Evaluation of Different Cannulation Strategies for Aortic Arch Surgery Using a Cardiovascular Numerical Simulator

Aortic disease has a significant impact on quality of life. The involvement of the aortic arch requires the preservation of blood supply to the brain during surgery. Deep hypothermic circulatory arrest is an established technique for this purpose, although neurological injury remains high. Additional techniques have been used to reduce risk, although controversy still remains. A three-way cannulation approach, including both carotid arteries and the femoral artery or the ascending aorta, has been used successfully for aortic arch replacement and redo procedures. We developed circuits of the circulation to simulate blood flow during this type of cannulation set up. The CARDIOSIM© cardiovascular simulation platform was used to analyse the effect on haemodynamic and energetic parameters and the benefit derived in terms of organ perfusion pressure and flow. Our simulation approach based on lumped-parameter modelling, pressure–volume analysis and modified time-varying elastance provides a theoretical background to a three-way cannulation strategy for aortic arch surgery with correlation to the observed clinical practice.


Introduction
Aortic disease has a significant impact on quality of life. Aneurysmatic dilatation and acute aortic dissection involving the aortic arch are the most devastating manifestations of aortic disease. Surgical or endovascular treatments for these conditions are currently available options. The outcome following successful total arch repair is satisfactory, but it still carries the risk of potentially devastating perioperative complications. Conventional aortic arch replacement can be offered to the majority of patients, although hybrid and endovascular techniques have gained popularity. Nevertheless, a less invasive approach can often be as technically challenging as open surgery, with stroke and endo-leaks as early limiting factors and less favourable mid-to long-term outcomes [1]. Mid-term outcomes and intra-operative complication rates with both hybrid and conventional aortic arch surgery remain heterogeneous and depend on centre experience and patient suitability [2]. Open repair with the elephant trunk technique [3] under hypothermic circulatory arrest is widely used with satisfactory long-term outcomes [4,5]. The frozen elephant trunk technique using the Thoraflex TM (Vascutek, Terumo, Inchinnan, Glasgow, UK) or the E-vita OPEN PLUS (JOTEC, Hechingen, Germany) hybrid device has recently become a popular choice, which allows subsequent endovascular procedures [6,7]. Hybrid repair involves Figure 1 shows four different cannulation strategies for aortic arch surgery as the main subject of this study. The centrifugal pump draws blood and ejects it according to the following connections [56][57][58] Bioengineering 2023, 10, 60 3 of 24 recent review has highlighted the critical role played by lumped-parameter models in patient-specific cardiovascular modelling [55]. Figure 1 shows four different cannulation strategies for aortic arch surgery as the main subject of this study. The centrifugal pump draws blood and ejects it according to the following connections [56][57][58]

The Heart and Circulatory Numerical Network
"In silico" study of cannulation for aortic arch surgery was performed developing new 0-D numerical modules of the cardiovascular network implemented in CARDIOSIM© platform [38,[59][60][61][62][63][64]. These modules allow the reproduction of the behaviour of native left and right ventricles; atria and septum; coronary circulation; ascending aorta and aortic arch; descending thoracic and abdominal aorta; upper limbs and head section; superior, inferior and abdominal vena cava section; renal and hepatic section; splanchnic and lower limbs sections; pulmonary circulation. Figure 2a shows an overview of the whole cardiovascular network. The native left and right ventricles, atria and septum and the ventricular, atrial and septal activity synchronized with the electrocardiographic (ECG) signal are implemented using the time-varying elastance concept [60,61,63,64]. This numerical representation allows the simulation of inter-ventricular and intra-ventricular dyssynchrony. The specific modules of coronary circulation are available in CARDIOSIM© platform [38]. The module of coronary circulation presented in [65] was selected for this study. Figure 2a shows two different electric analogues used to reproduce the behaviour of the aortic (AV), mitral (MV), tricuspid (TV) and pulmonary (PV) valves [38]. A representation with an ideal diode was chosen for this study: when the pressure across the valve was positive, the valve opened and allowed blood flow; when the pressure was less than or equal to zero, the valve closed, and blood flow was zero. The whole pulmonary circulation was modelled as described in the current literature [63,64,66,67]. Figure 2b shows the electric analogue of the following compartments: ascending and descending aorta, aortic arch and abdominal tract. All these sections were modelled with resistance, inertance and compliance (RLC) elements. The thoracic compartment was modelled with two resistances (R THOR and R ATI ), inertance (L ATI ) and compliance (C ATI ). The abdominal section was divided in two compartments both electrically represented with RLC elements. Finally, the behaviour of the superior vena cava section was reproduced with RC elements. The symbols used in Figure 2b are listed in Table 1.

