Transient flow prediction in an idealized aneurysm geometry using data assimilation

Transient flow prediction in an idealized aneurysm geometry using data assimilation

Journal Pre-proof Transient flow prediction in an idealized aneurysm geometry using data assimilation Franziska Gaidzik, Daniel Stucht, Christoph Rolo...

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Journal Pre-proof Transient flow prediction in an idealized aneurysm geometry using data assimilation Franziska Gaidzik, Daniel Stucht, Christoph Roloff, Oliver Speck, Dominique Thévenin, Gábor Janiga

PII: DOI: Reference:

S0010-4825(19)30371-3 https://doi.org/10.1016/j.compbiomed.2019.103507 CBM 103507

To appear in:

Computers in Biology and Medicine

Received date : 9 July 2019 Revised date : 27 September 2019 Accepted date : 12 October 2019 Please cite this article as: F. Gaidzik, D. Stucht, C. Roloff et al., Transient flow prediction in an idealized aneurysm geometry using data assimilation, Computers in Biology and Medicine (2019), doi: https://doi.org/10.1016/j.compbiomed.2019.103507. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Conflict of Interest Statement

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None.

Highlights (for review)

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Computers in Biology and Medicine

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“Transient flow prediction in an idealized aneurysm geometry using data assimilation“

Highlights:

 Improved flow prediction in intracranial aneurysms by merging PC-MRI data with CFD simulations

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 Transient data assimilation using a localization approach with Ensemble Kalman Filtering to reduce ensemble size requirements

 Two different update processes for the data assimilation experiment are evaluated

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(time-dependent, time-independent)

 Unique validation of PC-MRI and DA with high-resolution transient PIV data to generate ground-truth flow data

Keywords

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4D Flow; Data assimilation; Ensemble Kalman Filter; intracranial aneurysm; PC-MRI

Journal Pre-proof *Revised manuscript (clean) Click here to download Revised manuscript (clean): Revised_Clean.tex

Transient flow prediction in an idealized aneurysm geometry using data assimilation

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Franziska Gaidzika , Daniel Stuchtb,c , Christoph Roloffa , Oliver Speckb,d , Dominique Th´evenina , G´abor Janigaa,∗ a

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Lab. of Fluid Dynamics and Technical Flows, University of Magdeburg “Otto von Guericke” b Institute of Experimental Physics, University of Magdeburg “Otto von Guericke” c Institute of Biometry and Medical Informatics, University of Magdeburg “Otto von Guericke” d Leibniz Institute for Neurobiology, Magdeburg, Germany

Abstract

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Hemodynamic simulations are restricted by modeling assumptions and uncertain initial and boundary conditions, whereas Phase-Contrast Magnetic Resonance Imaging (PC-MRI) data is affected by measurement noise and artifacts. To overcome the limitations of both tech-

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niques, the current study uses a Localization Ensemble Transform Kalman Filter (LETKF) to fully incorporate noisy, low-resolution Phase-Contrast MRI data into an ensemble of highresolution numerical simulations. The analysis output provides an improved state estimate of the three-dimensional blood flow field in an intracranial aneurysm model. Benchmark measurements are carried out in a silicone phantom model of an idealized aneurysm under pulsatile inflow conditions. Validation is ensured with high-resolution Particle Imaging Velocimetry (PIV) obtained in the symmetry plane of the same geometry. Two data assimilation approaches are introduced, which differ in their way to propagate the ensemble members in time. In both cases the velocity noise is significantly reduced over the whole cardiac cycle. Quantitative and qualitative results indicate an improvement of the flow field prediction in

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comparison to the raw measurement data. Although biased measurement data reveal a systematic deviation from the truth, the LETKF is able to account for stochastically distributed errors. Through the implementation of the data assimilation step, physical constraints are ∗

Corresponding author Email address: [email protected], Universit¨ atsplatz 2, Magdeburg, D-39106, Germany (G´abor Janiga)

Preprint submitted to Computers in Biology and Medicine

September 27, 2019

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introduced into the raw measurement data. The resulting, realistic high-resolution flow field can be readily used to asses further patient-specific parameters in addition to the velocity distribution, such as wall shear stress or pressure.

