1 Introduction

Intracranial aneurysm (IA) is a structural and irreversible deformation of a cerebral artery wall. The estimated prevalence of unruptured intracranial aneurysm (UIA) is 2-5 % in the global population [1], which makes UIA a public health issue. The yearly IA rupture risk differs from 1 to 4 % in the carrier population [2]. The IA rupture leading to subarachnoid haemorrhage has a fatality rate of 30-40 % [3]. Among patients who have survived to this haemorrhagic stroke, 1 in 5 may be functionally dependent and even patients who obtain functional independence may have psychological and neurological sequelae [4, 5]. When the aneurysm is detected casually by a routine examination with clinical imagery such as MRI, several factors must be considered by the clinician to identify the optimum approach to management. The risk of the IA rupture without intervention has to be compared to the risks associated with endovascular treatment or a surgical clipping [6]. Several factors can be considered, either associated with the aneurysm (location, size, shape, presence of multiple lobes), or associated with the patient (age, medical history, personal or family history of aneurysm) [7,8,9,10]. Recommendations are mainly based on qualitative expert opinion and guidelines, most of which are in agreement that patients with large UIA and those who are asymptomatic should be considered for interventional treatment [11]. There is nevertheless no consensus about the optimum management of all other kinds of aneurysms. There is currently no method to accurately predict the rupture risk based on the quantitative patient-specific determination of the aneurysm wall stress state. Most current studies focus on the numerical modelling of Fluid–Structure Interaction (FSI) [12,13,14] that considers several anatomical variations and sizes of aneurysm and the brain circulatory system [15, 16]. Experimental investigations of UIA mechanical characterisation have been done only on ex vivo tissue samples and measured the rheological behaviour of the arterial wall using uniaxial tensile tests [17, 18], biaxial tensile tests [19] and indentation tests [20].

Investigating the mechanical behaviour of aneurysmal tissue in vivo is mandatory to improve the understanding and management of UIAs. This work forms an essential component of a large scale project that focuses on this topic. The purpose of this project is to provide a non-invasive patient-specific decision tool to clinicians to predict the aneurysm rupture from a standard clinical image. Artificial Intelligence (AI) (machine learning algorithm) based on the in vivo mechanical characterisation of the aneurysm wall is used to estimate a rupture criterion. The mechanical properties of the UIA wall is determined using an experimental arterial wall deformation device coupled with a medical imaging system. The anatomical image of the deformed aneurysm is numerically processed to quantify the aneurysm wall stress state by inverse analysis. The device is tested using in vitro studies and in vivo studies (on small animals) and the results will be the keystone of the AI analyses. In the first step, the deformation device was calibrated on an experimental study with an artificial phantom artery. The main objective of this calibration step was to determine the device parameters such as flow rate to achieve an observable and significant experimental deformation of the aneurysm. For practical reasons, emphasis was placed on working on a phantom artery instead of an ex vivo sample. Hence, the most critical element was to obtain an artificial artery with an arterial wall thickness and stiffness close to those of human arteries. The targeted values were in the range of 0.4\(-\)1.6 MPa for the aneurysm local stiffness [21] and in the range of 200-600 \(\mu \)m for the aneurysm wall thickness [22, 23]. This work explores the techniques available in this study to fabricate the phantom artery from a CAD file. 3D printing using carefully chosen material was used, as it is has been shown to be the most useful technique for creating phantom arteries in biomedical and pre-surgical studies [21, 24, 25]. The Polyjet and the Stereolithography (SLA) 3D printing techniques were tested with the associated materials [26]. Moreover, the phantom artery was also fabricated using an injection moulding technique. It was challenging with these methods to design a biofidelic artery in terms of mechanical properties and wall thickness. Due to their respective technical limitations, none of the techniques cited allowed obtaining a perfectly accurate patient-specific artery with an aneurysm. To overcome these technical limitations, a numerical model associated with the experimental study was developed. This model helped to address the following problem statements: (1) if the 3D printing techniques were applied, should the experimenter emphasised the lowest artery wall thickness (obtained with SLA 3D printing) or the lowest local stiffness (obtained with Polyjet 3D printing) ?; (2) if the injection moulding was applied, can the problem of heterogeneous thickness be overcome only with the local assessment of the patient-specific properties in the studied area only ?

Moreover, as it was impossible to precisely estimate the experimental location of the deformation device in the aneurysm, separately from the investigation of these techniques, the numerical model was used to quantify the sensitivity of the deformation device location in the aneurysm. The model helped to define a margin of error regarding immutable experimental criteria with the study of several deformation device locations and adjustments.