The Heart and Circulatory Numerical Network
"In silico" study of cannulation for aortic arch surgery was performed developing new 0-D numerical modules of the cardiovascular network implemented in CARDI-OSIM© platform [38,[59][60][61][62][63][64]. These modules allow the reproduction of the behaviour of native left and right ventricles; atria and septum; coronary circulation; ascending aorta and aortic arch; descending thoracic and abdominal aorta; upper limbs and head section; superior, inferior and abdominal vena cava section; renal and hepatic section; splanchnic and lower limbs sections; pulmonary circulation. Figure 2a shows an overview of the whole cardiovascular network. The native left and right ventricles, atria and septum and the ventricular, atrial and septal activity synchronized with the electrocardiographic (ECG) signal are implemented using the time-varying elastance concept [60,61,63,64]. This numerical representation allows the simulation of inter-ventricular and intra-ventricular dyssynchrony. The specific modules of coronary circulation are available in CARDI-OSIM© platform [38]. The module of coronary circulation presented in [65] was selected for this study. Figure 2a shows two different electric analogues used to reproduce the behaviour of the aortic (AV), mitral (MV), tricuspid (TV) and pulmonary (PV) valves [38]. A representation with an ideal diode was chosen for this study: when the pressure across the valve was positive, the valve opened and allowed blood flow; when the pressure was less than or equal to zero, the valve closed, and blood flow was zero. The whole pulmonary circulation was modelled as described in the current literature [63,64,66,67].
(a)  Figure 2b shows the electric analogue of the following compartments: ascending and descending aorta, aortic arch and abdominal tract. All these sections were modelled with resistance, inertance and compliance (RLC) elements. The thoracic compartment was modelled with two resistances (RTHOR and RATI), inertance (LATI) and compliance (CATI). The abdominal section was divided in two compartments both electrically represented with RLC elements. Finally, the behaviour of the superior vena cava section was reproduced with RC elements. The symbols used in Figure 2b are listed in Table 1.  The circuit representation of the upper and lower limbs and the head sections with the four different cannulation strategies implemented in CARDIOSIM© are illustrated in Figure 2c. The nomenclature of the symbols is listed in Table 1. The lower limb section consists of two parts reproducing left and right limb circulation. The left (right) arterial circulation of the lower limb is modelled with a variable resistance R LFA (R RFA ), the resistance R LFV (R RFV ) reproduces the venous circulation, and the capacitor C LLE (C RLE ) represents the left (right) compliance. The upper limbs are modelled with two resistances R ARM1 (variable resistance) and R ARM2 reproducing arterial and venous circulation, respectively, and compliance C ARM . The arterial and venous cerebral circulation is reproduced with a variable resistance R HD1 and a resistance R HD , respectively; the capacitor C HD mimics the vessel compliance. Figure 2d shows the electrical analogue of the second abdominal tract modelled with RLC elements. The variable resistance R SP1 and the resistance R SP2 allow the simulation of the arterial and venous circulation of the splanchnic tract; the capacitor C SP mimics its compliance.
The behaviour of the arterial and venous renal (hepatic) section is simulated with a variable resistance R KID1 (R HEP1 ) and the resistance R KID2 (R HEP2 ), respectively. The compliance of the renal and hepatic vessels is modelled with the capacitors C KID and C HEP , respectively.
