[1,3]Rizwan Ahmad

1]Biomedical Engineering, Ohio State University, Columbus, OH, USA 2]Davis
Heart and Lung Research Institute, Ohio State University Wexner Medical
Center, Columbus, OH, USA 3]Electrical and Computer Engineering, Ohio State
University, Columbus, OH, USA 4]Kinesiology, Ohio State University,
Columbus,OH, USA 5]Cardiovascular Medicine, Ohio State University Wexner
Medical Center, Columbus, OH, USA

# Accelerated Real-time Cine and Flow under In-magnet Staged Exercise

Preethi Chandrasekaran<EMAIL_ADDRESS>Chong Chen
<EMAIL_ADDRESS>Yingmin Liu<EMAIL_ADDRESS>Syed Murtaza Arshad
<EMAIL_ADDRESS>Christopher Crabtree<EMAIL_ADDRESS>Matthew Tong
<EMAIL_ADDRESS>Yuchi Han<EMAIL_ADDRESS><EMAIL_ADDRESS>[  [  [
[  [

###### Abstract

Background: Cardiovascular magnetic resonance imaging (CMR) is a well-
established imaging tool for diagnosing and managing cardiac conditions. The
integration of exercise stress with CMR (ExCMR) can enhance its diagnostic
capacity. Despite recent advances in CMR technology, ExCMR remains technically
challenging due to motion artifacts and limited spatial and temporal
resolution.

Methods: This study investigates the feasibility of biventricular functional
and hemodynamic assessment using real-time (RT) ExCMR during a staged exercise
protocol in 26 healthy volunteers. We introduce a coil reweighting technique
to minimize motion artifacts. In addition, we identify and analyze heartbeats
from the end-expiratory phase to enhance the repeatability of cardiac function
quantification. To demonstrate clinical feasibility, qualitative results from
five patients are also presented.

Results: Our findings indicate a consistent decrease in end-systolic volume
(ESV) and stable end-diastolic volume (EDV) across exercise intensities,
leading to increased stroke volume (SV) and ejection fraction (EF). Coil
reweighting effectively reduces motion artifacts, improving image quality in
both healthy volunteers and patients. The repeatability of cardiac function
parameters, demonstrated by scan-rescan tests in nine volunteers, improves
with the selection of end-expiratory beats.

Conclusions: The study demonstrates that RT ExCMR with in-magnet exercise is a
feasible and effective method for dynamic cardiac function monitoring during
exercise. The proposed coil reweighting technique and selection of end-
expiratory beats significantly enhance image quality and repeatability.

###### keywords:

cardiac MRI, exercise stress, accelerated imaging

##  1 Introduction

Cardiovascular magnetic resonance imaging (CMR) is an established tool with
proven diagnostic and prognostic value. It is considered a gold standard for
biventricular volume quantification and detecting myocardial scar [1]. When
paired with pharmacological or exercise stress, CMR-based first-pass perfusion
can accurately diagnose coronary artery disease [2, 3] by identifying
myocardial ischemia [4]. More recently, cardiac stress imaging has emerged as
an attractive option to investigate conditions other than ischemic heart
disease. For instance, the myocardial contractile reserve’s response to stress
is known to be a reliable prognostic indicator in conditions such as
cardiomyopathies and heart failure [5, 6].

Stress CMR investigations commonly involve the use of vasodilators or
inotropic agents [7], which fail to mimic the hemodynamic alterations or
provide the prognostic data associated with physical exercise [8, 9]. Due to
its ability to correlate symptoms (e.g., shortness of breath) and functional
capacity with workload, CMR with exercise stress (ExCMR) is able to offer a
more comprehensive and dynamic assessment of cardiovascular function. In
pulmonary artery hypertension, for example, ExCMR has shown a superior ability
to capture dynamic changes in the right heart than any other modality [10].

Several studies have demonstrated the use of CMR immediately after in-room
treadmill exercise [11]. In multiple clinical trials, this strategy has
demonstrated success in detecting myocardial ischemia [12]. However, the need
to transfer the patient from the treadmill to the MRI scanner creates a time
delay between peak exercise and imaging, and MRI-compatible treadmills are not
readily available. In contrast, the commercially available ergometers that
enable supine exercise on the MRI patient table are more desirable as they
enable simultaneous in-magnet exercise and imaging.