2 Materials and Method

2.1 Overview of the experimental deformation device modeled

The deformation device developed is a flux guidance system. It pumps out a miscible fluid (water for the experimental study, expected to be physiological liquid for the application to a living being) on the aneurysm wall that leads to its mechanical strain. The fluid flow rate is controlled by a pump system (Harvard Apparatus, Standard PHD ULTRA\(^\textrm{TM}\) CP Syringe Pump). The device was developed and calibrated on an experimental study with a polymeric artificial artery as a step prior to using an animal model. The deformation device was calibrated using a simplified Y-shaped of an artery with a bifurcation and a saccular aneurysm (figure 1). On the one hand, a simplified shape was chosen in order to avoid artefacts potentially related to a biofidelic arterial geometry. On the other hand, saccular aneurysms are responsible for most of the morbidity and mortality caused by subarachnoid haemorrhage and most frequently form in first and second order arteries originating from the cerebral arterial circle of Willis [27]. To facilitate the motion and the visualisation of the deformation device, the phantom artery had a diameter twice as large as that of the human cerebral artery diameter (8 mm instead of 4 mm, [28]). The aneurysm neck size studied was 8 mm, its height 12.5 mm and the aspect ratio (AR), 1.56 which was in line with the reported AR of ruptured aneurysms [29]. It was first essential to determine whether it was possible with the device to produce a significant deformation of the aneurysm even with an oversized artery. For these experiments, the extremities of the phantom artery were embedded in a tank containing water for the artery outer environment. The MatchID Digital Image Correlation (DIC) was used to measure, experimentally, the displacements and strains induced by the deformation device. The displacements were captured using two high-resolution cameras (Blackfly S USB3, Fujinon lens 50 mm; 2448*2048 pixels at 10FPS; 5.0MP; Sony IMX250 CCD; Global Shutter; Mono; C-mount).

2.2 Fabrication of the phantom artery: techniques and materials

Regarding the 3D printing techniques, the phantom artery was designed using the Formlabs SLA 3D printer with the elastic 50A Resin, while the Polyjet Stratasys J850\(^\textrm{TM}\) Digital Anatomy\(^\textrm{TM}\) 3D Printer was used with the GelMatrix. Regarding the injection moulding process, the Prevent Polyaddition Silicone FLUIDE FA5420N was used with a mould specially designed and built for the simplified shape of this phantom artery.

Several tensile tests were experimentally performed on controlled Formlabs elastic 50 A resin tensile specimens. A Young’s modulus mean value of 1.57 ± 0.16 MPa was found. For the Prevent Polyaddition Silicone FLUIDE FA5420N, a Young’s modulus of 0.7 ± 0.04 MPa was found. A Young’s modulus value of 1.1 MPa was provided by the Polyjet phantom artery supplier [30]. Each process presented a limitation regarding the reachable minimum wall thickness [31, 32]. For the SLA 3D printer and the injection moulding, it was possible to adjust the Artery Body (AB) and the Aneurysm (A) wall thicknesses separately. The experimentally reachable mean values of the Young’s modulus linked to the fabrication process are summarised in table 1 as is the Artery Body-Aneurysm (AB-A) thickness couple of the associated artery.

Table 1 Phantom artery mechanical and geometrical properties linked to the fabrication process in terms of Artery Body-Aneurysm (AB-A) thickness and the mean value of the Young’s moduli experimentally identified
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2.3 Numerical modelling of the deformation device operation

The numerical model associated with the experimental study and computed with the finite element method was implemented on COMSOL Mutliphysics, specifically using the Fluid–Structure Interaction (FSI) module. The artery CAD geometry file and the pump parameters were used to implement the numerical modelling. The numerical model incorporated the phantom artery, the deformation device, the inner main flow in the artery representing the human blood flow, the flow pulsed by the device and the outer environment surrounding the artery. The numerical simulation was run for 20 phantom arteries encompassing the experimental arteries studied: 4 wall thicknesses for the couple Artery body-Aneurysm (AB-A) and 5 Young’s moduli for each couple. It was also essential to consider different locations of the device into the aneurysm due to its flexibility. From one experiment to another, it was impossible to target the same deformation device location. However, it was essential to obtain a numerical range of results relating to it.

Fig. 1
figure 1

COMSOL Mutliphysics numerical modelling of the deformation device and the phantom artery from the artery CAD file of the experimental study

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2.3.1 Numerical modelling of the phantom artery and the deformation device

Two main solid mechanics components were considered in the numerical modelling: the phantom artery with a geometry linked to the artery CAD file from the experimental study; the deformation device corresponding to a cylindrical tube which delivered the fluid in the aneurysm. The artery COMSOL domain was imported from the associated STL file regarding the study considered. The deformation device domain was built using COMSOL geometry tools as a hollow cylinder with the appropriate length regarding the artery. Its location was defined regarding the impact area of the device pulsed flow (figure 1).

