Estimating absolute aortic pressure using MRIand a one-dimensional model

Maya Khalifé a,n, Astrid Decoene b, Filipa Caetano b, Ludovic de Rochefort a,Emmanuel Durand a, Dima Rodríguez a

a Imagerie par Résonance Magnétique Médicale et Multi-Modalités (IR4M), Université Paris-Sud - CNRS, UMR 8081, Orsay, Franceb Laboratoire de Mathématiques d'Orsay, Université Paris-Sud - CNRS, UMR 8628, Orsay, France

a r t i c l e i n f o

Article history:Accepted 22 July 2014

Keywords:Cardiovascular imagingMRIAortaBlood pressureOne-dimensional modelComplianceNon-invasive

a b s t r a c t

Aortic blood pressure is a strong indicator to cardiovascular diseases and morbidity. Clinically, pressuremeasurements are done by inserting a catheter in the aorta. However, imaging techniques have beenused to avoid the invasive procedure of catheterization. In this paper, we combined MRI measurementsto a one-dimensional model in order to simulate blood flow in an aortic segment. Absolute pressure wasestimated in the aorta by using MRI measured flow as boundary conditions and MRI measuredcompliance as a pressure law for solving the model. Model computed pressure was compared tocatheter measured pressure in an aortic phantom. Furthermore, aortic pressure was estimated in vivo inthree healthy volunteers.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

To date, blood pressure (BP) is one of the most useful clinicalindicator of cardiovascular disease. Hypertension, more specifi-cally, is a leading predictor of death in atherosclerosis diseasesworldwide (Cohn et al., 2004; P.S. Collaboration, 2002). Therefore,measuring BP is of great interest for diagnosis and risk preventionof cardiovascular events. An elevated pressure gives informationabout the aortic state, the presence of atherosclerotic plaques,stenosis, calcification or aneurisms. In a clinical routine, a sphyg-momanometer is used to measure systolic and diastolic brachialpressure. However, due to reflexion in the distal arteries, the aorticpressure waveform is altered while traveling through the vascularsystem. Thus distortion of the wave shape as well as systolicamplification occurs on the systolic pressure measured in thebrachial artery (O'rourke et al., 1968; Park and Guntheroth, 1970;Salvi, 2012). Although models and transfer functions to linkbrachial BP to aortic pressure exist (Chen et al., 1996, 1997;Liang, 2014), wave reflection in the arterial system makes itdifficult to reproduce the wave contour with great fidelity fromsuch methods. Until now, the gold-standard of aortic pressuremeasurement is catheterization, which is invasive and not repea-table in a routine procedure (Murgo et al., 1980; Skinner and

Adams, 1996). In recent years, imaging techniques have been usedto assess pressure gradients from velocity or acceleration mapsand its combination with fluid mechanics equations has beenexploited in order to measure the BP non-invasively. DopplerUltrasound (US) used to measure blood velocity in the arterieswas associated to the standard simplified Bernoulli equation inorder to determine pressure differences between two measure-ment sites. This latter technique is reported to be user-dependentand error-prone due to the wave angle of incidence (Zananiri et al.,1993; Muhler et al., 1993), hence its accuracy in determining themaximum velocity in the artery is debatable. Also, extending US toother situations is limited by the inapplicability of Bernoulliequation to unsteady flows (Yang et al., 1996). Phase-Contrast(PC) MRI allows accurate encoding of the blood velocity in thearteries in the three directions; hence, it has been largely used fornon-invasive pressure estimation. Some authors computed pres-sure differences using the Poisson equation (Yang et al., 1996),others integrated the Navier–Stokes (NS) equations using MRIvelocity maps (Tyszka et al., 2000; Thompson and McVeigh, 2003;Bock et al., 2011; Ebbers et al., 2001)or acceleration maps to avoidcomputational errors arising from velocity derivation (Buyenset al., 2005). These methods compute a pressure gradient, and toestimate an absolute pressure, require a zero-pressure referencepoint which has to be measured with a catheter, or user-defined ina gross assumption. Consequently, these methods are not analternative to catheter measurements, which remain more accu-rate. In this work, we propose a non-invasive technique to extract

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Journal of Biomechanics

http://dx.doi.org/10.1016/j.jbiomech.2014.07.0180021-9290/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author.E-mail address: mayakhal@gmail.com (M. Khalifé).