With the recent advances in highly accelerated real-time (RT) imaging, ExCMR
with in-magnet exercise has become an increasingly feasible and informative
tool for assessing cardiac function under stress conditions. In 2009, Lurz et
al. demonstrated the feasibility of accelerated RT ExCMR in 12 healthy
subjects [13]. More recently, several other studies have shown the feasibility
of RT ExCMR in small cohorts of healthy subjects or patients [14, 15] and the
benefit of applying advanced processing techniques [16, 17]. Despite these
advances, biventricular functional and hemodynamic assessment with ExCMR
remains technically challenging [18]. In particular, motion artifacts,
through-plane motion, and high acceleration rates can render RT ExCMR images
difficult to interpret and analyze, potentially affecting the diagnostic
accuracy and overall utility of this technique in clinical and research
settings.

In this study, we demonstrate the feasibility of biventricular hemodynamic and
functional assessment of the heart from highly accelerated RT cine and flow
data collected during a staged exercise protocol from 26 healthy subjects. To
improve image quality, we introduce a coil reweighting technique that
effectively suppresses motion artifacts due to the torso movement. To suppress
the impact of respiratory motion on cardiac function assessment, heartbeats
from the end-expiratory phase are isolated and analyzed [19], resulting in
improved repeatability of imaging parameters. To demonstrate the clinical
applicability of RT ExCMR, qualitative results from five patients are also
presented.

##  2 Methods

###  2.1 Data acquisition

Twenty-eight healthy volunteers over the age of 18 were recruited to
participate in an institutional review board-approved ExCMR study. Written
informed consent was obtained from each subject. Data from two volunteers were
excluded owing to their non-compliance with instructions. Twenty-six
volunteers ($17$ male, $9$ female; age = $29\pm 7$ years; $\text{BMI}=24.6\pm
4.1$ kg/m2; $\text{BSA}=1.41\pm 0.21$ m2) were imaged on a 3T scanner
(MAGNETOM Vida, Siemens Healthcare, Erlangen, Germany) fitted with an in-
magnet supine ergometer (MR Ergometer Pedal, Lode, The Netherlands). The
imaging protocol included a free-breathing RT short-axis cine stack (11 to 14
slices) covering the whole heart and at least two long-axis cine slices.
Additionally, it comprised two sets of three closely spaced phase-contrast CMR
(PC-CMR) slices to measure flow at the root of the aortic and pulmonic
arteries, as well as 4D flow with whole-heart coverage. The data for cine were
collected using balanced steady-state free-precession (bSSFP), while the data
for PC-CMR and 4D flow were collected using gradient echo (GRE) sequences. For
cine, the data were collected using a pseudo-random Cartesian sampling
pattern, called GRO [20]. For flow, the data were collected using a different
Cartesian sampling pattern, called CAVA [21]. The protocol used to collect
data is shown in Figure 1, and the imaging parameters for RT cine and flow are
summarized in Table 1.

Parameter/Sequence | RT Cine | RT Flow
---|---|---
Sequence | bSSFP | GRE
Acquisition time (s/slice) | $3-6$ | $3-6$
Acceleration rate | $8-9$ | $16-19$
Repetition Time (ms) | $2.60-3.16$ | $3.58$
Echo Time (ms) | $1.10-1.42$ | $2.13$
Spatial Resolution (mm2) | $(1.67-2.40)\times(1.67-2.40)$ | $(2.0-2.5)\times(2.0-2.5)$
Temporal Resolution (ms) | $34.8-41.6$ | $35.8-42.9$
Flip Angle (deg) | $24-45$ | $10$
Slice thickness (mm) | $6$ | $6$
Sampling pattern | Cartesian [20] | Cartesian [21]
VENC (cm/s) | N/A |  $100-150$ (rest), $200-300$ (stress)
Table 1: MRI acquisition parameters for RT cine and RT flow sequences.

After data acquisition at rest, the volunteers were asked to pedal the
ergometer at a rate of 60 cycles per minute by matching their cadence to a
metronome played through the headphones. The initial workload was set to $20$
W and was increased in increments of $20$ W to the maximum of $60$ W. The same
set of sequences was executed at each exercise stage after allowing a ramp-up
time of one minute for the heart rate (HR) to stabilize. Each exercise stage
lasted for 8 to 10 minutes. Due to exhaustion or leg fatigue, not all subjects
were able to complete all three exercise stages. In total, we have $n=26$,
$n=25$, $n=24$, and $n=17$ volunteers at rest, $20$ W, $40$ W, and $60$ W,
respectively. In one of the volunteers, an incidental finding during the
resting stage of the protocol prompted termination of the experiment after the
resting stage. For a subset of volunteers, we repeated RT cine and flow
acquisitions in quick succession (within 10 minutes) at rest ($n=11$) and at
the $40$ W exercise stage ($n=9$) to assess repeatability. This study is
focused on the analysis of RT cine and flow images to track hemodynamic
response to exercise. The data from the 4D flow acquisition are being
processed and will be reported separately [22].

