#### Algorithm implementation

On constructing a numerical fortitude of a liquid Navier–Stokes equation, we

assume that a quickness margin and a range conditions during a given time present are known. For this initial set of data, a numerical fortitude for a subsequent time

step is constructed, and it converges toward a fortitude when a smoothness equation

is confident (step 7 in Algorithm ۱). Starting with a fortitude for , a same iterative procession is solid to obtain a fortitude for , and so forth. In this manner, a time-dependent upsurge margin is computed.

In liquid upsurge simulations of a Navier–Stokes equations by substantial numerical

routines, a estuary of a transitory SIMPLER iterative procedures is homogeneous to

steady-state SIMPLER calculations applied, until convergence, for any time instant ۴۵]. In other words, elucidate transitory problems regulating SIMPLER is homogeneous to solving

unbroken steady-state problems. Moreover, steady-state calculations might be interpreted

as pseudo-transient solutions with spatially-varying time stairs 45]. In other words, a steady-state solutions are, in practice, liquid solutions,

deliberation a practical time step with bound range conditions and initial data. This proceed has been used by many

new authors 13], 16], 37]–۳۹]. While this was a numerical plan adopted in this work, a proceed proposed

here can also be used for liquid upsurge predictions.

The steady-state fortitude analogous to a given cardiac proviso (a temporal support within a cardiac cycle)

is distributed regulating a MRI-measured estuary and opening velocities for that cardiac

phase. CFD calculations start with an initial theory for , and simulations are carried brazen in time until convergence, i.e.:

(۸)

given a suficiently tiny ( denotes pattern magnitude) and suficiently tiny time step . Note that, here, time *t* is a simulation-only parameter, and is separate to time instants within a cardiac

cycle. It is also not compared with a iteration stairs (*n*) of a SIMPLER algorithm (Algorithm ۱), given a whole algorithm—with multiple

iterations until joining pattern is satisfied—is achieved during any time present *t*, until a joining pattern shown in Eq. ۸ is satisfied. At this point, is obtained. If mixed cardiac phases were to be reconstructed, afterwards would be exclusively distributed for any cardiac phase.

Our doing of a SIMPLER algorithm was certified with a bidimensional lid-driven

form upsurge problem, famous in a novel as a benchmark for contrast CFD algorithms ۴۶]–۴۸]. All algorithms were implemented in Matlab (The MathWorks, Inc., Natick, MA, USA).

Linear systems were solved regulating a biconjugate gradients stabilized method.

#### Proposed numerical solution

In this paper, we solve for a unnatural quickness field, , that is tighten adequate to a MRI-measured pattern margin , and satisfies a liquid dynamics equations, Eqs. (۲) and (3).

Let *M* be a sum series of voxels in a reconstructed 3D quickness field, i.e., , where , , and paint a series of voxels along a *x*, *y*, and *z* axes, respectively. Consider , , and as a built mainstay vectors with a PC-MRI measurements. Since a numerical fortitude of the

Navier–Stokes continuity-equation complement is formed on a fortitude of linear systems,

we introduce that a numerical optimal fortitude is achieved by minimizing, for each

quickness component, during iteration *n*, a following equations:

(۹)

(۱۰)

(۱۱)

The initial tenure on a right palm side of Eqs. (۹)–(۱۱) is compared to a numerical fortitude of a Navier–Stokes smoothness equations,

and a second tenure is compared to a comparison between a numerical fortitude and

a PC-MRI quickness field. The matrices and vectors are tangible in Eqs. (۴)–(۶), though note that we forsaken a “” subscripts for morality and clarity; these are updated by quickness and pressure

values distributed in a prior iteration. Coefficients , , and are regularization factors, that weight consistance with PC-MRI information opposite conformance

with a movement equations. Matrices , , and are of distance , and indication a blurring effects due to calculable k-space coverage in PC-MRI (this is

serve discussed below), while adjusting a series of points on a CFD grid in

sequence to concede a comparison between and . In this approach, a series of grid points in a CFD and MRI grids are not necessarily

a same; we can use a finer grid in CFD than in MRI, for example. The optimal solutions

for Eqs. (۹)–(۱۱) are straightforward ۴۲], and given by

(۱۲)

(۱۳)

(۱۴)

To know a construction of a matrices, cruise that in a deficiency of noise, artifacts, and distortions, the

MRI-measured pattern margin is a confused chronicle of a loyal pattern margin . For a *u* component, for example, we can write:

(۱۵)

where denotes convolution, and blurring heart is a point-spread duty compared with a k-space coverage that was used when

measuring . Similarly, we can write and . If all 3 PC-MRI quickness components are totalled regulating a same k-space coverage,

afterwards . For a 3DFT acquisition, these spatial blurring kernels are equal to

(۱۶)

where , and are a spatial resolutions of along a *x*, *y*, and *z* axis, respectively.

