Accelerating 4D flow MRI by exploiting low-rank matrix structure and hadamard sparsity

Hemodynamic effects of entry and exit tear size in aortic dissection evaluated with in vitro magnetic resonance imaging and fluid-structure interaction simulation

Understanding the complex interplay between morphologic and hemodynamic features in aortic dissection is critical for risk stratification and for the development of individualized therapy. This work evaluates the effects of entry and exit tear size on the hemodynamics in type B aortic dissection by comparing fluid-structure interaction (FSI) simulations with in vitro 4D-flow magnetic resonance imaging (MRI). A baseline patient-specific 3D-printed model and two variants with modified tear size (smaller entry tear, smaller exit tear) were embedded into a flow- and pressure-controlled setup to perform MRI as well as 12-point catheter-based pressure measurements. The same models defined the wall and fluid domains for FSI simulations, for which boundary conditions were matched with measured data. Results showed exceptionally well matched complex flow patterns between 4D-flow MRI and FSI simulations. Compared to the baseline model, false lumen flow volume decreased with either a smaller entry tear (- 17.8 and - 18.5%, for FSI simulation and 4D-flow MRI, respectively) or smaller exit tear (- 16.0 and - 17.3%). True to false lumen pressure difference (initially 11.0 and 7.9 mmHg, for FSI simulation and catheter-based pressure measurements, respectively) increased with a smaller entry tear (28.9 and 14.6 mmHg), and became negative with a smaller exit tear (- 20.6 and - 13.2 mmHg). This work establishes quantitative and qualitative effects of entry or exit tear size on hemodynamics in aortic dissection, with particularly notable impact observed on FL pressurization. FSI simulations demonstrate acceptable qualitative and quantitative agreement with flow imaging, supporting its deployment in clinical studies.

Low-Rank Matrix Denoising Algorithm-Based MRI Image Feature for Therapeutic Effect Evaluation of NCRT on Rectal Cancer

This study aimed to explore the therapeutic effects of neoadjuvant chemoradiotherapy (NCRT) on rectal cancer patients using the MRI based on low-rank matrix denoising algorithm, which was then compared with the postoperative pathological examination to evaluate its application value in tumor staging after NCRT treatment. 15 patients with rectal cancer who met the requirements of radiotherapy and chemotherapy after conventional MRI were selected as the research subjects. The conventional MRI images before and after NCRT treatment were divided in two groups. One group was not processed and set as the conventional group; the other group was processed with low-rank matrix denoising algorithm and set as the optimized group. The two groups of images were observed for the changes in the ADC value and length and thickness of the tumor before and after NCRT treatment. The two groups were compared with the pathological examination for the complete remission of pathology (pCR) after the NCRT treatment and the tumor stage results. The results showed that Root Mean Square Error (RMSE) and Peak Signal to Noise Ratio (PSNR) (18.9121 and 74.9911 dB) after introducing the low-rank matrix denoising algorithm were significantly better than those before (20.1234 and 70.1234 dB) (P < 0.05); there were notable differences in the tumor index data within the two groups before and after NCRT treatment (P < 0.05), indicating that the NCRT treatment was effective. The pathological examination results of pCR data of the two groups were not much different (P > 0.05); the examination results between the two groups were different, but no notable difference was noted (P < 0.05); in the optimized group, there was no notable difference between the MRI results and the pathological examination results (P < 0.05), while in the conventional group, there were notable differences in the MRI results and pathological examination results (P < 0.05). In conclusion, MRI images based on low-rank matrix denoising algorithm are clearer, which can improve the diagnosis rate of patients and better display the changes of the microenvironment after NCRT treatment. It also indicates that NCRT treatment has significant clinical effects in the treatment of rectal cancer patients, which is worth promoting.