Upon completing the scan, volunteers were requested to assess their level of
exertion during each stage of the exercise using the Borg rating of perceived
exertion (BRPE) scale [23]. This scale categorizes exertion into four distinct
levels: “no to extremely light exertion” (scores $6–8$), “light exertion”
(scores $9–12$), “somewhat hard to hard” (scores $13–16$), and “very hard to
maximal exertion” (scores $17–20$). It is important to note that the scale’s
range from $6$ to $20$, when multiplied by 10, corresponds closely with HR
measurements, underscoring its effectiveness in quantifying exertion levels.

After completing the healthy subject study, five patient volunteers ($2$ male,
$3$ female; age = $59\pm 15$ years; $\text{BMI}=31.3\pm 8$ kg/m2;
$\text{BSA}=1.8\pm 0.2$ m2) were also imaged. In contrast to the healthy
subjects, a $10$ W workload increment was employed when the $20$ W increase
was considered excessively challenging by the patients. The maximum exercise
intensity achieved by the five patients was $60$ W, $30$ W, $40$ W, $30$ W,
and $20$ W.

###  2.2 Image reconstruction

The image reconstruction for RT cine and flow was performed inline with
Gadgetron-based [24] implementation of SCoRe [25], which is a parameter-free
compressive sensing method. Coil sensitivity maps were estimated from the
temporally averaged k-space using ESPIRiT [26]. In the case of RT cine, three-
dimensional undecimated wavelet transform (UWT) was applied to sparsify both
the spatial and temporal dimensions. For RT flow, in addition to UWT, temporal
principal components [27], inferred after stacking flow-encoded and flow-
compensated images, were used as data-driven sparsifying transform. The
reconstruction was performed inline on a dedicated GPU workstation equipped
with NVIDIA GeForce RTX 3090. For a 6 s per slice acquisition, the
reconstruction time was 10 seconds per slice for RT cine and 15 seconds per
slice for RT flow.

Although subjects were instructed to firmly hold onto the handles connected to
the table’s rails, periodic bulk motion was observed during paddling in almost
all cases. For some subjects, this resulted in significant degradation of
image quality due to the temporally varying sensitivity maps of the body coil
array. We attempted to use dynamic sensitivity maps, estimated from a sliding
window of a small number of frames, but this approach was unsuccessful.
Specifically, a smaller window (less than 8 frames) did not yield a fully
sampled region of sufficient size or quality for reliable estimation of coil
sensitivity maps, while a larger window (greater than 8 frames) introduced
artifacts similar to those from time-averaged sensitivity maps. To mitigate
these artifacts, we employed our recently proposed coil reweighting method
[28], designed to automatically suppress contributions from artifact-inducing
coil elements. This preprocessing step is computationally efficient and can be
integrated with any reconstruction method.

###  2.3 Cardiac function quantification

For both RT cine and flow imaging, multiple heartbeats were collected from
each slice. The electrocardiogram (ECG) signal facilitated the segregation of
frames into individual heartbeats. Arrhythmic heartbeats, defined as those
deviating from the mean R-wave to R-wave interval by more than 15%, were
excluded. Several studies have highlighted the impact of respiration-induced
variation in cardiac output quantification, which is attributed to changes in
intrathoracic pressure and through-plane motion [29]. This variation is
expected to be more pronounced during exercise. To minimize the impact of
respiratory motion on cardiac function assessment, heartbeats from the RT cine
and flow captured during the end-expiratory phase were specifically isolated
for analysis. The respiratory signal, critical for identifying this phase, was
derived from RT images using a recently proposed principal component analysis-
based method [19]. In cases involving RT images captured during exercise, the
ECG signal proved unreliable. As illustrated in Supporting Movie S1, the
respiratory signal from [19] was superimposed at the bottom of the exercise RT
image series. A white cross marked the respiratory location for each frame.
This signal served as a reference to manually identify systolic and diastolic
frames from the end-expiratory phase. The steps of isolating the end-
expiratory beat (for rest images) and superimposing the respiratory signal
(for exercise images) were performed offline.

