4D Flow MRI Estimation of Boundary Conditions for Patient Specific Cardiovascular Simulation

Abstract

Accurate image based cardiovascular simulations require patient specific boundary conditions (BCs) for inlets, outlets and vessel wall mechanical properties. While inlet BCs are typically determined non-invasively, invasive pressure catheterization is often used to determine patient specific outlet BCs and vessel wall mechanical properties. A method using 4D Flow MRI to non-invasively determine both patient specific outlet BCs and vessel wall mechanical properties is presented and results for both in vitro validation with a latex tube and an in vivo pulmonary artery stenosis (PAS) stent intervention are presented. For in vitro validation, acceptable agreement is found between simulation using BCs from 4D Flow MRI and benchtop measurements. For the PAS virtual intervention, simulation correctly predicts flow distribution with 9% error compared to MRI. Using 4D Flow MRI to noninvasively determine patient specific BCs increases the ability to use image based simulations as pressure catheterization is not always performed.

Introduction

Image based computational fluid dynamics (CFD) has the capability to enhance existing diagnostic medical imaging data, investigate basic biomechanics science and inform clinical decision making^22,25,26,43 Image based CFD has been applied to numerous cardiovascular diseases including coronary artery stenosis⁴³, congenital heart defects^25,26,38, cerebral aneurysms²² and aortic aneurysms⁵. Patient specific volumetric flow is typically prescribed as an inlet BC and temporally varying inlet velocity fields measured from PC-MRI are increasingly being used for increased accuracy^28,33. Advances in numerical techniques and computational resources allow the use of patient specific vessel wall mechanical properties in fluid-structure interaction (FSI)^7,10,23 and patient specific reduced order modeling of the distal vasculature^27,45 not immediately available from medical imaging. Using FSI with patient specific vessel mechanical properties is important for simulation accuracy as CFD simulations assuming a rigid wall have been shown to overestimate wall shear stresses and fundamentally alters the physics of pulsatile flow as waves will be propagated at infinite speed³. Reduced order modelling of the distal vasculature has been shown to improve velocity field accuracy compared to simply assigning outlet flow³². Given the importance of patient specific vessel wall³ and distal vasculature characteristics^4,32,39 for the accuracy of model results, several studies have used pressure measurements to define these patient specific boundary conditions (BCs)^49,51,39. In Xiao et al⁴⁹, mean pressure was used to determine distal resistance and pulse pressure to calculate distal compliance. This approach has also been combined with PC-MRI measured vascular area change over the cardiac cycle to determine vessel stiffness as shown in Figure 1A⁵¹. While tonometry can be used to measure pressure in the carotid, brachial and iliac arteries for estimating pressures in aortic simulations, invasive catheterization is required to measure pressure in other vascular territories such as the pulmonary or hepatic vasculature.

Figure 1:

A. Flow chart describing the algorithm to determine patient specific conditions from pressure. An internal loop is used to match pressure data, then an external loop modulates vessel stiffness to match RAC. Every time vessel stiffness is changed resistance and compliance must be re-fit. B. Flow chart describing the algorithm to determine patient specific conditions from 4D Flow MRI. An initial iterative loop is used to rapidly match PWV and then a second loop matches RAC and ⊺.

Our group has previously performed non-invasive MRI based virtual surgery planning in the hepatic vasculature of living donor liver transplant⁴¹. Moving forward we aim to further utilize MRI, particularly 4D Flow MRI, to non-invasively define patient specific BCs for computational simulations to enhance treatment planning. A particular focus is virtual planning of catheter based stent intervention for treating the congenital heart defect (CHD) pulmonary artery stenosis (PAS) in conjunction with ongoing preclinical studies in a large animal model² PAS is a common post-operative complication of CHD surgeries^{12,14,18,31,47} and is associated with acute and chronic morbidity and mortality^9,13,36.