We wish a CFD-estimated pattern margin to be an accurate illustration of a loyal pattern margin . If this is so, afterwards we should design , , and . The discretization of these equations yields 3 linear systems. Then, regulating the

same footnote introduced earlier, for a *n*th iteration of a CFD algorithm, we can write:

(۱۷)

(۱۸)

(۱۹)

The coefficients of , , and are distributed from , , and , respectively. If all 3 PC-MRI quickness components are totalled regulating a same

k-space coverage, and reconstructed onto matching grids, afterwards .

The MRI-guided CFD guess analogous to one cardiac proviso was distributed as a

steady-state fortitude . All 3 components of a PC-MRI quickness margin totalled during a *z* positions during a bounds of a calculation domain were used as estuary and outlet

range conditions for that cardiac phase. Note that this solid state fortitude is a closest fit in a least-squares clarity to a proceed PC-MRI measurements that

infer both movement equation (Eq. ۲) and smoothness equation (Eq. ۳). This is guaranteed by a fact that a optimal solutions Eqs. (۱۲)–(۱۴) are solved in any iteration of a SIMPLER algorithm (steps 2 and 4, in Algorithm ۱),

and by a joining pattern (step 7).

In any of a experiments, all 3 PC-MRI quickness components were totalled using

a same k-space coverage, and reconstructed onto matching grids. In a phantom

experiments, we used a same grid distance for both and , since a haunt information were totalled with high spatial resolution. In these experiments,

was reconstructed but zero-padding, i.e., onto voxels, and a CFD grid points were tangible during a core of any of ’s voxels. Hence, was tangible as an temperament matrix. In a in vivo experiments, was reconstructed regulating 2-fold zero-padding along any of a spatial axes, since

a information was acquired with low spatial resolution. Then, was an symmetric matrix, with coefficients distributed from a indicate widespread duty , tangible in Eq. (۱۶). This gigantic support indicate widespread duty was truncated by mathematics with

a box function

(۲۰)

#### Experimental setup: haunt demonstration

PC-MRI information of a pulsatile carotid upsurge haunt (Phantoms by Design, Inc., Bothell,

WA) (Fig. ۲) were achieved with high spatial fortitude and high signal–to–noise ratio, from

4 time-resolved 3DFT FGRE picture volumes (three acquired any with a quickness encoding

bipolar slope on one of a 3 axes, and one but a bipolar gradient). The

indicate parameters were: mm spatial resolution; FOV cm; TR 11.4 ms; flip angle 8.5; temporal fortitude 91.2 ms; VENC 50 cm/s; 40 min per scan; 9 NEX. The information were

acquired on a GE Discovery MR750 3T complement (50 mT/m and 200 T/m/s max slope amplitude

and slew rate), with a 32-channel receive-only conduct curl array (Nova Medical, Inc.,

Wilmington, MA, USA). The through-slab (*z*) pivot was oriented along a S/I direction. The phantom’s beat cycle was set to

۶۰ bpm.

**Fig. 2.** **a** Pulsatile carotid upsurge haunt (Phantoms by Design, Inc., Bothell, WA, USA) used

to countenance a due method; **b** siphon controller that regulates upsurge bulk and gating signal; **c** atmosphere siphon that generates a upsurge inside a phantom

Only a temporal support analogous to rise upsurge was reconstructed. PC-MRI velocity

member maps and were distributed regulating information from all channels of a accept curl array. The lumen

was segmented by manually surveying a vessel borders from a smoke-stack of 2D axial images,

achieved from a reconstructed 3D volume. A few voxels presented phase-wrap artifacts;

these voxels were manually identified and their velocities were corrected by adding

to their values.

The total solver calculations insincere liquid flexibility of = 0.005 Pa s and firmness of = 1100 kg/m (these values were supposing by a haunt manufacturer). Calculations were performed

with time step = 0.1 ms on a Cartesian grid of mm voxel size.

The CFD make-believe domain was rectangular, of distance . Each iteration compulsory about 10 seconds of mathematics time on an Intel Core i7

processor regulating during 2.8 GHz.

Three unnatural steady-state quickness fields were obtained:

۱٫ regulating a compulsory SIMPLER algorithm, i.e., not regulating a PC-MRI information to constrain

a CFD fortitude ( was used usually as estuary and opening velocities for a geometry);

۲٫ regulating a quickness member compared with a categorical upsurge pivot (*z*) totalled with PC-MRI () to constrain a CFD fortitude (*u* and *v* components were dynamic usually from a liquid production model); and

۳٫ regulating all 3 quickness components totalled with PC-MRI (, , and ) to constrain a CFD solution.

The initial proceed is homogeneous to creation ; in a second approach, we used and ; in a third approach, we used . All 3 approaches used all 3 components of a PC-MRI quickness margin totalled during a *z* positions during a bounds of a calculation domain as estuary and opening boundary

conditions. The series of iterations until joining for a above simulations was

۸۹, ۴۰ and 5 iterations, respectively.

