# A novel method for non-invasively detecting the severity and location of aortic aneurysms

## Abstract

The influence of an aortic aneurysm on blood flow waveforms is well established, but how to exploit this link for diagnostic purposes still remains challenging. This work uses a combination of experimental and computational modelling to study how aneurysms of various size affect the waveforms. Experimental studies are carried out on fusiform-type aneurysm models, and a comparison of results with those from a one-dimensional fluid–structure interaction model shows close agreement. Further mathematical analysis of these results allows the definition of several indicators that characterize the impact of an aneurysm on waveforms. These indicators are then further studied in a computational model of a systemic blood flow network. This demonstrates the methods’ ability to detect the location and severity of an aortic aneurysm through the analysis of flow waveforms in clinically accessible locations. Therefore, the proposed methodology shows a high potential for non-invasive aneurysm detectors/monitors.

## Keywords

Aneurysm detection Experimental models Numerical models One-dimensional modelling Systemic circulation Waveforms## 1 Introduction

Cardiovascular disease is responsible for the death of over eight million people worldwide every year. Among these deaths, aortic aneurysms alone are responsible for more than 100,000 deaths, with about 6000 occurring in England and Wales as a result of rupture. An aortic aneurysm is a dilation of the aorta, usually exceeding the normal diameter by more than 50%. The abdominal aortic aneurysm (AAA) is the most prevalent type of aortic aneurysm, and it is often asymptomatic. As it increases in size, the aneurysm is more likely to rupture and becomes a life-threatening condition. The symptoms are rarely noticed before rupture of the aneurysm, which prompted the healthcare systems to investigate various screening programs based on ultrasound and MRI. A Multicentre Aneurysm Screening Study (MASS) was one of the first large screening programmes for AAA in the UK. The results from the MASS programme showed that detection of AAA reduced risk of death in a 4-year period from 0.33 to 0.19% (Ashton et al. 2002).

Opportunistic detection of asymptomatic AAAs during clinical examination is the most common way of diagnosis. An abdominal palpation has only a moderate overall sensitivity for detecting AAAs (unless they are large enough to warrant elective intervention), especially in overweight people (see for instance the work by Fink et al. (2000)). Symptoms usually only occur near to or at the point of rupture. Although scanning the elderly population for aneurysms provides an excellent opportunity to reduce potential mortalities, developing cheaper and faster methods is an important challenge for medical and biomedical engineering.

In a clinical setting, ultrasound (US) is currently the most practical, non-invasive and inexpensive modality in screening for and surveillance of AAA, with a sensitivity and specificity of more than 98% (see among others Ashton et al. (2002); Wilmink et al. (2002); Sprouse et al. (2004); Barkin and Rosen (2004); Walker et al. (2004); Fleming et al. (2005); Catalano and Siani (2005); Brekken et al. (2011)). Some common limitations of US diagnosis include: suboptimal imaging, attenuation and inaccurate measurements, often due to bowel gas, obesity, artery tortuosity and/or calcification. Inter-observer variability can also be a problem. The shape and size of the aneurysm can be determined most accurately by means of 3D CT or MRI (Sparks et al. 2002; Lee et al. 1984; Litmanovich et al. 2009; McBride et al. 2015) methods.

The goal of the paper is to develop cost-effective AAA detection methods based on accessible measurement and analysis of human pressure and/or velocity waveforms. These waveforms can be easily computed using a framework of 1D systemic circulation models.

The modelling, analysis and measurements carried out in the past (Swillens et al. 2008; Low et al. 2012) clearly indicate that an accurate analysis and decomposition of measured pressure waveforms may provide an indication of existence of an aortic aneurysm. Thus, in the present work we determine and analyse the pressure waveform changes caused by reflection from an aneurysm. We study dependence of the reflection coefficient on aneurysm size and also compare the results with the pressure waveform in a healthy vessel. We also propose a method of determination of parameters indicating existence of an aneurysm depending on its location and its rate of change in cross-sectional area. To experimentally model a blood vessel with and without aneurysm, we have developed an experimental set-up, representing a part of the cardiovascular system. The arrangements and the parameters of this set-up are also used in the numerical simulations. After a thorough analysis of the experimental and numerical results, the indicative parameters of an aneurysm, identified from the simplified experimental and numerical model, are put into practice in a full systemic circulation model.

The paper is organized into the following sections. In Sect. 2, we describe the experimental set-up, method of measuring the pressure and velocity waveforms and the experimental results. In Sect. 3 the governing equations are presented and the numerical scheme is briefly described as well as the definition of the boundary conditions. The numerical results are compared with the experimental data in order to validate the numerical model. The wave analysis is discussed in Sect. 4. Here, the waveform is decomposed into forward and backward waves, and the formulation of the aneurysm indicators is presented. The method developed is then applied to the experimental and numerical data described in Sects. 2 and 3, respectively, and then, in Sect. 5, to a human arterial model network. Section 6 draws some conclusions and discusses the challenges and unresolved problems. Some auxiliary material is presented in Appendix including details of the numerical scheme (“Appendix 1”) and the waveform generated by the pump (“Appendix 2”).