The artery inner radius was 4 mm and the deformation device’s inner radius was 0.84 mm. An isotropic linear elastic material was associated with both elements which is a usual hypothesis in the literature [33,34,35,36]. The deformation device was a hollow PTFE cylinder with a Young’s modulus of \(E = 0.41\) GPa, a density of \(\rho = 2160\) kg.m\(^{-3}\) and a Poisson’s ratio of \(\nu = 0.46\). The numerical model was built for the arteries with the following AB-A thickness couples: 600-400, 600-600, 1500-400 and 1500-1500 \(\mu \)m; for each couple, the following Young’s moduli were considered: 0.2, 0.5, 1, 1.5 and 2 MPa. In each case, the Poisson’s ratio was \(\nu = 0.49\) to represent the quasi incompressibility of the human artery and the density was \(\rho = 1350\) kg.m\(^{-3}\) [36].

The 20 numerical models of phantom arteries encompassed the mechanical properties of the phantom arteries identified experimentally in the framework of this study. The formulation used in the structural mechanics module was the Lagrangian formulation [37].

Concerning the boundary conditions, the extremities of the artery body were embedded in the surrounding tank to fit with the experimental conditions and to mimic the cerebral environment. Once its position was defined, the movement of the deformation device was also inhibited while its extremities were embedded. Data processing after the model computation was performed in order to consider the displacements and strains induced by the fluid delivered through the deformation device only, without accounting for displacements and strains induced by the blood flow.

Regarding the exploitation of the results, the displacement norm and the strain norm were studied. The displacement norm was defined as:

$$\begin{aligned} \vert \vert {\textbf {u}}\vert \vert = \sqrt{u_{X}^2 + u_{Y}^2 + u_{Z}^2}, \end{aligned}$$
(1)

and the strain norm based on the the linear Green Lagrange strain tensor was defined as:

$$\begin{aligned} \vert \vert \underline{\underline{\varepsilon }}\vert \vert = \sqrt{\varepsilon _{XX}^2 + \varepsilon _{YY}^2 + \varepsilon _{ZZ}^2 + 2\varepsilon _{XY}^2 + 2\varepsilon _{XZ}^2 + 2\varepsilon _{YZ}^2 }, \end{aligned}$$
(2)

where (\({\textbf {x}}, {\textbf {y}}, {\textbf {z}}\) ) is defined in figure 1.

2.3.2 Numerical modelling of the main flow and the deformation device flow

The experimental set-up included three main flow components. Firstly, a set-up replicating the outer environment of the artery: a steady liquid mass was poured in tank (Fluid 0). The phantom artery extremities were embedded in this tank. Secondly, a primary flow mimicking arterial blood flow (Fluid 1) was applied to the artery inlet. With the aim of simplifying the experimental set-up, this time-dependent boundary condition did not replicate a patient-specific flow profile. The profile of this steady flow included a one-second ramp function as loading phase before reaching a 500 mL/min flow rate (8 s experimentally maintained over time for this study). The maximum value of the associated parabolic flow velocity profile was 0.165 m/s accounting the artery diameter. Furthermore, this velocity value (accounting the oversized artery dimensions) closely matched the average blood speed measured in intracranial arteries [38,39,40,41,42,43,44]. Thirdly, a flow introduced directly into the aneurysm cavity through the device (Fluid 2). A flow rate of 150 mL/min was applied after the initial loading phase and the Fluid 1 stabilisation for a 6 s pulse. This value was based on equipment capabilities, particularly the syringe pump, and was selected to induce a visible deformation of the aneurysm wall and be detected by the DIC system. The flow velocity associated was 1.15 m/s accounting the device diameter. For this experimental study, water was used for both the blood flow modelling (Fluid 1) and the device pulsated flow (Fluid 2).

The three experimental flow components were numerically modelled and concatenated into two main fluid mechanics components: the liquid external environment which contain these whole set artery and deformation device assembly (Fluid 0), a fluid domain containing the main artery flow (Fluid 1) and the flow delivered by the deformation device (Fluid 2). For the latter, the following assumption was made: Fluid 1 and Fluid 2 were considered miscible and could be defined as a unique flow domain with the solid deformation device boundaries separating the two flows and defining two different inputs. All the fluid mechanics elements were defined as an incompressible laminar flow [34, 35, 45] according to the following Navier-Stokes equations:

$$\begin{aligned} \rho \frac{\partial {\textbf {u}}_{fluid}}{\partial t} + \nabla \cdot \eta (\nabla {\textbf {u}}_{fluid} + (\nabla {\textbf {u}}_{fluid})^{\textrm{T}} ) + \rho ({\textbf {u}}_{fluid}.\nabla ){\textbf {u}}_{fluid} + \nabla \textrm{p} = {\textbf {F}}, \end{aligned}$$
(3)
$$\begin{aligned} \nabla \cdot {\textbf {u}}_{fluid} = 0. \end{aligned}$$
(4)

With the velocity field, \(\textbf{u}_{fluid}\), the density, \(\rho \), the dynamic viscosity, \(\eta \), the pressure, p and a body force term such as gravity, \({\textbf {F}}\). Water was associated with all the fluid domains with \(\rho \) = 1000 kg.m\(^{-3}\) and \(\eta \) = 0.001 Pa.s. The external fluid environment was defined as a still flow with a null inlet and outlet speed. To be close to the experimental conditions and applied flow rates, the main artery inlet flow rate was 500 mL/min, the deformation fluid flow rate was 150 mL/min.