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Keywords: 4D Flow, Data assimilation, Ensemble Kalman Filter, intracranial aneurysm,

Abbreviations

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PC-MRI

CFD – Computational Fluid Dynamics; DA – Data assimilation; EnKF – Ensemble Kalman Filters; ICP – Iterative Closest Point; LETKF – Localization Ensemble Transform Kalman Filter; PC-MRI – Phase-Contrast Magnetic Resonance Imaging; PIV – Particle

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1. Introduction

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Image Velocimetry; SNR – signal-to-noise-ratio

Rupture of intracranial aneurysms can cause severe outcomes, such as irreversible dis-

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abilities or even death. To improve our understanding of cardiovascular diseases, hemody-

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namic studies are very helpful. Such a gain of knowledge can support physicians concerning

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outcome prediction and therapy planning. Flow-dependent parameters, such as wall shear

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stress or the oscillatory shear index, have been proposed as relevant quantities related to

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the growth and rupture probability of aneurysms [1, 2, 3]. Different modalities are available

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to assess patient-specific hemodynamic flow fields. Numerical investigations using Compu-

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tational Fluid Dynamics (CFD) have been frequently used to identify key parameters and

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flow characteristics. However, such simulations require accurate initial and boundary condi-

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tions, and depend on the assumptions used in the numerical model [4, 5, 6]. Although high

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temporal and spatial resolution can be reached, uncertain initial and boundary conditions

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lead to a limited clinical acceptance of the simulation results [7, 8, 9]. Phase-Contrast Mag-

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netic Resonance Imaging (PC-MRI) in-vivo measures blood flow quantitatively by encoding

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the velocities in the phase of the acquired MR signal [10, 11, 12]. However, in addition to

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measurement noise and artifacts, the currently achievable temporal and spatial resolution

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limits the extraction of clinically-relevant flow features from PC-MRI measurement data.

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Alternative measurement techniques, such as Particle Image Velocimetry (PIV), can acquire

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high resolution flow fields in-vitro, but a clinical application would be limited or even im-

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possible [13]. Being an optically-based technique, PIV requires fully transparent settings.

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Patient-specific silicone models allow PIV measurements but are expensive in terms of manu-

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facturing time and costs. Additionally, the surface of patient-based silicone models is usually

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extracted from MR images, which limits the accuracy of the geometry bounding the flow.

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This, together with the pulsatile nature of patient vessels, leads to differences between the

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physiological condition and the flow measured within silicone models. Therefore, highly re-

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solved PIV can be mainly used as a validation procedure for specific cases and benchmark

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configurations.

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Uncertainties associated with either numerical or experimental approaches naturally sug-

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gest the use of data assimilation to incorporate the inherently noisy measurement data of

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limited resolution into high-resolution numerical simulations. In this manner, accuracy and

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physical correctness of the measured velocity field can be improved. De-noising techniques,

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as well as divergence-free filtering approaches, can also improve the physical correctness

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of the velocity field; however, spatial and temporal resolution remain unchanged [14, 15].

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Hence, data assimilation seems to be a promising remedy to improve resolution while enforc-

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ing constraints, such as incompressibility and conservation laws. Contrary to most existing

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approaches that use available PC-MRI measurement data only as initial conditions for CFD

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simulations [4, 16], data assimilation uses a complete measured 4D flow field to be incor-

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porated into numerical simulations. In this manner, the reliability of the intracranial state

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estimate together, with the prediction of clinically relevant parameters such as wall shear

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stress (WSS), should be improved.

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For fluid dynamics applications regarding compressible flows promising results have been

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achieved with adjoint-based data assimilation [17, 18, 19], even if the resulting computational

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costs are in the order of 300 CPUh for one iteration loop. That is because solving the

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adjoint equations needs approximately twice as much time as solving the direct equation.

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More than 30 loops are often required to reach the desired convergence criteria [17]. D’Elia

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et al. [20, 21, 22] and Funke et al. [23] applied variational data assimilation approaches in

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intracranial aneurysms. As an alternative to the sequential technique they minimized the

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error between observations of a reference flow and a numerical estimation in terms of a

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cost function. The authors stated that the need for linear and adjoint models increases the

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computation time by a factor of 50 to 100 in comparison to one single CFD simulation. As

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a consequence of these high computational costs, most of the stated numerical studies on

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variational data assimilation in intracranial aneurysms currently only address steady-state

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flows and/or 2D geometries [17, 18, 19, 20, 21, 22]. Funke et al. [23] investigated transient

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3D flow fields, but with a reduced spatial resolution to maintain acceptable computational

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costs. Further limitations of previous hemodynamic DA studies result from 1) synthetically

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generated measurement data (replaced by real PC-MRI data in the present study), and/or 2)

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the lack of a ground truth, in particular for real 4D flow fields (delivered by PIV experiments

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in this work).