## 2 Experimental set-up and measurements

The experimental set-up includes a pump representing the heart and system of tubes characterizing the blood vessels. One of those tubes has a bulge to represent an aneurysm as shown in Fig. 1. Water is used as a working liquid. The pump, 1, generates pulses propagating through the system of tubes. It is directly connected to a rigid tube, 2, which through a rigid fitting, 7, is connected to the main tube, 3–4, located in a transparent box, 6. The tube segment with the artificial aneurysm, 5, is cut into the main tube. The outlet of the main tube through the fitting, 8, located in the wall of the box is connected to a 6.5-m-length pipe, 9, and its outlet is connected to a collecting reservoir, 10. This allows us to minimize the wave reflection. The rigid inlet tube has a branch, 11, through which the pressure measuring catheter is inserted.

The aneurysms are axially symmetric with the generatrix having the shape of a circular arc of radius \(R_A\) smoothly conjugated to the constant area parts of the tube. Conjugating radii are \(R_s = 10\) mm. The length of the aneurysm is \(L_A = 9\) cm. It is located in the middle of the tube segment having a length \(L_S = 14\) cm. Wall thickness, *h*, is kept uniform for all parts of the segment and \(h=2\) mm. The elastic properties of the aneurysm tube segment are measured to be close to those of the main tube. Besides the aneurysm, similar experiments and measurements have been conducted on a control setting: a uniform main pipeline. The fitting is an approximately rigid tube with an internal diameter of 12 mm.

Pressure was measured using a 6 F pressure transducer-tipped catheter (Gaeltec, Scotland, UK), which was inserted into the tube through a Y-junction at the inlet of the tube or outlet of the tube. The flow rate was measured using ultrasound flow probes (Transonic System Inc, Ithaca, NY, USA). All measurements of the pressure and flow rate are taken in tube segment 3 shown in Fig. 1 in four sites: at a distance 2, 10, 25 and 50 cm from the inlet of the main tube.

Repeated measurements for every aneurysm in every site are taken from 8 to 15 for every combination of aneurysm size/site, resulting in more than 200 measurements overall. The basic sampling frequency is 500 Hz, but 1 kHz is also used in some measurements to ensure that pressure waveforms are properly captured. First, the measured waveforms are smoothed by convolution with a Gaussian function at a width of two time steps. As the recording is not synchronized with the pump, it is necessary to align all waveforms to start the main pulse at the same instant. Then the measured waveforms are averaged in the sample rate 500 Hz in order to decrease the influence of noise further.

The averaged measured pressure waveforms are presented in Fig. 4 indicated by black curves for all 20 cases: five aneurysm sizes (including the control tube) times four sites. We can observe strong oscillations at sites downstream from the inlet in the set-up with aneurysms. The further from the aneurysm the stronger the oscillations are. For those tests without an aneurysm (the first row of plots), we observe much smaller oscillations of similar amplitude in all the sites. Also we see that the greater aneurysm size the lower is the maximal pulse amplitude. The next stage of experimental data processing is described in Sect. 4.2.

## 3 Numerical simulation

### 3.1 Governing equations

*t*and

*x*denote the partial derivatives with respect to time

*t*and coordinate along the pipe

*x*, respectively;

*A*is the cross-sectional area of the lumen; \(u, \rho and p\) are, respectively, the average velocity, density and pressure of the liquid, whilst \(\tau \) is the wall shear stress. Assuming cylindrical vessels and a Poiseuille flow we find:

*E*and \(\sigma \) are the Young modulus and Poisson ratio for the wall material, respectively; \(A_{0}\) is the initial tube inner area; \(E'\) is an analogue of the Young modulus for plates and shells. Now Eqs. (1)–(4) form a closed system.

### 3.2 Characteristic variables

*c*is the pulse wave speed in the tube. Under physiological flow conditions, it is known that the pulse wave speed is higher than the fluid velocity

*u*(Formaggia et al. 2002, 2003; Sherwin et al. 2003a, c). Therefore, the system is strictly hyperbolic and subcritical with \(\lambda _{f}>0\) and \(\lambda _{b}<0\). The characteristic variables of the system are well known to have the form

*u*and

*c*are amplitude dependent. If the characteristics are calculated, the physical variables can be easily restored via:

*n*denotes the

*n*th time step for any variable and \(\bar{w}_f\) is the characteristic variable computed from prescribed pressure or velocity.

### 3.3 Boundary conditions

At the connections between tubes the continuity of the flow rate \(Q = Au\) and the total pressure \(p + \frac{1}{2} \rho u^2\) are imposed (Mynard and Nithiarasu 2008). The boundary condition at the outlet (reservoir) is not essential as we are dealing with a single pulse at a time period (less than 1 s) shorter than the time required for a wave to propagate from the pump to the reservoir and back (about 1 s). Nevertheless, we modelled the reservoir as a wide tube with a non-reflecting outlet boundary condition.

One can see in Fig. 5 (green curve) that at 2 cm from the inlet, the velocity almost vanishes after \(t>0.6\), i.e. when the piston stops its motion. At the same time, the pressure tends to have a low-frequency harmonic oscillation. Its period is about 2 s that is four times that of the wave propagation time from the piston to the reservoir.