Inferior vena cava resistances (compliance) mmHg·cm −3 ·sec (mmHg −1 ·cm −3 ) R HEP1 and R HEP2 (C HEP ) Hepatic resistances (compliance) Left (right) ventricular end-systolic elastance mmHg/ml Ea S (Ea P ) Systemic (pulmonary) arterial elastance mmHg/ml Ea S /Ees Left (Ees Right /Ea P ) Left (right) ventricular-arterial coupling -- Figure 2c shows the four different cannulation strategies implemented in CARDIOSIM© platform. In the first one, the centrifugal pump draws blood from RA and ejects it in the ascending aorta and in the common carotid artery bilaterally (RA→AA&CC). In this case, the switches SW5 (which opens the input cannula connected to the right atrium), SW3 (which opens the output cannula connected to the common carotid artery) and SW6 (which opens the output cannula connected to the ascending aorta) are ON, and the other switches are OFF. When the second type of cannulation is activated, the centrifugal pump draws blood from the right atrium and ejects it into the FA and in the common carotid artery bilaterally (RA→FA&CC). This connection is obtained by switching SW5, SW3 and SW4 (which opens the output cannula connected to the femoral artery) ON, and setting SW1, SW2 and SW6 OFF. In the connection (FV→FA&CC), the pump draws blood from the femoral vein and ejects it into the FA and CC. The cannulation can be configured by setting SW1 (which opens the input cannula connected to the femoral vein), SW3 and SW4 ON. Finally, in the last connection, the centrifugal pump draws blood from the FV and the superior vena cava and ejects it into the FA and CC. The cannulation (FV&SVC→FA&CC) can be activated by switching SW1, SW2 (which opens the input cannula connected to the SVC), SW3 and SW4 ON, whilst SW5 and SW6 are switched OFF. All the cannulae are modelled with RLC elements [63].

Numerical Models of the Cannulae and Centrifugal Pump
The numerical model of the centrifugal pump implemented in CARDIOSIM© has been previously described in [61,68].

Simulation Protocol
The results of the simulations performed in this work were obtained starting from baseline conditions and setting heart rate (HR), left and right ventricular and septal elastance, arterial and venous resistances and compliances for all compartments to achieve minimal physiological conditions characterized by HR = 90 beat/min, systolic aortic pressure (AoP sys ) = 90 mmHg, LAP = 21 mmHg, RAP = 10 mmHg, PAP = 18 mmHg, CO = 4.3 l/min. Starting from baseline conditions, each cannulation configuration was activated by setting the centrifugal pump speed to 2000, 2500, 3000, 3500, 4000 and 4500 rpm. The effects induced by different cannulations on the left and right ventricular loop and on the left atrial loop are presented and discussed. Furthermore, the instantaneous waveform of the aortic and pulmonary arterial pressures for each cannulation configuration is included in the Results section. Finally, an analysis of the percentage variations in the haemodynamic and energetic variables calculated with respect to baseline conditions for each cannulation configuration and different speeds of the centrifugal pump was carried out in this study. Haemodynamic variables, such as mean aortic pressure, PAP, LPA, RAP, CO, left (right) ventricular-arterial coupling Ea S /Ees Left (Ees Right /Ea P ), cerebral, renal and superior vena cava flow, together with energetic variables, such as left and right ventricular external work (EW) and right ventricular pressure-volume area (PVA), were analysed.       