The goals of this paper are:

1. Present a method for defining patient specific BCs from non-invasive 4D Flow MRI

2. Validate this method using an in vitro model and compare simulated results to benchtop flow probe and high speed camera measurements

3. Demonstrate the feasibly of defining patient specific BCs from 4D Flow MRI in vivo by performing a virtual stent intervention in a porcine PAS and compare simulated results to experimental intervention outcomes

Materials and Methods

Boundary Condition Derivation

Our noninvasive method to determine patient specific BCs from 4D flow MRI is motivated by cardiovascular physics and physiology (Figure 1B). First, pulse wave velocity (PWV) is used to estimate the vessel wall stiffness (Eh, where E is elastic modulus and h is vessel wall thickness).

$$PW{V}^2=\frac{Eh}{{\rho }_{f}D}$$ (1)

$$Eh=PW{V}^2{\rho }_{f}D$$ (2)

PWV is calculated by taking the slope of flow versus vessel cross-sectional area curve for the initial time points of systole when there should be only forward travelling waves (Figure 2)³⁵. This is derived from wave intensity analysis where flow rate (Q) and vessel cross-sectional area (A) are decomposed into forward and backwards travelling waves.

$$dQ=d{Q}_{+}+d{Q}_{-}$$ (3)

$$dA=d{A}_{+}+d{A}_{-}$$ (4)

It can be shown from conservation of momentum and mass equations that PWV is equal to the ratio of incremental changes in either forward or backward traveling waves.

$$PWV=\pm \frac{d{Q}_{\pm }}{d{A}_{\pm }}$$ (5)

The difficulty is that to decompose waves into forward and backward travelling components PWV must be known³⁰.

Figure 2:

Plots showing how flow rate and vessel cross-sectional area are used in the calculation of PWV, RAC and ⊺.

Fortunately, in early systole it is appropriate to assume that there are only forward travelling waves and thus PWV is calculated as the ratio of incremental changes in Q and A.

$$PWV\approx \frac{dQ}{dA}{|}_{t\epsilon \mathit{early}\mathit{systole}}$$ (6)

From clinical MRI images, the spatial resolution is not sufficiently high to accurately measure vessel wall thickness. Using a thin-wall assumption though, the vessel wall thickness is only present in the term Eh for vessel stiffness. By determining the total vessel stiffness from PWV vessel wall thickness and elastic modulus do not need to be independently determined.

Once stiffness is estimated from PWV, the pulse pressure (ΔP) in the vessel can be calculated from the artery area change from systole to diastole using the Law of Laplace²⁹.

$$\Delta P=\frac{{\sigma }_{\theta \theta }h}{R_{\mathit{dia}}}=\frac{Eh{\epsilon }_{\theta \theta }}{R_{\mathit{dia}}}=\frac{Eh}{(1-v^2)R_{\mathit{dia}}}\left(1-\sqrt{\frac{A_{\mathit{dia}}}{A_{\mathit{sys}}}}\right)$$ (7)

where σ_θθ is circumferential stress, R_dia is diastolic radius, ϵ_θθ is circumferential strain and ν is Poisson’s ratio. Distal vasculature compliance (C) can then be calculated from Q and the pulse pressure.

$$C=\frac{\Delta V}{\Delta P}=\frac{\Delta Q\Delta t}{\Delta P}$$ (8)

Lastly, an exponential decay is fit to vessel area during diastole (Fig2). The exponential time constant (τ) is assumed to be the distal vasculature time constant and used to calculate resistance. While τ for area decay will not be the same as τ for pressure decay, vessel area is a function of pressure and thus should provide an estimate of τ from which to calculate distal vasculature resistance (R).

$$P=P_1{e}^{-\frac{t}{\tau }}$$ (9)

$$A=A_1{e}^{-\frac{t}{\tau }}+A_0$$ (10)

$$\tau =RC$$ (11)

While the values for Eh, C and R could be directly used for simulation the assumptions of circular cross-sections and straight vessel centerlines in the above derivation will likely limit accuracy. Instead, they will be used as initial guesses and over several simulation iterations Eh, C and R will be refined to decrease FSI - 4D Flow MRI error.