Therefore, the velocity waveform measured at the interval \([0,0.6~\mathrm{s}]\) can be taken as the actual load adjusted waveform. As this wave is forward propagating one, the generated pressure waveform can be calculated by multiplying the velocity by \(\rho c_0\) in the linear approximation. This pressure waveform is set as the inlet boundary condition in the numerical model. The somewhat uncommon two-humped shape is explained in “Appendix 1”. The load depends on the aneurysm size; therefore, for every aneurysm, the initial waveform is calculated independently. All the initial waveforms extracted from the experimental data and utilized in the numerical modelling are shown in Fig. 6.

### 3.4 Numerical results

The results of the numerical simulations are shown in Fig. 4 by red curves. One can see that despite some discrepancies, the waveforms generally coincide. The frequency and the phase of oscillation are in better agreement than the amplitude. The fact is that the frequencies are determined by the time of wave propagation from the pump to a reflecting object (aneurysm, second fitting) and back. The amplitudes of the oscillations depend mainly on the reflection coefficients which are more sensitive to the set-up parameters, and it is not easy to measure some of them accurately. The discrepancies can also be attributed to the lack of the 1D model’s ability to accurately describe the process in the pump, losses in the tube, parameters of the fittings, transverse motion of the tube and other phenomena.

## 4 Wave separation and reflection coefficient

### 4.1 Transposition into forward and backward waves

There are two approaches on the separation of the waveform into forward and backward travelling waves. The first approach is based on the fact that nonlinear and viscous effects are relatively small. The second is based on the water-hammer equation and is valid for finite amplitude waves (wave intensity analysis) (Parker and Jones 1990; Khir et al. 2001; Swillens et al. 2008; Hughes and Parker 2009). We actively use the first approach by assuming that nonlinear and viscous effects are weak and can cause a noticeable change in the wave amplitude and shape only after propagation over a sufficiently large distance, i.e. beyond the domain of interest.

### 4.2 The aneurysm as a localized reflector

*x*-coordinate is referenced from the aneurysm inlet. The cross-sectional compliance is defined as

*K*parameter is the ratio of the additional aneurysm compliance to that of the compliance of the healthy vessel of same length. It can easily be calculated numerically, but for the model aneurysms used in the experiment it can be calculated analytically as well by directly integrating Eq. (20). Both calculations give the same values as displayed in Table 1 for different aneurysm diameters (

*D*) studied here. We use these values as reference ones. They are compared below with the values of

*K*evaluated from the waveform analysis.

Comparison of reference parameters with parameters obtained by fitting \(p_f*R\) into numerical and experimental waveforms

| 24 mm | 34 mm | 44 mm | 50 mm | |
---|---|---|---|---|---|

\(\Delta t,\) ms | Refer. | 57.9 | 57.9 | 57.9 | 57.9 |

Numer. | 57.4 | 56.8 | 56.6 | 56.5 | |

Exper. | 61.8 | 52.0 | 53.1 | 55.6 | |

\(x_A,\) cm | Refer. | 55.0 | 55.0 | 55.0 | 55.0 |

Numer. | 54.5 | 53.9 | 53.8 | 53.6 | |

Exper. | 58.7 | 49.4 | 50.5 | 52.8 | |

\(\tau \), ms | Refer. | 2.62 | 9.29 | 20.5 | 30.0 |

Numer. | 3.98 | 11.4 | 26.2 | 40.6 | |

Exper. | 4.59 | 20.8 | 41.6 | 45.8 | |

| Refer. | 1.11 | 3.92 | 8.66 | 12.7 |

Numer. | 1.68 | 4.82 | 11.1 | 17.2 | |

Exper. | 1.94 | 8.79 | 17.6 | 19.3 | |

\(\Delta C_A, \frac{{\mathrm{cm}}^3}{{\mathrm{MPa}}}\) | Refer. | 6.27 | 22.2 | 49.0 | 71.6 |

Numer. | 9.50 | 27.3 | 62.6 | 97.1 | |

Exper. | 11.0 | 49.7 | 99.3 | 109 |

### 4.3 Reflected waves from an aneurysm

Now we consider the wave reflection from an aneurysm by treating it as a 0D object with compliance \(\Delta C_A\). As the wave changes its shape after the reflection, we consider first the reflection of a harmonic wave having an angular frequency, \(\omega \), and a wavenumber, \(k = \omega /c_0\).

*R*is the complex reflection coefficient that needs to be determined. To the right (exit), there will be a propagating wave \(S\exp \{ikx-i\omega t\}\) with the unknown amplitude

*S*. Omitting the common factor \(\exp \{-i\omega t\}\), we can write the solutions as

*R*and

*S*. Solving Eqs. (21) and (22) reflection coefficient

*R*can be written in the following form

*H*(

*t*) is the unit-step Heaviside function. This allows the computation of the reflected pulse directly in the time domain through the convolution,

Results of wave separation performed at the location 2 cm from the inlet are shown in Fig. 7 for both the numerical and experimental data. Here, the results for the least squares fit of the \(p_f(t)*R(t)\) function are also presented. Parameters \(\tau \) and \(\Delta t\) extracted from the fit are displayed in Table 1. In this table, one can also find other parameters calculated via \(\Delta t\): distance from the 2 cm location to the midpoint of the aneurysm (should be \(x_A = 55\) cm) and also parameters calculated via \(\tau \): the dimensionless parameter *K* (see (20)) and the aneurysm compliance measured in \(\hbox {cm}^3\) per MPa.