When the pump speed was set to 2000 rpm, the AoP and PAP waveforms (blue) moved downward compared to baseline conditions. Figure 6 shows the effects induced on the right ventricular pressure-volume loop (top-left panel) and instantaneous PAP waveform (bottom-left panel) by FV→FA&CC cannulation when the rotational pump speed was set to 2000, 2500, 3000 and 3500 rpm. This type of cannulation generated limited effects compared to baseline conditions (black dashed line) when the centrifugal pump speed was set to 2000 rpm (red line). When the pump rotational speed increased, the right ventricular pressure-volume loop shifted to the right increasing both ESV and EDV. Instantaneous PAP waveforms were also affected by high pump rotational speeds. When the pump speed was set to 2000 rpm, the AoP and PAP waveforms (blue) moved downward compared to baseline conditions. Figure 6 shows the effects induced on the right ventricular pressure-volume loop (top-left panel) and instantaneous PAP waveform (bottom-left panel) by FV→FA&CC cannulation when the rotational pump speed was set to 2000, 2500, 3000 and 3500 rpm. This type of cannulation generated limited effects compared to baseline conditions (black dashed line) when the centrifugal pump speed was set to 2000 rpm (red line). When the pump rotational speed increased, the right ventricular pressure-volume loop shifted to the right increasing both ESV and EDV. Instantaneous PAP waveforms were also affected by high pump rotational speeds.  The outcome of RA→AA&CC cannulation on the left atrial pressure-volume loop and on the aortic pressure (AoP) is shown in Figure 7. This type of cannulation produced negligible effects when the speed of the pump was set to 2000 rpm (red lines in the top-and bottom-left panels). At higher pump speeds, the loops (top left panel) move to the right for high values of the left atrial end-diastolic volume. AoP increased considerably when the pump rotational speed was set to 3000 rpm (lilac waveforms in the bottom panel). Figure 8 shows the initial analysis of percentage variation in AoP, CO and left (right) atrial pressure LAP (RAP) for the four different cannulation strategies compared to baseline conditions when the rotational pump speed was set to 2500 rpm. The percentage variation in mean AoP, CO and LAP decreased for RA→FA&CC and FV&SVC→FA&CC cannulations. A decrease in percentage variation in the right atrial pressure was observed for FV&SVC→FA&CC cannulation (bottom-right panel). Figure 9 shows the relative changes in cardiac output calculated in comparison to baseline conditions for the four methods of cannulation. The simulations were performed setting the pump rotational speed to 2000, 2500, 3000, 3500, 4000 and 4500 rpm. In the case of RA→AA&CC (FV→FA&CC) cannulation, CO increased up to 58% (52%) when the pump speed was set to 4500 rpm (top left panel and bottom right panel). On the contrary, in the case of RA→FA&CC cannulation, CO decreased by 17% when the pump speed was set to 4500 rpm (upper right panel). The ventricular loops and the instantaneous waveforms obtained using CARDIOSIM© were stored in excel files and subsequently plotted. In RA→AA&CC cannulation (right panel) the centrifugal pump draws blood from the right atrium and ejects it into the ascending aorta and common carotid artery bilaterally. Figure 8 shows the initial analysis of percentage variation in AoP, CO and left (right) atrial pressure LAP (RAP) for the four different cannulation strategies compared to baseline conditions when the rotational pump speed was set to 2500 rpm. The percentage variation in mean AoP, CO and LAP decreased for RA→FA&CC and FV&SVC→FA&CC cannulations. A decrease in percentage variation in the right atrial pressure was observed for FV&SVC→FA&CC cannulation (bottom-right panel). Figure 9 shows the relative changes in cardiac output calculated in comparison to baseline conditions for the four methods of cannulation. The simulations were performed setting the pump rotational speed to 2000, 2500, 3000, 3500, 4000 and 4500 rpm. In the case of RA→AA&CC (FV→FA&CC) cannulation, CO increased up to 58% (52%) when the pump speed was set to 4500 rpm (top left panel and bottom right panel). On the contrary, in the case of RA→FA&CC cannulation, CO decreased by 17% when the pump speed was set to 4500 rpm (upper right panel).     For FV&SVC→FA&CC cannulation, the outcome of the simulations showed that cardiac output took a positive turn when the rotational pump speed was set to 3000 rpm (lower left panel). A percentage increase in excess of 32% (compared to the baseline conditions) in CO was observed when the pump speed was set to 4500 rpm.

