Cardiovascular diseases are currently the leading cause of mortality in the population of developed countries, due to the constant increase in cardiovascular risk factors, such as high blood pressure, cholesterol, overweight, tobacco use, lack of physical activity, etc. Numerous prospective and retrospective studies have shown that arterial stiffening is a relevant predictor of these diseases. Unfortunately, the arterial stiffness distribution across the human body is difficult to measure experimentally. We propose a numerical approach to determine the arterial stiffness distribution of an arterial network using a subject-specific one-dimensional model. The proposed approach calibrates the optimal parameters of the reduced-order model, including the arterial stiffness, by solving an inverse problem associated with the noninvasive in vivo measurements. An uncertainty quantification analysis has also been carried out to measure the contribution of the model input parameters variability, alone or by interaction with other inputs, to the variation of clinically relevant hemodynamic indices, here the arterial pulse pressure. The results obtained for a lower limb model, demonstrate that the numerical approach presented here can provide a robust and subject-specific tool to the practitioner, allowing an early and reliable diagnosis of cardiovascular diseases based on a noninvasive clinical examination.

References

References
1.
Hallab
,
M.
,
Pichierri
,
S.
,
Chevalet
,
P.
, de
Decker
,
L.
, and
Berrut
,
G.
,
2012
, “
La Rigidité Artérielle, un Nouveau Facteur de Risque à Mesurer
,”
Gériatrie et Psychologie Neuropsychiatrie du Vieillissement
,
10
(
3
), pp.
235
243
.
2.
Laurent
,
S.
,
Katsahian
,
S.
,
Fassot
,
C.
,
Tropeano
,
A. I.
,
Gautier
,
I.
,
Laloux
,
B.
, and
Boutouyrie
,
P.
,
2003
, “
Aortic Stiffness is an Independent Predictor of Fatal Stroke in Essential Hypertension
,”
Stroke
,
34
(
5
), pp.
1203
1206
.
3.
Laurent
,
S.
, and
Boutouyrie
,
P.
,
2007
, “
Arterial Stiffness: A New Surrogate End Point for Cardiovascular Disease?
J. Nephrol.
,
20
(
6
), pp.
45
50
.http://www.sin-italy.org/web%5Ceventi%5CSIN%5Carchivio%5Cjn%5C2007%5Cjnephrol_2007_S12%5CS45-S50%20Laurent.pdf
4.
Korteweg
,
D. J.
,
1878
, “
Uber die Fortpflanzungesgechwindigkeit des Schalles in Elastischen Rohern
,”
Ann. Phys. Chem. (NS)
,
5
, pp.
525
527
.
5.
Antonini-Canterin
,
F.
,
Carerj
,
S.
,
Di Bello
,
V.
,
La Carrubba
,
S.
,
Vriz
,
O.
,
Pavan
,
D.
,
Balbarini
,
A.
, and
Nicolosi
,
G. L.
,
2008
, “
Arterial Stiffness and Ventricular Stiffness: A Couple of Diseases or a Coupling Disease? A Review From the Cardiologist's Point of View
,”
Eur. J. Echocardiography
,
10
(
1
), pp.
36
43
.
6.
Brown
,
A. G.
,
Shi
,
Y. B.
,
Marzo
,
A.
,
Staicu
,
C.
,
Valverde
,
I.
,
Beerbaum
,
P.
,
Lawford
,
P. V.
, and
Hose
,
D. R.
,
2012
, “
Accuracy vs. Computational Time: Translating Aortic Simulations to the Clinic
,”
J. Biomech.
,
45
(
3
), pp.
516
523
.
7.
Gallo
,
D.
,
De Santis
,
G.
,
Negri
,
F.
,
Tresoldi
,
D.
,
Ponzini
,
R.
,
Massai
,
D.
,
Deriu
,
M. A.
,
Segers
,
P.
,
Verhegghe
,
B.
,
Rizzo
,
G.
, and
Morbiducci
,
U.
,
2012
, “
On the Use of In Vivo Measured Flow Rates as Boundary Conditions for Image-Based Hemodynamic Models of the Human Aorta: Implications for Indicators of Abnormal Flow
,”
Ann. Biomed. Eng.
,
40
(
3
), pp.
729
741
.
8.
Quarteroni
,
A.
, and
Formaggia
,
L.
,
2003
, “
Mathematical Modelling and Numerical Simulation of the Cardiovascular System
,”
Modelling of Living Systems
(Handbook of Numerical Analysis Series), Elsevier, Amsterdam, The Netherlands.
9.
Reymond
,
P.
,
Perren
,
F.
,
Lazeyras
,
F.
, and
Stergiopulos
,
N.
, “
Patient-Specific Mean Pressure Drop in the Systemic Arterial Tree, a Comparison Between 1-D and 3-D Models
,”
J. Biomech.
,
45
(
15
), pp.
2499
2505
.
10.