Thus, the proposed method allows for detecting both the aneurysm location and its compliance. The accuracy of the procedure can be seen from the comparison of reference values, which are denoted as “Refer.” in Table 1, with the results following from the signal processing procedure. One can see from Table 1 that the distance to the aneurysm can be determined rather accurately from the described fitting procedure applied to both the numerical and experimental data. As for the aneurysm compliance, the procedure gives enhanced values, in some cases more than two times higher than the analytical values. The discrepancy is greater for the experimental results due to unavailability of precise values of elastic properties of the model aneurysms. The discrepancy between analytical and numerical data is less clear and needs additional analysis. Note that the fitting procedure is sensible to the wave separation and noise.

## 5 Application to human arterial system model

### 5.1 The model

Segments of the human arterial network selected for monitoring: *L* is the segment length, \(D_1,D_2\) are the segment inlet and outlet diameters, and \(c_0\) is the averaged wave speed

\(\hbox {N}^\circ \) | Name | | \(D_1,\hbox {mm}\) | \(D_2,\hbox {mm}\) | \(c_0,\hbox {m}/\hbox {s}\) |
---|---|---|---|---|---|

1 | Aortic Arch I | 7.44 | 31.9 | 25.9 | 4.03 |

3 | Aortic Arch II | 0.96 | 25.9 | 25.1 | 4.08 |

15 | Aortic Arch III | 0.70 | 25.1 | 24.6 | 4.09 |

19a | Aortic Arch IV | 4.31 | 24.6 | 21.1 | 4.12 |

19b | Thoracic Aorta I | 0.99 | 21.1 | 20.7 | 4.15 |

27 | Thoracic Aorta II | 0.79 | 20.7 | 20.4 | 4.15 |

29 | Thoracic Aorta III | 1.56 | 20.4 | 19.8 | 4.16 |

31 | Thoracic Aorta IV | 0.53 | 19.8 | 19.6 | 4.17 |

33a | Thoracic Aorta V | 12.16 | 19.6 | 15.1 | 4.23 |

33b | Thoracic Aorta VI | 0.32 | 15.1 | 15.0 | 4.30 |

35 | Abdominal Aorta I | 1.40 | 15.0 | 14.6 | 4.31 |

41 | Abdominal Aorta II | 0.43 | 14.6 | 14.5 | 4.32 |

43 | Abdominal Aorta III | 1.20 | 14.5 | 14.2 | 4.32 |

45 | Abdominal Aorta IV | \(10.60^*\) | 14.2 | 12.9 | 4.36 |

47 | Abdominal Aorta V | \(1.00^*\) | 12.9 | 11.8 | 4.43 |

5 | R. Common Carotid | 8.12 | 9.0 | 6.7 | 4.89 |

14 | L. Common Carotid | 12.13 | 9.0 | 6.7 | 4.89 |

*p*(

*t*), average velocity

*u*(

*t*) and area variation

*A*(

*t*) are monitored at the centre of the segments indicated in Table 2 (aorta and left and right common carotids). An example of the computed pressure waveform in segment 19a (Aortic Arch IV) is shown in Fig. 9.

As seen, the presence of an aneurysm results in a pressure drop at a certain time interval after the main peak. This is particularly pronounced in the case of the AAA-3 aneurysm. The graphs clearly show the main feature of the large AAA’s presence, i.e. a pressure drop just after the main peak followed by a distinct second peak after that.

### 5.2 AAA detection based on aortic waveform analysis

We apply the proposed aneurysm detection method to the sites located in aortic segments. First, eight segments listed in Table 2 are selected for that: from 1 to 31 (from Aortic Arch I to Thoracic Aorta IV). Equation (14) is applied to separate the forward \(p_f\) and backward \(p_b\) waves. Then the convolution \(p_f(t)*R(t)\) is calculated. This function has to be fitted into the backward wave \(p_b\) via the least squares method. In the experimental and numerical modelling described in previous sections, the analysis was easier when the pulses reflected from the aneurysm and other parts of the network are separated in time. As a result, the pulse reflected from the aneurysm was the first to reach the monitoring point as a backward wave. In the case of a full human arterial network, however, the pressure pulse exerts multiple reflections from bifurcations proximal to the aneurysm, and, hence, there is no such time interval where the only pulse reflected from the aneurysm exists. The reflections here are mixed with other reflections. Therefore, the aneurysm detection is more complicated, and the aneurysm detection procedure has to be modified.

Another challenge is that the background noise of the waves reflected from locations other than the AAA remains. Therefore, we fit the function \(p_r = p_f*R + B\) into the \(p'_b\) function where *B* is the parameter to be found. The *B* parameter approximates the background of the remaining reflections, noise and the forward wave pulse. A simple constant value for this parameter is sufficient here.

