Estimating absolute aortic pressure using MRI and a one-dimensional model

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Estimating absolute aortic pressure using MRI and a

one-dimensional model

Maya Khalifé, Astrid Decoene, Filipa Caetano, Ludovic de Rochefort,

Emmanuel Durand, Dima Rodríguez

To cite this version:

Estimating absolute aortic pressure using MRI and a

one-dimensional model

Maya Khaliféa_{, Astrid Decoene}b_{, Filipa Caetano}b_{, Ludovic de Rochefort}a_,

3

Emmanuel Duranda_{, Dima Rodríguez}a

4

a_{Imagerie par Résonance Magnétique Médicale et Multi-Modalités (IR4M), Université} 5

Paris-Sud - CNRS, UMR 8081, Orsay, France

6

b_{Laboratoire de Mathématiques d'Orsay, Université Paris-Sud - CNRS, UMR 8628, Orsay,} 7

France

8

Abstract

9

Aortic blood pressure is a strong indicator to cardiovascular diseases and mor-bidity. Clinically, pressure measurements are done by inserting a catheter in the aorta. However, imaging techniques have been used to avoid the invasive procedure of catheterization. In this article, we combined MRI measurements to a one-dimensional model in order to simulate blood ow in an aortic segment. Absolute pressure was estimated in the aorta by using MRI measured ow as boundary conditions and MRI measured compliance as a pressure law for solv-ing the model. Model computed pressure was compared to catheter measured pressure in an aortic phantom. Furthermore, aortic pressure was estimated in vivo in three healthy volunteers.

1. Introduction

10

To date, blood pressure (BP) is one of the most useful clinical indicator of

11

cardiovascular disease. Hypertension, more specically, is a leading predictor

12

of death in atherosclerosis diseases worldwide Cohn et al. [13], Collaboration

13

[14]. Therefore, measuring BP is of great interest for diagnosis and risk

pre-14

vention of cardiovascular events. An elevated pressure gives information about

15

the aortic state, the presence of atherosclerotic plaques, stenosis, calcication

or aneurisms. In a clinical routine, a sphygmomanometer is used to measure

17

systolic and diastolic brachial pressure. However, due to reexion in the distal

18

arteries, the aortic pressure waveform is altered while traveling through the

vas-19

cular system. Thus distortion of the wave shape as well as systolic amplication

20

occur on the systolic pressure measured in the brachial artery O'rourke et al.

21

[32], Park and Guntheroth [34], Salvi [37]. Although models and transfer

func-22

tions to link brachial BP to aortic pressure exist Chen et al. [12, 11], Liang [26],

23

wave reection in the arterial system makes it dicult to reproduce the wave

24

contour with great delity from such methods. Until now, the gold-standard of

25

aortic pressure measurement is catheterization, which is invasive and not

repeat-26

able in a routine procedure Murgo et al. [30], Skinner and Adams [40]. In recent

27

years, imaging techniques have been used to assess pressure gradients from

ve-28

locity or acceleration maps and its combination with uid mechanics equations

29

has been exploited in order to measure the BP non-invasively. Doppler

Ultra-30

sound (US) used to measure blood velocity in the arteries was associated to the

31

standard simplied Bernoulli equation in order to determine pressure dierences

32

between two measurement sites. This latter technique is reported to be

user-33

dependent and error-prone due to the wave angle of incidence Zananiri et al.

34

[51], Muhler et al. [29], hence its accuracy in determining the maximum velocity

35

in the artery is debatable. Also, extending US to other situations is limited

36

by the inapplicability of Bernoulli equation to unsteady ows Yang et al. [50].