Boundary Condition Estimation

PWV is a function of vessel stiffness (Eq 1) but not distal resistance or compliance. Vessel stiffness is thus estimated before resistance and compliance by modulating vessel stiffness until error between measured and simulated PWV is less than 10%. Additionally, PWV is not affected by initial transients so to more rapidly estimate vessel stiffness only the initial portion of systole is simulated.

Next, full heartbeats are simulated and compliance and resistance are modulated until relative area change (RAC, RAC = (A_sys – A_dia)/A_sys) and τ error is less than 10%.

An algorithmic approach⁴⁹ to varying vessel stiffness, compliance and resistance based on FSI-4D Flow MRI error was attempted. For example, vessel stiffness (Eh) was changed for FSI iteration n+1 based on PWV error for iteration n.

$$E{h}_{n+1}=E{h}_n-{\rho }_{f}D(PW{V}_{\mathit{FSI}}^2-PW{V}_{\mathit{MRI}}^2).$$ (12)

This approach successfully reduced FSI-4D Flow MRI error for PWV and τ below 10% but RAC error remained above 10% for many FSI iterations and was not decreasing. Parameters were instead varied in an ad hoc manner.

MRI Acquisition and Analysis

MRI data is obtained on a 3.0T imaging system using 4D Flow MRI PC-VIPR (Phase Contrast Vastly Under sampled Projection Imaging)^8,46. Flow rates are calculated from velocity fields using Ensight (Ansys, Canonsburg, PA). Vessel areas are calculated from time resolved velocity images using ImageJ. Flow rates and vessel areas are both measured on the same plane to calculate PWV, RAC and ⊺. For the latex tube this plane is located at the midpoint between the inlet and outlet. For the PAS this plane is located in the MPA.

Geometry Segmentation and Mesh Generation

Geometries for FSI are segmented using the open source software Simvascular⁴⁴. Vessel centerline points are manually placed. Vessel cross-sections are automatically segmented using a level-set method and then lofted together. Near the stenosis in the in vivo simulations vessel cross-sections needed to be manually delineated as the level-set method performed poorly. To simulate PAS intervention a stent is manually added to the originally segmented anatomy.

Simvascular is also used to created tetrahedral meshes for all three simulations (latex tube, PAS, PAS + stent). Mesh parameters are listed in Table 1. A mesh sensitivity study^17,34 found that the PAS mesh settings resolved wall shear stress at a location in the main pulmonary artery and within the stenosis.

Table 1:

Mesh Parameter

Geometry	Max Edge Size (mm)	Boundary Layers	Element #
Latex Tube	1.5	2	200k
PAS*	0.5	4	1M
PAS + Stent	0.5	4	1M

^*The PAS mesh is refined to a 0.3mm max edge size near the stenosis

FSI Implementation

FSI is performed with Simvascular⁴⁴ using a stabilized finite element method and linear shape functions for both velocity and pressure.. A coupled momentum method (CMM-FSI) approach is used for FSI⁷. We will briefly describe CMM-FSI and its assumptions, similar to the presentation of Boccadifuoco et al³. The differential (strong) form of the conservation of momentum equations for the vessel wall is

$${\rho }_S\frac{{\partial }^2{\mathit{d}}_s}{\partial t^2}=\mathbf{\nabla }\cdot {\mathit{\sigma }}_s+{\mathit{b}}_s,$$ (13)

where ρ_s is the solid density, d_s is the displacement field, σ_s is the solid stress tensor and b_s is the volume force. By employing a thin wall approximation for the vessel wall, b_s can be represented by the vessel wall thickness h and the surface traction acting on the wall t_s

$${\mathit{b}}_s=\frac{{\mathit{t}}_s}{h}.$$ (14)

Recognizing the relationship between the traction acting on the vessel wall and the traction acting on the fluid (t_s = −t_f), the solid and fluid problems may be coupled and a unified variational formulation for the fluid - thin elastic wall problem is obtained⁷. Linear elasticity and small-displacement kinematics are assumed for the vessel wall so that the fluid domain mesh remains static⁷. This reduces the computational cost of performing FSI and is similar to the analytical solution for pulsatile flow in an elastic tube⁴⁸.

