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Journal ArticleDOI

Patient Specific Haemodynamic Modeling after Occlusion Treatment in Leg

TL;DR: Pat specific reconstruction algorithm of 1D core network based on MRI data is proposed as a tool for analysis of postsurgical haemodynamics after femoral artery treatment of occlusive vascular disease and provides effective personalizing predictive method that is validated with clinical observations.
Abstract: In this work we propose a method for analysis of postsurgical haemodynamics after femoral artery treatment of occlusive vascular disease. Patient specific reconstruction algorithm of 1D core network based on MRI data is proposed as a tool for such analysis. Along with presurgical ultrasound data fitting it provides effective personalizing predictive method that is validated with clinical observations.

Summary (2 min read)

1. Introduction

  • Revascularization is the conventional method of surgical treatment which increases lifetime and life quality for many patients.
  • So far no method has been developed for presurgical assessment of revascularization impact to the postsurgical blood flow.
  • It is based on personalized 1D core network reconstruction algorithm, presurgical ultrasound patient specific data fitting and the 1D blood flow simulations.
  • The authors briefly introduce the mathematical model in section 2.1.

2.1.1. Core model

  • As a core model for the closed blood circulation the authors used 1D network dynamical model [15–17] accounting for arterial and venous parts of the thigh sub-network.
  • The model considers the flow of viscous incompressible fluid in the network of elastic tubes/vessels.
  • The system of equations (2.4)-(2.5) is closed by finite difference approximation of compatibility conditions along outgoing characteristics.
  • The order of the respective junction system can be reduced from 2M+1 to M equations and effectively solved by Newton method [15,17].
  • Details on numerical implementation of this model are discussed in [14,15,17].

2.1.2. Boundary conditions

  • Since the authors consider a local region of the thigh vasculature, they should address the boundary conditions at the vascular network inlet and outlet.
  • Patient specific data for this node were unavailable for us and the authors set the inlet boundary conditions as u(t, 0)S(t, 0) = αQH (t) , (2.8) where QH (t) is heart ejection profile simulated by four-chamber dynamical heart model [15].
  • Moreover, the outlet boundaries are placed downstream and blood flow may be changed significantly due to surgical intervention.
  • The authors connect the averaged venous network of the same structure to the terminal outlet points of the considered arterial region.

2.2. Reconstruction of patient specific 1D core network

  • 1D blood flow network simulations require 1D core network reconstruction.
  • The overall algorithm of reconstruction is divided into the following steps: 1) 3D volume segmentation of vascular structure, 2) meshing and centerlines extraction, 3) centerlines merging, and 4) network reconstruction.
  • These centerlines should be merged and segmented with junction points.
  • The other centerlines are checked for the intersection with the root centerline and branching points are determined for every intersection.
  • In this case only one node is added to the set of graph nodes.

2.3. Patient specific model fitting

  • The authors apply the above approach to 1D core network reconstruction to predict changes of local haemodynamics in thigh vasculature due to the occlusion treatment in the femoral artery.
  • Geometrical parameters of the network (vessels lengths and diameters) are defined by their 1D core network reconstruction algorithm and correspond to the patient specific morphology.
  • The authors assess their acceptable ranges basing on general physiological and anatomical data [7,11–13], as well as patient parameters such as height, weight, medical history and the following assumptions: 1. Vessels 1, 3, 4, 5, 7 and 9 are the parts of the femoral artery and its descendant popliteal artery.
  • These main leg arteries have relatively high stiffness ck and low resistance Rk since they carry large bulk of blood.
  • The authors assume these parameters to remain the same after the occlusion treatment since according to numerical evidence the Reynolds number is not changed significantly in the most vessels.

3. Results

  • Two series of numerical experiments have been performed.
  • They simulate haemodynamics in large thigh arteries before and after femoral artery occlusion treatment.
  • The authors note that even better matching can be achieved by further adjustment of the vessels parameters shifting them to non-physiological ranges.
  • The authors compare the computed model results and measured values presented in the column postsurgical in Table 2.
  • The largest error is observed in the distal part of the superficial femoral 95 “Ivanov˙mmnp2014˙6” —.