On the impact of vessel wall stiffness on quantitative flow dynamics in a synthetic model of the thoracic aorta

Aortic wall stiffening is a predictive marker for morbidity in hypertensive patients. Arterial pulse wave velocity (PWV) correlates with the level of stiffness and can be derived using non-invasive 4D-flow magnetic resonance imaging (MRI). The objectives of this study were twofold: to develop subject-specific thoracic aorta models embedded into an MRI-compatible flow circuit operating under controlled physiological conditions; and to evaluate how a range of aortic wall stiffness impacts 4D-flow-based quantification of hemodynamics, particularly PWV. Three aorta models were 3D-printed using a novel photopolymer material at two compliant and one nearly rigid stiffnesses and characterized via tensile testing. Luminal pressure and 4D-flow MRI data were acquired for each model and cross-sectional net flow, peak velocities, and PWV were measured. In addition, the confounding effect of temporal resolution on all metrics was evaluated. Stiffer models resulted in increased systolic pressures (112, 116, and 133 mmHg), variations in velocity patterns, and increased peak velocities, peak flow rate, and PWV (5.8–7.3 m/s). Lower temporal resolution (20 ms down to 62.5 ms per image frame) impacted estimates of peak velocity and PWV (7.31 down to 4.77 m/s). Using compliant aorta models is essential to produce realistic flow dynamics and conditions that recapitulated in vivo hemodynamics.

Accelerated aortic 4D flow MRI with wave-CAIPI

PURPOSE

The aim of this study was to investigate the acceleration potential of wave-CAIPI (controlled aliasing in parallel imaging) for 4D flow MRI, provided that image quality and precision of flow parameters are maintained.

METHODS

The 4D flow MRIs with acceleration factor R = 2 were performed on 10 healthy volunteers, using both wave-CAIPI and standard Cartesian/2D-CAIPI sampling for reference. In addition, 1 patient with known aortic valve stenosis was examined. The flow rate ( Q ), net flow ( Q net ), peak velocity v max , and net average through-plane velocity ( v ¯ ⊥ ) were calculated in eight analysis planes in the ascending and descending aorta. The acquisitions were retrospectively undersampled (R = 6), and deviations of flow parameters and hemodynamic flow patterns were evaluated.

RESULTS

Flow parameters measured with an undersampled wave-CAIPI trajectory showed considerably smaller deviations to the references than the 2D-CAIPI images. For v max , the mean absolute differences were 6.02 ± 2.08 cm/s versus 14.36 ± 5.68 cm/s; for Q net , the mean absolute differences were 3.67 ± 1.40 ml versus 5.87 ± 1.91 ml for wave-CAIPI versus 2D-CAIPI, respectively. Noise calculations indicate that the 2D-CAIPI sampling exhibits a 43 ± 38 % higher average noise level than the wave-CAIPI technique. Qualitative discrepancies in hemodynamic flow patterns, visualized through streamlines, particle traces and flow velocity vectors, could be reduced by using the undersampled wave-CAIPI trajectory.

CONCLUSION

Use of wave-CAIPI instead of 2D-CAIPI sampling in retrospectively 6-fold accelerated 4D flow MRI enhances the precision of flow parameters. The acquisition time of 4D flow measurements could be reduced by a factor of 3, with minimal differences in flow parameters.

Highly accelerated aortic 4D flow MRI using compressed sensing: Performance at different acceleration factors in patients with aortic disease

PURPOSE

To systematically assess the feasibility and performance of a highly accelerated compressed sensing (CS) 4D flow MRI framework at three different acceleration factors (R) for the quantification of aortic flow dynamics and wall shear stress (WSS) in patients with aortic disease.

METHODS

Twenty patients with aortic disease (58 ± 15 y old; 19 M) underwent four 4D flow scans: one conventional (GRAPPA, R = 2) and three CS 4D flows with R = 5.7, 7.7, and 10.2. All scans were acquired with otherwise equivalent imaging parameters on a 1.5T scanner. Peak-systolic velocity (V_max ), peak flow (Q_max ), and net flow (Q_net ) were quantified at the ascending aorta (AAo), arch, and descending aorta (DAo). WSS was calculated at six regions within the AAo and arch.

RESULTS

Mean scan times for the conventional and CS 4D flows with R = 5.7, 7.7, and 10.2 were 9:58 ± 2:58 min, 3:40 ± 1:19 min, 2:50 ± 0:56 min, and 2:05 ± 0:42 min, respectively. V_max , Q_max , and Q_net were significantly underestimated by all CS protocols (underestimation ≤ -7%, -9%, and -10% by CS, R = 5.7, 7.7, and 10.2, respectively). WSS measurements showed the highest underestimation by all CS protocols (underestimation ≤ -9%, -12%, and -14% by CS, R = 5.7, 7.7, and 10.2).