Journal of Biomechanics 47 (2014) 3390–3399

absolute pressure in a straight artery from MR velocimetry using abiomechanical one-dimensional (1D) model as proposed byFormaggia et al. (2003). Although a three-dimensional (3D) modelgives a more complete and realistic reproduction of the aortic flow,1D models are able to describe the non-linear flow behavior inlarge elastic vessels (Hughes and Lubliner, 1973). As these modelsare reasonably accurate, they are widely used for aortic flowsimulations. Their accuracy has been assessed by comparison withexperimental data acquired in a tube phantom (Bessems et al.,2008), in a distributed arterial model (Olufsen et al., 2000;Alastruey et al., 2011) and in vivo (Reymond et al., 2009; Alastrueyet al., 2009). Furthermore, the 1D model relies on the establishmentof a pressure law consisting of a relation between pressure andvessel section area. The pressure laws used in 1D model equationsare determined experimentally with invasive measurements orestimated using complex algorithms. Additionally, these pressurelaws are complicated and involve the determination of multipleparameters. Here, we propose a pressure law based on the aorticcompliance which reflects arterial elasticity and can be determinednon-invasively with MRI. Using this pressure law, we coupled the1D model with realistic boundary conditions measured by MRI toestimate absolute pressure in the aortic segment. The derivedmodel was tested on a straight compliant phantom and computedpressure was compared to experimental pressure measurementsrecorded simultaneously with the MRI acquisition. The model wasalso tested on a real-sized compliant aortic phantom. Then, themodel is used to estimate BP on healthy volunteers.

2. Methods

The 1D-model, studied in Formaggia et al. (2003), is a reduced modeldescribing blood flow in arteries and its interaction with wall motion. The arteryis considered as a cylindrical compliant tube of length L and radius R (R{L). Themodel derivation approach consists of integrating the NS equations on a genericsection S. Some simplifying assumptions are made:

� the model assumes axisymmetry� the wall displacement is supposed to solely be in the radial direction� pressure is assumed to be uniform in each section� the axial velocity uz is predominant.

For large arteries such as the aorta, it is a safe assumption to consider a flat velocityprofile for the boundary layer which is very thin compared to the vessel radius(Olufsen et al., 2000).

The main variables of the problem are (Fig. 1)

� axial section area A

Aðt; zÞ ¼ZSðt;zÞ

dσ ð1Þ

� mean flow Q

Q ðt; zÞ ¼ZSuz dσ ð2Þ

� blood pressure Pðt; zÞ,

where dσ denotes the area element. Their evolution is described by the momentumconservation and the mass conservation equations, while considering a constantviscosity:

∂A∂t

þ∂Q∂z

¼ 0 ð3Þ

∂Q∂t

þ ∂∂z

Q2

A

!þAρ∂P∂z

þKrQA

� �¼ 0 ð4Þ

Kr is the friction coefficient; for a flat profile in blood flow problems, Kr ¼ 22πν,where ν is the vessel wall kinematic viscosity (Formaggia and Veneziani, 2003), ρ isthe blood density.

2.1. Pressure law

To close the system, a relation between the section area A and the pressure P isdefined. This pressure law depends on section area A0ðzÞ ¼ πR2

0ðzÞ at time t¼0 andon a set of parameters β¼ ðβ0;β1 ;…;βnÞ related to the vessel wall's physical andmechanical properties. Pext is the external pressure exerted by the environment ofthe vessel whereas P is the intravascular pressure. The pressure law should meetthese conditions:

� ∂P∂A

40

� when A¼ A0, P ¼ Pext .

Some pressure laws are proposed in the literature to link the pressure behavior to thesection area. A frequently used relation between A and P is stated as follows (Quarteroniand Formaggia, 2004; Alastruey, 2006):

P�Pext ¼ β0

ffiffiffiA

p�

ffiffiffiffiffiffiA0

pA0

ð5Þ

β0 ¼ffiffiffiffiπ

ph0E=ð1�ξ2Þ using Young's modulus E and the vessel thickness h0 and

Poisson's ratio ξ¼ 0:5 for an incompressible material deformed elastically at smallstrains.