4. Discussion

  • The proposed method of 1D core network reconstruction is competitive to the methods implemented in the well-known commercial software [20].
  • Formally, boundary conditions at the junction nodes should be derived from limiting ratios in (2.1), (2.2) that reduce to the total pressure conservation in pseudo-steady approximation.
  • This approach was discussed in more detail in [15].
  • The authors have to fit them in physiological range to conform with Doppler ultrasound data.
  • In this work the authors focused on the haemodynamic analysis of the atherosclerotic occlusion treatment in the femoral artery.

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Math. Model. Nat. Phenom.
Vol. 9, No. 6, 2014, pp. 85–97
DOI: 10.1051/mmnp/20149607
Patient Specific Haemodynamic Modeling after
Occlusion Treatment in Leg
T. Gamilov
1
, Yu. Ivanov
2
, P. Kopylov
3
, S. Simakov
1
, Yu. Vassilevski
1,2
1
Moscow Institute of Physics and Technology, 141700, Dolgopru dny, 9 Ins ti tu ts ki i Lane, Russia
2
Institute of Nu mer ic al Mathematics RAS, 119333, Moscow, 8 Gubkina St., Russia
3
I.M. Sechenov First Moscow State Medical University, 2-4 Bolshaya Pirogovskaya st.
119991 Moscow, Russia
Abstract. In this wo r k we propose a method for analysis of postsurgical haemodynamics after
femoral artery treatment of occlusive vascular disease. Patient specific reconstruction algorithm
of 1D core network based on MRI data is proposed as a tool for such analysis. Along with
presurgical ultr a so u n d data fitting it provides effective personalizing predictive method that is
validated with clinical observations.
Keywords and phrases: haemodynamics, femoral artery occlusion, math ema tic a l model-
ing, patient specific, 1D core network reconstruction algorithm, presurgical and postsu rg ic a l
modeling
Mathematics Subject Classification: 35Q92, 76Z05, 65M06
1. Introducti on
Revascularization is the conventional method of surgical treatment which increases lifetime and life quality
for many patients. Balloon angioplasty and stenting are the most advanced techn iq ue s of revascularization
nowadays. So far no method has been developed f or presurgical assessment of revascularization impact to
the postsurgical blood flow. Such method would p r ovide a basis for patient specific surgery tactics devel-
opment. In this work we propose a method of presurgical patient specific analysis of the haemodynamic
changes in lower extremities due to atherosclerotic occlusion treatment. It is based on personalized 1D
core network reconstruction algorit hm, pr es u r gical ultrasound patient specific data fitting and the 1D
blood flow simulations.
The basics of our approach to 1D haemodynamic simulation are presented in [15,17]. For other works
in this field we refer to [1, 5, 9, 10]. In this work we address the boundar y conditions for the local part
of the network. The problem is that haemodynamic parameters in the downstream outlets can not be
correctly evaluated presurgically. To cope with this, we impose remote downstream conditions in veins
which are dis tu r bed slightly by surgical treatment.
The 1D blood flow simulations require 1D core network representation of the 3D patient specific
vascular network. This motivate s importance of an algorithm which produces the set of 1D edges (cores)
Corresp ondin g author. E-mail: simakov@crec.mipt.ru
c
EDP Sciences, 2014
Article published by EDP Sciences and available at http://www.mmnp-journal.org or http://dx.doi.org/10.1051/mmnp/20149607