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Although promising results have been achieved with this technique for other fluid dynam-

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ical applications [24, 25], little attention has been paid to the sequential Kalman Filters in

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intracranial aneurysm modeling. Bakhshinejad et al. [26] implemented an Extended Kalman

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Filter for a pulsatile cardiac flow. However, the lack of localization requires a large amount of

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ensemble members to ensure filter convergence. The resulting, high computing times reveal

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the need for a sequential data assimilation technique that can converge with a limited amount

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of ensemble simulations. A possible approach relies on Proper Orthogonal Decomposition

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(POD) to reduce the needed ensemble members [27]; POD can also be used to quantitatively

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analyze PC-MRI flow data [28]. As an alternative, the present study predicts an improved

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flow field for measured transient intracranial aneurysm flow using a Localization Ensemble

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Kalman Filter (LETKF). In general, Ensemble Kalman Filters (EnKF) sample the system

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state and covariance matrices by an ensemble of model states. Thus, the covariance ma-

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trices are not calculated directly, but estimated through an ensemble and replaced by the

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sample covariance. The uncertainty of the numerical model (background) is estimated and

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the Ensemble Kalman Filter can be seen as a Monte-Carlo approximation of the original

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Kalman Filter [29]. Usually, a large ensemble size is required to sample the system state in

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a statistically representative manner. With the implementation of a localization procedure

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in the Ensemble Kalman Filter, the amount of needed ensemble members can be reduced

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significantly. Building on top of previous investigations [13], the current study relies on

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transient high-resolution PIV data to provide a ground truth, allowing comparisons with

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PC-MRI and CFD data and – even more important – a validation of the developed data

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assimilation procedure.

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2. Material and Methods

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(a) numerical model

(b) set-up

(c) cardiac cycle trajectory

Figure 1: (a) Surface model of the idealized aneurysm used for phantom manufacturing and CFD discretization; (b) MRI setup with the gear pump (1), wave guide through RF-shield (2), flow meter (3), MR-scanner (4) and the 32-channel head coil with the phantom model (5); (c) pulsatile flow profile provided by the programmable pump [13].

PC-MRI and PIV data are obtained from a two-component silicone model (Wacker RT

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601, Burghausen, Germany) of an idealized intracranial aneurysm, with proximal and distal

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vessels of 4 mm. To enable a fully developed flow profile, a long (length 40 cm, diame-

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ter 4mm), straight, rigid pipe was installed upstream of the aneurysm inlet. Exactly the

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same geometry is used for the calculation of the forward model, i.e., the ensemble of CFD

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simulations. To mimic realistic flow conditions, a blood substitute is used in all experiments.

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It matches approximately the fluid dynamic properties of real blood (ρ = 1222 kg/m3 ;

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η = 4.03 mPa s), and simultaneously the refractive index (=1.4122 at 22◦ C) of the silicone

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block, which ensures optimal optical access for PIV measurements. The pulsatile flow, rep-

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resenting a cardiac cycle (Figure 1 (c)), is provided by a programmable micro-gear pump

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(HNP Mikrosysteme, Schwerin, Germany). To prevent any difference concerning the flow,

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the experimental set-up was held completely identical (same construction, pipe length, fit-

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tings...) for both PIV and PC-MRI measurements. This allows a proper comparison and

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cross-validation between the different modalities. Additionally, the resulting data was regis-

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tered with the implementation of an Iterative Closest Point (ICP) algorithm, which enables

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rigid body transformation. Difficulties in the registration process arose due to geometric

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distortions in the PC-MRI data. These artifacts increase with growing distance from the

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measurement center, which was chosen in the center of the aneurysm sack. The distorted

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parts of distal and proximal vessels are removed prior to the registration step, to ensure

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proper matching of the velocity fields in the center of the aneurysm, as acquired from the

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different modalities.

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Figure 2 shows the flowchart that is used to get an improved state estimate of the intracra-

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nial velocity field. A detailed description of the main steps regarding the overall workflow is

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given in the following sections.

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2.1. Phase-Contrast Magnetic Resonance Imaging (PC-MRI)

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The 4D flow data was acquired on a 7 Tesla whole-body MRI system (Siemens Health-

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ineers, Forchheim, Germany) in a 32-channel head coil (Nova Medical, Wilmington, MA)

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using 4D phase-contrast magnetic resonance imaging (PC-MRI). Hereby, the acquisition se-

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quence is based on an RF-spoiled gradient echo with quantitative flow encoding in all three

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spatial dimensions [30, 31]. The data was acquired with an imaging matrix of 224x224 px,

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a 128x128 mm FoV and a slice thickness of 0.57 mm resulting in an isotropic voxel size of

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0.57 mm. The total scan time for 11 temporal phases was approx. 109 minutes. The data

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of one time step is acquired in the interval between two subsequent time points and results

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in a temporal resolution of 82.8 ms. To correct for eddy currents a reference measurement

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was acquired, without activated pump and thus without flow inside the aneurysm phantom

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model. The phase of the reference data was subtracted from the phase of the flow data to

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Figure 2: Technical flow chart of the current data assimilation technique. The proposed workflow is conducted for every time point at which a phase-contrast MR image is available. For time-independent data assimilation the workflow can be performed for all 11 time steps individually; whereas for time-dependent data assimilation the ensemble simulations of the subsequent time step are initialized by the analysis outcome of the previous time point.

obtain purely flow-related phase differences. As the flow information is encoded in the phase

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of the complex MR-Signal, a velocity encode parameter (venc) specifies the highest velocity,

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uniquely encoded in one phase. For the acquired data, this parameter was set to 1.5 m/s.