Results
The analysis of the two parameters that measure the coupling between the circulatory arterial network and the two ventricles and of the left and right ventricular external work is summarized in Figure 10 for all the cannulations with the pump rotational speed set to 3500 rpm. For FV&SVC→FA&CC cannulation, the outcome of the simulations showed that cardiac output took a positive turn when the rotational pump speed was set to 3000 rpm (lower left panel). A percentage increase in excess of 32% (compared to the baseline conditions) in CO was observed when the pump speed was set to 4500 rpm.
The analysis of the two parameters that measure the coupling between the circulatory arterial network and the two ventricles and of the left and right ventricular external work is summarized in Figure 10 for all the cannulations with the pump rotational speed set to 3500 rpm. An increase in EaS/EesLeft (top-left panel) for both RA→FA&CC and FV&SVC→ FA&CC cannulations was observed when the pump rotational speed was set to 3500 rpm. On the contrary, the right ventricular-arterial coupling showed a percentage reduction

An increase in Ea S /Ees Left (top-left panel) for both RA→FA&CC and FV&SVC→FA&CC
cannulations was observed when the pump rotational speed was set to 3500 rpm. On the contrary, the right ventricular-arterial coupling showed a percentage reduction compared to the baseline values for all types of cannulations (top-right panel). Only the RA→FA&CC cannulation configuration underwent a reduction in left ventricular EW of less than 2%. An EW increase of up to 7% was observed for the RA→AA&CC cannulation method (bottom-left panel). When the pump speed was set to 3500 rpm, a modest percentage increase in right ventricular EW compared to baseline conditions (bottom right panel) was observed for all the cannulation configurations.
An analysis of the percentage changes in left ventricular EW (right ventricular PVA) for FV→FA&CC, FV&SVC→FA&CC and RA→AA&CC cannulation configurations when the pump rotational speed was set to 2000, 2500, 3000, 3500, 4000 and 4500 rpm is shown in the top-left (right) panel of Figure 11. When the pump speed was set to 4500 rpm, an increase of more than 15% in left ventricular EW and about 25% in right ventricular PVA was observed for the RA→AA&CC cannulation configuration.  Figure 12 shows the effects induced by the four different cannulation strategies on cerebral blood flow when the pump rotational speed was set to 3500 rpm (top left panel).
The top-right panel shows the effect induced by RA→FA&CC cannulation on cerebral blood flow, superior vena cava flow and CO when the pump speed was set to 2500 rpm. The relative changes in cerebral blood flow*, SVC* and CO* were evaluated when a little increase in the resistance of the ascending aorta tract was simulated.  Figure 12 shows the effects induced by the four different cannulation strategies on cerebral blood flow when the pump rotational speed was set to 3500 rpm (top left panel).
The top-right panel shows the effect induced by RA→FA&CC cannulation on cerebral blood flow, superior vena cava flow and CO when the pump speed was set to 2500 rpm.
The relative changes in cerebral blood flow*, SVC* and CO* were evaluated when a little increase in the resistance of the ascending aorta tract was simulated. The effects induced by different cannulation strategies on SVC and hepatic flow were further analysed. Figure 13 shows the relative changes in SVC (top-left panel) and hepatic flow (bottom-left panel) obtained when the rotational pump speed was set to 3500 rpm. A percentage decrease was observed during the RA→FA&CC cannulation configuration. The top (bottom)-right panel shows the relative changes in cerebral (kidney, hepatic and SVC) blood flow obtained when the FV&SVC→FA&CC cannulation method was simulated with different rotational pump speeds. A trend reversal was observed when the pump speed is greater than or equal to 3000 rpm. The effects induced by different cannulation strategies on SVC and hepatic flow were further analysed. Figure 13 shows the relative changes in SVC (top-left panel) and hepatic flow (bottom-left panel) obtained when the rotational pump speed was set to 3500 rpm. A percentage decrease was observed during the RA→FA&CC cannulation configuration. The top (bottom)-right panel shows the relative changes in cerebral (kidney, hepatic and SVC) blood flow obtained when the FV&SVC→FA&CC cannulation method was simulated with different rotational pump speeds. A trend reversal was observed when the pump speed is greater than or equal to 3000 rpm.