For PAS and PAS intervention simulations using realistic material properties for the Gore-Tex graft and stent prohibitively decreased the stable time-step size. The graft material in the stenosis region and the stent were instead given a stiffness two orders of magnitude greater than the vessel wall. A Poisson ratio of ½ is used for all materials.

In Vitro: Latex Tube

This novel BC estimation method is applied to an in vitro phantom of pulsatile flow in a latex tube (Figure 3)⁴⁰. The tube inner diameter is 15.9mm and the tube length is 130mm. The experimental fluid was water and modeled as an incompressible, Newtonian fluid with a density of 1g/cm³ and a viscosity of 0.009 dyne sec cm⁻². The Reynold’s number for this experiment is 4200.

Figure 3:

A. Simulated latex tube displacements show wave propagation from the inlet (left) to outlet (right). The tube geometry is curved downwards due to gravity. B. FSI-4D Flow MRI error plots of the in vitro latex tube showing the convergence of PWV and RAC over the course of 3 FSI simulation iterations.

The experimental flow loop⁴⁰ did not have distal compliance, only distal resistance so for FSI a resistance outlet BC is used. Without distal compliance the RC time constant should be zero therefore τ is not used to determine BCs, only PWV and RAC. PWV determination of initial vessel stiffness remains the same and ΔP from area change (Eq. 9) is now used to calculate initial resistance,

$$R=\frac{\Delta P}{\Delta Q}$$ (15)

FSI simulation results are then compared against benchtop measurements of flow and displacement for validation⁴⁰. Flowrates measured with ultrasonic flow probes (16PXL, Transonic, NY) at the tube inlet and outlet are used to calculate benchtop PWV. PWV is calculated using a cross-correlation (XCOR) method where a cross-correlation of the flow upstroke is used to determine the inlet-to-outlet time lag. The tube length divided by the time lag gives PWV. PWV is also calculated using time-to-mid (TTM) where the 50% point of maximum flow is used to calculate inlet-to-outlet time lag. The last benchtop PWV calculation is time-to-foot (TTF) where the time axis intercept of a line fit to the flow upstroke is used to calculate inlet-to-outlet time lag. Complete details of benchtop PWV calculation may be found in Ruesink et al⁴⁰. High-speed camera (Phantom v341, Vision Research, NJ) measurements with strain tracking (StrainMaster, LaVision, MI) were also used to calculate benchtop maximum displacement and RAC assuming a circular tube cross-section. FSI maximum displacement is the maximum wall displacement over one heartbeat.

In Vivo: Pulmonary Artery Stenosis

In vivo subject specific simulation of a porcine pulmonary artery stenosis (PAS) (Fig3) is performed next. Isolated left PAS was created in a neonatal piglet by suturing a short segment of 4.0mm Gore-Tex tube graft around the proximal left PA (LPA). Cardiac catheterizations with pre and post stent intervention MRI occurred at 6 weeks of age (4.5 kg). The Institutional Animal Care and Use Committee of the University of Wisconsin reviewed and approved this protocol. Blood is modeled as an incompressible - Newtonian fluid with a density of 1.06 g/cm3 and a viscosity of 0.04 dyne sec cm⁻². Cardiac output is 0.9 L/min and the Reynold’s numbers in the MPA and the stenosis are 430 and 440 respectively.