The DICOM images from the short-axis cine and flow acquisitions from the
ascending aorta (AAo) and main pulmonary artery (MPA) were imported into
suiteHEART (NeoSoft, Pewaukee, WI, USA) for hemodynamic and cardiac function
quantification. For the RT data collected at rest, only one end-expiratory
beat was imported. However, for RT data acquired during exercise stress, the
entire cine and flow series, superimposed with the respiratory signal, were
imported. Analyses were conducted on expiratory heartbeats, aided by the
reference respiratory signal. For an initial cohort of volunteers imaged with
an acquisition time of 3 s/slice, all effort was made to conduct the analyses
on the end-expiratory beat where possible. The protocol was consequently
modified to an acquisition time of 6 s/slice to accommodate slower breathing
patterns at rest. Biventricular segmentation was initially performed using
suiteHEART software and manually adjusted where necessary. The following
parameters were derived for cardiac function analysis: left ventricular (LV)
and right ventricular (RV) end-diastolic volume (EDV), end-systolic volume
(ESV), stroke volume (SV), and ejection fraction (EF). From the flow images,
peak velocity (Vmax), net forward flow (NFF), and SV were measured for both
major arteries. To account for the exaggerated through-plane motion, three PC-
CMR slices were acquired from AAo and MPA. From the three slices available, a
slice closest to but above the aortic valve was used for AAo flow
quantification, and a slice closest to the pulmonic valve but away from the
bifurcation was used for MPA flow analysis. The HR at rest and during each
exercise stage was extracted from both RT cine and flow images and converted
to a percentage of the age-predicted maximal heart rate (APHR). Cardiac output
(CO), computed from both cine and flow images, was calculated as the product
of SV (mL/beat) and HR (beats/min). In repeat imaging at rest and during a 40
W workload, quantification was performed on a heartbeat from an arbitrary
respiratory phase (AP) and on a heartbeat from end-expiration (EE).

##  3 Results

###  3.1 Results from healthy subjects

In Figure 2A, the cardiac function parameters extracted from the RT short-axis
cine analysis are depicted, tracing the progression from rest to three stages
of exercise for each volunteer. In summary, the EDV remained relatively
consistent from rest through the exercise stages in both ventricles. In
contrast, the ESV decreased as exercise intensity and HR increased, indicating
more vigorous cardiac contractions during exercise stress. Consequently, the
SV typically increased under stress. However, the pattern of increase in SV
varied among volunteers, ranging from a rapid initial rise to plateauing in
some, and a more gradual increase in others. One of the healthy volunteers had
an incidental finding of congenital cardiovascular abnormality, resulting in
dilated RV and a mismatch between aortic and pulmonic flow. The consistent
elevation in CO was attributable to both the increased SV and HR. Both HR and
APHR trends extracted from the cine images (not shown) were similar to the
ones extracted from the flow images.

In Figure 2B, the aortic and pulmonic flow parameters, tracked from rest
through three stages of exercise for each volunteer, are depicted. The NFF
values demonstrate an overall increase with exercise, although this rise is
less pronounced compared to the biventricular stroke volumes. However, CO
exhibits a similar trend of increasing with exercise intensity. The peak
velocities for both arteries also show an upward trend with exercise. Heart
rates consistently rose with increasing stress, with each volunteer reaching
up to 70% of their APHR on average. Representative cine and flow images at
rest and various stages of exercise are shown in Figure 3.

The perceived exertion, measured using BRPE, ranged from 6 to 16, with an
average value of 9.3. For the subjects who quit before reaching the 60 W
workload, the most commonly expressed reasons were shortness of breath, leg
fatigue, and overall discomfort. Some of the subjects also found it
challenging to ramp up their cadence back to 60 cycles per minute if they fell
below 40 cycles per minute. This is because, for a given workload, the
ergometer automatically increases the resistance to account for the lower
frequency. We addressed this issue by periodically reminding the volunteers to
maintain a cadence of around 60 cycles per minute.