A more general law proposed in Hayashi et al. (1980) and Smith et al. (2000) iswritten as

P�Pext ¼ β0AA0

� �β1

�1

" #: ð6Þ

The parameters’ β0 and β1 values can be obtained either by fitting experimentalpressure vs. section measurements (Smith, 2004) or by solving an inverse problemwith a 3D-model solution (Martin et al., 2005; Dumas, 2008). Hence, these lawscannot be determined non-invasively, and need knowledge about the vesselproperties. Additionally, they seem too complex to determine during a clinicalapplication. Finding a simple non-invasive way to determine a pressurelaw is of great interest, consequently, we turned to the compliance. Indeed, underphysiological conditions, the aorta section deformation is commonly assumedto be linked to the intravascular pressure by the aortic compliance (Langewouterset al., 1984).

Aortic compliance establishes a linear relation between the pressure and thesection area; it represents the arterial wall's ability to deform in response to apressure variation (Conrad, 1969). Also, it is clinically used and can be estimatednon-invasively by measuring the pulse wave velocity (PWV) in MRI. Hence, itprovides a simple and non-invasive pressure law that can be applied in vivo and, asit includes compliance changes, is patient-specific.

In fact, compliance is given by the ratio of section variation to pressurevariation:

C¼ dAdPt

ð7Þ

where Pt ¼ P�Pext is the transmural pressure. The compliance is considered as alocal constant on an arterial segment. By integrating Eq. (7) and knowing thatwhen Pt ¼ 0, i.e. P ¼ Pext , A¼ A0, we write A¼ CPtþA0 where A0 is the section areaat the equilibrium state.

We write Eq. (7) as C¼ ðA�A0Þ=ðP�PextÞ, thus deriving the pressure law:

P�Pext ¼ A0

CAA0

�1� �

ð8ÞFig. 1. The 1D model simplified geometry. It assumes that the artery is a straightcylinder of length L with a circular cross section Aðt; zÞ that deforms with respect tothe radial vector.

M. Khalifé et al. / Journal of Biomechanics 47 (2014) 3390–3399 3391

In the limit of small displacements, this pressure law is equivalent to the commonlyused linear law (5) for

β0 ¼2A3=2

0

C ; ð9Þ

In fact, by linearizing (5) (see also Alastruey et al., 2012), we have

β0

ffiffiffiA

p�

ffiffiffiffiffiffiA0

pA0

¼ β0

2A3=20

ðA�A0ÞþOðA�A0Þ2

2.2. Numerical approximation

The equations of the system (3) and (4) are written in the conservative form:

∂U∂t

þ∂FðUÞ∂z

¼ SðUÞ ð10Þ

where

� U¼ ½AQ�� F¼ ½ QQA

2 þC1�, C1ðAÞ ¼

R A0 c21ðτÞ dτ and c1 ¼ c1ðAÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðA=ρÞ∂P=∂A

p,

� S¼ ½ 0�Kr

QA�

Since A40, the matrix DF possesses two real distinct eigenvalues λ7 ¼ ðQ=AÞ7c1and system (10) is a hyperbolic system of partial differential equations (Formaggia andVeneziani, 2003). The blood flow is assumed to be sub-critical, thus λ� o0, and λ7

have opposite signs.Following Formaggia and Veneziani (2003), system (10) is written in diagonal

form, which is more suitable to understand the solution behavior and for numericalpurpose:

∂W∂t

þΛ∂W∂z

¼ L � S ð11Þ

where W ¼ ðW1 ;W2Þ are the characteristic variables, Λ¼ ½λþ0

0λ�

� and L¼ ½ℓTþ

ℓT�� is a

matrix of left eigenvalues of DF such that L �DF¼Λ � L. In the case of the law (8),we have

c1 ¼ffiffiffiffiffiffiAρC

s;

W1 ¼QAþ2

ffiffiffiffiffiffiAρC

s; W2 ¼

QA�2

ffiffiffiffiffiffiAρC

s: ð12Þ

Inversely, the primitive set of variables (A,Q) is written in terms of the diagonalvariables ðW1 ;W2Þ as