*B*:

*T*/ 3 for the width of the interval gives satisfactory results in all the cases considered. Note that the procedure is more sensitive to the time interval \([t_1,t_2]\).

Comparison of the reference parameters of a model AAA and parameters evaluated through waveform analysis: \(\Delta t\), ms and \(\Delta C_A\), \(\hbox {cm}^3/\hbox {MPa}\)

\(\hbox {N}^\circ \) | \(\Delta t^{\mathrm{ref}}\) | AAA-2 | AAA-3 | ||
---|---|---|---|---|---|

\(\Delta C_A^{\mathrm{ref}}{=}22.3\) | \(\Delta C_A^{\mathrm{ref}}{=}75.8\) | ||||

\(\Delta t\) | \(\Delta C_A\) | \(\Delta t\) | \(\Delta C_A\) | ||

1 | 163 | 178 | 13 | 220 | 48 |

3 | 143 | 137 | 19 | 179 | 58 |

15 | 139 | 126 | 21 | 163 | 72 |

19a | 126 | 105 | 26 | 132 | 90 |

19b | 113 | 90 | 32 | 109 | 83 |

27 | 109 | 84 | 33 | 103 | 83 |

29 | 104 | 76 | 35 | 95 | 84 |

31 | 99 | 69 | 38 | 87 | 86 |

5 | 194 | 151 | 13 | 172 | 48 |

14 | 189 | 145 | 16 | 161 | 71 |

### 5.3 AAA detection based on carotid waveform analysis

Finally, we consider the case that is most relevant when targeting this technology for non-invasive assessments in clinical practice, i.e. the determination of the AAA parameters by measuring the waveforms at more accessible sites such as the carotid artery. Here we will use the waveforms computed in the middle of the right common carotid (RCA) (segment 3) and left common carotid (LCA) (segment 14). This case is more complicated because the wave separation does not help, as the forward pulse and the pulse reflected from the AAA propagate in the same direction. Therefore, aneurysm detection based on the carotid waveform requires a more sophisticated approach of signal processing. The approach should be based on the possibility to calculate the shape of pulse reflected from the aneurysm and then to recognize it on the background of the main signal, other reflections and noise. Our proposed signal processing solution to this problem is described next.

We propose to apply the above method to the velocity waveform. This makes sense as the velocity waveform peak has a shorter duration, and therefore, the reflected wave from the AAA pulse should be easier to distinguish on the background signal consisting of the direct pulse and other reflections. Thus, we take the velocity (or flow rate) waveform, filter it using \(u' = u - G*u\) where *G* is given by Eq. (32) and fit the function \(u_r = u'*R + B\) using least squares with respect to the same three parameters: aneurysm characteristic time \(\tau \), time lag \(\Delta t\) and pulse background *B*.

In the case of the LCA, the time lag is the time of pulse propagation from the LCA inlet to the AAA and back. In the case of the RCA, the time lag is the duration of pulse propagation from the brachiocephalic trunk inlet (segment 2 in (Boileau et al. 2015)) to the AAA and back. Calculated values of the time lag for these cases are listed in the second column in Table 3. The results of the least squares fit of the \(u_r\) function are shown in Fig. 11. From Table 3 we see that the proposed method underestimates the \(\tau \) parameter and the time lag \(\Delta t\) but nevertheless gives reasonable values for the AAA parameters. Note that in the absence of the aneurysm, the method gives a value of \(\Delta C < 2\) \(\hbox {cm}^3\hbox {/MPa}\). This indicates the method is highly sensitive to the presence of an aneurysm.

## 6 Discussion and conclusion

A new method is developed to detect and characterize aortic aneurysms using the pulse reflections caused by those aneurysms. An experimental set-up is used to: 1) investigate waveform changes caused by aneurysms of various severity, 2) to validate our 1D computational model in the presence of aneurysms. The results obtained allow us to develop a new method of aneurysm detection based on waveform analysis. These results are then successfully employed to a numerical model of a human arterial system to evaluate the potential for detecting aneurysms. Analysis of the waveforms observed in the carotid arteries shows that aneurysms can be detected in terms of location and severity through our new method.

The proposed method, which incorporates the reflected waveform computation \(p_f*R\) or \(u*R\) with the subsequent least squares fitting procedure, looks promising for aneurysm detection and determination of its main parameters. The novel parameter introduced is the aneurysm compliance as defined by Eq. (19) and can be determined via this procedure. Moreover, if we are working in the frequency range in which the 1D theory is applicable and accurate enough, the \(\Delta C_A\) parameter looks to be the only parameter of an aneurysm that can be determined through the waveform analysis. To extract finer details of an aneurysm (its more detailed geometrical and elastic parameters) it is necessary to register and process the waveform in the higher frequency range where the wavelength is comparable to the aneurysm dimension. The 1D theory may be inadequate for this.

The aneurysm compliance given by Eq. (15) is a very useful parameter. As it is proportional to the integral of the vessel diameter cubed (see Eq. (19)), the wider parts of the aneurysm strongly contribute to its value. This can help to evaluate the aneurysm diameter for most of aneurysm geometrical shapes.