37

Phase-Contrast (PC) MRI allows accurate encoding of the blood velocity in the

38

arteries in the three directions; hence, it has been largely used for non-invasive

39

pressure estimation. Some authors computed pressure dierences using Poisson

40

equation Yang et al. [50], others integrated the Navier-Stokes (NS) equations

41

using MRI velocity maps Tyszka et al. [45], Thompson and McVeigh [44], Bock

42

et al. [8], Ebbers et al. [17] or acceleration maps to avoid computational errors

arising from velocity derivation Buyens et al. [10]. These methods compute a

44

pressure gradient, and to estimate an absolute pressure, require a zero-pressure

45

reference point which has to be measured with a catheter, or user-dened in

46

a gross assumption. Consequently, these methods are not an alternative to

47

catheter measurements, which remain more accurate. In this work, we propose

48

a non-invasive technique to extract absolute pressure in a straight artery from

49

MR velocimetry using a biomechanical one-dimensional (1D) model as proposed

50

by Formaggia et al. Formaggia et al. [18]. Although a three-dimensional (3D)

51

model gives a more complete and realistic reproduction of the aortic ow, 1D

52

models are able to describe the non-linear ow behavior in larger elastic vessels

53

Hughes and Lubliner [22]. As these models are reasonably accurate, they are

54

widely used for aortic ow simulations. Their accuracy has been assessed by

55

comparison with experimental data acquired in a tube phantom Bessems et al.

56

[7], in a distributed arterial model Olufsen et al. [31], Alastruey et al. [2] and in

57

vivo Reymond et al. [36], Alastruey et al. [3]. Furthermore, the 1D model relies

58

on the establishment of a pressure law consisting of a relation between pressure

59

and vessel section area. The pressure laws used in 1D model equations are

deter-60

mined experimentally with invasive measurements or estimated using complex

61

algorithms. Additionally, these pressure laws are complicated and involve the

62

determination of multiple parameters. Here, we propose a pressure law based

63

on the aortic compliance which reects arterial elasticity and can be determined

64

non-invasively with MRI. Using this pressure law, we coupled the 1D model with

65

realistic boundary conditions measured by MRI to estimate absolute pressure

66

in the aortic segment. The derived model was tested on a straight compliant

67

phantom and computed pressure was compared to experimental pressure

mea-68

surements recorded simultaneously with the MRI acquisition. The model was

69

also tested on a real-sized compliant aortic phantom Then, the model is used to

estimate BP on healthy volunteers.

71

2. Methods

72

The 1D-model, studied in Formaggia et al. [18], is a reduced model

describ-73

ing blood ow in arteries and its interaction with wall motion. The artery is

74

considered as a cylindrical compliant tube of length L and radius R (R  L).

75

The model derivation approach consists of integrating the NS equations on a

76

generic section S. Some simplifying assumptions are made:

77

• the model assumes axisymmetry

78

• the wall displacement is supposed to solely be in the radial direction

79

• pressure is assumed to be uniform in each section

80

• the axial velocity uzis predominant.

81

For large arteries such as the aorta, it is a safe assumption to consider a at

82

velocity prole for the boundary layer is very thin compared to the vessel radius

83

Olufsen et al. [31].

84

The main variables of the problem are (Figure 1):

85

• axial section area A

86 A(t, z) = ˆ S(t,z) dσ (1) • mean ow Q 87 Q(t, z) = ˆ S uzdσ (2) • blood pressure P (t, z), 88

where dσ denotes the area element. Their evolution is described by the

mo-89

mentum conservation and the mass conservation equations, while considering a

constant viscosity: 91 ∂A ∂t + ∂Q ∂z = 0 (3) ∂Q ∂t + ∂ ∂z  Q2 A  +A ρ ∂P ∂z + Kr  Q A  = 0 (4)

Kris the friction coecient; for a at prole in blood ow problems, Kr= 22πν,

92

where ν is the vessel wall kinematic viscosityFormaggia and Veneziani [19]. ρ is

93

the blood density.

94

Pressure law

95

To close the system, a relation between the section area A and the pressure

96

Pis dened. This pressure law depends on section area A0(z) = πR2

0(z)at time

97

t = 0 and on a set of parameters β = (β0, β1, ...βn) related to the vessel wall

98

physical and mechanical properties. Pextis the external pressure exerted by the

99

environment of the vessel whereas P is the intravascular pressure. The pressure

100

law should meet these conditions:

101

• ∂P ∂A > 0

102

• when A = A0, P = Pext.