A¼ffiffiffiffiffiffiρC

p W1�W2

4

� �2

; Q ¼ AW1þW2

2: ð13Þ

For system (10) approximation, a finite volume scheme is defined by a grid over½0; L� � ½0; T�, with uniform mesh-spacing Δz and time step Δtn . The cells are givenby ½zi ; ziþ1½�½tn ; tnþ1½, where zi ¼ iΔz, i¼ 0;…;M, with MΔz¼ L, and tn ¼ nΔtn ,n¼ 0;…;N, with NΔtN ¼ T . We seek Un

i approximating ð1=ΔzÞ R ziþ 1zi

Uðx; tnÞ dz.The numerical scheme is a classical finite volume scheme of Rusanov type for

hyperbolic problems:

1Δtn

ðUnþ1i �Un

i Þþ1Δz

ðFniþ1=2�Fni�1=2Þ�Sni ¼ 0; ð14Þ

F iþ1=2 ¼12

FðUni ÞþFðUn

iþ1Þ�λiþ1=2ðUnþ1i �Un

i Þh i

with λiþ1=2 ¼maxfλ� ðUni Þ; λþ ðUn

iþ1Þg, and where Sni ¼ ½ 0

�KrQni

Ani

�. Time-space Δtn must

verify the well-known stability condition 2maxijλiþ1=2jΔtnrΔz.

Numerical boundary conditions are prescribed, thus Unin ¼Un

�1 and

Unout ¼Un

Mþ1 are given to compute Unþ10 and Unþ1

M at z¼0 and z¼L respectively.Since flows involved in this model are sub-critical, the numerical (14) requires onecondition at each boundary. An admissible condition imposes the incomingcharacteristic and allows the wave corresponding to the outgoing characteristicto leave the domain. As W2 remains constant along the characteristic curve definedby dz=dt ¼ λ� ðUðz; tÞÞ, if we approximate λ� ðUðz; tÞÞ by its numerical valueλ� ðUðz; tnÞÞ in ½tn; tnþ1½, we consider that

W2ðz; tnþ1ÞCW2ðz�fλ� gn0Δtn; tnÞ;thus

Wnþ12;in ¼ΔzþλΔt

ΔzWn

2;�1�λΔtΔz

Wn2;0

where λ¼ λn�0 and Δt ¼Δtn . Similarly, we impose

Wnþ11;out ¼

Δz�λΔtΔz

W1nMþ1þ

λΔtΔz

W1nM

where λ¼ fλþ gnM .

2.3. Boundary conditions

We used MRI measured boundary conditions on the inlet of the tube phantomor the arterial segment. Flow measured on the entrance was imposed as acondition on the first mesh. We extrapolated, as explained above, the inlet valueof W2, and A was computed by using relations (12) and (13). The outlet boundarycondition is obtained by coupling the 1D model with a 0D model that consists of asystem of differential equations linking pressure to flow at the 1D model exit. Infact, as shown in Alastruey et al. (2008), the behavior happening beyondthe 1D-modeled arteries can be represented by a 0D lumped-parameters model,also known as the Windkessel model, that simulates hemodynamics by electricalcircuit analogy. Pressure gradient is represented by a potential difference,blood flow by the electrical current and hydraulic impedance by an electricalimpedance. Hydraulic impedance combines friction loss, arterial wall elasticity andblood flow inertia, which are modeled by a resistance, a capacitor and aninductance respectively. A three-element (RCR) Windkessel model is a goodcompromise accounting for wall compliance and resistance in the peripheralnetwork as well as proximal aortic impedance (Wetterer, 1954; Westerhof et al.,1969) . It is a combination of an R-model and an RC-model that improvesconsiderably the behavior of the original two-element model proposed by Frank(1899) by removing pressure and flow oscillations (Stergiopulos et al., 1999;Alastruey, 2006). The first resistance Zc is the proximal aorta characteristicimpedance, the compliance C in parallel to the resistance R2 simulates the volumecompliance and the resistance of the vascular network found downstream theabdominal aorta (Fig. 2).