Figure 4 displays the repeat assessment of biventricular functional parameters
from cine images at rest and a workload of $40$ W. Similarly, Figure 5
presents the repeat assessment of hemodynamic parameters from flow images at
the same conditions. The repeatability comparison is conducted for both AP and
EE beats. Overall, both AP and EE beats demonstrate a high level of
repeatability. Quantitatively, the cardiac function parameters for AP and EE
beats are compared in Figure 6. In terms of the concordance correlation
coefficient (CCC) and normalized mean absolute error (NMAE), EE beats show a
marginal but consistent improvement over AP beats. NMAE, inferred from the two
repeats, $x_{1}$ and $x_{2}$, is calculated as
$\mathrm{NMSE}=100\times(|x_{1}-x_{2}|)/(\frac{1}{2}|x_{1}+x_{2}|)$.

###  3.2 Results from patients

The RT cine and flow data from five patients were reconstructed and analyzed.
From each subject, one mid-ventricular short-axis cine slice, one long-axis
cine slice, one PC-CMR slide from AAo, and one PC-CMR slice from MPA were
scored by an expert reader on a five-point Likert-type scale (5-excellent,
4-good, 3-adequate, 2-fair, 1-non-diagnostic) in terms of overall image
quality. The perceived exertion, measured using BRPE, ranged from 6 to 19,
with an average value of 9.3 All images, after coil reweighting, received a
score of 3 or higher. Figure 7 shows representative images from one of the
patients. The adequate quality of cine and flow images allowed us to perform
cardiac function and hemodynamic quantification on all five patients. Due to
the small sample size and lack of reference values, the cardiac function and
hemodynamic parameters are not reported.

###  3.3 Artifact suppression

The artifact suppression method based on coil reweighting was applied to all
RT cine data. In cases, where images had little to no artifact, coil
reweighting had minimal impact on the image quality or quantification.
However, in cases where image artifacts were severe, image interpretation or
quantitative analysis was not feasible without coil reweighting. Figure 8
shows one such example, where images without coil reweighting exhibit severe
artifacts, which are mostly mitigated after coil reweighting.

##  4 Discussion

ExCMR with in-magnet exercise allows dynamic monitoring of the quantitative
cardiac function during exercise. Since breath-holding is generally not
feasible during exercise, free-breathing RT imaging is often used to collect
the data. Some of the technical challenges associated with ExCMR with in-
magnet exercise include image quality degradation due to motion artifacts and
limited spatial and temporal resolution. Recent studies have shown the promise
of applying highly accelerated RT imaging techniques to enable ExCMR during
exercise. In this work, we demonstrate the feasibility of ExCMR for measuring
biventricular cardiac function and hemodynamics under staged exercise for both
healthy subjects and patients.

Figure 2 summarizes the changes in cardiac function and hemodynamic parameters
from rest through various stages of exercise. The values of ESV, both for LV
and RV, consistently decrease due to more vigorous contractions under stress.
The values of EDV remain relatively stable across different exercise
intensities. As a result, both SV and EF gradually increase with exercise.
Compared to SV, the increase in CO is more pronounced due to the increased HR.
This biventricular increase is observed in both cine and flow images. Figure 3
shows an example of RT cine and flow images from a healthy subject. The cine
images show a clear delineation of the blood-myocardium boundary, with some
marginal blurring at 60 W. The end-systolic frames exhibit stronger
contractility, especially at 60 W. The flow images are mostly free of motion
artifacts, with the 60 W magnitude images showing marginal blurring.

Figure 4 and Figure 5 demonstrate the scan-rescan repeatability of measured
parameters at rest and 40 W from eleven and nine subjects, respectively. With
few exceptions, most parameter values are consistent across the scans. Figure
6 shows that selecting an EE beat, compared to an AP beat, results in a modest
but consistent improvement in repeatability. With the sole exception of MPA
CO, NMAE from all parameters from the EE beat is less than 10%. In contrast,
the NMAE exceeds the 10% threshold four times for AP. A similar trend is
observed in CCC. Relatively poor CCC and NMAE values for MPA CO can be
attributed to HR variations between the two repeats and the small sample size.

Figure 7 shows a representative frame from a patient who could only exercise
at the workload of $20$ W. Both RT cine images show clear delineation of blood
and myocardium during the systolic and diastolic phases. The flow images are
also mostly free of artifacts, with the 20 W magnitude image showing slight
blurring. The phase images, however, clearly capture the flow in the ascending
and descending aorta. Note that a higher value of VENC was used at 20 W;
therefore, the velocities are scaled differently in the 20 W image compared to
the resting image. Finally, Figure 8 highlights the impact of coil reweighting
on image quality. Without applying coil reweighting, some of the cine images
are not interpretable and amenable to quantification. With coil reweighting,
most of the artifacts are eliminated. Note, the coil reweighting comes at the
cost of suppressing contribution from certain coil elements, which can lead to
partial loss of signal in some areas of the image. However, we did not observe
a case where the benefit of suppressing artifacts was outweighed by some loss
of intensity.