CONCLUSIONS

Highly accelerated aortic CS 4D flow at R = 5.7, 7.7, and 10.2 showed moderate agreement with the conventional 4D flow, despite systematically underestimating various hemodynamic parameters. The shortened scan time may enable the clinical translation of CS 4D flow, although potential hemodynamic underestimation should be considered when interpreting the results.

Pseudo-spiral sampling and compressed sensing reconstruction provides flexibility of temporal resolution in accelerated aortic 4D flow MRI: A comparison with k-t principal component analysis

INTRODUCTION

Time-resolved three-dimensional phase contrast MRI (4D flow) of aortic blood flow requires acceleration to reduce scan time. Two established techniques for highly accelerated 4D flow MRI are k-t principal component analysis (k-t PCA) and compressed sensing (CS), which employ either regular or random k-space undersampling. The goal of this study was to gain insights into the quantitative differences between k-t PCA- and CS-derived aortic blood flow, especially for high temporal resolution CS 4D flow MRI.

METHODS

The scan protocol consisted of both k-t PCA and CS accelerated 4D flow MRI, as well as a 2D flow reference scan through the ascending aorta acquired in 15 subjects. 4D flow scans were accelerated with factor R = 8. For CS accelerated scans, we used a pseudo-spiral Cartesian sampling scheme, which could additionally be reconstructed at higher temporal resolution, resulting in R = 13. 4D flow data were compared with the 2D flow scan in terms of flow, peak flow and stroke volume. A 3D peak systolic voxel-wise velocity and wall shear stress (WSS) comparison between k-t PCA and CS 4D flow was also performed.

RESULTS

The mean difference in flow/peak flow/stroke volume between the 2D flow scan and the 4D flow CS with R = 8 and R = 13 was 4.2%/9.1%/3.0% and 5.3%/7.1%/1.9%, respectively, whereas for k-t PCA with R = 8 the difference was 9.7%/25.8%/10.4%. In the voxel-by-voxel 4D flow comparison we found 13.6% and 3.5% lower velocity and WSS values of k-t PCA compared with CS with R = 8, and 15.9% and 5.5% lower velocity and WSS values of k-t PCA compared with CS with R = 13.

CONCLUSION

Pseudo-spiral accelerated 4D flow acquisitions in combination with CS reconstruction provides a flexible choice of temporal resolution. We showed that our proposed strategy achieves better agreement in flow values with 2D reference scans compared with using k-t PCA accelerated acquisitions.

On the limitations of echo planar 4D flow MRI

PURPOSE

To compare EPI and GRE readout in high-flow velocity regimes and evaluate their impact on measurement accuracy in silico and in vitro.

THEORY AND METHODS

Phase-contrast sequences for EPI and GRE were simulated using CFD velocity data to assess displacement artifacts as well as effective spatial resolution. In silico findings were validated experimentally using a steady flow phantom.

RESULTS

For EPI factor 5 and simulated stenotic flow with peak velocity of 2.2 m s - 1 , displacement artifacts resulted in misregistration of 7.3 mm at echo time and the effective resolution was locally reduced by factors 5 and 8 compared to GRE for flow along phase and frequency encoding directions, respectively. In vitro, a maximum velocity difference between EPI factor 5 and GRE of 0.97 m s - 1 was found.

CONCLUSIONS

Four-dimensional flow MRI using EPI readout results not only in considerable velocity misregistration but also in spatially varying degradation of resolution. The proposed work indicates that EPI is inferior to standard GRE for 4D flow MRI.

Highly accelerated 4D flow cardiovascular magnetic resonance using a pseudo-spiral Cartesian acquisition and compressed sensing reconstruction for carotid flow and wall shear stress

BACKGROUND

4D flow cardiovascular magnetic resonance (CMR) enables visualization of complex blood flow and quantification of biomarkers for vessel wall disease, such as wall shear stress (WSS). Because of the inherently long acquisition times, many efforts have been made to accelerate 4D flow acquisitions, however, no detailed analysis has been made on the effect of Cartesian compressed sensing accelerated 4D flow CMR at different undersampling rates on quantitative flow parameters and WSS.