The inlet flow of the R model, which is the 1D-model outlet flow(Qin�0D ¼Qout�1D), is

Qin�0D ¼ Pin�0D�PC

Zcð15Þ

As for the RC-model:

CdPC

dt�Pout�0D�PC

R2�Qin�0D ¼ 0 ð16Þ

Pout�0D is the pressure at the 0D-model exit. We compute PC at each time step byexplicitly discretizing equation (16):

PNþ1C ¼ f ðPN

C ; QNin�0D; Pout�0DÞ

then use the relation at the 1D-model exit:

QNþ1in�0D ¼ PNþ1

in�0D�PNþ1C

Zc

2.4. MRI experiments

A straight flexible silicone tube was used as a phantom to validate the 1D-model.The MRI setup (Fig. 3), described in a previous work (Khalife et al., 2012), consisted ofthe phantom connected to a non-magnetic CardioFlow 5000MR pump (ShelleyAutomation, Toronto, Canada) that can generate a programmable flow pattern. Weused a blood mimicking fluid (BMF, 62% water and 38% glycerol) with similarmagnetic and physical properties as blood. It has a Newtonian behavior which is asafe assumption for blood circulating in large arteries (Ottesen et al., 2004). TwoMRI-compatible optical pressure sensors (Opsens, Quebec, Canada) were introduced

Fig. 2. Coupling the 1D model outlet with the 0D model: a three-element Wind-kessel model (RCR) forms the outlet boundary condition of the 1D model.ðQin�0D; Pin�0DÞ ¼ ðQout�1D; Pout�1DÞ and ðQout�0D ; Pout�0DÞ are flow and pressureat the inlet and the outlet of the RCR model respectively. To solve the RCR model,two sub-models, RC and R, are considered and then combined. PC is the pressureacross C at the exit of the RC model and at the entrance of the R model.

M. Khalifé et al. / Journal of Biomechanics 47 (2014) 3390–33993392

into the phantom. These sensors allow simultaneous pressure measurements used asvalidation for the simulation.

To test the 1D-model application on a curved segment and estimate the error,the tube phantom was replaced with an aortic phantom (Elastrat, Geneva,Switzerland) comprising an ascending aorta branch (AAo) and a descending aortabranch (DAo). Pressure sensors were introduced at the AAo entrance and at theDAo exit.

A sinusoidal waveform of amplitude 150 mL � s�1 was applied in the tubephantom setup. In the aorta phantom setup, a flow wave similar to the onemeasured at the left ventricle exit was programmed on the pump control systemwith an amplitude of 250 mL � s�1 and a period of 0.8 s (75 bpm).

Experiments were performed on a 1.5 T MRI system (Achieva; Philips, Best, theNetherlands) using SENSE Flex-L coils. Phase-Contrast (PC) flow-encoded gradient-echo multiphase sequences were performed to assess through-plane velocity ontwo slices, one situated at the phantom entrance with the first pressure sensor andthe second at the phantom exit, with the second pressure transducer. Scan para-meters: spatial resolution: 1�1�8 mm3, echo time/repetition time TE/TR¼5.75/9.59 ms, velocity encoding Venc ¼ 80 cm � s�1 for a flow rate of 150 mL � s�1 andVenc ¼ 100 cm � s�1 for a flow rate of 250 mL � s�1, 40 time frames per cardiac cyclewith ECG-triggering on the pump signal.

For the in vivo feasibility of the 1D-model, 3 healthy subjects (male, ages 27, 30and 33) were included in our study after having filled out a written consent form.All measurements were performed on a 1.5 T MRI system with SENSE cardiac coil.PC flow-encoded sequence covering the thoracic aorta was acquired during breath-hold and ECG-gated gradient echo sequence (spatial resolution: 1.1�1.1�8 mm3,TE/TR¼3.0/5.1 ms, Venc ¼ 180 cm � s�1, 40 time frames per cardiac cycle). Data wereacquired in an axial plane to measure through-plane velocity in the descending andthe abdominal aorta (AbAo). A coronal plane of the aorta was acquired to measurethe distance between velocity encoding sites.

Arm pressure was measured before and after MRI acquisitions and used as areference for the 1D-model pressure simulations.

2.5. Compliance measurements

MRI images were processed using an in-house code. Aorta section wassegmented semi-automatically and visually assessed by the operator on themagnitude image then flow waveforms were computed from velocity dataprovided by the corresponding phase images. The inlet flow is thus used as themodel input condition.