##  5 Conclusion

In this preliminary study, we demonstrated the feasibility of ExCMR with in-
magnet exercise for quantitative cardiac function evaluation in both healthy
volunteers and patients under multi-stage exercise. The highly accelerated RT
imaging was facilitated by parameter-free compressed sensing, overcoming the
traditional challenges of limited resolution. Additional key innovative
aspects of our technique are the incorporation of a coil reweighting technique
and the selection of end-expiratory heartbeats, which resulted in
significantly improved image quality and enhanced repeatability of cardiac
function measurements. Data from both healthy subjects and patients
underscored ExCMR’s ability to dynamically capture physiological changes in
key cardiac parameters under stress. The advancement in ExCMR technology can
be pivotal for the enhanced diagnosis and management of cardiovascular
diseases, particularly for conditions where functional impairment is not
elucidated on resting cardiovascular exams.

Figure 1: A volunteer exercising inside the MRI bore (left). The multi-stage
ExCMR protocol used to scan 26 healthy subjects (right).

A

B

Figure 2: Biventricular cardiac function and hemodynamic assessment from RT
cine (A) and PC-CMR (B). Each trace represents a volunteer, and the thicker
black line represents the average value. The HR and APHR values from cine (not
shown) were similar to that from flow. The line that is above the rest for the
RV parameters corresponds to the healthy volunteer that had an incidental
finding of dilated RV.

Figure 3: Representative cine and flow frames from a healthy volunteer at rest
and across three stages of exercise. Stronger contractility under exercise
stress is observed from end-systolic frames (second row). The brighter phase
(bottom row) in the resting image is due to the lower value of VENC.

Figure 4: Repeatability of the cine parameters at rest and 40 W for beats from
arbitrary respiratory phase (AP) and beats from end-expiratory phase (EE).
Here, “Rest-R” and “40 W-R” represent the repeated measurements.

Figure 5: Repeatability of the flow parameters at rest and 40 W for beats from
arbitrary respiratory phase (AP) and beats from end-expiratory phase (EE).
Here, “Rest-R” and “40 W-R” represent the repeated measurements.

Figure 6: Concordance correlation coefficient (CCC) and normalized mean
absolute error (NMAE) for cine and flow parameters for AP and EE beats at rest
and 40 W.

Figure 7: Representative cine and flow frames from a patient at rest and 20 W
exercise. Stronger contractility under exercise stress is observed from end-
systolic frames. The brighter phase in the resting image is due to the lower
value of VENC.

ABCD Figure 8: Cine images from one of the patients without coil reweighting
(A and C) and with coil reweighting (B and D). Strong motion artifacts are
observed in A and C, which are effectively removed in B and D. The images at
the bottom represent $\mathrm{x}$-$\mathrm{t}$ profile along the dashed red
line passing through the heart.

Supporting Movie S1: The movie shows a 6 s cine image series from a healthy
volunteer collected under 60 W of exercise. The curve at the bottom represents
the respiratory motion, with the moving cross representing the current frame.

Acknowledgments and Funding This work was funded by NIH grants R01-EB029957,
R01-HL151697, and R01- HL135489.

## Ethics Declarations

  * •

Competing interests: The authors declare no competing interests.

  * •

Ethics approval and consent: For the human subject data, approval was granted
by the Institutional Review Board (IRB) at The Ohio State University
(2019H0076). Informed consent to participate in the study and to publish
results was obtained from all individual participants.

  * •

Data and code availability: MRI data and the code to generate results are
available upon request

  * •

Author contribution: P. Chandrasekaran assisted with data acquisition and
processing as well as manuscript writing, C. Chen implemented data processing
techniques, Y. Liu assisted with data acquisition and pulse sequence
programming, S.M. Arshad assisted with experiment planning, C. Crabtree
assisted with subject monitoring during exercise, M. Tong and Y. Han assisted
with experiment design and results interpretation, and R. Ahmad supervised all
aspects of the study.