METHODS

We implemented a retrospectively triggered 4D flow CMR acquisition with pseudo-spiral Cartesian k-space filling, which results in incoherent undersampling of k-t space. Additionally, this strategy leads to small jumps in k-space thereby minimizing eddy current related artifacts. The pseudo-spirals were rotated in a tiny golden-angle fashion, which provides optimal incoherence and a variable density sampling pattern with a fully sampled center. We evaluated this 4D flow protocol in a carotid flow phantom with accelerations of R = 2-20, as well as in carotids of 7 healthy subjects (27 ± 2 years, 4 male) for R = 10-30. Fully sampled 2D flow CMR served as a flow reference. Arteries were manually segmented and registered to enable voxel-wise comparisons of both velocity and WSS using a Bland-Altman analysis.

RESULTS

Magnitude images, velocity images, and pathline reconstructions from phantom and in vivo scans were similar for all accelerations. For the phantom data, mean differences at peak systole for the entire vessel volume in comparison to R = 2 ranged from - 2.3 to - 5.3% (WSS) and - 2.4 to - 2.2% (velocity) for acceleration factors R = 4-20. For the in vivo data, mean differences for the entire vessel volume at peak systole in comparison to R = 10 were - 9.9, - 13.4, and - 16.9% (WSS) and - 8.4, - 10.8, and - 14.0% (velocity), for R = 20, 25, and 30, respectively. Compared to single slice 2D flow CMR acquisitions, peak systolic flow rates of the phantom showed no differences, whereas peak systolic flow rates in the carotid artery in vivo became increasingly underestimated with increasing acceleration.

CONCLUSION

Acquisition of 4D flow CMR of the carotid arteries can be highly accelerated by pseudo-spiral k-space sampling and compressed sensing reconstruction, with consistent data quality facilitating velocity pathline reconstructions, as well as quantitative flow rate and WSS estimations. At an acceleration factor of R = 20 the underestimation of peak velocity and peak WSS was acceptable (< 10%) in comparison to an R = 10 accelerated 4D flow CMR reference scan. Peak flow rates were underestimated in comparison with 2D flow CMR and decreased systematically with higher acceleration factors.

Accelerated aortic 4D flow cardiovascular magnetic resonance using compressed sensing: applicability, validation and clinical integration

BACKGROUND

Three-dimensional time-resolved phase-contrast cardiovascular magnetic resonance (4D flow CMR) enables the quantification and visualisation of blood flow, but its clinical applicability remains hampered by its long scan time. The aim of this study was to evaluate the use of compressed sensing (CS) with on-line reconstruction to accelerate the acquisition and reconstruction of 4D flow CMR of the thoracic aorta.

METHODS

4D flow CMR of the thoracic aorta was acquired in 20 healthy subjects using CS with acceleration factors ranging from 4 to 10. As a reference, conventional parallel imaging (SENSE) with acceleration factor 2 was used. Flow curves, net flows, peak flows and peak velocities were extracted from six contours along the aorta. To measure internal data consistency, a quantitative particle trace analysis was performed. Additionally, scan-rescan, inter- and intraobserver reproducibility were assessed. Subsequently, 4D flow CMR with CS factor 6 was acquired in 3 patients with differing aortopathies. The flow patterns resulting from particle trace visualisation were qualitatively analysed.

RESULTS

All collected data were successfully acquired and reconstructed on-line. The average acquisition time including respiratory navigator efficiency with CS factor 6 was 5:02 ± 2:23 min while reconstruction took approximately 9 min. For CS factors of 8 or less, mean differences in net flow, peak flow and peak velocity as compared to SENSE were below 2.2 ± 7.8 ml/cycle, 4.6 ± 25.2 ml/s and - 7.9 ± 13.0 cm/s, respectively. For a CS factor of 10 differences reached 5.4 ± 8.0 ml/cycle, 14.4 ± 28.3 ml/s and - 4.0 ± 12.2 cm/s. Scan-rescan analysis yielded mean differences in net flow of - 0.7 ± 4.9 ml/cycle for SENSE and - 0.2 ± 8.5 ml/cycle for CS factor of 6.

CONCLUSIONS

A six- to eightfold acceleration of 4D flow CMR using CS is feasible. Up to a CS acceleration rate of 6, no statistically significant differences in measured flow parameters could be observed with respect to the reference technique. Acquisitions in patients with aortopathies confirm the potential to integrate the proposed method in a clinical routine setting, whereby its main benefits are scan-time savings and direct on-line reconstruction.