MRI non-invasive measurements were used to determine phantoms and aorticcompliance through PWV. The method, described in a previous work (Khalife et al.,2012), is based on the transit time (TT) method commonly used for the PWVestimation (Murgo et al., 1980; Avolio et al., 1983) (Fig. 4).

For each experiment, the linear compliance is determined from PWV based onthe Bramwell–Hill theory: C¼ A=ρðPWVÞ2 (Bramwell and Hill, 1922; Vulliémozet al., 2002), where A is the vessel lumen area and ρ is the fluid density.

2.6. Simulations

Simulations were performed on an in-house program using Matlab (theMathworks, Natick, USA) by setting the domain geometry, fluid properties andMRI measured compliance for the pressure law. Flow Q, section area A and pressure

P are computed on the whole domain as functions of time and space, thencompared to measured data on the corresponding nodes.

The values of the input parameters were the following:

� Tube phantom: L¼40 cm, C¼ 0:0012 cm2 mmHg�1. The RCR-model componentswere estimated from measurements: Zc ¼ 1:24� 108 Pa � s �m�3 was computedusing the approach proposed in Mitchell et al. (2001), R2 ¼ 1:63� 108 Pa � s �m�3,Cper ¼ 0:288 cm3 �mmHg�1 and Pout�0D ¼ 0.

� Aortic phantom: L¼ 21:5 cm, C¼ 0:051 cm2 mmHg�1, Zc ¼ 2:63� 106 Pa � s �m�3,R2 ¼ 1:07� 108 Pa � s �m�3, Cper ¼ 0:137 cm3 mmHg�1.

� In vivo: the RCR parameters at the outlet were taken from Alastrueyet al. (2008) because of the absence of invasive measurements. The characteristicimpedance is Z0 ¼ ρc0=A0. The peripheral resistance is R2 ¼ RT �Zc whereRT ¼ 1:89� 108 Pa � s �m�3 is the complete vascular system resistance.

The simulations were performed on a 100 mesh grid on a scope of 10 periods, withan average simulation time of 328 s. A and Q are computed at each time step,Δt ¼ 10�4 s. To evaluate results, the root mean square error in % (RMS) wascalculated between computed and measured pressure waves.

Furthermore, to evaluate the equivalence between the pressure law (5) and thecompliance pressure law (8) proposed in this work, two simulations were carriedout with the same flow conditions, but with different pressure laws. The flowwaveform and the model parameters used were those of a physiological scenario asproposed in Alastruey et al. (2008): A0 ¼ π � 10�4 m2, β0 ¼ 4

3

ffiffiffiffiπ

phE for the β0

pressure law, with E¼ 0:4� 106 Pa, h¼ 1:5� 10�3 m, C ¼ 7:85� 10�9 m3=Pa, andcompliance of 2A3=2

0 =β0 for the compliance pressure law.

Fig. 3. Experimental MRI setup with the MR-compatible pump and the aortaphantom. Optical pressure sensors are introduced in two locations on the phantom.

∆t0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

50

100

150

200

250

300

350

400

450

Time(s)

Flowrate (mL/s)

DAoAAo

AAo

DAo

AAo

DAo

Fig. 4. Transit time estimated on MR flow waves in subject A. (a) Magnitude imageof the acquired axial plane. (b) AAo and DAo are visible in hypersignal; through-plane velocity image in AAo and DAo: flow waveforms are extracted from theencoded velocity. (c) The TT method for PWV measurement. The TT, which is thedelay Δt that a waveform needs to travel a distance L, is used to compute the wavevelocity: PWV ¼ L=Δt. Here, Δt was measured between the feet of flow waveformsextracted at two sections: tangent lines are obtained by the linear regression of theupslope segment on each waveform.

M. Khalifé et al. / Journal of Biomechanics 47 (2014) 3390–3399 3393

3. Results

The results of the equivalence test between the pressure law(5) and the compliance pressure law (8) are depicted in Fig. 5.

Based on these curves, we can say that the two pressure lawsprovide nearly identical results which means that the β0 pressurelaw agree well with the linear pressure law with C¼ 2A3=2

0 =β0, asin Eq. (9). The RMS between both pressure curves is 6.2%. Hence,

Fig. 5. Results of the 1D model using pressure laws with β0 in Eq. (5) (straight line) vs. with compliance C¼ 2A3=20 =β0 (dotted line). Simulations are performed with a pulsed

ventricular flow rate and parameters given in Alastruey et al. (2008). Plotted results are such as (a) flow waveform, (b) mean velocity and (c) pressure waveform simulated atthree sites of the artery, at the inlet (0 m), in the middle (0,2 m) and at the exit (0,4 m).