Discussion
The management of aortic arch disease either in the context of acute type A aortic dissection or in the presence of aneurysmatic disease is quite challenging and still a matter of significant debate [69,70]. Open repair for aortic arch disease remains the standard of care in high volume centres, although endovascular treatment has become quite an established approach in the presence of significant comorbidities [71]. The protection of cerebral blood flow remains the key issue during aortic arch replacement. Experimental evidence confirms the benefit of selective antegrade cerebral perfusion compared to retrograde perfusion and hypothermic circulatory arrest [72]. The effect of arterial filtration on cerebral auto-regulation [73,74] and the choice of pH management [75] has also been addressed. A combined experimental and computational approach may have a role to play with the optimisation of cerebral blood flow on cardiopulmonary bypass in terms of a reduction in neurological events [76,77]. The presence of carotid or intracranial atherosclerotic disease does not seem to affect the incidence of stroke during aortic arch surgery, whilst its perioperative cause remains predominantly embolic [78]. Impaired autoregulation of cerebral blood flow during CPB may predispose to hypoperfusion and increase the risk of perioperative stroke following cardiac surgery [79,80], although a threshold remains difficult to determine [81]. Continuous monitoring with near-infrared spectroscopy (NIRS) is an established approach [82,83], with particular application during aortic arch surgery. The aortic arch is relatively free of atherosclerosis due to the helical flow pattern induced by aortic torsion, which stabilizes blood flow in the aorta and compensates the adverse effects related to aortic curvature [84]. The computational fluid dynamics modelling of blood flow has given insights into the physiological flow interactions in the health and disease of the circulatory system [85,86]. Adequate perfusion remains critical for organ preservation during cardiopulmonary bypass with particular reference to cerebral circulation. Our approach aimed to give a theoretical and quantitative background to an

Discussion
The management of aortic arch disease either in the context of acute type A aortic dissection or in the presence of aneurysmatic disease is quite challenging and still a matter of significant debate [69,70]. Open repair for aortic arch disease remains the standard of care in high volume centres, although endovascular treatment has become quite an established approach in the presence of significant comorbidities [71]. The protection of cerebral blood flow remains the key issue during aortic arch replacement. Experimental evidence confirms the benefit of selective antegrade cerebral perfusion compared to retrograde perfusion and hypothermic circulatory arrest [72]. The effect of arterial filtration on cerebral auto-regulation [73,74] and the choice of pH management [75] has also been addressed. A combined experimental and computational approach may have a role to play with the optimisation of cerebral blood flow on cardiopulmonary bypass in terms of a reduction in neurological events [76,77]. The presence of carotid or intracranial atherosclerotic disease does not seem to affect the incidence of stroke during aortic arch surgery, whilst its perioperative cause remains predominantly embolic [78]. Impaired autoregulation of cerebral blood flow during CPB may predispose to hypoperfusion and increase the risk of perioperative stroke following cardiac surgery [79,80], although a threshold remains difficult to determine [81]. Continuous monitoring with near-infrared spectroscopy (NIRS) is an established approach [82,83], with particular application during aortic arch surgery. The aortic arch is relatively free of atherosclerosis due to the helical flow pattern induced by aortic torsion, which stabilizes blood flow in the aorta and compensates the adverse effects related to aortic curvature [84]. The computational fluid dynamics modelling of blood flow has given insights into the physiological flow interactions in the health and disease of the circulatory system [85,86]. Adequate perfusion remains critical for organ preservation during cardiopulmonary bypass with particular reference to cerebral circulation. Our approach aimed to give a theoretical and quantitative background to an established technique in clinical practice [37]. Although right atrial cannulation to aortic return remains the standard and most appropriate approach in routine cardiac surgery, it may not be completely applicable to aortic surgery on a regular basis. Therefore, an alternative configuration is necessary to address the needs of aortic arch surgery. Our findings suggest that RA→AA&CC and FV→FA&CC cannulation configurations increase and maintain cerebral perfusion appropriately up to 27% and 14% (Figure 12), confirming their adaptability