The schematic layout of the developed numerical model developed is displayed in figure 2.

Fig. 2
figure 2

Schematic layout of the COMSOL numerical model: miscible fluids and associated flow rates of the fluid mechanics elements, simplified Y-shaped of the phantom artery with a AB-A thickness couple of 600-400 \(\mu \)m. Visible markers in the Region of Interest (ROI) were used in the validation procedure

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2.3.3 Numerical modelling of the interaction between the deformation device flow and the aneurysm wall

To capture the interaction between the fluid and the solid structure, Fluid–Structure Interaction (FSI) multiphysics coupling was used combining the fluid flow with structural mechanics. The FSI couplings appeared on the inner wall of the artery (the boundaries between the fluid and the solid). No interactions between the flow and the device outer boundaries were considered. An arbitrary Lagrangian-Eulerian (ALE) method was used to combine the fluid flow formulation using an Eulerian description with a solid mechanics formulation using a Lagrangian description [33,34,35,36,37, 45].

The fluid flow was described by Navier–Stokes equations (3) and (4), which provided a solution for the velocity field \(\textbf{u}_{fluid}\). The total force exerted on the artery inner wall boundary was:

$$\begin{aligned} {\textbf {f}} = {\textbf {n}} \cdot [-\textrm{p} \underline{\underline{\textrm{I}}} + (\eta (\nabla {\textbf {u}}_{fluid} + ({\textbf {u}}_{fluid})^T ) - \frac{2}{3}\eta (\nabla \cdot {\textbf {u}}_{fluid})\underline{\underline{\textrm{I}}}], \end{aligned}$$
(5)

where \(\textbf{n}\) denotes the inward normal to the boundary, p pressure, \(\eta \) the dynamic viscosity for the fluid, and \(\underline{\underline{\textrm{I}}}\) the identity matrix.

The one-way coupled models solved sequentially for the fluid flow, computed the load from (5), and then applied it in the solution for the solid displacement. The fluid flow was solved with a time dependent study: the main artery flow was computed for a 8 s flow, the catheter flow was considered as a 6 s flow speed disruption in addition to the main flow at a controlled and well known time.

2.4 Towards the computational model validation versus the experimental results

2.4.1 Modelling of the experimental deformation device locations

To model the experimental uncertainty of the deformation device location, a set of locations was considered: each initial location of the device led to a specific mechanical stress area (figure 3). For each location shown in figures 3a, 3b and 3c the study was performed for distance D between the deformation device exit and the artery wall equal to 1 mm and 1.5 mm. For the location shown in figure 3d, the angle \(\theta \) was defined in relation to the normal axis of the central location; \(0.5^{\circ }\) and \(0.8^{\circ }\) angles were studied for an identical deformation device length inserted in the artery/aneurysm (associated with a deviation of the device tip of 1.2 mm and 1.9 mm compared to the initial central location, respectively). Due to the deviation of the deformation device, the arterial wall distance D was therefore smaller than 1 mm in relation to the angle studied.

Fig. 3
figure 3

The deformation device locations leading to various solicitations of the aneurysm wall, studied to model the experimental uncertainties

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2.4.2 Development of an original validation method

An original procedure was implemented to validate the COMSOL Multiphysics computational model in relation to the MathID DIC experimental results. For each experimental test series of the device on a phantom artery, several specific computational models were built to enfold the device location experimentally identified. Variable locations and distances to the aneurysm wall were considered as embodied in 2.4.1. The mesh coordinates and associated displacements of the aneurysm top area were extracted from the numerical study. This Region of Interest (ROI) can be seen in figure 2. This area was defined in relation to the experimental DIC zone visualised. These data were processed on the initial experimental correlated image, with a spatial calibration performed using experimentally and numerically visible markers on the area (figure 2). The strain computation was performed on the newly built correlated area based on the numerical data and compared with the experimental results.

3 Results

3.1 Validation of the computational models in relation to the experimental results

The results for the device tests performed on an injection moulded phantom artery are shown in figure 4. The mesh displacements were well conserved after building a correlated area based on these data; the experimental strain norm was merged with the results from several deformation device locations considered numerically. This procedure was applied on both the SLA and injection moulded phantom arteries. The results led to the validation of the computational model.