103

Some pressure laws are proposed in the literature to link the pressure behavior

104

to the section area. A frequently used relation between A and P is stated as

105

follows Quarteroni and Formaggia [35], Alastruey [1]:

106 P − Pext = β0 √ A −√A0 A0 (5) β0= √ πh0E

1−ξ2 using Young's modulus E and the vessel thickness h0and Poisson's 107

ratio ξ = 0.5 for an incompressible material deformed elastically at small strains.

108

A more general law proposed in Hayashi et al. [21], Smith and Hunter [41]

109

is written as:

P − Pext = β0 "  A A0 β1 − 1 # . (6)

The parameters β0 and β1 values can be obtained either by tting

experi-111

mental pressure versus section measurements Smith [42], or by solving an inverse

112

problem with a 3D-model solution Martin et al. [27], Dumas [16]. Hence, these

113

laws cannot be determined non-invasively, and need knowledge about the vessel

114

properties. Additionally, they seem too complex to determine during a clinical

115

application. Finding a simple non-invasive way to determine a pressure law is of

116

great interest, consequently, we turned to the compliance. Indeed, in

physiolog-117

ical conditions, the aorta section deformation is commonly assumed to be linked

118

to the intravascular pressure by the aortic compliance Langewouters et al. [24].

119

Aortic compliance establishes a linear relation between the pressure and

120

section area; it represents the arterial wall ability to deform in response to a

121

pressure variation Conrad [15]. Also, it is clinically used and can be estimated

122

non-invasively by measuring the pulse wave velocity (PWV) in MRI. Hence, it

123

provides a simple and non-invasive pressure law that can be applied in vivo and,

124

as it includes compliance changes, is patient-specic.

125

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

126

variation :

127

C = dA

dPt (7)

where Pt= P − Pext is the transmural pressure. The compliance is considered

128

as a local constant on an arterial segment. By integrating equation (7) and

129

knowing that when Pt = 0, i.e. P = Pext, A = A0, we write A = CPt+ A0

130

where A0 is the section area at the equilibrium state.

We write equation 7 as C = A−A0

P −Pext, thus deriving the pressure law: 132 P − Pext=A0 C  A A0 − 1  (8)

In the limit of small displacements, this pressure law is equivalent to the

com-133

monly used linear law ((5)) for

134

β0 =

2A3/2₀

C , (9)

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

135 β0 √ A −√A0 A0 = β0 2A3/2₀

(A − A0) + O(A − A0)2

Numerical approximation

136

System (3)-(4) is written in the conservative form :

137 ∂U ∂t + ∂F(U) ∂z = S(U) (10) where 138 • U =    A Q    139 • F =    Q Q A 2 + C1   ,C1(A) = ´A 0 c 2 1(τ ) dτ and c1= c1(A) = q A ρ ∂P ∂A, 140 • S =    0 −KrQ_A    141

Since A > 0, the matrix DF possesses two real distinct eigenvalues λ±= Q_A± c1

142

and system (10) is a hyperbolic system of partial dierential equations

For-143

maggia and Veneziani [19]. The blood ow is assumed to be sub-critical, thus

144

λ− < 0,and λ± have opposite signs.

Following Formaggia and Veneziani [19], system (10) is written in

diago-146

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

147 numerical purpose: 148 ∂W ∂t + Λ ∂W ∂z = L · S (11)

where W = (W1, W2)are the characteristic variables, Λ =    λ+ 0 0 λ−    and 149 L =    `T + `T₋  

 is a matrix of left eigenvalues ofDFsuch that L · DF = Λ · L. In

150

the case of the law (8), we have

151 c1= s A ρC, W1= Q A + 2 s A ρC, W2= Q A − 2 s A ρC. (12)

Inversely, the primitive set of variables (A, Q) is written in terms of the

152 diagonal variables (W1, W2)as 153 A =  p ρCW1− W2 4 2 , Q = AW1+ W2 2 . (13)

For system (10) approximation, a nite volume scheme is dened by a grid over

154

[0, L] × [0, T ],with uniform mesh-spacing ∆z and time step ∆tn_{. The cells are}

155

given by [zi, zi+1[×[tn_{, t}n+1_{[, where zi = i∆z, i = 0, . . . , M, with M∆z = L,}