according to the clinical setting. RA→AA&CC cannulation would be quite appropriate for an elective aortic arch replacement due to severe aneurysmatic dilatation, whilst FV→FA&CC would be more appropriate in the presence of significant haematoma and dissection involving the whole aortic arch. An additional arterial cannula can be included during the setting for cardiopulmonary bypass and used later to restore antegrade flow through the side branch of the Dacron graft after completion of distal anastomosis during the reconstruction of the aortic arch. RA→FA&CC and FV&SVC→FA&CC are suitable alternatives but more dependent on pump setting and flow conditions (Figures 9 and 13). RA→AA&CC and FV→FA&CC cannulation configurations increase cardiac output up to 58% and 52% (Figures 8 and 9). The two cannulation configurations are consistent with increased cardiac output following a stepwise increase in pump rotational speed. The PV-loop analysis shows the appropriate unloading of the left ventricle whilst the right ventricle seems less affected ( Figure 4). This would be addressed by suction, as required in routine cardiac surgery. FV&SVC→FA&CC is consistent with increased cardiac output when the pump rotational speed is at least 3000 rpm or above. A progressive decrease in cardiac output was observed with stepwise increase in pump rotational speed following the RA→FA&CC cannulation configuration. Despite its criticism, 0-D (lumped-parameter) models rely on the geometrical and mechanical properties of the constitutive elements and require low computational cost. The advantages of a 0-D model consist of a global representation of cardiovascular dynamics in the whole circulatory system and the evaluation of pressure and flow changes in a local circulation loop. The disadvantages are related to an inability to simulate the effect of high frequency components of arterial impedance and accurately match the aortic pressure and flow waveforms. Pressure and flow changes in specific segments and venous pressure fluctuations cannot be described; the pulse wave transmission effect cannot be simulated. Nevertheless, 0-D models remain an excellent tool that provides a simplified and effective representation of the cardiovascular system and strongly contributes to our understanding of cardiovascular physiology with great potential for clinical application [87][88][89][90]. A key aspect is the achievement of a compromise between the level of complexity and the pragmatic limitation of data collection in a clinical setting to allow appropriate personalised management [91]. In addition, a non-invasive and patient-specific approach may be beneficial for the estimation of haemodynamic variables that cannot be evaluated directly from clinical measurements [92,93]. Needless to say, 0-D,-1-D and 0-D-3-D coupling remains an important modelling application [94,95]. Although we acknowledge the limitations of our work, initial results confirm the clinical observation. The quantitative evaluation is quite reliable considering that our software has been previously validated in terms of accuracy and predictive performance [96]. Bilateral carotid artery cannulation through Dacron tubular grafts anastomosed in an end-to-side fashion allows continuous cerebral perfusion with great flexibility and regardless of the completion of the circle of Willis [37]. Stroke remains the most feared complication whether an open, hybrid or total endovascular approach is used, and its cause is multi-factorial. The increased use of selective antegrade cerebral perfusion (SACP) with warmer circulatory arrest (24-28 • C) has contributed significantly to improved stroke and lower death rates following open surgery, although there is no clear consensus about the optimal protection strategy. The three-way cannulation approach here described is a sophisticated approach, which may have a role to play for complex aortic arch and redo procedures requiring safe and prolonged antegrade cerebral perfusion to allow more surgical time. Needless to say, a close co-operation between cardiac and vascular surgeons and interventional radiologists remains essential to address aspects that are not completely within the domain of individual specialists.

Conclusions
The outcome of our simulations suggests adequate haemodynamics and organ perfusion pressure when the three-way cannulation approach is used in RA→AA&CC and FV→FA&CC cannulation configurations. This is consistent with the observed clinical practice, which may be considered an indirect validation of our simulation approach. The other two cannulation configurations, RA→FA&CC and FV&SVC→FA&CC, may be considered as alternative option, although less attractive, due to their more pronounced dependence on pump settings and reduced beneficial effect on cerebral circulation. Nevertheless, further work is required to address other aspects of the cannulation set up here described, which may help improve its clinical application.