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(1+1) dimensional scalar field theory on q-deformed space

Poula Tadros

Department of Applied Physics, Aalto University School of Science, FI-00076
Aalto, Finland.

<EMAIL_ADDRESS>
###### Abstract

We study scalar field theory in one space and one time dimensions on a
q-deformed space with static background. We write the Lagrangian and the
equation of motion and solve it to the first order in $q-1$ where $q$ is the
deformation parameter of the space.

##  1 Introduction

Non-commutative geometry was first introduced in string theory in reference
[1], where it was shown that the coordinates of the endpoints of strings on
D-branes in the presence of a Neveu-Schwartz field are non-commutative. Non-
commutative field theories have also been defined, as they can be derived from
string theories and have interesting features, as described in references [2]
and [3].

The introduction of non-commutative spacetime in field theory is motivated by
the Heisenberg uncertainty principle in quantum mechanics, which states that
at small distance scales, there is a large uncertainty in momentum
measurement. This means that energy can reach very high values in a small
spatial distance, approaching the Planck scale. However, according to the
general theory of relativity, high energy in a small spatial distance creates
a black hole, which prevents the position from being fully certain. To
reconcile these two phenomena, it is necessary to introduce non-commutativity
in spacetime, which implies non locality in the theory. This is explained in
references [4] and [5].

In this paper we study (1+1) dimensional classical scalar field theory with
static spacetime on a q-deformed space, we present both analytical and
numerical analysis of the resulting theory. In section 2, we review the some
types of non-commutativity on space times and motivate the choice of
q-deformation non-commutativity as the subject of the study . In section 3. we
study the scalar field theory on q-deformed space time, we write the
Lagrangian and deduce the equation of motion, we also truncate the equation of
motion to the linear order in $q-1$ and solved the resulting equation. In
section 4, we study the numerical solutions of the truncated equation of
motion showing that the solutions grow exponentially with $x$ and $t$ meaning
that the equation is stiff and there are instabilities in the theory. In
section 5, we conclude the study and suggest topics for further research.

##  2 Types of non-commutativity

Here, we briefly review three of the most popular types of non-commutativity
relations and justify our motivation to use the q-deformation type

  1. 1.

Canonical non-commutativity It is the simplest type which used in physics
literature, it was introduced in [6], it is defined by imposing the following
commutation relations

$[x^{\mu},x^{\nu}]=i\theta^{\mu\nu},$

where $x^{\mu}$ are the spacetime coordinates and $\theta^{\mu\nu}$ is a
constant, anti-symmetric matrix.

The idea of canonical non-commutativity involves smearing the structure of
space-time in a particular way, regardless of the specific mathematical
details of the space. In order to incorporate non-commutative geometry
capturing the mathematical structures on the manifold, it is necessary to
consider more complex forms of non-commutativity beyond just this basic
version.

  2. 2.

Lie-type non commutativity

In this case the coordinates has a Lie algebra structure i.e. the commutation
relations can capture a Lie algebra structures [7]. The commutation relations
are given by

$[x^{\mu},x^{\nu}]=if^{\mu\nu}_{\rho}x^{\rho},$

where $f^{\mu\nu}_{\rho}$ are the structure constants of the defined Lie
algebra. However, this type is not useful because Lie structures are rigid
i.e. any small deformation of a Lie algebra is isomorphic to the original Lie
algebra.

  3. 3.

q-deformations

A solution to the rigidity problem for Lie algebras is to replace Lie group
with a flexible structure called quantum groups [8-10]. The term quantum group
used in this context refers to the deformations of the universal enveloping
algebra of a given group, these objects have Hopf algebra structures which are
flexible structures unlike Lie groups and algebras.

The commutation relations are given by

$x^{\mu}x^{\nu}=\frac{1}{q}R^{\mu\nu}_{\sigma\tau}x^{\sigma}x^{\tau},$

where $q$ is a parameter and $R^{\mu\nu}_{\sigma\tau}$ is the R-matrix of the
quantum group defined on the space.

In this space a Lie algebra is replaced by a non-commutative Hopf algebra with
deformation parameter q. The resulting space is deformed according to the Lie
group on the space and on the parameter q, this is the simplest way to deform
a space time while capturing the full algebraic structure of the space.