156

and tn _{= n∆t}n_{, n = 0, . . . , N, with N∆t}N _{= T. We seek U}n

i approximating 157 1 ∆z ´zi+1 zi U (x, t n_)dz. 158

The numerical scheme is a classical nite volume scheme of Rusanov type

159

for hyperbolic problems:

161 F_i+1 2 = 1 2 h F (U_in) + F (U_i+1n ) − λ_i+1 2(U n+1 i − U n i ) i with λi+1 2 = max{λ−(U n i), λ+(U n

i+1)}, and where S n i =    0 −KrQni An i   . Time-162

space ∆tn_{must verify the well-known stability condition 2 max} i |λi+

1 2|∆t

n_{≤ ∆z.}

163

Numerical boundary conditions are prescribed, thus Un

in= U−1n and Uoutn =

164

Un

M +1 are given to compute U n+1 0 and U

n+1

M at z = 0 and z = L respectively.

165

Since ows involved in this model are sub-critical, the numerical scheme (14)

166

requires one condition at each boundary. An admissible condition imposes the

167

incoming characteristic and allows the wave corresponding to the outgoing

char-168

acteristic to leave the domain. As W2 remains constant along the characteristic

169

curve dened by dz

dt = λ−(U (z, t)),if we approximate λ−(U (z, t))by its

numer-170

ical value λ−(U (z, tn_{))in [t}n_{, t}n+1_{[,we consider that}

171 W2(z, tn+1) ' W2(z − {λ−}n0∆t n_{, t}n_), thus 172 W_2,inn+1=∆z + λ∆t ∆z W n 2,−1− λ∆t ∆z W n 2,0 where λ = λn

−0 and ∆t = ∆tn. Similarly, we impose

173 W_1,outn+1 = ∆z − λ∆t ∆z W1 n M +1+ λ∆t ∆z W1 n M where λ = {λ+}n M. 174 Boundary conditions 175

We used MRI measured boundary conditions on the inlet of the tube

phan-176

tom or the arterial segment. Flow measured on the entrance was imposed as

177

a condition on the rst mesh. We extrapolated, as explained above, the inlet

178

value of W2,and A was computed by using relations (12) and (13). The outlet

boundary condition is obtained by coupling the 1D model with a 0D model that

180

consists of a system of dierential equations linking pressure to ow at the 1D

181

model exit. In fact, as shown in Alastruey et al. [4], the behavior happening

182

beyond the 1D-modeled arteries can be represented by a 0D lumped-parameters

183

model, also known as the Windkessel model, that simulates hemodynamics by

184

electrical circuit analogy. Pressure gradient is represented by a potential

dif-185

ference, blood ow by the electrical current and hydraulic impedance by an

186

electrical impedance. Hydraulic impedance combines friction loss, arterial wall

187

elasticity and blood ow inertia, which are modeled by a resistance, a capacitor

188

and an inductance respectively. A three-element (RCR) Windkessel model is

189

a good compromise accounting for wall compliance and resistance in the

pe-190

ripheral network as well as proximal aortic impedance Wetterer [48], Westerhof

191

et al. [47]. It is a combination of an R-model and an RC-model that improves

192

considerably the behavior of the original two-element model proposed by Otto

193

Frank Frank [20] by removing pressure and ow oscillations Stergiopulos et al.

194

[43], Alastruey [1]. The rst resistance Zc is the proximal aorta characteristic

195

impedance, the compliance C in parallel to the resistance R2 simulate

respec-196

tively the volume compliance and the resistance of the vascular network found

197

downstream the abdominal aorta (Figure 2).