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The signal-to-noise-ratio (SNR) of the acquired images was calculated using the mean of the

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signal inside the aneurysm and the signal density of the background noise and was found to

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be SNR ≈ 90. The data was post processed using MeVisLab 2.3.1 (MeVis, Bremen, Ger-

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many) and the automated tool described in [32]. This includes noise masking, antialiasing

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and conversion to the format of the commercial software package EnSight (ANSYS Inc.,

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Canonsburg, PA, USA). The acquired flow data serve as an input for the data assimilation

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step in the form of observation data.

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2.2. Particle Image Velocitmetry (PIV)

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The generation of a quantitative gold standard is one of the main challenges in the

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formulation of a suitable data assimilation experiment. Particle Image Velocitmetry (PIV) is

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currently the most established flow field measurement technique allowing for highly resolved

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flow field data. In this study, a high-speed stereoscopic PIV system was used to obtain

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time-resolved flow data at the sagittal plane of the idealized aneurysm geometry. By that, a

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ground truth is formulated, which is used to validate the data assimilation experiment. The

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entire fluidic setup used for the PC-MRI measurements was used without any modification

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as the basis for the PIV study. A thin laser light sheet illuminated small resin tracer particles

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doped with Rhodamine B (diameter d = 10.46 ± 0.18 µm, density ρ = 1510 kg/m3 ), which

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were given to the fluid. Two high-speed cameras recorded double-frame images (inter-frame

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delay of 200 µs) of the particles’ fluorescent light at 500 Hz recording frequency. Hence, after

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processing (Davis software 8.4.1, LaVision, Goettingen, Germany), a velocity field snapshot

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was available every 20 ms during the entire cardiac cycle. A total of 36 independent cardiac

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cycles were covered by the PIV recordings; the final phase-averaged velocity field was used

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as validation data for this study.

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2.3. Background model

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In the current study, CFD simulations are used as the numerical background model; it

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is needed as an input for the formulation of the data assimilation algorithm. To calculate

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the background model, an ensemble of CFD simulations is propagated forward in time to

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sample the system state and to provide an estimate of the error covariance matrix related

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with the numerical model. A block-structured hexahedral mesh is first generated from the

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underlying geometry using ANSYS IcemCFD (ANSYS Inc., Canonsburg, PA, USA) result-

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ing in approx. 750,000 finite-volume cells. The results of a mesh-independence analysis did

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not show any noticeable changes in flow parameters, so that numerical stability with respect

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to the mesh size is ensured. Ensemble simulations are carried out using the open source soft-

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ware OpenFOAM 4.0 (OpenCFD Ltd., Bracknell, UK). Blood is treated as an isothermal,

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incompressible fluid, with density and viscosity prescribed as in the experimental measure-

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ments. The vessel walls are assumed to be rigid and no-slip boundary conditions as well

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as a zero-pressure outlet are implemented. To ensure computational stability, the time-

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step of the background needs to be smaller than the time step provided from the PC-MRI

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measurements; it has been chosen to be 1 ms as suggested by [33].

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2.4. Data Assimilation

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(a) Time-independent DA

(b) Time-dependent DA

Figure 3: For an easier visualization, only 10 ensemble members are shown: (a) Ensemble trajectories for the time-independent data assimilation. Here, the analysis states are independently calculated at the measurement times, indicated by the red points; (b) Schematic illustration of the time-dependent data

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assimilation case.

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By using data assimilation, numerical model data and PC-MRI measurement data are

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merged together for an improved state estimate of the flow field in the intracranial aneurysm.

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The inflow condition throughout the cardiac cycle, as well as the three-dimensional ve-

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locity field, serve as the uncertain quantities in the present data assimilation experiment.

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This provides an interesting alternative to approaches frequently used in past studies, in 9

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which PC-MRI data were only used to extract inlet and outlet flow rates applied as initial

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and/or boundary conditions in a single CFD calculation. The Local Ensemble Transform

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Kalman Filter (LETKF) applied in this paper was originally introduced by Harlim and Hunt

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[34, 35] in the field of meteorology. It combines the localization method of the Local En-

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semble Kalman Filter (LEKF) of Ott et al. [36] and the Ensemble Transform Kalman Filter

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(ETKF) of Bishop et al. [37]. The analysis ensemble is formed as a weighted average of

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the background ensemble mean and the observations. Using background and observation

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uncertainties, the weights are determined in such a way that the analysis ensemble mean

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fits in an optimal manner the given background and observation probability distributions.