M. Khalifé et al. / Journal of Biomechanics 47 (2014) 3390–33993394

even if the pressure to section area relation is not perfectly linearin the range of physiological pressure, the error committed byconsidering it linear is negligible.

In phantom experiments, simulated pressure was compared topressure sensor measurements.

The inlet flow and the simulated pressure within the tubephantom are shown in Fig. 6 for the sinusoidal flow waveform ofamplitude 150 mL � s�1. Computed pressure curves show goodagreement with the measured pressure in the tube phantom.RMS error computed between both is 8.15% and confirms theagreement of the model results with measurements.

Furthermore, the simulated pressure at the DAo outlet in theaortic phantom was compared to pressure sensor measurementsacquired at the same location (Fig. 7b). RMS between bothpressure curves is 4.97%. Velocity measurement noise causedoscillations on the MR measured flow which were reported onthe simulated pressure. This is due to the relatively low velocitymeasured in the descending branch. Thus, the flow was filteredbefore simulations (Fig. 7a).

Beyond this, pressure curves simulated by the 1D-model on astraight segment formed by the DAo and the AbAo in healthy

volunteers are plotted in Fig. 8. Their contour is in good agreementwith pressure curves reported in vivo in the literature (Schnabelet al., 1952; Murgo et al., 1980). Also, according to the reflectionprinciple described in Salvi (2012), the systolic pressure increasesalong the arterial tree and consequently the brachial systolicpressure overestimates the aortic systolic pressure up to 20 mmHg(Levick, 2003). Thus in Fig. 8, the simulated systolic pressure in theaorta is smaller than the systolic pressure measured on eachvolunteer. Mean arterial pressure (MAP) computed on the simu-lated and measured pressure is added in Fig. 8.

4. Discussion

In this work, we showed that by combining a 1D aorta model tonon-invasive MRI velocity measurements, we were able to esti-mate aortic pressure. Flow MRI had commonly been used to derivepressure differences, so estimating a local pressure without havingto set a null pressure point is a step forward in cardiovasculardiagnosis.

Fig. 6. Pressure curves in the tube phantomwith a sinusoidal flow waveform of amplitude 150 mL s�1. Boundary conditions used are the measured flow at the inlet (a) andgiven by the 1D–0D coupled model at the outlet. (b) Simulated pressure curves at the sensors locations are compared to pressure measurements.

M. Khalifé et al. / Journal of Biomechanics 47 (2014) 3390–3399 3395

The MR-measured flow was used as an input for the 1D-modeland a coupled 0D-model was used as an output condition. Thepressure law, which relies on invasive measurements in formalstudies, is replaced by a non-invasive pressure law based on theaortic compliance. This compliance was measured from MRIassessed PWV. The model simulations were successfully validatedon an MRI setup with a straight phantom, an aortic arch phantomand in vivo.

At first, the simulations were applied on a simple straight tubeto respect the model conditions and simplifications. Pressurevalues obtained with simulations were in good agreement withmeasurements and showed satisfactory results as in previousstudies (Alastruey et al., 2011). Then, simulations were appliedon an aortic phantom and results showed a negligible error whencompared to pressure measurements, meaning that, in the phy-siological range of blood velocity, the 1D-model coupled with MRIflow measurements and subject-specific compliance pressure lawwould be enough to describe the aortic flow behavior. Resultsin vivo showed that it can estimate realistic pressure curves in theDAo, in agreement with the brachial pressure measured in theclinical routine. A bias was observed between the two brachialmeasurements carried out before and after the MRI exam due to

the subject changing state going from agitated prior to the exam torested state after the exam was completed. Another explanationcould be the difference in the subjects position (seated vs. supine)during measurements. Consequently, the measured arm pressuresmight have been different from what the simulations would yield.