198

The inlet ow of the R model, which is the 1D-model outlet ow (Qin−0D=

199 Qout−1D), is: 200 Qin−0D = Pin−0D− PC Zc (15)

As for the RC-model:

Pout−0D is the pressure at the 0D-model exit. We compute PC at each time

202

step by explicitly discretizing equation (16):

203

P_CN +1 = f (P_CN, QN_in−0D, Pout−0D)

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

204 QN +1_in−0D = P N +1 in−0D− P N +1 C Zc MRI experiments 205

A straight exible silicone tube was used as a phantom to validate the

206

1D-model. The MRI setup (Figure 3), described in a previous work Khalife

207

et al. [23], consisted of the phantom connected to a non-magnetic CardioFlow

208

5000MR pump (Shelley Automation, Toronto, Canada) that can generate a

pro-209

grammable ow pattern. We used a blood mimicking uid (BMF, 62% water

210

and 38% glycerol) with similar magnetic and physical properties as blood. It

211

has a Newtonian behavior which is a safe assumption for blood circulating in

212

large arteries Ottesen et al. [33]. Two MRI-compatible optical pressure sensors

213

(Opsens, Québec, Canada) were introduced into the phantom. These sensors

214

allow simultaneous pressure measurements used as validation for the simulation.

215

To test the 1D-model application on a curved segment and estimate the

216

error, the tube phantom was replaced with an aortic phantom (Elastrat, Geneva,

217

Switzerland) comprising an ascending aorta branch (AAo) and a descending

218

aorta branch (DAo). Pressure sensors were introduced at the AAo entrance and

219

at the DAo exit.

220

A sinusoidal waveform of amplitude 150mL.s−1_{was applied in the tube}

phan-221

tom setup. In the aorta phantom setup, a ow wave similar to the one measured

at the left ventricle exit was programmed on the pump control system with an

223

amplitude of 250mL.s−1 _{and a period of 0.8s (75bpm).}

224

Experiments were performed on a 1.5T MRI system (Achieva; Philips, Best,

225

the Netherlands) using SENSE Flex-L coils. Phase-Contrast (PC) ow-encoded

226

gradient-echo multiphase sequences were performed to assess through-plane

ve-227

locity on two slices, one situated at the phantom entrance with the rst pressure

228

sensor and the second at the phantom exit, with the second pressure transducer.

229

Scan parameters: spatial resolution: 1 × 1 × 8mm3_{, echo time/repetition time}

230

TE/TR = 5.75/9.59ms, velocity encoding Venc = 80cm.s−1 for a owrate of

231

150mL.s−1 and Venc = 100cm.s−1 for a owrate of 250mL.s−1, 40 time frames

232

per cardiac cycle with ECG-triggering on the pump signal.

233

For the in vivo feasibility of the 1D-model, 3 healthy subjects (male, ages 27,

234

30 and 33) were included in our study after having lled out a written consent

235

form. All measurements were performed on a 1.5T MRI system with SENSE

236

cardiac coil. PC ow-encoded sequence covering the thoracic aorta was acquired

237

during breath-hold and ECG-gated gradient echo sequence (spatial resolution:

238

1.1 × 1.1 × 8mm3_{, TE/TR = 3.0/5.1 ms, Venc= 180cm.s}−1_{, 40 time frames per}

239

cardiac cycle). Data were acquired in an axial plane to measure through-plane

240

velocity in the descending and the abdominal aorta (AbAo). A coronal plane of

241

the aorta was acquired to measure the distance between velocity encoding sites.

242

Arm pressure was measured before and after MRI acquisitions and used as

243

a reference for the 1D-model pressure simulations.

244

Compliance measurements

245

MRI images were processed using an in-house code. Aorta section was

seg-246

mented semi-automatically and visually assessed by the operator on the

magni-247

tude image then ow waveforms were computed from velocity data provided by

248

the corresponding phase images. The inlet ow is thus used as the model input

condition.

250

MRI non-invasive measurements were used to determine phantoms and

aor-251

tic compliance through PWV. The method, described in a previous work Khalife

252

et al. [23], is based on the transit time (TT) method commonly used for the

253

PWV estimationMurgo et al. [30], Avolio et al. [6] (Figure 4).

254

For each experiment, the linear compliance is determined from PWV based

255

on the Bramwell-Hill theory: C = A

ρ(P W V )2 Bramwell and Hill [9], Vulliémoz 256

et al. [46], where A is the vessel lumen area and ρ the uid density.