#### Experimental setup: in vivo demonstration

PC-MRI information of a carotid tributary of one healthy proffer were achieved from

4 time-resolved 3DFT FGRE picture volumes (three acquired any with a quickness encoding

bipolar slope on one of a 3 axes, and one but a bipolar gradient). The

indicate parameters were: mm spatial resolution; FOV cm; TR 7.0 ms; flip angle 15; temporal fortitude 56 ms; VENC 160 cm/s; 7 min per scan; 1 NEX. The information were acquired

on a GE Signa 3T EXCITE HD complement (40 mT/m and 150 T/m/s max slope width and

slew rate), with a 4-channel neck accept curl array. The through-slab (*z*) pivot was oriented along a S/I direction. The institutional examination house of the

University of Southern California authorized a imaging protocols. The theme was

screened for MRI risk factors and supposing sensitive agree in suitability with institutional

policy.

Only a cardiac proviso analogous to rise upsurge was reconstructed. PC-MRI velocity

member maps and were distributed regulating information from usually one channel of a accept curl array. Residual

linear quickness offsets in any quickness member map (e.g., due to eddy-currents)

were private by behaving a linear fit to manually tangible 3D regions containing

usually still tissue. The lumen was segmented by manually surveying a vessel borders

from a smoke-stack of 2D axial images, achieved from a reconstructed 3D volume.

The total solver calculations insincere blood flexibility = 0.0032 Pa s and firmness of = 1060 kg/m ۴۹]. Calculations were achieved with time step = 0.25 ms on a Cartesian grid of mm voxel size. The CFD make-believe domain was rectangular, of distance mm (the PC-MRI information was cropped to compare this grid size). Each iteration compulsory about

۱۸۰ s of mathematics time on an Intel Core i7 processor regulating during 2.8 GHz.

Three unnatural steady-state quickness fields were obtained, regulating a same 3 approaches used in a haunt experiment. The

series of iterations until joining for a simulations was 1058, 190 and 6, respectively.

#### Quantitative evaluation

The CFD-simulated quickness fields were quantitatively compared with a PC-MRI measurements

by means of a signal-to-error ratio (SER). The SER measures a ratio between the

appetite of a vigilance and a appetite of a determination error. We deliberate a PC-MRI

quickness field, , as a ground-truth “signal”; consequently, a determination blunder is a pattern difference

between a CFD-estimated quickness field, , and a ground-truth field, . Thus, a SER is distributed (in decibels) as:

(۲۱)

where integers *i*, *j*, and *k* paint grid-point indexes along a *x*, *y*, and *z* axes, respectively. Similarly, a SER was also distributed away for each

of a quickness components, as:

(۲۲)

(۲۳)

(۲۴)

Using these SER values, a 3 CFD approaches—pure CFD, CFD driven by one PC-MRI

quickness component, and CFD driven by all 3 PC-MRI quickness components—were quantitatively

evaluated and compared.

#### Evaluation of denoising properties

Under a hypothesis, CFD simulations yield a smooth, noise-free upsurge field. Therefore,

we design that a due proceed can be used as a denoising resource for PC-MRI

upsurge assessment. In sequence to determine a denoising effects of a total solver,

we combined zero-mean Gaussian sound with customary flaw 8 cm/s to a phantom’s

totalled quickness field, .

This loud upsurge margin was used to constrain a CFD calculations, regulating a approach

in that all 3 quickness components totalled with PC-MRI are used. In this experiment,

we used ; and 4 opposite values of were evaluated: , , , and . The SER between a due proceed and a strange PC-MRI measurements was calculated,

and compared with a SER of a loud upsurge field. The pristine CFD approach, in which

a loud PC-MRI information are used usually as estuary and opening velocities for a geometry,

was also evaluated (this is homogeneous to creation ).

The haunt information was used in this denoising experiment, since it was acquired using

۹ NEX—which formula in high signal-to-noise ratio (SNR), while a in vivo information was

acquired regulating usually 1 NEX. The sound levels on a phantom’s totalled quickness components—estimated

as a customary flaw in regions of uniform meant velocity—are reduce than 3 cm/s;

while a SNR of a bulk images exceeds 26 dB. The velocity-to-noise ratio (VNR) ۵۰], ۵۱] for a *u* and *w* components strech 28 and 31 dB, respectively (the VNR for a *v* member was not calculated, since *v* is approximately nothing over a whole geometry).

Finally, in sequence to clear a denoising experiment, we investigate a sound distribution

in PC-MRI images. We note that, from a limit odds perspective, Eqs. (۹)–(۱۱) assume that a PC-MRI information is degraded by Gaussian noise. Under certain conditions,

one can infer that quickness margin sound in PC-MRI satisfies a zero-mean Gaussian distribution ۵۲]. Therefore, a addition sound behaving on a quickness fields can be insincere to be

Gaussian distributed ۲۷], ۵۲]. Hence, a due minimization is befitting in terms of a MR sound distribution.