Three-element Windkessel outlet BCs were used with a 1 to 9 ratio of proximal to distal resistance. Total resistance (R_T) and compliance (C_T) are split equally between the right and left pulmonary arteries (R/LPA) and individual outlet values are then set based on outlet area⁴⁹,

$$R^{\mathit{LPA}}=R^{\mathit{RPA}}=2R_T;\frac{1}{R^i}=\left(\frac{A^i}{A_T}\right)\left(\frac{1}{2R_T}\right)$$ (15)

$$C^{\mathit{LPA}}=C^{\mathit{RPA}}=\frac{C_T}{2};C{}^i=\left(\frac{A^i}{A_T}\right)\left(\frac{C_T}{2}\right)$$ (16)

where R^LPA, R^RPA, Rⁱ, C^LPA, C^RPA, Cⁱ, Aⁱ, A_T denote the total resistance of the left and right lungs, the total resistance of outlet i, the total compliance of the left and right lung, the compliance of outlet i, the area of outlet i and the total are of all right or left lung outlets.

The same vessel stiffness, compliance and resistance BCs that were estimated for the pre-intervention PAS are then applied to the anatomical model with a virtually placed stent to simulate catheter based intervention (Fig4A). The results of the virtual intervention are compared to 4D Flow MRI measurements taken immediately before and after intervention in the porcine PAS.

Figure 4:

A. PAS stenosis geometry and stented geometry. Black arrows point to regions where the stiffness was increased (colored red) to represent the Gore-Tex graft (stenosis) and stent. B. FSI-4D Flow MRI error plots of the in vitro latex tube showing the convergence of PWV and RAC over the course of 8 FSI simulation iterations.

Results

In Vitro: Latex Tube

FSI displacements at subsequent time points show wave propagation (Figure 3A). A plot of error convergence between the FSI and 4D Flow MRI measurements for in vitro latex tube (Figure 3B) shows that the 10% error criterion is met by PWV at the second iteration and RAC at the third iteration. Table 2 shows the errors between validation measurements for benchtop, 4D Flow MRI and FSI. The greatest discrepancy between FSI and benchtop measurements is 52% error for TTF PWV. A prior validation study of CMM-FSI found 13% error between FSI and 2D PC-MRI¹⁹ and reported 10% error between 2D PC-MRI and ultrasonic flow probe measurements²⁰. Given that 4D Flow MRI was found to have 6-12% error compared to benchtop measurements, all FSI measurements except TTF PWV show acceptable agreement with benchtop measurements. A source of error for PWV FSI-benchtop error is that experimental flows were recorded in rigid tubing sections before and after the latex tube. The flow waveform shape will distort between rigid and flexible sections. The location of the “foot” used in the TTF PWV calculation is particularly sensitive to waveform shape and could explain why TTF PWV error is greater.

Table 2:

FSI Validation Against Benchtop Measurements

Measurement	Benchtop40	MRI40	FSI	MRI Error	FSI Error
XCOR PWV (cm/sec)	110 ± 3	103	83	6%	25%
TTM PWV (cm/sec)	93 ±2	86	102	7%	10%
TTF PWV (cm/sec)	126 ±6	134	192	7%	52%
RAC*	0.32 ± 0.003	0.28	0.28	12%	12%
Max Displacement (mm)	1.73 ± 0.02	--	1.98	--	15%

^*FSI wall mechanical properties and distal resistance were tuned to match RAC from MRI

In Vivo: Pulmonary Artery Stenosis

A plot of error convergence between the FSI and 4D Flow MRI measurements for in vivo PAS (Figure 4B) shows that the 10% error criterion is met by PWV at the second iteration, τ at the fourth iteration and RAC at the eighth iteration. This is a comparable number of total iterations to determining BCs using pressure⁴⁹.

Virtual intervention accurately predicts post-intervention LPA flow percentage within 9% of 4D Flow MRI (Figure 5). There is larger error between FSI and 4D Flow MRI for the stenosis which could be due to difficulty segmenting an accurate stenosis geometry. Intervention is predicted to pulse pressure by 2mmHg while experimentally pulse pressure also decreased by 2mmHg. Simulation predicts that intervention decreases vessel wall displacement and thus RAC (Figure 5). There is 20% error between FSI and 4D Flow MRI post-intervention RAC. FSI simulations over predict the intervention effectiveness for both LPA flow percentage and RAC. Virtual intervention accuracy is shown in Table 3.