257

Simulations

258

Simulations were performed on an in-house program using Matlab (the

Math-259

works, Natick, USA) by setting the domain geometry, uid properties and MRI

260

measured compliance for the pressure law. FlowQ, section area A and

pres-261

sure P are computed on the whole domain as functions of time and space, then

262

compared to measured data on the corresponding nodes.

263

The values of the input parameters were the following:

264

• Tube phantom : L = 40cm, C = 0.0012 cm2_.mmHg−1_{. The RCR-model}

265

components were estimated from measurements: Zc= 1.24 · 108_Pa.s.m−3

266

was computed using the approach proposed in Mitchell et al. [28], R2 =

267

1.63 · 108 _Pa.s.m−3_{, Cper= 0.288 cm}3_.mmHg−1 _{and Pout−0D= 0.}

268

• Aortic phantom: L = 21.5cm, C = 0.051cm2_.mmHg−1_{, Zc = 2.63 ·}

269

106_Pa.s.m−3_{, R2= 1.07 · 10}8_Pa.s.m−3_{, Cper = 0.137 cm}3_.mmHg−1 _.

270

• In vivo : the RCR parameters at the outlet were taken from Alastruey

271

et al. [4] because of the absence of invasive measurements. The

character-272

istic impedance is Z0= ρc0

A₀. The peripheral resistance is R2= RT − Zc

273

where RT = 1.89 · 108Pa.s.m−3is the complete vascular system resistance.

The simulations were performed on a 100 meshgrid on a scope of 10 periods,

275

with an average simulation time of 328 s. A and Q are computed at each time

276

step,∆t = 10−4_{s. To evaluate results, the root mean square error in % (RMS)}

277

was calculated between computed and measured pressure waves.

278

Furthermore, to evaluate the equivalence between the pressure law (5) and

279

the compliance pressure law (8) proposed in this work, two simulations were

280

carried out with the same ow conditions, but with dierent pressure laws. The

281

ow waveform and the model parameters used were those of a physiological

282

scenario as proposed in Alastruey et al. [4]: A0= π · 10−4m2_{, β0=} 4 3

√

πhEfor

283

the β0pressure law, with E = 0.4·106

Pa, h = 1.5·10−3m, C = 7.85·10−9m3_/Pa,

284

and compliance of 2A3/2₀

β0 for the compliance pressure law. 285

3. Results

286

The results of the equivalence test between the pressure law (5) and the

287

compliance pressure law (8) are depicted in gure 5. Based on these curves, we

288

can say that the two pressure laws provide nearly identical results which means

289

that the β0 pressure law agree well with the linear pressure law with C = 2A3/2₀ β0 , 290

as in equation (9). The RMS between both pressure curves is 6.2% . Hence,

291

even if the pressure to section area relation is not perfectly linear in the range of

292

physiological pressure, the error committed by considering it linear is negligible.

293

In phantom experiments, simulated pressure was compared to pressure

sen-294

sor measurements.

295

The inlet ow and the simulated pressure within the tube phantom are

296

shown in gure 6 for the sinusoidal ow waveform of amplitude 150mL.s−1_.

297

Computed pressure curves show good agreement with the measured pressure in

298

the tube phantom. RMS error computed between both is 8.15% and conrms

299

the agreement of the model results with measurements.

300

Furthermore, the simulated pressure at the DAo outlet in the aortic phantom

was compared to pressure sensor measurements acquired at the same location

302

(Figure 7b). RMS between both pressure curves is 4.97%. Velocity measurement

303

noise caused oscillations on the MR measured ow which were reported on the

304

simulated pressure. This is due to the relatively low velocity measured in the

305

descending branch. Thus, the ow was ltered before simulations (Figure 7a).

306

Beyond this, pressure curves simulated by the 1D-model on a straight

seg-307

ment formed by the DAo and the AbAo in healthy volunteers are plotted in

308

gure 8. Their contour is in good agreement with pressure curves reported in

309

vivo in the literature Schnabel et al. [38], Murgo et al. [30]. Also, according

310

to the reection principle described in Salvi [37], the systolic pressure increases

311

along the arterial tree and consequently the brachial systolic pressure

overesti-312

mates the aortic systolic pressure up to 20mmHg Levick [25]. Thus in gure 8,

313

the simulated systolic pressure in the aorta is smaller than the systolic pressure

314

measured on each volunteer. Mean arterial pressure (MAP) computed on the

315

simulated and measured pressure is added in gure 8.

