graft, the right territory is better
perfused and the pressures in the right area become
higher than those of the left territories. This has been
also demonstrated by other authors: Miyamoto et al. [16]
have shown that revascularization of the receiving artery
can reverse the pressure gradient across the collateral
network, establishing collateral flow in the opposite di-
rection.
Qcol1 and Qcol2, in the case 0G, are slightly higher than
Qcol1 and Qcol2, in the case 2G (mean value of Qcol1 in the
case (0G) = 5.0 ± 3.2, and in the case (2G) = 4.2 ± 2.2).
This is consistent with the fact that, in the presence of
the left grafts, the flow in the native artery decreases,
and consequently, Qcol1 and Qcol2 decrease.
For patients with moderate lesions on LMCA and se-
vere stenoses on LAD and LCx (for example, patient 1
and 5), the blood is forced to flow through Rcol1 and Rcol2,
yielding values of Qcol1 and Qcol2 in the case (0G) higher
than those obtained for other patients. For these patients,
Qcol3 is high also, because Rcol is low. For the same rea-
son, the collateral flows of the case (2G) for these pa-
tients are somewhat higher than those obtained for other
patients.
We also notice that, in the case (0G) and (2G), Qcol3
directly varies as the pressure drop (Pao-Pw).
In the case (3G), the collateral flows become negligi-
ble. Loss of collateral flow after revascularization agrees
with the findings of previous studies: Wang et al. found
that collateral flow disappeared in 12/14 patients after
grafting of the RCA [17]. Werner et al. also demon-
strated regression of collateral function after recanaliza-
tion of chronic total coronary occlusions [18].
3.6. Total Flow Qt
The results obtained for Qt are shown in Ta bl e 8. It ap-
pears that the better perfusion will be obtained with the
three grafts. In the case (2G), there is an improvement of
Qt, compared to the pathological case (0G) (mean value of
Qt in the (2G) case = 91.4 ± 26.4, and in the (0G) case =
75.3 ± 24.7), but this improvement is related to an in-
crease of QLAD and QLCx, and the presence of the left
grafts does not really improve QRCAc. Conversely, in the
case (1G), QRCAc is improved by the presence of the right
graft, but QLAD and QLCx remain rather the same as in the
pathological case. This demonstrates that, for the pa-
tients considered in this study, complete revasculariza-
tion (with the 3 grafts) is fully justified. However, for
Patient 8 and 10 (and in a lesser extent, 6), the revascu-
larization yields a very small improvement of Qt.
Table 8. Values of the total flow rates Qt (ml/min), for each
patient, in the four different revascularization situations.
Patient Qt (0G) Qt (1G) Qt (2G) Qt (3G)
1 51.9 102.2 82.3 122.9
2 55.9 84.5 64.1 89.4
3 57.1 113.3 94.5 141.9
4 75.8 98.7 108.1 129.4
5 94.5 135.3 100.2 142.8
6 49.3 62.5 54.2 69.6
7 120.2 155.6 139.9 156.4
8 67.7 74.2 70.2 69.7
9 108.8 134.9 119.6 175.7
10 71.7 78.5 81.2 79.4
Mean ± σ75.3 ± 24.7104 ± 30.5 91.4 ± 26.4 117.7 ± 38.2
4. DISCUSSION
To the best of our knowledge, the current study is unique
because of the specific three vessel disease situation, and
because our simulations take into account simultane-
ously the effect of revascularization, the grade of native
arteries stenoses and the collaterality.
The values presented in this paper for the flow rates,
pressures, capillary and collateral resistances, are in
complete agreement with the results previously obtained
with a less sophisticated version of the model [1]. The
range of these values and the main assumptions and
limitations of the model have already been discussed in
[1].
In spite of these limitations, the clinical relevance of
our results seem good; the results confirm the surgeons
professional experience and agree with the data and
analysis that can be found in the literature.
The most important features shown by the calcula-
tions can be summarized as follows:
 Important variability between individuals for their
capillary resistances and collateral resistances
(RLADc, RLCxc, RRCAc, Rcol). This supports the moti-
vation of our work to develop patient’s specific
simulations.
 The complete revascularization is fully justified
for these patients because neither the right graft
alone, nor the left grafts alone can ensure a suffi-
cient perfusion improvement for the heart. The left
grafts mainly contribute to a better perfusion of the
left territories, instead of the collateralized right
region. In all cases, the contribution of the collat-
eral flows remain low (a few ml/min).

M. Maasrani et al. / J. Biomedical Science and Engineering 4 (2011) 34-45
43
 When the three grafts are functional, the collateral
channels play no more role and a severe reduction
of the flow rates in the native left arteries is dem-
onstrated (depending on their degree of occlusion).
The left grafts could thus promote progression of
the native disease.
 The values of the capillary resistances and collat-
eral resistances have a major impact on the flows
and pressures everywhere in the network and it
appears that each variable depends on all the oth-
ers. For example, it is difficult to study separately
the influence of the LAD stenosis or LCx stenosis,
because the flow in the corresponding branch also
depends on the LMCA stenosis, on the capillary
resistance RLADc or RLCxc, on the collateral resis-
tance Rcol, on the pressure difference between the
left and right branches, on the aortic and venous
pressures, …
However, further work is necessary to improve the
physiological relevance of the present model:
 Improve the representation of collateral vessels
(the assumption that all the collateral resistances
are the same is probably not very realistic)
 Take into account the capillary resistance variation
during the cardiac cycle, due to myocardial con-
traction
In a further step, the model could then be transformed
in order to work with pre-operative clinical measure-
ments and thus become a full predictive model.
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NOMENCLATURE
LMCA: left main coronary artery
LAD: left anterior descending artery
LCx: left circumflex branch
RCA: right coronary artery
Pao: aortic pressure
Pv: central venous pressure
Pw: pressure distal to the RCA occlusion
RLADc: resistances of the capillaries vascularized by the
LAD artery
RLCXc: resistances of the capillaries vascularized by the
LCx artery
RRCAc: resistances of the capillaries vascularized by the
RCA artery
QRCAg: flow rate in the RCA graft
QLADg: flow rate in the LAD graft
QLCxg: flow rate in the LCx graft
QLAD1: flow rate in the native stenosed LAD
QLCx1: flow rate in the native stenosed LCx
QLADc: blood flow rate across the LAD capillaries
QLCXc: blood flow rate across the LCx capillaries
QRCAc: blood flow rate across the RCA capillaries
Qcol1: collateral flow rate from LAD towards RCA before
LAD stenosis
Qcol4: collateral flow rate from LAD towards RCA after
LAD stenosis
Qcol2: collateral flow rate from LCX before LCx stenosis
Qcol5: collateral flow rate from LCX after LCX stenosis
Qcol3: collateral flow rate from the aorta towards the RCA
R: resistance
C: capacitance
L: inductance
IMAG: internal mammary artery graft
SVG: saphenous vein graft
ITA: internal thoracic artery