Figure 5:

Virtual intervention A. predicts that pressure is decreased, B. predicts increased LPA flow percentage with 9% accuracy, C. predicts decreased vessel wall displacement, and D. predicts decreased RAC with 20% accuracy.

Table 3:

Virtual Intervention Accuracy

Measurement	Experiment	FSI	Error
LPA Flow Percentage	37%	40%	9%
RAC	0.24	0.19	20%

Cut-planes show simulated velocity magnitude in the stenosis and stent at three time points of systole (Figure 6). Intervention decreases velocity in both the LPA and RPA and eliminates the jet impinging on the LPA wall distal to the stenosis. Simulations predict that time averaged wall shear stress (TAWSS) and oscillatory shear index (OSI) are both decreased by stent intervention (Figure 7), although TAWSS is only slightly decreased.

Figure 6:

FSI velocity magnitude is shown on cut planes in MPA, LPA and RPA for time points in early, peak and late systole. The 3D geometry is shown as transparent. A jet is seen impinging on the LPA wall distal to the stenosis which stent intervention is predicted to eliminate.

Figure 7:

FSI TAWSS and OSI are shown before and after intervention. Stent intervention is predicted to decrease both TAWSS and OSI.

Discussion

Using 4D flow MRI to estimate FSI boundary conditions highlights an avenue for implementing advantages of patient specific simulation in enhancing information from medical imaging, surgery and intervention planning, and basic science research without the risks associated with catheterization. Realistically, a cardiologist will only perform invasive catheterization when necessary for treatment or diagnosis. For patient specific simulation to be widely applicable, methods for non-invasively determining BCs need to be developed and validated. This work represents a significant step towards these aims.

When performing validation against benchtop latex tube measurements, we found errors of 10%-25% between benchtop measurements and FSI for all metrics except TTF PWV (52% error). FSI-benchtop error is the compounded error of FSI-MRI and MRI-benchtop, each of which is approximately 10%. In particular, uncertainty regarding MRI flow measurements is a concern as 4D Flow MRI velocity fields will have non-trivial conservation of mass errors³⁷. Error in numerical techniques should also be considered and as a rough estimate different groups have reported a 6% discrepancy²³ for the Greenshields–Weller FSI benchmark problem of flow in a deformable tube¹¹. Given these various compounded errors we find the agreement of FSI and benchtop measurements to be acceptable.

We found good agreement between FSI simulation and 4D Flow MRI when virtually performing a stent intervention in a porcine PAS model. Not using realistic graft and stent material properties could cause error in our FSI simulations. However, small displacement values are seen in the stenosis and stent regions compared to the PA tissue (Figure 5C) suggesting that using a stiffness two orders of magnitude greater than the vessel wall is sufficient. FSI simulation of PAS treatment has been performed before⁵⁰. They simulated surgical repair, not a catheter based stent intervention and used pressure catheter measurements to determine BCs. They reported 2% and 6% error for two patients between a lung perfusion scan 1-2 weeks after surgery and FSI using an adaptive outlet BC algorithm to maintain constant WSS in distal pulmonary arteries. Using pre-operative BCs they reported 23% and 18% error. Our LPA flow error was 9% between FSI and 4D Flow MRI performed immediately after intervention. Their improved accuracy using adaptive BCs for a 1-2 week post-operative measurement and our accuracy using pre-intervention BCs for an immediately post-intervention measurement suggest that the distal vasculature is adapting to the new hemodynamic conditions in that 1-2 week time frame. While the prior study and our current study both used CMM-FSI and three element Windkessel outlets in SimVascular, the small sample sizes (n=2 and n=1) and differences in simulated treatment (surgical graft vs catheter based stent), species (human vs porcine) and clinical imaging modality (lung perfusion scan vs 4D Flow MRI) highlight that this is speculative and needs for further investigation.