316

4. Discussion

317

In this work, we showed that by combining a 1D aorta model to non-invasive

318

MRI velocity measurements, we were able to estimate aortic pressure. Flow

319

MRI had commonly been used to derive pressure dierences, so estimating a

320

local pressure without having to set a null pressure point is a step forward in

321

cardiovascular diagnosis.

322

The MR-measured ow was used as input for the 1D-model and a coupled

323

0D-model was used as an output condition. The pressure law, which relies on

324

invasive measurements in formal studies, is replaced by a non-invasive pressure

325

law based on the aortic compliance. This compliance was measured from MRI

326

assessed PWV. The model simulations were successfully validated on an MRI

327

setup with a straight phantom, an aortic arch phantom and in vivo.

At rst, the simulations were applied on a simple straight tube to respect the

329

model conditions and simplications. Pressure values obtained with simulations

330

were in good agreement with measurements and showed satisfactory results as

331

in previous studies Alastruey et al. [2]. Then, simulations were applied on an

332

aortic phantom and results showed a negligible error when compared to pressure

333

measurements, meaning that, in the physiological range of blood velocity, the

334

1D-model coupled with MRI ow measurements and subject-specic compliance

335

pressure law would be enough to describe the aortic ow behavior. Results in

336

vivo showed that it can estimate realistic pressure curves in the DAo, in

agree-337

ment with the brachial pressure measured in the clinical routine. A bias was

338

observed between the two brachial measurements carried out before and after

339

the MRI exam due to the subject changing state going from agitated prior to

340

the exam to rested state after the exam was completed. Another explanation

341

could be the dierence in the subjects position (seated vs. supine) during

mea-342

surements. Consequently, the measured arm pressures might have been dierent

343

than what the simulations would yield.

344

This 1D-model, which is frequently used to model the aorta, is based on

345

strong assumptions, considering only radial and axisymmetric wall

deforma-346

tions, constant pressure on a section along the tube axis, thus only depending

347

on time and on the axial coordinate z, dominance of axial velocity, absence of

348

turbulence and neglecting the vessel wall inhomogeneities. Nevertheless, the

349

model serves its initial role of estimating an absolute pressure in a simple and

350

fast non-invasive manner, thus it is widely used in large artery modeling for

351

its fair reproduction of the physiological conguration. Even if the model is

352

one-dimensional and must be applied to a straight segment, its application to a

353

curved aortic phantom did not greatly aect the agreement of simulated

pres-354

sure and measured pressure at the phantom outlet. However, the eect of an

inertance in the 0D-model represented by an inductor in the electrical circuit

356

could be considered. The inertance element as proposed in Segers et al. [39]

357

could absorb the oscillations which appear on the pressure curve.

358

The compliance based pressure law is simple, non-invasive and widely

an-359

chored in clinical practice and its association to the 1D-model may be more

360

advantageous than the previously used pressure laws reported here. The model

361

showed satisfactory results and good reproduction of invasive measurements

362

recorded on a tube phantom. Additionally, we have shown that this

simpli-363

ed linear pressure law is in good agreement with the√Alaw proposed in the

364

literature.

365

Some improvements could be made to the 1D-model in the future. The model

366

could be made more patient-specic by adapting the 0D-model parameters to the

367

patient at the outlet boundary condition. Since these parameters are taken from

368

the literature and are not patient-specic, previous work Willemet et al. [49] has

369

proposed to measure peripheral resistance by introducing pressure catheters in

370

the femoral artery. However, this is a disadvantage to a method which oers to

371

be strictly non-invasive. Further studies are needed to propose a non-invasive

372

way to measure the RCR parameters. Also, the model supposes a constant

373

compliance along the artery which is a non-realistic condition when studying

374

aortic disease or stenosis. This problem will be addressed in future studies.