Multipoint 5D flow cardiovascular magnetic resonance - accelerated cardiac- and respiratory-motion resolved mapping of mean and turbulent velocities

BACKGROUND

Volumetric quantification of mean and fluctuating velocity components of transient and turbulent flows promises a comprehensive characterization of valvular and aortic flow characteristics. Data acquisition using standard navigator-gated 4D Flow cardiovascular magnetic resonance (CMR) is time-consuming and actual scan times depend on the breathing pattern of the subject, limiting the applicability of the method in a clinical setting. We sought to develop a 5D Flow CMR framework which combines undersampled data acquisition including multipoint velocity encoding with low-rank image reconstruction to provide cardiac- and respiratory-motion resolved assessment of velocity maps and turbulent kinetic energy in fixed scan times.

METHODS

Data acquisition and data-driven motion state detection was performed using an undersampled Cartesian tiny Golden angle approach. Locally low-rank (LLR) reconstruction was implemented to exploit correlations among heart phases and respiratory motion states. To ensure accurate quantification of mean and turbulent velocities, a multipoint encoding scheme with two velocity encodings per direction was incorporated. Velocity-vector fields and turbulent kinetic energy (TKE) were obtained using a Bayesian approach maximizing the posterior probability given the measured data. The scan time of 5D Flow CMR was set to 4 min. 5D Flow CMR with acceleration factors of 19 .0 ± 0.21 (mean ± std) and velocity encodings (VENC) of 0.5 m/s and 1.5 m/s per axis was compared to navigator-gated 2x SENSE accelerated 4D Flow CMR with VENC = 1.5 m/s in 9 subjects. Peak velocities and peak flow were compared and magnitude images, velocity and TKE maps were assessed.

RESULTS

While net scan time of 5D Flow CMR was 4 min independent of individual breathing patterns, the scan times of the standard 4D Flow CMR protocol varied depending on the actual navigator gating efficiency and were 17.8 ± 3.9 min on average. Velocity vector fields derived from 5D Flow CMR in the end-expiratory state agreed well with data obtained from the navigated 4D protocol (normalized root-mean-square error 8.9 ± 2.1%). On average, peak velocities assessed with 5D Flow CMR were higher than for the 4D protocol (3.1 ± 4.4%).

CONCLUSIONS

Respiratory-motion resolved multipoint 5D Flow CMR allows mapping of mean and turbulent velocities in the aorta in 4 min.

Two-Minute k-Space and Time-accelerated Aortic Four-dimensional Flow MRI: Dual-Center Study of Feasibility and Impact on Velocity and Wall Shear Stress Quantification

Purpose

To investigate the two-center feasibility of highly k-space and time (k-t)-accelerated 2-minute aortic four-dimensional (4D) flow MRI and to evaluate its performance for the quantification of velocities and wall shear stress (WSS).

Materials and Methods

This cross-sectional study prospectively included 68 participants (center 1, 11 healthy volunteers [mean age ± standard deviation, 61 years ± 15] and 16 patients with aortic disease [mean age, 60 years ± 10]; center 2, 14 healthy volunteers [mean age, 38 years ± 13] and 27 patients with aortic or cardiac disease [mean age, 78 years ± 18]). Each participant underwent highly accelerated 4D flow MRI (k-t acceleration, acceleration factor of 5) of the thoracic aorta. For comparison, conventional 4D flow MRI (acceleration factor of 2) was acquired in the participants at center 1 (n = 27). Regional aortic peak systolic velocities and three-dimensional WSS were quantified.

Results

k-t-accelerated scan times (center 1, 2:03 minutes ± 0:29; center 2, 2:06 minutes ± 0:20) were significantly reduced compared with conventional 4D flow MRI (center 1, 12:38 minutes ± 2:25; P < .0001). Overall good agreement was found between the two techniques (absolute differences ≤15%), but proximal aortic WSS was significantly underestimated in patients by using k-t-accelerated 4D flow when compared with conventional 4D flow (P ≤ .03). k-t-accelerated 4D flow MRI was reproducible (intra- and interobserver intraclass correlation coefficient ≥0.98) and identified significantly increased peak velocities and WSS in patients with stenotic (P ≤ .003) or bicuspid (P ≤ .04) aortic valves compared with healthy volunteers. In addition, k-t-accelerated 4D flow MRI-derived velocities and WSS were inversely related to age (r ≥-0.53; P ≤ .03) over all healthy volunteers.