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T. Gamilov, Yu. Ivanov, P. Kopylov, S. Simakov, Yu. Vassilevski Patient specific haemodynamic modeling
and junction nodes on the basis of 3D vascular domain extracted from MRI data. A few software tools
provide partly such functionality. The most popular tools are Amira
R
[20] and VMTK [19]. In this work
we use VMTK for 3D volume extraction from MRI data for meshing and reconstruction vessel centerlines.
The final 1D core network is produced by a new algorithm capable to process general networks with loops.
The outline of the paper is as follows. We br ie fly introduce the mathematical model in section 2.1.
Patient specific network reconstruction tools are pr e s ented in section 2.2. Additional patient specific
fitting issues are particularly addressed in s ec tion 2.3. The results of patient spec ifi c haemodynamic
modeling after occlusion treatment in leg are pres ented in se ct ion 3. The discussion of the presented
approach and the future work constitutes section 4.
2. Methods
2.1. 1D network model for haemodynamics
2.1.1. Core model
As a core model for the closed blood circulation we used 1D network dynamical model [15–17] accounting
for arterial and venous parts of the thigh sub-network. The model considers the flow of viscous incom-
pressible fluid in the network of elastic tubes/vessels. The flow in every vessel is described by mass and
momentum balance equations
S/∂t + (Su) /∂x = 0, (2.1)
u/∂t +
u
2
/2 + p/ρ
/∂x = f
tr
(S, u) , (2.2)
where t is the time; x is the coordinate along the vessel; ρ is the blood density (constant); S(t, x) is the
vessel cross-section area; S
0
is the unstres se d cross-section area; u(t, x) is t h e linear velocity of blood flow
averaged over the cross-section; p is the blood pressure; f
tr
is the friction force given in the k-th vessel
by
f
tr
(S
k
, u
k
) =
4πµu
k
S
k
˜
S
k
˜
S
k
+
˜
S
1
k
, (2.3)
µ is the coefficient of friction,
˜
S
k
= S
k
/S
0
k
.
At the vessels junctions the Poiseuille pressure drop and the mass conser vation conditions [15] are
applied (for motivation we refer to section 4):
p
k
(S
k
(t, ˜x
k
)) p
l
node
(t) = ε
k
R
l
k
S
k
(t, ˜x
k
) u
k
(t, ˜x
k
) , k = k
1
, k
2
, . . . , k
M
, (2.4)
X
k=k
1
,k
2
,...,k
M
ε
k
S
k
(t, ˜x
k
) u
k
(t, ˜x
k
) = 0, (2.5)
where M is the number of vessels meeting at node with index l; {k
1
, . . . , k
M
} is the range of the indexes
of th es e vessels; p
node
(t) is the pressure at the junction node; ε = 1, ˜x
k
= L
k
for in comin g vessels,
ε = 1, ˜x
k
= 0 for outgoing vessels; R
l
k
is the hydraulic resistance. The system of equations (2.4)-(2.5) is
closed by finite difference approximation of compatibility conditions along outgoing characteristics. The
order of the respective junction system can be reduced from 2M + 1 to M equations and effectively solve d
by Newton method [15, 17].
The e last ic properties of the vessel wall are given by the state equation:
p(S
k
) p
k
= ρc
2
k
f(S
k
) , (2.6)
where S-like function f(S
k
) is approximated as
f(S
k
) =
(
exp
˜
S
k
1
1,
˜
S
k
> 1
ln
˜
S
k
,
˜
S
k
6 1,
(2.7)
86