375

In the future, it would be interesting to compare the pressure estimation

376

found in our model with catheter measurements in an animal experiment. Also,

377

more in vivo experiments would allow assessing the reliability and the

repeata-378

bility of such estimation techniques in the aorta and to study pressure changes

379

in patients suering from cardiovascular diseases or under various stress

condi-380

tions.

Conict of interest statement

382

The authors declare that there are no conicts of interest.

383

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Figure 1: The 1D model simplied geometry. It assumes that the artery is a straight cylinder of length L with a circular cross section A(t, z) that deforms in respect to the radial vector.

Figure 2: Coupling the 1D model outlet with the 0D model: a three-element Windkessel model (RCR) forms the outlet boudary condition of the 1D model. (Qin−0D, Pin−0D) =

(Qout−1D, Pout−1D) and (Qout−0D, Pout−0D) are ow and pressure at 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 pressure across C at the exit of the RC model

and at the entrance of the R model.

AAo DAo (a) AAo DAo (b) ∆t 0 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) Flowra te (mL /s) DAo AAo (c)

Figure 4: Transit time estimated on MR ow waves in subject A. (a) magnitude image of the acquired axial plane. (b) AAo and DAo are visible in hypersignal; through-plane velocity image in AAo and DAo : ow waveforms are extracted from the encoded velocity. (c) The TT method for PWV measurement. The TT, which is the delay ∆t that a waveform needs to travel a distance L, is used to compute the wave velocity: P W V = L

∆t. Here, ∆t was

6.4 6.5 6.6 6.7 6.8 6.9 7 7.1 0 50 100 150 200 250 300 Time (s) F l o w r a t e ( m L / s ) β 0law 0m β 0law 0,2m β 0law 0,4m C law 0m C law 0,2m C law 0,4m (a) 6.4 6.5 6.6 6.7 6.8 6.9 7 7.1 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time (s) V e l o c i t y ( m / s ) β₀law 0m β₀law 0,2m β₀law 0,4m C law 0m C law 0,2m C law 0,4m (b) 6.4 6.5 6.6 6.7 6.8 6.9 7 7.1 70 80 90 100 110 120 130 Time (s) P r e s s u r e ( m m H g ) β₀law 0m β₀law 0,2m β₀law 0,4m C law 0m C law 0,2m C law 0,4m (c)

Figure 5: Results of the 1D model using pressure laws with β0 in equation (5) (straight

line) vs. with compliance C =2A3/2₀

β0 (dotted line). Simulations are performed with a pulsed

0.9 1 1.1 1.2 1.3 1.4 1.5 50 0 50 100 150 Time(s) Flowra te(mL/ s) (a) 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 20 0 20 40 60 80 100 120 140 Pressu re (mm Hg) Simulated-sensor 1 Simulated-Mesured Mesured (b)

Figure 6: Pressure curves in the tube phantom with a sinusoidal ow waveform of amplitude 150 mL.s−1_{. Boundary conditions used are the measured ow at the inlet (a) and given by}

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) measured fitted (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 −2 0 2 4 6 8 10 12 14 16 18 Time (s) Pressure (mmHg) simulated− DAo measured− DAo (b)

Figure 7: (a) Flow mesured 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 the measure-ments curves and are reported in the 1D model simulation results. Hence, the tted measured ow (dashed line) is used for pressure calculation. (b) Simulated vs. measured pressure at the exit of the aorta phantom with a ventricular ow of amplitude 250 mL.s−1_{. Boundary}

7 7.2 7.4 7.6 7.8 8 8.2 60 70 80 90 100 110 Time (s) Pressu re (mm Hg) DAo 0cm DAo AbA

(a) subject A, 27 years old

6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 75 80 85 90 95 100 105 110 115 120 Time (s) Pressu re (mm Hg) DAo 0cm DAo AbA

(b) subject B, 30 years old

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) Pressu re (mm Hg) DAo 0cm DAo AbA -- 9cm - 18cm

(c) subject C, 33 years old

Figure 8: Healthy volunteers' aortic pressure simulated in a straight segment going from the descending aorta (DAo) to the abdominal aorta(AbAo). The horizontal dashed lines in orange show the averaged brachial diastolic and systolic pressure (Pdiastand Psyst) measured for each