Note that in this work we focus on the effect of aneurysm geometries on the pulse wave reflections. Therefore, the artificial aneurysms have been produced to keep the complexities to a minimum by adapting constant thickness and stiffness. We use identical settings in the numerical simulations. If the wall thickness *h* and its elastic modulus \(E'\) also vary along the vessel, then Eq. (18) should be used instead of Eq. (19).

Observe that the wall stiffness \(h_A(x)E'_A(x)\) is in the denominator in (18). Therefore, the \(\Delta C_A\) parameter is very sensitive to the local wall softness: the softer/thinner parts of the wall strongly contribute to the integral (18). Hence, rapid increase in time of the aneurysm compliance \(\Delta C_A\) can indicate that some parts of the wall are very thin and soft. This can potentially allow aneurysm monitoring, especially when the proposed method is employed in parallel to other modalities, for example, ultrasound (US). This may provide an opportunity to evaluate the elastic parameters of an aneurysm and predict its rupture.

One of the advantages of the proposed method is that it can be implemented in the time domain where it is easier to distinguish pulses reflected from different parts in the cardiovascular system. Another advantage is that there is no need to calculate the reflection coefficient as the pulse reflected from an aneurysm definitely changes its shape. Therefore, if we define the reflection coefficient, for example, as a ratio of peak values of reflected and incident pulses, it will depend on the shape of the incident pulse, i.e. it will not be invariant with respect to incident pulse shape and parameters. The same remark can be directed to the pulse intensity analysis(Parker and Jones 1990; Khir et al. 2001; Swillens et al. 2008; Hughes and Parker 2009). In the frequency domain, at any particular frequency all the reflected pulses are mixed together, and it is difficult to outline the contribution of the aneurysm. So those approaches can at best help to detect the presence of an aneurysm, but they can hardly help to determine its geometrical/elastic properties accurately.

The results of the work are very promising and show that the proposed method has a real potential to be further developed into a powerful technique that will be adopted in a clinical setting some day.

### 6.1 Limitations

In this subsection, the limitations of the proposed method of detecting aneurysms are briefly highlighted. Although the limitations are less severe when comparing the proposed model to the in vitro experimental data, the limitations become prominent when the model is applied to in vivo data of patients. This is due to the fact that the in vitro experimental parameters are controlled but the patient data come with many unknowns. Some of the specific limitations of the work are briefed below.

The ability of the model to correctly determine the aneurysm size depends on the aneurysm wall stiffness *Eh* as demonstrated by Eq. (18). Although a large number publications on aneurysm wall properties are constantly appearing in the literature, the uncertainty related to patient specificity will be extremely difficult to eliminate. The question on relationship between the aneurysm size and material property is also not completely answered with contradicting reports in the literature. For example, Sekhri et al. (2004) found that the aneurysm wall becomes stiffer with an increase in aneurysm size whilst Kolipaka et al. (2016) did not confirm this. Therefore, if the vessel wall stiffens with the an increase in aneurysm size, then the aneurysm compliance grows at a slower rate than that is described by equation (19), which will underestimate the aneurysm diameter. Moreover, if the wall stiffness grows proportionally to the cube of the aneurysm diameter, then the aneurysm compliance will remain very small and the proposed method may not detect the aneurysm. On the other hand, if the aneurysm is very compliant locally, which can occur just before the rupture, then the pronounced reflection will immediately indicate the presence of an aneurysm. In addition to these limitations, assumed parameters of the healthy vessels can also contribute to the inaccuracy in predictions.

It is also important to mention that Eqs. (24) and (28) are derived for a tube with a constant cross-sectional area and an embedded aneurysm. Such conditions can be easily reproduced using an in vitro experimental set-up. Nevertheless, for accurate aneurysm detection, these equations need generalization to the in vivo cases that have tapered blood vessels and varying vessel stiffness. We will consider such generalizations in a subsequent, future work. However, if the tapering and stiffness variations are small, then the effects of such variations on the results will be much lower than that of the uncertainties due to other parameters.

## Notes

### Acknowledgements

The study was founded by the A4B Welsh Government (Grant Number HE09151024).

### Compliance with Ethical Standards

### Conflict of interests

The authors declare that they have no conflict of interests.