Fig. 4
figure 4

Validation of the computational model based on the comparison of the strain norm with the experimental results on the aneurysm ROI. The results associated with the device tested on an injection moulded artery. (A) Displacements norm (\(\mu \)m) at the top of the aneurysm for several deformation device locations studied numerically to enfold the experimental location. (B) Numerical mesh data export used to build a correlated area on a DIC reference image: mesh spatial calibration based on visible marker locations on the reference image (left side). Displacement norm (\(\mu \)m) for each device location on the newly built correlated area: data retention in the procedure (right side). (C) Experimental strain norm enfolded with two deformation device locations considered numerically

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3.2 Highlighting of the the aneurysm wall deformation related to the device and of the most solicited area

For further in vivo studies, it was essential to determine if the influence of the deformation device was significant in terms of displacements and strains of the aneurysm wall compared to the main flow. The displacement norm is shown in figure 5 for an AB-A thickness couple of 600-400 \(\mu \)m with a Young’s modulus of 1.5 MPa, a central location of the deformation device and a 1 mm distance to the arterial wall. The flow parameters were outlined in 2.3.2.

The influence of the device was highlighted with significant additional displacements compared to that of the main flow. The maximum aneurysm deformation was observed at its top, in an area directly linked to the deformation device location. For the 20 simulations with the different AB-A thickness couples and Young’s moduli, the same behaviour was observed and the same area was identified for the data extraction. This area was considered for each case study.

Fig. 5
figure 5

Displacement norm (\(\mu \)m) of the arterial wall for a AB-A thickness of 600-400 \(\mu \)m, a Young’s modulus E= 1.5 MPa and a deformation device central location: highlighting of the most solicited aneurysm area

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3.3 Support for dimensioning the phantom artery

The numerical model was used to address the issue of designing a phantom artery mimicking the in vivo characteristics of an aneurysm in terms of thickness and local stiffness. In particular, for the design of the phantom artery, it was necessary to identify the phantom artery, ensuring the judicious choice of the wall thickness and Young’s modulus that was also feasible regarding the technique applied and the material. The displacement vector norm and the strain tensor norm were considered in the core of the impact area associated with the deformation device location.

3.3.1 Estimating the best ratio thickness/local stiffness for the 3D printing application

3D printing is the technique most used to create phantom arteries in biomedical and pre-surgical studies [21, 24, 25]: the Polyjet and the Stereolithography (SLA) 3D printing techniques were applied experimentally with the associated materials [26]. The objectives were to determine whether the device had a significant influence in terms of displacements and strains for each phantom artery and identify a displacement/strain range linked to the technique and material. Experimentally, with the Formlabs SLA 3D printer and the elastic 50A Resin, it was possible to reach an AB-A thickness of 600-600 \(\mu \)m and a Young’s modulus of 1.57 MPa [31]. With the Polyjet Stratasys J850\(^\textrm{TM}\) Digital Anatomy\(^\textrm{TM}\) 3D Printer and the GelMatrix, it was possible to reach an AB-A thickness of 1500-1500 \(\mu \)m and a Young’s modulus of 1 MPa [32]. The numerical study performed on these thickness couples covered these characteristics. The results of the maximum displacement norm and strain norm in the area of interest for these arteries are shown in figure 6.

Fig. 6
figure 6

a Displacement norm (\(\mu \)m) and b strain tensor norm for 600-600 \(\mu \)m and 1500-1500 \(\mu \)m phantom artery with variable Young’s modulus (MPa): ranges associated with the SLA and Polyjet 3D printing techniques

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For both thickness couples, it was possible to identify a deformation of the aneurysm due to the device in addition to the main flow. Nevertheless, the highest displacement/strain norms were reached with the SLA phantom artery compared to the Polyjet phantom artery. For an AB-A thickness couple of 1500-1500 \(\mu \)m, the Young’s modulus should be at least around 0.5 MPa to have displacement and strain norms equivalent to the SLA phantom artery: this was experimentally impossible with the techniques applied. To reach the highest displacements and strains in the impact area it was necessary to favour a thin thickness couple (600-600 \(\mu \)m) and therefore the Formlabs SLA 3D printer and the elastic 50A Resin.

3.3.2 Local assessment of the thickness: benefits of the injection moulding application compared to the 3D printing

As previously indicated in section 1, the Young’s modulus values targeted were in the range of 0.4\(-\)1.6 MPa [21] in order to be close to human local stiffness. In the experimental study, with the injection moulding process and the Prevent Polyaddition Silicone FLUIDE FA5420N it was possible to reach a Young’s Modulus around 0.7 MPa but it was technically impossible to ensure a constant artery wall thickness: the AB-A thickness couple was 1500-400 \(\mu \)m. With the Formlabs SLA 3D printer and the elastic 50A Resin it was possible to reach an AB-A thickness couple of 600-400 \(\mu \)m and a Young’s modulus of 1.57 MPa. The objective was to identify whether it was relevant to locally adjust the aneurysm wall thickness in terms of reachable displacements and strains and determine a range associated with each techniques. The numerical study was performed on phantom arteries with AB-A thickness couples of 1500-400 and 600-400 \(\mu \)m and with a Young’s modulus between 0.2 MPa and 2 MPa covering the experimental range.