This 1D-model, which is frequently used to model the aorta, isbased on strong assumptions, considering only radial and axisym-metric wall deformations, constant pressure on a section along thetube axis, thus only depending on time and on the axial coordinatez, dominance of axial velocity, absence of turbulence and neglect-ing the vessel wall inhomogeneities. Nevertheless, the modelserves its initial role of estimating an absolute pressure in a simpleand fast non-invasive manner, thus it is widely used in large arterymodeling for its fair reproduction of the physiological configura-tion. Even if the model is one-dimensional and must be applied toa straight segment, its application to a curved aortic phantom didnot greatly affect the agreement of simulated pressure andmeasured pressure at the phantom outlet. However, the effect ofan inertance in the 0D-model represented by an inductor in theelectrical circuit could be considered. The inertance element asproposed in Segers et al. (2008) could absorb the oscillationswhich appear on the pressure curve.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−50

0

50

100

150

200

250

300

Time (s)

Flowrate (mL/s)

measuredfitted

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.7−2

0

2

4

6

8

10

12

14

16

18

Time (s)

Pressure (mmHg)

simulated−DAomeasured− DAo

0.8

Fig. 7. (a) Flow measured at the entrance of the aorta phantom (solid line). Due to the motion of the phantom and the low velocity amplitude, oscillations are visible at themeasurements curves and are reported in the 1D model simulation results. Hence, the fitted measured flow (dashed line) is used for pressure calculation. (b) Simulated vs.measured pressure at the exit of the aorta phantomwith a ventricular flow of amplitude 250 mL � s�1. Boundary conditions used are the flow at the entrance of the ascendingaorta (AAo) and given by the 1D–0D coupled model at the outlet of the phantom.

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The compliance based pressure law is simple, non-invasive andwidely anchored in clinical practice and its association to the1D-model may be more advantageous than the previously used

pressure laws reported here. The model showed satisfactoryresults and good reproduction of invasive measurements recordedon a tube phantom. Additionally, we have shown that this

7 7.2 7.4 7.6 7.8 8

60

70

80

90

100

110

Time (s)

Pressure (mmHg)

6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.675

80

85

90

95

100

105

110

115

120

Time (s)

Pressure (mmHg)

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8

70

80

90

100

110

120

130

Time (s)

Pressure (mmHg)

8.2

7.7

Fig. 8. Healthy volunteers' aortic pressure simulated in a straight segment going from the descending aorta (DAo) to the abdominal aorta (AbAo). The horizontal dashed linesin orange (upper and lower lines) show the averaged brachial diastolic and systolic pressure (Pdiast and Psyst) measured for each subject. In green, the corresponding meanarterial pressure (MAP) lines (middle lines) are computed on the simulated pressure (solid line) and the arm measurements (dashed line). The inlet condition is the flowmeasured at the base of the descending aorta and the outlet condition is given by the 1D–0D coupled model. Pressure law based on compliance measurement (Eq. (8)) isused. Mean arterial pressure is given by MAP ¼ Pdiastþ1

3 ðPsyst�Pdiast Þ. (a) Subject A, 27 years old, (b) subject B, 30 years old, (c) subject C, 33 years old. (For interpretation ofthe references to color in this figure caption, the reader is referred to the web version of this paper.)

M. Khalifé et al. / Journal of Biomechanics 47 (2014) 3390–3399 3397

simplified linear pressure law is in good agreement with theffiffiffiA

p

law proposed in the literature.Some improvements could be made to the 1D-model in the

future. The model could be made more patient-specific by adapt-ing the 0D-model parameters to the patient for the outletboundary condition. Since these parameters are taken from theliterature and are not patient-specific, previous work (Willemetet al., 2013) has proposed to measure peripheral resistance byintroducing pressure catheters in the femoral artery. However, thisis a disadvantage to a method which offers to be strictly non-invasive. Further studies are needed to propose a non-invasiveway to measure the RCR parameters. Also, the model supposes aconstant compliance along the artery which is a non-realisticcondition when studying aortic disease or stenosis. This problemwill be addressed in future studies.

In the future, it would be interesting to compare the pressureestimation found in our model with catheter measurements in ananimal experiment. Also, more in vivo experiments would allowassessing the reliability and the repeatability of such estimationtechniques in the aorta and to study pressure changes in patientssuffering from cardiovascular diseases or under various stressconditions.

Conflict of interest statement

The authors declare that there are no conflicts of interest.

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