Conclusion

k-t-accelerated aortic 4D flow MRI providing 2-minute scan times was feasible and reproducible at two centers. Although consistent healthy aging- and disease-related changes in aortic hemodynamics were observed, care should be taken when considering WSS, which can be underestimated in patients.© RSNA, 2019See also the commentary by François in this issue.

On the limitations of partial Fourier acquisition in phase-contrast MRI of turbulent kinetic energy

PURPOSE

To investigate limitations of partial Fourier acquisition in phase-contrast MRI of turbulent kinetic energy (TKE).

METHODS

To assess the validity of partial Fourier reconstruction of TKE and phase images, computational fluid dynamics data of mean and turbulent velocities in a stenotic U-bend phantom was used. Partial Fourier acquisition with 75% k-space coverage was simulated and TKE data were reconstructed using zero-filling, homodyne reconstruction, and the method of projections onto convex sets (POCS). Results were compared to data from fully sampled k-space and 75% symmetric sampling. In addition, compressed sensing (CS) reconstruction was compared for a standard variable density sampling pattern and a variable density sampling pattern combined with 75% partial Fourier. For illustration purposes, in vivo examples of velocity magnitude and TKE maps of aortic flow reconstructed with the different methods are provided.

RESULTS

In accordance with theory, partial Fourier reconstruction of TKE maps from phase-contrast data results in artifacts relative to fully sampled data. It is demonstrated that neither homodyne reconstruction nor POCS can improve reconstruction of TKE data with respect to zero-filling reconstruction when compared to ground-truth (RMS error: 4.70%, 4.34%, and 2.45% for homodyne, POCS, and zero-filling reconstruction of in vivo data, respectively). CS reconstruction from data acquired with partial Fourier did not recover the resolution loss incurred by partial Fourier sampling.

CONCLUSION

Partial Fourier reconstruction of TKE maps from phase-contrast data does not yield a benefit over zero-filling reconstruction. In consequence, symmetric sampling is preferred over partial Fourier acquisition for a given number of phase-encodes in phase-contrast MRI.

Phase Contrast Magnetic Resonance Imaging in the Rat Common Carotid Artery

Phase contrast magnetic resonance imaging (PC-MRI) is a noninvasive approach that can quantify flow-related parameters such as blood flow. Previous studies have shown that abnormal blood flow may be associated with systemic vascular risk. Thus, PC-MRI can facilitate the translation of data obtained from animal models of cardiovascular diseases to pertinent clinical investigations. In this report, we describe the procedure for measuring blood flow in the common carotid artery (CCA) of rats using cine-gated PC-MRI and discuss relevant analysis methods. This procedure can be performed in a live, anesthetized animal and does not require euthanasia after the procedure. The proposed scanning parameters yield repeatable measurements for blood flow, indicating excellent reproducibility of the results. The PC-MRI procedure described in this article can be used for pharmacological testing, pathophysiological assessment, and cerebral hemodynamics evaluation.

Accelerated Cardiac Diffusion Tensor Imaging Using Joint Low-Rank and Sparsity Constraints

OBJECTIVE

The purpose of this paper is to accelerate cardiac diffusion tensor imaging (CDTI) by integrating low-rankness and compressed sensing.

METHODS

Diffusion-weighted images exhibit both transform sparsity and low-rankness. These properties can jointly be exploited to accelerate CDTI, especially when a phase map is applied to correct for the phase inconsistency across diffusion directions, thereby enhancing low-rankness. The proposed method is evaluated both ex vivo and in vivo, and is compared to methods using either a low-rank or sparsity constraint alone.

RESULTS

Compared to using a low-rank or sparsity constraint alone, the proposed method preserves more accurate helix angle features, the transmural continuum across the myocardium wall, and mean diffusivity at higher acceleration, while yielding significantly lower bias and higher intraclass correlation coefficient.

CONCLUSION

Low-rankness and compressed sensing together facilitate acceleration for both ex vivo and in vivo CDTI, improving reconstruction accuracy compared to employing either constraint alone.

SIGNIFICANCE

Compared to previous methods for accelerating CDTI, the proposed method has the potential to reach higher acceleration while preserving myofiber architecture features, which may allow more spatial coverage, higher spatial resolution, and shorter temporal footprint in the future.