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By implementing localization, purely local analyses at each model grid point are obtained,

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so that only observations within a local region surrounding the grid point are taken into ac-

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count. The localization scheme enables efficient parallel computations of the analysis model

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state; additionally, it limits the number of needed ensemble members. Beyond model (CFD

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simulations) and noisy measurement (PC-MRI) data, following parameters are used as input

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for the DA study:

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• A localization radius of 3 mm, taking only observations into account when they are found within this radius.

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• An observational operator H to map the state variables from the m-dimensional simu-

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lation space to the s-dimensional observation space. The current observation operator

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H is a spatial binning operator, which downsamples the ensemble velocity vectors to

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the MRI grid resolution.

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• An s × s dimensional observation error covariance matrix R based on the noise char√ 2 venc acteristics of the measurement data: σv = . π SN R

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With an randomly large ensemble size, the error covariance matrix of the underlying numer-

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ical model can be sampled correctly. Nevertheless, this results in unacceptably high com-

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putational costs when running a large amount of multiple CFD resolutions. By introducing

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the localization procedure of the LETKF, a smaller ensemble size can be used to sample 10

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the subdomains and to determine local weights, which are used to calculate the analysis.

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Using 25 ensemble members the filter was able to converge throughout the whole cardiac

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cycle. To calculate an improved state estimate, two different data assimilation experiments

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are performed, denoted in what follows time-independent (or Case 1), resp. time-dependent

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– with cardiac modeling (or Case 2):

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1. Case 1: From the raw PC-MRI data the inflow rate at the 11 measurement time steps

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can be extracted. Based on the noise characteristics of the PC-MRI measurements

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the reconstructed flow rates at every time point are perturbed to generate 25 differ-

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ent inflows. Cubic interpolation randomly connects one of the 25 flow rates of the 11

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different time points and calculates a trajectory in between the time steps to gener-

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ate the ensemble members. Thus, the resulting CFD ensemble simulations differ in

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the underlying inflow trajectory throughout the cardiac cycle. Each trajectory is in-

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dividually propagated forward in time throughout the cardiac cycle (Figure 3a). For

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every measurement time point all ensemble member states are stored separately and

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11 different time-independent analyses are calculated. In this case, the analysis is a

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function of the ensemble inflow conditions and the measurement data, but is indepen-

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dent from the analysis state of a previous time step. No forecast step in the classical

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sense can be performed, because no background model for the cardiac cycle trajectory

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is implemented.

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2. Case 2: Similar to Case 1, the ensemble members for the analysis of the first time

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point are generated using the noise characteristics of the measurement data. After the

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calculation of the first analysis, the cardiac cycle trajectory in between the current and

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next time step is reconstructed using a forecast model in the form of Xt = f (Xt−1 ) +

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t . Currently, the simple forward model is a linear function, which serves as initial

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condition for the ensemble CFD simulations. Thus, each analysis ensemble velocity

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field is propagated forward in time, using the previously calculated analysis velocity

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field and a linear inflow trajectory as initial conditions.

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2.5. Validation The high-resolution PIV measurements of the center slice inside the aneurysm are set as

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a ground truth criteria and used to validate the calculated analyses. Quantitative validation

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and comparison is based on the interpolation of velocity values obtained by PIV, PC-MRI, or

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data assimilation, respectively, onto a random point cloud distributed over the center slice.

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The Root-Mean Square Error (RMSE) between the different modalities is calculated and

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normalized (NRMSE) to enable a better comparison of the values throughout the cardiac

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cycle.

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v u m u1 X t RMSE = (vAnalysis − vPIV )2 m i=1 NRMSE =

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RMSE ¯ PIV v

The Similarity Index (SI) measures the similarity of two independent vector sets and

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takes deviations of the angle between both vectors (ASI), as well as the differences in the

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normalized magnitudes (MSI) into account. For validation purposes the SI, together with

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ASI and MSI, between PIV and either PC-MRI or analyses flow fields is evaluated. Equation

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(2) shows for example the SI calculation between PIV and analysis:      |vAnalysis | vAnalysis · vPIV |v | PIV SI = ASI · MSI = 0.5 · 1 + · 1 − − |vAnalysis | · |vPIV | max (|vAnalysis |) max (|vPIV |) (2) 3. Results

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3.1. Qualitative flow field validation

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First, for a qualitative flow visualization the velocity distribution along the center slice

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inside the aneurysm is shown at every time point for which measurement data are available

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(Figure 4). PC-MRI and calculated analyses flow fields are compared with the PIV mea-

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surements for either time-independent or time-dependent data assimilation. Overall, a good

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agreement can be observed between the analyses flow fields and the measured ground truth. 12

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Figure 4: Qualitative comparison of the transient flow field prediction for both data assimilation experiments together with the PC-MRI and PIV data. The velocity magnitude of a cross-section cut through the center of the aneurysm model is displayed.