Steinman
,
D. A.
,
2002
, “
Image-Based Computational Fluid Dynamics Modeling in Realistic Arterial Geometries
,”
Ann. Biomed. Eng.
,
30
(
4
), pp.
483
497
.
11.
Steinman
,
D. A.
, and
Taylor
,
C. A.
,
2005
, “
Flow Imaging and Computing: Large Artery Hemodynamics
,”
Ann. Biomed. Eng.
,
33
(
12
), pp.
1704
1709
.
12.
Alastruey
,
J.
,
2006
, “
Numerical Modelling of Pulse Wave Propagation in the Cardiovascular System: Development, Validation and Clinical Applications
,”
Ph.D. thesis
, Imperial College London, University of London, London.http://ethos.bl.uk/OrderDetails.do?did=1&uin=uk.bl.ethos.431761
13.
Formaggia
,
L.
,
Lamponi
,
D.
, and
Quarteroni
,
A.
,
2003
, “
One Dimensional Models for Blood Flow in Arteries
,”
J. Eng. Math.
,
47
(
3–4
), pp.
251
276
.
14.
Matthys
,
K. S.
,
Alastruey
,
J.
,
J.
Peirò
,
Khir
,
A. W.
,
Segers
,
P.
,
Verdonck
,
P. R.
,
Parker
,
K. H.
, and
Sherwin
,
S. J.
,
2007
, “
Pulse Wave Propagation in a Model Human Arterial Network: Assessment of 1-D Numerical Simulations Against In Vitro Measurements
,”
J. Biomech.
,
40
(
15
), pp.
3476
3486
.
15.
Sherwin
,
S. J.
,
Formaggia
,
L.
,
Peirò
,
J.
, and
Franke
,
V.
,
2003
, “
Computational Modelling of 1D Blood Flow With Variable Mechanical Properties and Its Application to the Simulation of Wave Propagation in the Human Arterial System
,”
Int. J. Numer. Methods Fluids
,
43
(
6–7
), pp.
673
700
.
16.
Reymond
,
P.
,
Merenda
,
F.
,
Perren
,
F.
,
Rufenacht
,
D.
, and
Stergiopulos
,
N.
,
2009
, “
Validation of a One-Dimensional Model of the Systemic Arterial Tree
,”
Am. J. Physiol.: Heart Circ. Physiol.
,
297
(
1
), pp.
208
222
.
17.
Alastruey
,
J.
,
2011
, “
Numerical Assessment of Time-Domain Methods for the Estimation of Local Arterial Pulse Wave Speed
,”
J. Biomech.
,
44
(
5
), pp.
885
891
.
18.
Reymond
,
P.
,
Bohraus
,
Y.
,
Perren
,
F.
,
Lazeyras
,
F.
, and
Stergiopulos
,
N.
,
2011
, “
Validation of a Patient-Specific One-Dimensional Model of the Systemic Arterial Tree
,”
Am. J. Physiol.: Heart Circ. Physiol.
,
301
(
3
), pp.
1173
1182
.
19.
Bollache
,
E.
,
Kachenoura
,
N.
,
Redheuil
,
A.
,
Frouin
,
F.
,
Mousseaux
,
E.
,
Recho
,
P.
, and
Lucor
,
D.
,
2014
, “
Descending Aorta Subject-Specific One-Dimensional Model Validated Against in Vivo Data
,”
J. Biomech.
,
47
(
2
), pp.
424
431
.
20.
Chen
,
P.
,
Quarteroni
,
A.
, and
Rozza
,
G.
,
2013
, “
Simulation-Based Uncertainty Quantification of Human Arterial Network Hemodynamics
,”
Int. J. Numer. Methods Biomed. Eng.
,
29
(
6
), pp.
698
721
.
21.
Xiu
,
D.
, and
Sherwin
,
S. J.
,
2007
, “
Parametric Uncertainty Analysis of Pulse Wave Propagation in a Model of a Human Arterial Network
,”
J. Comput. Phys.
,
226
(
2
), pp.
1385
1407
.
22.
Sandor
,
Z.
, and
Andras
,
P.
,
2004
, “
Alternative Sampling Methods for Estimating Multivariate Normal Probabilities
,”
J. Econometrics
,
120
(
2
), pp.
207
234
.
23.
Cameron
,
R.
, and
Martin
,
W.
,
1947
, “
The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals
,”
Ann. Math.
, pp.
385
392
.
24.
Wiener
,
N.
,
1938
, “
The Homogeneous Chaos
,”
Am. J. Math.
, pp.
897
936
.
25.
Beyer
,
H. G.
, and
Schwefel
,
H. P.
,
2002
, “
Evolution Strategies: A Comprehensive Introduction
,”
Nat. Comput.
,
1
(
1
), pp.
3
52
.
26.
Thiriet
,
M.
, 2007,
Biology and Mechanics of Blood Flows
,
Springer
, New York.
27.
Alastruey
,
J.
,
Parker
,
K. H.
,
Peirò
,
J.