## References

- Alastruey J, Parker KH, Peiro J, Byrd SM, Sherwin SJ (2007) Modelling the circle of Willis to assess the effects of anatomical variations and occlusions on cerebral flows. J Biomech 40(8):1794–1805CrossRefGoogle Scholar
- Ashton HA, Buxton MJ, Day NE, Kim LG, Marteau TM, Scott RA, Thompson SG, Walker NM (2002) The multicentre aneurysm screening study (MASS) into the effect of abdominal aortic aneurysm screening on mortality in men: a randomised controlled trial. Lancet 16(360(9345)):1531–1539Google Scholar
- Avolio AP (1980) Multi-branched model of the human arterial system. Med Biol Eng Comput 18(6):709–718CrossRefGoogle Scholar
- Barkin AZ, Rosen CL (2004) Ultrasound detection of abdominal aortic aneurysm. Emerg Med Clin N Am 22:675–682CrossRefGoogle Scholar
- Barnard AC, Hunt WA, Timlake WP, Varley E (1966) A theory of fluid flow in compliant tubes. Biophys J 6(6):717–724CrossRefGoogle Scholar
- Blanco PJ, Trenhago PR, Fernandes LG, Feijo RA (2012) On the integration of the baroreflex control mechanism in a heterogeneous model of the cardiovascular system. Int J Numer Methods Biomed Eng 28:412–433MathSciNetCrossRefGoogle Scholar
- Blanco PJ, Watanabe SM, Passos MARF, LP A, Feijo RA (2015) An anatomically detailed arterial network model for onedimensional computational hemodynamics. IEEE Trans Biomed Eng 62:736–753CrossRefGoogle Scholar
- Boileau E, Nithiarasu P, Blanco PJ, Müller LO, Fossan FE, Hellevik LR, Donders WP, Huberts W, Willemet M, Alastruey J (2015) A benchmark study of numerical schemes for one-dimensional arterial blood flow modelling. Int J Numer Methods Biomed Eng. doi: 10.1002/cnm.2732
- Brekken R, Dahl T, Hernes TAN (2011) Ultrasound in abdominal aortic aneurysm. In: Grundmann R (ed) Diagnosis, screening and treatment of abdominal, thoracoabdominal and thoracic aortic aneurysms. InTech, Rijeka, pp 103–124Google Scholar
- Catalano O, Siani A (2005) Ruptured abdominal aortic aneurysm: categorization of sonographic findings and report of 3 new signs. J Ultrasound Med 24:1077–1083CrossRefGoogle Scholar
- Chen P, Quarteroni A, Rozza G (2013) Simulation-based uncertainty quantification of human arterial network hemodynamics. Int J Numer Methods Biomed Eng 29:698–721MathSciNetCrossRefGoogle Scholar
- Fink H, Lederle FA, Roth CS, Bowles CA, Nelson DB, Haas MA (2000) The accuracy of physical examination to detect abdominal aortic aneurysm. Arch Intern Med 160(6):833–836CrossRefGoogle Scholar
- Fleming C, Whitlock EP, Beil TL, Lederle FA (2005) Screening for abdominal aortic aneurysm: a best-evidence systematic review for the U.S. preventive services task force. Ann Intern Med 142(3):203–211CrossRefGoogle Scholar
- Formaggia L, NobileAlfio F, Quarteroni Veneziani A (1999) Multiscale modelling of the circulatory system: a preliminary analysis. Comput Vis Sci 2(2):75–83CrossRefzbMATHGoogle Scholar
- Formaggia L, Gerbeau JF, Nobile F, Quarteroni A (2001) On the coupling of 3D and 1D Navier–Stokes equations for flow problems in compliant vessels. Comput Methods Appl Mech Eng 191(6–7):561–582MathSciNetCrossRefzbMATHGoogle Scholar
- Formaggia L, Nobile F, Quarteroni A (2002) A one-dimensional model for blood flow: application to vascular prosthesis. In: Babus̆ka I, et al (eds) Mathematical modeling and numerical simulation in continuum mechanics, lecture notes in computational science and engineering, vol 19, pp 137–153Google Scholar
- Formaggia L, Lamponi D, Quarteroni A (2003) One-dimensional models for blood flow in arteries. J Eng Math 47:251–276MathSciNetCrossRefzbMATHGoogle Scholar
- Franke V, Peiró J, Sherwin S, Parker K, Ling W, Fisk NM (2002) Computational modelling of 1D blood flow and its applications. In: Thiriet M (ed) ESAIM Proceedings, vol 12, pp 48–54Google Scholar
- Huang PG, Muller LO (2015) Simulation of one-dimensional blood flow in networks of human vessels using a novel tvd scheme. Int J Numer Methods Biomed Eng. doi: 10.1002/cnm.2701
- Hughes AD, Parker KH (2009) Forward and backward waves in the arterial system: impedance or wave intensity analysis? Med Biol Eng Comput 47:207–210CrossRefGoogle Scholar
- Hughes TJ, Lubliner J (1973) On the one-dimensional theory of blood flow in the larger vessels. Math Biosci 18(1–2):161–170CrossRefzbMATHGoogle Scholar
- Khir AW, O’Brien A, Gibbs JSR, Parker KH (2001) Determination of wave speed and wave separation in the arteries. J Biomech 34:1145–1155CrossRefGoogle Scholar
- Kolipaka A, Illapani VC, Kenyhercz W, Dowel JD, Go MR, Starr JE, Vaccaro PS, White RD (2016) Quantification of abdominal aortic aneurysm stiffness using magnetic resonans elastography and its comparison to aneurysm diameter. J Vasc Surg 64(4):966–974CrossRefGoogle Scholar