However, the velocity magnitude seems to be underestimated by PC-MRI and analysis in

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comparison to the PIV data, especially in the high velocity areas. Additionally, the analysis

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velocity field calculated in Case 1 seems to be smoother than the flow field in Case 2, in which

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a simplified cardiac model was used to propagate the ensemble members forward in time.

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In both cases, the LETKF de-noises the PC-MRI data and increases the spatial resolution

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of the three-dimensional flow field. The resulting spatial resolution matches the number of

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grid points from the underlying ensemble CFD simulations. Thus, the walls of the analyses

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cross-section cuts appear to be smoother and better resolved compared to the raw measure-

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ment data of PC-MRI. Thanks to this improvement in spatial resolution and extraction of

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a smooth surface boundary, clinically relevant parameters like the Wall Shear Stress (WSS)

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can be computed. Figure 5 shows the calculated WSS at five different time-points based

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on the time-independent, three-dimensional analysis flow field. An area of high shear stress

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at the transition from the aneurysm sack to the distal vessel can be clearly identified at all

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time-steps.

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Figure 5: WSS distribution based on the time-independent three-dimensional analysis flow field. The spatial

3.2. Quantitative flow field validation

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resolution matches the resolution from the ensemble CFD simulations.

Figure 6: Velocity magnitudes of PIV are plotted against PC-MRI and analysis magnitudes, respectively.

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The data is displayed in a 2D histogram density plot.

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The flow fields obtained from the different modalities are interpolated onto a randomly

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generated point cloud to enable a quantitative comparison. Measurement time point 4

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(velocity peak) has been chosen to exemplarily show the performance of the data assimilation

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steps, as well as the accuracy of the PC-MRI measurement. The velocity magnitudes of 14

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Figure 7: Histograms displaying the relative frequency of the similarity index SI (top row), magnitude similarity index MSI (middle row) and angular similarity index ASI (bottom row) over all compared data points for the different modalities. The first quartile (Q1 ) is marked with a vertical blue line.

the different modalities are first plotted against each other (Figure 6). For an illustrative

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quantification, a two-dimensional histogram grid is provided, which is additionally color-

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coded by the amount of data points corresponding to a specific location; the brighter the

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grid cell, the more data points lie there. The red line indicates a perfect match between both

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compared modalities. Regarding the agreement between PIV and PC-MRI, a significant

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(a)

(b)

Figure 8: NRMSE (a) and Q1 (b) values of the three different modalities throughout the cardiac cycle.

difference concerning the scatter around the red line is observed in the low and high velocity

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ranges. For velocity magnitudes below 0.25 m/s, many points are found and appear to

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be equally distributed around the red line (σ = 0.014); the scatter represents the noise of

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the measurements. At higher velocities the points become more and more shifted toward

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the area above the red line; thus, the MRI measurement increasingly underestimates the

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velocities. About 98% of the velocity magnitudes higher than 0.5 m/s are underestimated

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by PC-MRI. A similar result is observed for the velocities calculated in the data assimilation

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experiments. Although the spread around the red line and thus the noise of the data is

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reduced in comparison to PC-MRI (σ = 0.01), the deviation from the ground truth again

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increases at higher velocities. For the DA analyses only 83% (Case 1) and 86% (Case 2) of the

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velocity values higher than 0.5 m/s are underestimated. Figure 7 shows the relative frequency

275

of the similarity index (SI), angular similarity index (ASI) and magnitude similarity (MSI)

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found at the random sample points. All histogram plots contain a line representing the Q1

277

value, which quantifies the point that separates the lower 25% of the frequency distribution

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from the higher 75%. The indices are calculated between analysis or PC-MRI data compared

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to the PIV measurements. For the PC-MRI data, the SI is higher than 0.75 in 75% of the

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data points. A much better agreement is found for DA, with a Q1 equal to 0.9 (Case 1) or

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0.88 (Case 2), respectively. A more detailed insight in the comparison of the vector fields

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can be achieved by splitting the SI into its angular (Figure 7, middle row) and magnitude

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part (Figure 7, bottom row), respectively. For both indices the Q1 value is increased thanks

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to DA. The MSI is higher than 0.96 in 75% of the data points for both analyses, while the

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corresponding value is only 0.92 for the PC-MRI data. This effect is even more pronounced

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for the angular differences. While the Q1 value for the PC-MRI data is relatively low (0.85),

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the analyses have very high Q1 values equal to 0.98 (Case 1) and 0.96 (Case 2), respectively.