It was also relevant to numerically compare the injection moulding phantom artery to a theoretical phantom artery, impossible to obtain with the techniques applied in the experimental study presented. This theoretical artery had an AB-A thickness couple of 600-400 \(\mu \)m and a Young’s modulus of 0.5 MPa, which would be the phantom artery with properties closest to human ones. The displacement and strain norms associated with each case are shown in figure 7 with the associated ranges.

Fig. 7
figure 7

a Displacement norm (\(\mu \)m) and b strain tensor norm for 600-400 \(\mu \)m and 1500-400 \(\mu \)m phantom artery with a variable Young’s modulus (MPa): local assessment of the aneurysm wall thickness

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Regarding the injection moulded artery, the deformation of the aneurysm due to the device in addition to the main flow was significant, with a maximum displacement norm around 200 \(\mu \)m in the core of the area solicited. The displacement and strain norms were higher than those observed for the phantom artery printed using Formlabs SLA 3D. The comparison with the theoretical phantom artery was relevant. Regarding the displacement norm results, the maximum displacement norm in the core of the impact area for the theoretical phantom artery was higher than the injection moulded phantom artery and were not comparable. Nevertheless, regarding the strain norm results, the maximum strain norm in the core of the impact area was similar to the injection moulded artery with 2.8 % differences between the associated strain norms.

3.3.3 Overview of reachable displacements and strains for each design technique

The ranges of displacement and strain norms for the thicknesses reachable with Formlabs SLA or Polyjet 3D printers and injection moulding with the associated Young’s modulus are summarised in figure 8. The higher displacement and strain norms were reached in the range covered by the injection moulded artery.

Fig. 8
figure 8

a Displacement norm (\(\mu \)m) and b strain tensor norm of phantom arteries feasible with Formlabs SLA, Polyjet 3D printing and injection moulding

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3.4 Study of the influence of the deformation device location

Several adjustments of the device into the aneurysm were considered numerically to quantify the impact of the location on the displacements and strains induced. Due to the device’s flexibility, it was indeed complex to precisely mimic and identify the experimental location of the deformation device. The aim was to define a displacement and strain range associated with the location variations. The study was done in the core of the impact area. The displacement norm regarding the angle of the deformation device to the normal axis for a central location of the deformation device is shown in figure 9a; the strain tensor norm for the same positions is shown in figure 9b. For each location of the deformation device, the same artery parameters were considered with an AB-A thickness couple of 600-400 \(\mu \)m and a Young’s modulus E = 1.57 MPa: these parameters were linked to the SLA phantom artery but could be extended to the other phantom arteries. For an identical deformation device length inserted in the artery/aneurysm, the displacement value was between 108 \(\mu \)m for a central location without an angle and almost 130 \(\mu \)m with an angle of \(0.8^{\circ }\). The same tendency was observed with the strain tensor norm with a minimum value around 2.5 % for a central location with no angle and a maximum value of 3.6 % with an angle of \(0.8^{\circ }\). For an identical deformation device length inserted in the artery/aneurysm, it was thus possible to quantify the influence of this angle on the mechanical load of the aneurysm wall.

Fig. 9
figure 9

Displacements (\(\mu \)m) and strain tensor norm of the arterial wall for an AB-A thickness of 600-400 \(\mu \)m and a Young’s modulus E= 1.5 MPa versus the deformation device angle

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Fig. 10
figure 10

Displacements (\(\mu \)m) and strain tensor norm of the arterial wall for an AB-A thickness of 600-400 \(\mu \)m and a Young’s modulus E= 1.5 MPa versus the deformation device location

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Fig. 11
figure 11

Strain tensor norm regarding the deformation device location for a Young’s modulus of E = 1.57 MPa, D = 1 mm and \(\theta \) = \(0.5^{\circ }\). A Central location B Upper location C Lower location D Central location with angle

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For the lower and upper locations of the deformation device, the displacement norm regarding the device location is shown in figure 10a; the strain tensor norm for the same locations is shown in figure 10b. For each location, two distances between the deformation device exit and the arterial wall were considered: 1 mm and 1.5 mm. The figure 11 condenses the strain tensor norm results for each location and D = 1 mm.

Regarding the 8 models considered, the maximum displacement value in the core of the impact area was 108 \(\mu \)m, the minimum was 93 \(\mu \)m. The maximum strain tensor norm in the same conditions was around 2.5 % and the minimum 2 %. These simulations allowed to defining confidence limits for each experimental analysis. For known conditions and parameters of the experimental study, it was possible to observe differences in the displacements and strains induced due to the location of the device.