This method also provides MRI quantification of distal vascular beds, which cannot be directly imaged with MRI, by calculating the distal vascular compliance, resistance and time constant from flow and area measured in the proximal vasculature. 4D Flow MRI has previously estimated proximal physiological biomarkers like pressure drop over stenosis⁴⁶, vessel stiffness through PWV⁴⁰, wall shear stress¹ and kinetic energy^15,16. 2D PC MRI with wave intensity analysis identifies distal vascular beds properties like wave reflection sites³⁵ but not the more commonly used clinical markers of vascular resistance and compliance.

This method may not give as accurate of simulated pressures, particularly mean pressure, as approaches which use pressure catheterization to determine BCs. For incompressible liquids such as blood only pressure gradients drive flow not the mean pressure. For applications looking at local flow quantities such as velocity, wall shear stress or pressure gradients fitting BCs to vessel wall motion could provide more accurate results as the flow will now be solved for more accurate time resolved geometry. For other applications such as those quantifying of the effect of pathological remodeling or an intervention on ventricular afterload in a closed-loop simulation²¹ simulating accurate mean pressures is important.

For CFD studies using pressure data, patient specific vessel stiffness could be determined from PWV using partial simulations (Figure 8). Resistance and compliance would then be set to match pressures. RAC could be used to validate the simulation, as it was not used in determining BC parameters. Most patient specific simulations do not validate their in vivo simulations against additional clinical measurements due to all clinical measurements being used to determine BCs.

Figure 8:

Flow chart describing an algorithm to determine patient specific conditions from PWV and pressure measurements. An initial iterative loop is used to rapidly match PWV, then a second loop matches pressures. RAC would then remain as a validation measurement as it was not used to determine BCs.

A key limitation of CMM-FSI is that it solves the solid governing equations in an Eulerian reference frame so accuracy decreases with increasing deformation magnitude. This method for determining BCs could be implemented using more accurate (but more computationally expensive) numerical techniques for FSI such as arbitrary Lagrangian-Eulerian approaches or the immersed boundary method. The solid in this work was also assumed to be isotropic-linearly elastic and the unloaded geometry chosen in diastole. Determining unloaded geometries of blood vessels is an ongoing area of research^6,24 but even if the unloaded geometry could be determined a non-linear and anisotropic material model is still needed for accurate mechanical behavior⁴². Determining anisotropic properties could possibly be done from longitudinal movement of vessels measured with MRI. It is unclear how non-linear mechanical properties would be determined. Exercise or other stimuli could potentially be used to alter loading conditions. Blood vessels are relatively linear elastic in normal physiological pressure ranges so determining non-linear mechanical properties may not be necessary.

Lastly, this method could be implemented using 2D PC MRI, as these images would also provide time resolved flow rates and cross-section areas. Advantages of using a radially acquired 4D Flow MRI sequence like PC-VIPR include: no need to prospectively set planes perpendicular to flow, simultaneous acquisition of flow and anatomy, free breathing acquisition, minimization of motion artifacts and isotropic spatial resolution.

Conclusion

Flow and areas measured from 4D Flow MRI can estimate vessel wall mechanical properties and distal vasculature bed resistance and compliance for patient specific BCs. The ability to estimate BCs for patient-specific simulations without invasive catheterization increases the ability to couple FSI and 4D Flow MRI for translational and basic science studies of cardiovascular disease and eventually clinical decision making. Future work will further validate this method in additional in vitro phantoms as well as apply this technique to other vascular anatomies like the aorta and portal vein.

Abbreviations

CFD

Computational Fluid Dynamics

FSI

Fluid-Structure Interaction

BC

Boundary Condition

PAS

Pulmonary Artery Stenosis

RAC

Relative Area Change

PWV

Pulse Wave Velocity

CMM-FSI

Coupled Momentum Method Fluid-Structure Interaction

XCOR

Cross Correlation

TTM

Time to Mid

TTF

Time To Foot