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Figure 8a compares the calculated NRMSE values over the considered cardiac cycle. For

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all time points, the PC-MRI data has a noticeably higher NRMSE than both DA analyses.

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When comparing the two different data assimilation experiments, the time-independent anal-

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ysis flow field shows a better agreement with the PIV measurements. Although the RMSEs

292

have been normalized by the velocity, higher values are still observed at time points that are

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associated to higher peak flows (see also Figure 4). The Q1 values calculated in Figure 8b

294

support the previous discussion. The Q1 values are the lowest (i.e., poorest) for the raw

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PC-MRI data, while they are almost always maximal (i.e., best) for the time-independent

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DA analysis.

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4. Discussion

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Due to recent advances in computational resources as well as medical imaging techniques,

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the patient-specific investigation of intracranial blood flows has become increasingly popu-

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lar [30, 31]. Limitations in numerical and experimental methods impair the acceptance of

301

intracranial flow investigations by the clinical community [8, 9, 38]. By introducing the tran-

302

sient data assimilation approach a better prediction of the velocity field becomes possible; this

303

has been validated by comparison with high-resolution PIV data. Previous studies have al-

304

ready tried to assimilate PC-MRI data and fluid dynamics simulations [20, 21, 22, 23, 26, 27];

305

however, validation was difficult due to the lack of a ground truth – or sometimes of any

306

real measurement data. A qualitative comparison of the velocity magnitude reveals a high

307

similarity between the ground-truth PIV data and the calculated DA analyses. In both

308

cases, the noise is reduced in comparison to the raw MRI data. The time-independent data

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assimilation approach results in smooth velocity flow fields and continuous ensemble trajec-

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tories throughout the whole cardiac cycle. From the calculated analysis flow field, the WSS

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distribution can be calculated successfully, as shown in Figure 5. These observations are

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further supported by the quantitative results. The spread of the velocity values in Figure 6

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around the optimal red line is reduced from σ = 0.014 to σ = 0.01. At higher velocity values

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a systematic error appears in the PC-MRI measurement data, which increases the velocity

315

differences of the corresponding data points. The LETKF can only account for random-

316

based error sources, but cannot improve systematic deviations. Therefore, the calculated

317

DA analyses are still biased by the systematic error sources of the original measurement

318

data. Nevertheless, NRMSE and Q1 values indicate that the analysis flow fields are much

319

closer to the ground truth than the raw PC-MRI measurement data. The noise reduction in-

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dicates a better agreement with the general conservation laws (mass and momentum), while

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PC-MRI data do not have any physical constraints implemented in the acquisition sequence.

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In previous studies, the error sources for 4D flow data have been mainly attributed to

323

temporal and spatial downsampling, as well as acquisition artifacts such as partial volume

324

effects [39]. For our own PC-MRI measurement data, we hypothesize a relative dependency

325

of the errors from the absolute velocity, which could perhaps be reduced by introducing

326

a scaling factor. We further assume that the value of this factor might be associated to

327

gradient inhomogeneities or to the Lorentz force. These effects possibly influence the phase

328

of the acquired MR signal and thus the quantitative flow data. Currently, a follow-up study

329

is being performed to quantify such possible artifacts and to identify the physical origin of

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such a scaling factor.

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The spatial resolution can be increased for both, time-dependent and time-independent

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data assimilation. However, the need and possibility for a temporal improvement remains

333

unclear. To calculate the pressure and WSS field, an increase in temporal resolution is not

334

really needed, but an improved spatial resolution is crucial. In terms of rupture or growth

335

prediction, it could be sufficient to only improve the spatial resolution and physical correct-

336

ness at selected time-points. Thus, the time-independent data assimilation approach should

337

currently be preferred, as possible parallel analyses calculations further reduce the compu-

338

tational time. Additionally, continuous ensemble inflow trajectories throughout the whole

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cardiac cycle for all ensemble members lead to a smoother ensemble solution and smooth

340

analysis flow fields. With the introduction of a simplified cardiac model, as it was attempted

341

in the time-dependent approach, abrupt changes in the inflow trajectory after an analysis

342

calculation are introduced. This might distort the forward propagation of the individual en-

343

semble members. To ensure a proper reconstruction of the cardiac cycle trajectory, a better

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cardiac model forecast must be introduced.