4 Discussion

3D printing is commonly considered as the most used and useful technique for creating phantom arteries in biomedical and pre-surgical studies [21, 24, 25]. Thus, the Polyjet and the SLA 3D printing techniques were first applied experimentally with the associated materials [26]. Regarding the latest improvements in the 3D printing of a surgical phantom, the Polyjet 3D printer was mainly used for mimicking aorta and cardiac tissues [46] and not for cerebral arteries. These phantom arteries are mostly used for surgical training and flow modelling without necessarily the aim of targeting a biofidelic local stiffness. As expressed in [47], in the 3D printing of a surgical phantom the artery thickness is a critical issue and has an influence on the artery’s mechanical response. With the SLA 3D printer it was possible to reach thin AB-A thickness couples such as 600-400 \(\mu \)m or 600-600 \(\mu \)m and a Young’s modulus around 1.5 MPa [31]. With the Polyjet 3D printer it was possible to reach a lower Young’s modulus of 1.1 MPa with a higher AB-A thickness couple of 1500-1500 \(\mu \)m [32]. The aim was to identify whether with a simplified flow and a monitored pulse from the deformation device, it was possible to observe the displacements and strains of the aneurysm wall induced and to define a range associated with each technique. The numerical analyses of the displacements and strains highlighted that the mechanical load associated with the deformation device flow overshadowed that associated with the main flow. This load was also maximal in the aneurysm area compared to the other surfaces in the artery body.

In this numerical study, the displacement/strain range associated with the SLA phantom artery was higher than that associated with the Polyjet phantom artery. Nevertheless, with an homogeneous distribution of the artery wall thickness, this effect observed was expectable. Indeed, the aneurysm wall deformation was a combination of tension and bending of the wall. The stiffness in tension is proportionally of both wall thickness and Young’s modulus [48] while in bending it is linearly dependant on Young’s modulus and proportional to the 3\(^{rd}\) power of the thickness. Thus, the effect of the device was predominant with the lower thickness couple associated with the SLA phantom artery. In the framework of the experimental study, the aneurysm deformation was studied using the MatchID Digital Image Correlation (DIC) system. Based on the numerical results, the SLA phantom artery should be emphasised to ensure capturing the aneurysm deformation with the DIC system. Moreover, the Formlabs SLA 3D printer and the elastic 50A Resin ensured the transparency of the phantom artery and allowed fabricating a complex shape with marked tortuosity. It could be experimentally interesting to visualise the deformation device location in the aneurysm and test its behaviour with a patient-specific shape using this 3D printing technique.

Although injection moulding has seldom been used in bio engineering compared to 3D printing, it was investigated to fabricate the phantom artery to benefit from the properties of silicone. Indeed, it allowed reaching a local stiffness closer to the properties of human arteries [21, 23]. To overcome technical difficulties concerning the thickness heterogeneity, the influence of the local variation thickness was also investigated: the aim was to determine if the thickness of the artery could be significantly lowered on the aneurysm wall compared to the artery body. Indeed, with the injection moulding technique, a 400 \(\mu \)m thickness was not ensured for the entire artery. The aim of the study was to identify if the behaviour of a phantom artery with an AB-A thickness couple of 1500-400 \(\mu \)m (feasible with injection moulding) was comparable to a phantom artery with an AB-A thickness couple of 600-400 \(\mu \)m with a Young’s modulus between 0.5 and 1 MPa (theoretical case not feasible with SLA 3D printing or injection moulding). The spatial variations of the material properties and thickness of a membrane were studied previously in [49]; however, the framework and application of this study was extended to a more complex shape and an FSI problem. The displacement study in the core of the impact area showed that the two cases were not equivalent when the displacement norm was considered. For a 600 \(\mu \)m thickness of the artery body wall, the mechanical behaviour induced by the deformation device flow upstream of the aneurysm (in the structure studied at the crossing between the outlet arms and the aneurysm) was different for a 1500 \(\mu \)m thickness of the artery body wall. However, the two cases were equivalent in the strain study with very small differences. When studying the strains studied in the artery analysis, it is not prohibitive to consider an artery with a thickness couple of 1500-400 \(\mu \)m instead of 600-400 \(\mu \)m. Moreover, the displacement/strain range obtained with the injection moulding was higher than that of the SLA 3D printer (AB-A thickness couples of 600-400 \(\mu \)m and a Young’s modulus around 1.5 MPa). In the framework of the experimental study with the MatchID Digital Image Correlation (DIC) system, the injection moulded phantom artery should be emphasised when the aim is to reach the greatest deformation of the aneurysm. Nevertheless, the fabrication of a patient specific artery mould remained complicated: for these complex artery shapes the SLA 3D printer is the most appropriate. However, it was also intended to perform the testing of the deformation device coupled with the use of clinical imagery. The deformed aneurysm anatomical image will be processed numerically to quantify the aneurysm wall stress state by inverse analysis. The aim was to set the study in a real and plausible medical environment, without risk, before testing the deformation device on small animals on further studies. The Spectral Photon Counting Computed Tomography (SPCCT) was chosen to perform these studies. The spatial resolution of the SPCCT was around 200 \(\mu \)m [50] which was not reached with the SLA 3D printing displacement analyses. The use of the injection moulded artery appears more appropriate for this testing.