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T. Gamilov, Yu. Ivanov, P. Kopylov, S. Simakov, Yu. Vassilevski Patient specific haemodynamic modeling
p
k
is the pressure in the tissues surrounding the vessel, c
k
is the velocity of small disturbance propagation
in the vessel wall which also has meaning of pulse wave velocity (PWV) in the unstressed vessel [18]
characterizing material stiffness.
Details on numerical impleme ntation of this model are discussed in [14, 15, 17].
2.1.2. Boundary conditions
Since we consider a local region of the thigh vasculature, we should address the boun dar y conditions at
the vascular network inlet and outlet.
The entry point of t he simulated vesse ls network corresponds to the beginning of common iliac artery
(see Figure 1 and vessel 1 in Figure 9). Patient specifi c data for this node were unavailable for us and we
set t he inlet boundar y conditions as
u(t, 0) S(t, 0) = αQ
H
(t) , (2.8)
where Q
H
(t) is heart ejection profile simulated by four-chambe r dynamical heart model [15]. We ap-
proximate common iliac artery blood flow inlet profile by scaling heart ejection profile with coefficient α.
The lower estimate of α for the common iliac artery is based on the estimates for blood flow through the
femoral artery 635 ml/min [6], heart rate 60 strokes/min and stroke volume 60 ml:
α >
635ml/min
60str/min · 60ml/str
1
6
. (2.9)
The lower estimate is in a good agreement with th e value computed in the model of global systemic
circulation [15]. In this work we take α = 0.21 as it provides the best fit between calculated and
measured presurgical velocity maximum in the femoral arte r y (see Table 2). The same inlet boundar y
condition is applied for postsurgical state simulations as it is distanced upstream from the surgical region.
The outlet boundary conditions at the terminal points of the vessels 6, 8, 9, 11, 13, 14 (Figure 9)
can be based on presurgically measured velocities in the particular p oints given in Table 2. However,
our objective is to predict haemodynamic changes presurgically and we can not rely on postsu r gical
measurements. Moreover, the outlet boundaries are placed downstream and blood flow may be changed
significantly due to surgical intervention. Therefore, we have to introduce conditions which are slight ly
disturbed by postsurgical blood flow redistribution. We connect the averaged venous network of the same
structure (see Figure 9) to the terminal outlet points of the considered arterial region. Vein d iamet er s are
set two times greater and coefficients c
k
are set two times less than the respective values for the arteries.
This resul ts in accumulation of f our times more blood in venous part and makes the venous wall mor e soft
and flexible. Since we neglect pulsations in veins, we set at the terminal point of the venous network a
steady blood flow as the boundary c ondi ti on. In order to maintain the mass balance in the total ne twork,
we make this flow equal to the time-averaged blood flow at the entry point of th e arterial part
u(t, L) S(t, L) = α
t+T
H
Z
t
Q
H
(t) dt , (2.10)
where Q
H
(t) is taken from (2.8), T
H
is th e per i od of the cardiac cycle.
2.2. Reconstru ct ion of patient specific 1D core network
1D b lood flow network simulations require 1D core network reconstruction. The overall algorithm of
reconstruction is divided into the following steps: 1) 3D volume segmentation of vascular structure, 2)
meshing and centerlines extraction, 3) centerlines merging, and 4) network reconstruction.
We should note that commercial software Amira
R
provides similar functionality. It helps to perform
semi-automatic blood vessels segmentation, to produce skeletonisation based on distance map and thin-
ning methods for a connected set of voxels, and to gener ate centerlines on the basis of calculation of
Euclidean distance to th e nearest boundar y for every point in the vascular 3D domain.
87

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T. Gamilov, Yu. Ivanov, P. Kopylov, S. Simakov, Yu. Vassilevski Patient specific haemodynamic modeling
In this work we propose a method which is based on the open sour ce library VMTK. This library
can be modified and can be easily extended with new methods. We use VMTK to produce centerlines
of 3D vascular domain extracted from MRI data and exte nd VMTK with the new algorithm for graph
reconstruction. To perform vascular domain extraction, we filter input data and eliminate bones, air
and surrounding tiss ue s from the original image by thresholding. Level set method is us ed for tracking
vascular branches and marching cubes method is used to extract the surface of vessels. The result of this
preprocessing of patient specific MRI data is presented in Figure 1.
1D core network is produced on the basis of centerlines which represent the 3D vascular domain. Sever al
methods of centerlines generation are reviewed and compared in [4,8]. We use the method implemented i n
VMTK which is based on Voronoi diagrams [2]. The results of meshing by marching cube s and centerlines
computation are presented in Figur e 2.
A
B
Figure 1. 3D segmentation based on MRI data: A vessels with bones, B vessels.
The resulting 1D core network is described by a set of nod es and a set of edges (cores) connecting
nodes. These data can be produced from centerline representation of the initial 3D vascular domain.
Every centerline is described by an ordered set of pairs {(a
i
, r
i
)}
n
i=1
where a
i
= (x
i
, y
i
, z
i
) i s the radius
vector of the central point in the vessel cross-section and r
i
is the mean radius of the vessel at this point.
These centerlines should be me r ged and segmented with junction points.
To present algorithms of 1D core network reconstruction, we take advantage of the similarity between
the vascular network and a tree-like structure with branches, junctions and root. However, our c or e
networks may have closed loops and graph terminology seems to be more appropriate for our algorithmic
description. Further in this sect ion we shall deal with sets of graph nodes and sets of graph branches. We
ascribe to every graph node a point A
i
in the 3D space with corresponding radius vector a
i
= (x
i
, y
i
, z
i
).
The branch object contains a pair of incident no d es , the length of the correspondin g vessel segment (along
the centerline) and averaged radiu s.
Every centerline computed with VMTK goes from every chosen inlet to every outlet. For further
purposes we should split t he center l ine s by e limi natin g coincided parts. In Figure 3 we show a set of
88