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The computational time needed to calculate one CFD simulation is ≈ 32 CPU hours. This

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is about the same time needed to calculate one analysis time step, without any paralleliza-

347

tion. By distributing the calculation of local analyses on 6 nodes, the overall computational

348

time was reduced by a factor of 4. The localization step not only enables parallel compu-

349

tations, but also reduces the ensemble size from 50 to 25 members in comparison to that

350

needed in [26]. Fine-tuning of parameters such as the localization radius or ensemble size

351

will be considered in further studies. After obtaining an assimilated high-resolution velocity

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field, clinically relevant parameters like pressure distribution and WSS can be derived.

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Limitations of the Study. The whole process aims to point out the feasibility of the

355

LETKF to merge CFD and PC-MRI data together. Therefore, several assumptions have

356

been made to reduce the complexity of the study, mainly with respect to the numerical

357

model and measurement data. Fluid-structure interactions are not considered, because the

358

influence of the time-dependent pressure on the walls of the silicone phantom appears visually

359

to be negligible. However, it might be important to consider the detection of the vessel

360

walls in future studies, in particular when using a more flexible material or direct phantom

361

models with thin walls. Patient-specific configurations have not been considered in this

362

study. However, the use of an idealized aneurysm model ensures well-controlled laminar

363

flow and, therefore, a better validation of the data assimilation step. The complexity of the

364

data assimilation steps has been further reduced by using underlying PC-MRI measurement

365

data that have a relatively high resolution, in order to achieve reliable state estimates for

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the data assimilation step. Future studies will systematically decrease the resolution of the

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measurement data to reduce scan times, but at the same time ideally keep reliable state

368

estimates with the help of data assimilation. Finally, as only a single configuration has been

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considered in this initial study, it is obvious that the present findings should be confirmed

370

by later systematic investigations involving a sufficient number of patient-specific vessel

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structures.

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5. Conclusion

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The current study describes an implementation of LETKF to merge PC-MRI measure-

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ment data together with numerical flow simulations of an artificial intracranial aneurysm

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geometry. A unique validation is ensured by high-resolution PIV measurements of the same

376

geometry. Both qualitative and quantitative validation demonstrate that the data assimila-

377

tion step works as a filter and increases the spatial resolution while reducing the noise of the

378

measurements. Conservation laws are enforced by DA onto the measurement data by the

379

Navier-Stokes equations, which increases the physical correctness of the data assimilation

380

results. Using a simplified cardiac forecast model, a forecast step in the classical sense has

381

been introduced to reconstruct the inflow conditions for the ensemble propagation. However,

382

the underlying PC-MRI measurements apparently show a systematic deviation of the true

383

values for high velocities; this systematic bias cannot be improved by LETKF. In order to

384

further improve the data assimilation results, a bias correction of the measurement data is

385

crucial. These findings emphasize the need for an improved quantification of uncertainty

386

for the underlying PC-MRI measurements. In this way, the observation error covariance

387

matrix can also be adjusted to better account for uncertainties in the measurement data.

388

Additionally, an improved model for the cardiac cycle could enhance the forecast of the

389

aneurysm flow in-between available measurement time steps. Finally, uncertain geometric

390

boundaries should be additionally taken into account within the data assimilation step for

391

future applications. Including fluid-structure interaction in the background model for future

392

patient-specific simulations would be a straightforward implementation, as this procedure

393

is already established in hemodynamic CFD simulations. Furthermore, PC-MRI is able to

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detect wall movements, even though the corresponding uncertainty will probably be in the or-

395

der of the spatial resolution. Accordingly, the determination of the true geometric boundary

396

after the assimilation step is one of the main tasks for the use of LETKF in patient-specific

397

applications. One possibility would consist of detecting the zero-contour of the calculated

398

analysis, considering that the near-wall velocities need to be zero.

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To increase the reliability of the data assimilation algorithm a follow-up study will con-

400

sider systematic changes of the spatial and temporal resolution of the underlying measure-

401

ment data. The goal is to enable lower resolution PC-MRI measurements to reduce scan

402

times, while keeping at the same time reliable state estimates thanks to data assimilation.

403

Fine tuning of current data assimilation parameters is also important.

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405

Acknowledgements

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This work was in part conducted within the context of the International Graduate

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School MEMoRIAL at Otto von Guericke University (OVGU) Magdeburg, Germany, kindly

408

supported by the European Structural and Investment Funds (ESF) under the program

409

“Sachsen-Anhalt WISSENSCHAFT Internationalisierung” (project no. ZS/2016/08/80646).

410

Fruitful discussions with the research team of Professor Sebastian Reich (University of Pots-

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dam), in particular with Dr. Sahani Pathiraja, is warmly acknowledged.

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Conflict of interest

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The authors declare no conflict of interest. References

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