In addition to central location of the deformation device in the aneurysm, several locations were studied. Experimentally, the deformation device was positioned without a precise control of the solicited area which was visualised with the MatchID Digital Image Correlation (DIC). The experimental deformation device was modified to set the total length of the inserted device and to ensure a minimal distance of 1 mm to the aneurysm arterial wall. Due to the device’s flexibility, it remained impossible to precisely repeat the deformation device location with the numerical modelling: several numerical locations were studied; the cumulative mechanical load area contained the location of the experimental deformation device. It was also necessary to consider several distances to the wall: for each location, the distances of 1 mm and 1.5 mm were considered. As shown in figure 3a, 3b and 3c, the deformation device remained normal to the aneurysm wall in the central location but this was no longer the case for the lower and upper locations. For these locations, the flow behaviour of the deformation device and its mechanical load on the aneurysm wall were different. This shows that the displacements and strains were lower for the upper and lower locations compared to the central location for distances of 1 mm an 1.5 mm to the arterial wall. The displacements and strains of the arterial wall were higher when the deformation device remained normal to the arterial wall (as shown in figures 10a and 10b). The interaction between the surrounding principal flow and the deformation device was not taken into account in the numerical model: for each location the deformation device was considered as steady with embedded extremities. Thus, this makes it all the more important to consider several device locations numerically in each comparison with the experimental results, as a movement of the deformation device linked to the flow was observed experimentally.

One case studied numerically was the addition of an angle to the central location as shown in figure 3d. With the different locations, it was noticed that without the deformation device normal to the aneurysm wall, the flow behaviour was different and the displacements and strains were modified. With an angle, the deformation device stayed more normal to the aneurysm wall and the influence of the angle was significant, even with small angles such as \(0.5^{\circ }\) or \(0.8^{\circ }\) as shown in figures 9a and 9b, thus higher displacements and strains were found. In the experimental study, there was however no way to approximate the angulation of the deformation device into the aneurysm cavity. Regarding the numerical results, the angulation of the deformation device and its location had an influence and constituted an immovable uncertainty. This uncertainty was not precisely quantifiable in an experimental way but must be considered and kept as important data in the comparison between the numerical and experimental results.

Nevertheless, this study led on phantom arteries displayed several limits. Even though it was applied in several numerical studies [23, 36, 51,52,53], the linear elastic material law applied to the phantom artery can be considered as an oversimplification compared to the human arteries ones. This assumption was adequate for the in vitro study on phantom arteries as a first step of this large scale project. A simplified shape of the phantom artery was chosen to focus on the interaction between the deformation device and the phantom aneurysm wall and avoid artefacts potentially related to a patient-specific arterial geometry. The oversized dimensionning of the artery eased the handle of the device into the aneurysm. Furthermore, water was used instead of a liquid with a viscosity similar to the blood one. The implementation of a pulsatile profile mimicking an artery blow flow instead of the steady flow used would also be a necessary improvement to this study [36, 54].

The aim of this first step was to implement a favourable setting to the first deformation device tests, the handling and the observation of the device influence in a controlled and simplified environment. Therefore, the presented numerical study is linked to this particular set-up. The next steps of this project encompassed the improvement and the complexification of this experimental set-up to draw near patient-specific conditions. The in vivo application of the device required eventually a complexification of the numerical model prior to the inverse analysis in terms of artery material law and boundary conditions.

5 Conclusion

A Fluid–Structure Interaction numerical model based on the finite elements method was developed on COMSOL Mutltiphysics, associated with an experimental study aimed at developing a mechanical characterisation device of intracranial aneurysm wall. The numerical study provided helpful information regarding the scientific challenges of the experimental study: the optimisation of the artificial phantom artery design linked to the technique used (3D printing of injection moulding) and the quantification of the mechanical influence of the location of the deformation device. If the 3D printing is used to obtain the phantom artery, a thin thickness should be emphasised in order to reach the highest displacements and strains on the aneurysm and thus obtain significant data for further inverse analysis to determine the aneurysm stress state. If the injection moulding is used, the local assessment of the aneurysm wall thickness is not prohibitive in a strain study. The numerical model also quantified the uncertainties of the experimental study such as the location and the angulation of the deformation device in the aneurysm cavity.

This numerical model should be applied to patient-specific studies that associate biological parameters for solid and fluid mechanics studies. This next step encompasses experimental trials of the deformation device on small animals and an associated numerical study that will consist of an inverse analysis to determine of the mechanical stress state of the aneurysm wall.