- Lee JK, Ling D, Heiken JP, Glazer HS, Sicard GA, Totty WG, Levitt RG, Murphy WA (1984) Magnetic resonance imaging of abdominal aortic aneurysms. Am J Roentgenol 143(6):1197–1202CrossRefGoogle Scholar
- Litmanovich D, Bankier AA, Cantin L, Raptopoulos V, Boiselle PM (2009) CT and MRI in diseases of the aorta. Am J Roentgenol 193(4):928–940CrossRefGoogle Scholar
- Low K, van Loon R, Sazonov I, Bevan RLT, Nithiarasu P (2012) An improved baseline model for a human arterial network to study the impact of aneurysms on pressure-flow waveforms. Int J Numer Methods Biomed Eng 28:1224–1246MathSciNetCrossRefGoogle Scholar
- McBride OMB, Berry C, et al (2015) MRI using ultrasmall superparamagnetic particles of iron oxide in patients under surveillance for abdominal aortic aneurysms to predict rupture or surgical repair: MRI for abdominal aortic aneurysms to predict rupture or surgery—the \(\text{MA}^3\text{ RS }\) study. Open Heart 2Google Scholar
- Müller LO, Toro EF (2013) Well-balanced high-order solver for blood flow in networks of vessels with variable properties. Int J Numer Methods Biomed Eng 29:1388–1411MathSciNetCrossRefGoogle Scholar
- Müller LO, Toro EF (2014) A global multiscale mathematical model for the human circulation with emphasis on the venous system. Int J Numer Methods Biomed Eng 30:681–725MathSciNetCrossRefGoogle Scholar
- Mynard JP, Nithiarasu P (2008) A 1D arterial blood flow model incorporating ventricular pressure, aortic valve and regional coronary flow using the locally conservative Galerkin (LCG) method. Commun Numer Methods Eng 24:367–417MathSciNetCrossRefzbMATHGoogle Scholar
- Parker KH, Jones CJH (1990) Forward and backward running waves in the arteries—analysis using the method of characteristics. J Biomech Eng 112:322–326CrossRefGoogle Scholar
- Sekhri AR, Lees WR, Adiseshiah M (2004) Measurement of aortic compliance in abdominal aortic aneurysms before and after open and endoluminal repair: preliminary results. J Endovasc Ther 11(4):472–482CrossRefGoogle Scholar
- Sherwin SJ, Formaggia L, Peiró J, Franke V (2003a) Computational modelling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system. J Numer Methods Fluids 43:673–700MathSciNetCrossRefzbMATHGoogle Scholar
- Sherwin SJ, Franke V, Peiró J, Parker K (2003b) One-dimensional modelling of a vascular network in space-time variables. J Eng Math 47:217–250MathSciNetCrossRefzbMATHGoogle Scholar
- Sherwin SJ, Franke V, Peiró J, Parker KH (2003c) One-dimensional modelling of a vascular network in space–time variables. J Eng Math 47:217–250MathSciNetCrossRefzbMATHGoogle Scholar
- Sparks AR, Johnson PL, Meyer MC (2002) Imaging of abdominal aortic aneurysms. Am Fam Physician 65(8):1565–1570Google Scholar
- Sprouse LR, Meier GH, Parent FN, DeMasi RJ, Glickman MH, Barber GA (2004) Is ultrasound more accurate than axial computed tomography for determination of maximal abdominal aortic aneurysm diameter? Eur J Vasc Endovasc Surg 28(1):28–35CrossRefGoogle Scholar
- Steele BN, Olufsen MS, Taylor CA (2007) Fractal network model for simulating abdominal and lower extremity blood flow during resting and exercise conditions. Comput Methods Biomech Biomed Eng 10(1):39–51CrossRefGoogle Scholar
- Stergiopulos N, Young D, Rogge T (1992) Computer simulation of arterial flow with applications to arterial and aortic stenoses. J Biomech 25:1477–1488CrossRefGoogle Scholar
- Swillens A, Lanoye L, De Backer J, Stergiopulos N, Verdonck PR, Vermassen F, Segers P (2008) Effect of an abdominal aortic aneurysm on wave reflection in the aorta. IEEE Trans Biomed Eng 55(5):1602–1611CrossRefGoogle Scholar
- Urquiza SA, Blanco PJ, Vénere MJ, Feijóo RA (2006) Multidimensional modelling for the carotid artery blood flow. Comput Methods Appl Mech Eng 195:4002–4017MathSciNetCrossRefzbMATHGoogle Scholar
- Walker A, Brenchley J, Sloan JP, Lalanda M, Venables H (2004) Ultrasound by emergency physicians to detect abdominal aortic aneurysms: a uk case series. Emerg Med J 21:257–259CrossRefGoogle Scholar
- Wang JJ, Parker KH (2004) Wave propagation in a model of the arterial circulation. J Biomech 37(4):457–470CrossRefGoogle Scholar
- Watanabe SM, Blanco PJ, Feijo RA (2013) Mathematical model of blood flow in an anatomically detailed network of the arm. ESAIM. Math Model Numer Anal 47:961–985MathSciNetCrossRefzbMATHGoogle Scholar
- Wilmink ABM, Forshaw M, Quick CRG, Hubbard CS, Day NE (2002) Accuracy of serial screening for abdominal aortic aneurysms by ultrasound. J Med Screen 9:125–127CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.