, and
Sherwin
,
S. J.
,
2008
, “
Lumped Parameter Outflow Models for 1-D Blood Flow Simulations: Effect on Pulse Waves and Parameter Estimation
,”
Commun. Comput. Phys.
,
4
(
2
), pp.
317
336
.https://www.researchgate.net/profile/Jordi_Alastruey/publication/228661502_Lumped_parameter_outflow_models_for_1-D_blood_flow_simulations_effect_on_pulse_waves_and_parameter_estimation/links/0912f5106c9e2af60e011002.pdf
28.
Milnor
,
W. R.
,
1989
,
Hemodynamics
,
2nd ed.
,
Williams and Wilkins
,
Baltimore, MD
.
29.
Nichols
,
W. W.
, and
O'Rourke
,
M. F.
,
1998
,
McDonald's Blood Flow in Arteries: Theoretical, Experimental and Clinical Principles
, Arnold, London.
30.
Cockburn
,
B.
,
Karniadakis
,
G.
, and
Shu
,
C. W.
,
2000
,
Discontinuous Galerkin Methods Theory, Computation and Applications
(Lectures Notes in Computational Science and Engineering), Vol.
11
, Springer, Berlin, Heidelberg, pp.
3
50
.
31.
Karniadakis
,
G.
, and
Sherwin
,
S. J.
,
2005
,
Spectral/HP Element Methods for Computational Fluid Dynamics
, 2nd ed., Oxford University Press, New York, pp.
288
299
.
32.
Bollache
,
E.
,
2014
, “
Caractérisation Hémodynamique de l'aorte Thoracique par IRM, Tonométrie D'applanation et Simulations Numériques
,” Ph.D. thesis, Université Pierre et Marie Curie, Paris.
33.
El Bouti
,
T.
,
2015
, “
Optimisation Robuste et Application à la Reconstruction du Réseau Artériel Humain
,” Ph.D. thesis, LMV, Université de Versailles Saint Quentin en Yvelines, Versailles, France.
34.
Hansen
,
N.
, and
Ostermeier
,
A.
,
2001
, “
Completely Derandomized Self-Adaptation, Evolution Strategies
,”
Evol. Comput.
,
9
(
2
), pp.
159
195
.
35.
Safar
,
M. E.
, and
London
,
G. M.
,
1994
, “
The Arterial System in Human Hypertension
,”
Textbook of Hypertension
,
J. D.
Swales
, ed.,
Blackwell Scientific
,
Hoboken, NJ
, pp.
85
102
.
36.
Xiu
,
D.
, and
Karniadakis
,
G.
,
2002
, “
The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations
,”
SIAM J. Sci. Comput.
,
24
(
2
), pp.
619
644
.
37.
Evans
,
G.
,
1993
,
Practical Numerical Integration
,
Wiley
,
Chichester, UK
.
38.
Clenshaw
,
C. W.
, and
Curtis
,
A. R.
,
1960
, “
A Method for Numerical Integration on an Automatic Computer
,”
Numer. Math.
,
2
(
1
), pp.
197
205
.
39.
Gerstner
,
T.
, and
Griebel
,
M.
,
1998
, “
Numerical Integration Using Sparse Grids
,”
Numer. Algorithms
,
18
(
3–4
), pp.
209
232
.
40.
Smolyak
,
S. A.
,
1963
, “
Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions
,”
Soviet Math. Dokl.
,
4
, pp.
240
243
.
41.
Sobol
,
I. M.
,
1993
, “
Sensitivity Estimates for Nonlinear Mathematical Models
,”
Math. Model. Comput. Exp.
,
1
(4), pp.
407
414
.
42.
Novak
,
E.
, and
Ritter
,
K.
,
1996
, “
High Dimensional Integration of Smooth Functions Over Cubes
,”
Numer. Math.
,
75
(
1
), pp.
79
97
.
43.
Novak
,
E.
, and
Ritter
,
K.
,
1997
, “
The Curse of Dimension and a Universal Method for Numerical Integration
,”
Multivariate Approximation and Splines
, Vol. 125, Birkhauser, Basel, Switzerland, pp.
177
187
.
44.
Alastruey
,
J.
,
Parker
,
K.
, and
Sherwin
,
S.
,
2012
, “
Arterial Pulse Wave Haemodynamics
,”
11th International Conference on Pressure Surges
, S. Anderson, eds., Virtual PiE Led t/a BHR Group, pp.
401
442
.http://wwwf.imperial.ac.uk/ssherw/spectralhp/papers/PulseSurges_2012.pdf
45.
Brault
,
A.
,
Dumas
,
L.
, and
Lucor
,
D.
,
2016
, “
Uncertainty Quantification of Inflow Boundary Condition and Proximal Arterial Stiffness Coupled Effect on Pulse Wave Propagation in a Vascular Network
,” e-print
arXiv
:1606.06556.https://arxiv.org/abs/1606.06556
You do not currently have access to this content.