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T. Gamilov, Yu. Ivanov, P. Kopylov, S. Simakov, Yu. Vassilevski Patient specific haemodynamic modeling
A
B
Figure 2. 3D vascular domain: A polygonal mesh, B computed centerlines.
centerlines corresponding to branch ends [O, A], [O, B], [O , C], [O, D] befor e splitting and new s pli t and
merged set [O, A], [O
1
, B], [O
2
, C], [O
3
, D] .
Centerlines intersection is determined as follows. Centerline C
= {(a
i
, r
i
)}
n
i=1
intersects centerline
C = {(a
i
, r
i
)}
n
i=1
if
a
{a
1
, a
n
}, k {1, .., n 1} : |a
a
| r
k+0.5
(a
)
where a
is th e projection of a
onto [a
k
, a
k+1
] and r
k+0.5
is lin ear ly interp olated radi us
r
k+0.5
(a
) =
r
k
(1 λ
k+0.5
(a
)) + r
k+1
λ
k+0.5
(a
) , a
[a
k
, a
k+1
]
0 , a
/ [a
k
, a
k+1
],
λ
k+0.5
(a
) =
|a
a
k
|
|a
k+1
a
k
|
.
We shall refer t o a
as the branching point (Figure 4).
89

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TL;DR: The Anatomical Nomenclature of the Nervous System and Systemic Overview of the Endocrine System, Principles of Hormone Production and Secretion and Development of the Cardiovascular and Lymphatic Systems are presented.
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Book
22 Dec 2011
TL;DR: The author explains the background mechanics of blood vessel walls, solid mechanics and the properties of blood vessels, and an introduction to mass transfer.
Abstract: Continuing demand for this book confirms that it remains relevant over 30 years after its first publication. The fundamental explanations are largely unchanged, but in the new introduction to this second edition the authors are on hand to guide the reader through major advances of the last three decades. With an emphasis on physical explanation rather than equations, Part I clearly presents the background mechanics. The second part applies mechanical reasoning to the component parts of the circulation: blood, the heart, the systemic arteries, microcirculation, veins and the pulmonary circulation. Each section demonstrates how an understanding of basic mechanics enhances our understanding of the function of the circulation as a whole. This classic book is of value to students, researchers and practitioners in bioengineering, physiology and human and veterinary medicine, particularly those working in the cardiovascular field, and to engineers and physical scientists with multidisciplinary interests.

1,174 citations

Book
01 Jan 2009
TL;DR: This book provides a set of well described and reproducible test cases and applications of cardiovascular physiopathology, focusing on the main characteristics of the different flow regimes encountered in the cardiovascular system.
Abstract: Chapter 1 introduces the most important terms and concepts of cardiovascular physiopathology ,while Chapter 2 illustrates the basic mathematical models for blood flow and biochemical transfer. The derivation of the equations that governs blood flow is covered in Chapter 3, while Chapter 4 is devoted to the treatment of medical images to obtain geometries suitable for numerical computations. Chapter 5 illustrates the important relationship between geometry and type of flow, focusing on the main characteristics of the different flow regimes encountered in the cardiovascular system. Mathematical models for blood rheology are discussed in Chapter 6. In Chapter 7 mathematical and numerical models of biochemical transport are explained in detail, with practical examples. The mathematical analysis of coupled models for fluid-structure interaction is addressed in Chapter 8, while Chapter 9 focuses on numerical methods for the mechanical coupling between blood flow and the vessel structure. Reduced models play an important role in cardiovascular modelling to enable the simulating of large parts of (or even th whole) vascular system. Their derivation is presented in Chapter 10. The intertwining of such models with more complex three dimensional ones is the foundation of the so called geometric multiscale approach, illustrated in detail in Chapter 11. Finally, Chapter 12 provides a set of well described and reproducible test cases and applications.

526 citations


"Patient Specific Haemodynamic Model..." refers background in this paper

  • ...For other works in this field we refer to [1, 5, 9, 10]....

    [...]

Journal ArticleDOI
TL;DR: This paper presents an automatic technique for the objective comparison of distributions of geometric and hemodynamic quantities over the surface of bifurcating vessels and demonstrates how similar results are obtained over similar geometries, allowing for proper model-to-model comparison.
Abstract: Computational modeling of human arteries has been broadly employed to investigate the relationships between geometry, hemodynamics and vascular disease. Recent developments in modeling techniques have made it possible to perform such analyses on realistic geometries acquired noninvasively and, thus, have opened up the possibility to extend the investigation to populations of subjects. However, for this to be feasible, novel methods for the comparison of the data obtained from large numbers of realistic models in the presence of anatomic variability must be developed. In this paper, we present an automatic technique for the objective comparison of distributions of geometric and hemodynamic quantities over the surface of bifurcating vessels. The method is based on centerlines and consists of robustly decomposing the surface into its constituent branches and mapping each branch onto a template parametric plane. The application of the technique to realistic data demonstrates how similar results are obtained over similar geometries, allowing for proper model-to-model comparison. Thanks to the computational and differential geometry criteria adopted, the method does not depend on user-defined parameters or user interaction, it is flexible with respect to the bifurcation geometry and it is readily extendible to more complex configurations of interconnecting vessels.

242 citations


"Patient Specific Haemodynamic Model..." refers methods in this paper

  • ...Discussion The proposed method of 1D core network reconstruction is competitive to the methods implemented in the well-known commercial software [3]....

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Journal ArticleDOI
TL;DR: In this article, the authors present a non-linear 1D model of coronary and systemic arterial circulations, as well as ventricular pressure and an aortic valve that opens and closes independently based on local haemodynamics.
Abstract: There is an important interaction between the pumping performance of the ventricle, arterial haemodynamics and coronary blood flow. While previous non-linear 1D models have focused only on one of these components, the model presented in this study includes coronary and systemic arterial circulations, as well as ventricular pressure and an aortic valve that opens and closes ‘independently’ and based on local haemodynamics. The systemic circulation is modelled as a branching network of elastic tapering vessels. The terminal element applied at the extremities of the network is a single tapering vessel which is shown to adequately represent the input characteristics of the downstream vasculature. The coronary model consists of left and right coronary arteries which both branch into two ‘equivalent’ vessels that account for the lumped characteristics of subendocardial and subepicardial flows. As contracting heart muscle causes significant compression of the subendocardial vessels, a time-varying external pressure proportional to ventricular pressure is applied to the distal part of the equivalent subendocardial vessel. The aortic valve is modelled using a variable reflection coefficient with respect to backward-running aortic waves, and a variable transmission coefficient with respect to forward-running ventricular waves. A realistic ventricular pressure is the input to the system; however, an afterload-corrected ventricular pressure is calculated and results in pressure gradients between the ventricle and aorta that are similar to those observed in vivo. The 1D equations of fluid flow are solved using the locally conservative Galerkin method, which provides explicit element-wise conservation, and can naturally incorporate vessel branching. Each component of the model is verified using a number of tests to ensure accuracy and reveal the underlying processes that give rise to complex pressure and flow waveforms. The complete model is then implemented, and simulations are performed with input parameters representing ‘at rest’ and exercise states for a normal adult. The resulting waveforms contain all of the important features seen in vivo, and standard measures of haemodynamic state are found to be normal. In addition, one or several characteristics of some common diseases are imposed on the model and are found to produce haemodynamic changes that agree with experimental observations in the published literature. Using a patient-specific carotid bifurcation geometry, 1D velocity waveforms are also compared with waveforms obtained from a three-dimensional model. The 1D and 3D results show good agreement. Copyright © 2008 John Wiley & Sons, Ltd.

213 citations


"Patient Specific Haemodynamic Model..." refers background in this paper

  • ...For other works in this field we refer to [1, 5, 9, 10]....

    [...]

Frequently Asked Questions (2)
Q1. What are the contributions in "Patient specific haemodynamic modeling after occlusion treatment in leg" ?

In this work the authors propose a method for analysis of postsurgical haemodynamics after femoral artery treatment of occlusive vascular disease. 

The same method can be applied to the other parts of the vascular system and other angiosurgical procedures such as cava filter implantation and artificial embolisation of arterial-venous malformations that will be parts of their future work.