##  3 Lagrangian and the equation of motion

We begin with the Lagrangian of the scalar field on the commutative manifold
then introduce non-commutativity by replacing the derivatives by Jackson
derivatives, since the symmetry group is $U(1)$, the deformations of its
universal enveloping algebra gives a commutative algebra. Thus, we do not have
to worry about defining a product of functions on the new space. The
Lagrangian is then

$\mathcal{L}=\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}\rightarrow\mathcal{L}_{q}=D_{\mu
q}\phi D^{\mu}_{q}\phi-m^{2}\phi^{2},$

where $\mu=0,1$ with $x^{0}=t$ and $x^{1}=x$.

Assuming the field is defined everywhere and is infinitely differentiable and
the deformations are small i.e. $q\approx 1$, we can relate the theory on the
non commutative topological space to the theory on the commutative manifold
(i.e. transforming the non-commutative theory back to the commutative
manifold) using the formulae

$D_{xq}(f(x))=\partial_{x}f+\sum_{k=1}^{\infty}\frac{(q-1)^{k}}{(k+1)!}x^{k}f^{(k+1)}(x),$

where $f^{(k)}$ is the k-th ordinary derivative of f with respect to x.

$D_{tq}(f(x))=\partial_{t}f+\sum_{k=1}^{\infty}\frac{(q-1)^{k}}{(k+1)!}x^{k}f^{(k+1)}(x),$

where $f^{[k]}$ is the k-th ordinary derivative of f with respect to t.

The resulting Lagrangian on the commutative manifold is

$\mathcal{L}_{q}=\frac{1}{2}\partial\phi\partial\phi-\frac{1}{2}m^{2}\phi^{2}+2\partial\phi\sum_{k=1}^{\infty}\frac{(q-1)^{k}}{(k+1)!}x^{k}\phi^{(k+1)}$
$+\sum_{l,m=1}^{\infty}\frac{(q-1)^{(l+m)}}{(m+1)!(l+1)!}x^{k+l}\phi^{(l+1)}\phi^{(m+1)}+(x\rightarrow
t).$

where $(x\rightarrow t)$ means the same terms but with $x$ replaced by $t$
including in the derivatives.

The Lagrangian has an infinite series of derivatives, in this case the Euler-
Lagrange equation will be

$\frac{\partial\mathcal{L}_{q}}{\partial\phi}+\sum_{k=1}^{\infty}(-1)^{k}\frac{d^{k}}{dx^{k}}(\frac{\partial\mathcal{L}_{q}}{\partial\phi^{(k)}})+\sum_{k=1}^{\infty}(-1)^{k}\frac{d^{k}}{dt^{k}}(\frac{\partial\mathcal{L}_{q}}{\partial\phi^{[k]}})=0,$
(1)

where $k=2,3,...$.

The Lagrangian is clearly non local as expected from a non-commutative theory.

The derivatives of the Lagrangian are given by

$\frac{\partial\mathcal{L}_{q}}{\partial\phi}=-m\phi,$

$\frac{\partial\mathcal{L}_{q}}{\partial(\partial\phi)}=\partial_{x}\phi+2\sum_{n=1}^{\infty}\frac{(q-1)^{n}}{(n+1)!}x^{n}\phi^{(n+1)}$
(2)

$\rightarrow\frac{d}{dx}(\frac{\partial\mathcal{L}_{q}}{\partial(\partial_{x}\phi)})$
$=\partial_{x}\partial_{x}\phi+2\sum_{n=1}^{\infty}(\frac{n(q-1)^{n}}{(n+1)!}x^{n-1}\phi^{(n+1)})+2\sum_{n=1}^{\infty}(\frac{(q-1)^{n}}{(n+1)!}x^{n}\phi^{(n+2)}),$
(3)

$\frac{\partial\mathcal{L}_{q}}{\partial(\phi^{(k)})}=\frac{2(q-1)^{k-1}x^{k-1}}{k!}\sum_{n=0}^{\infty}\frac{(q-1)^{n}}{(n+1)!}x^{n}\phi^{(n+1)}$

$\rightarrow\frac{d}{dx}(\frac{\partial\mathcal{L}_{q}}{\partial(\phi^{(k)})})$
$=\frac{2(q-1)^{k-1}x^{2k-1}}{k!}\sum_{m,n=0}^{\infty}\binom{k}{m}\frac{(q-1)^{n}}{(n+1)!}\frac{(n+k+1)!}{(n+2k-m-1)!}x^{n-m}\phi^{(n+k+1)},$
(4)

with similar formulae for derivatives with respect to t.

Putting all together from (2), (3), (4) in (1) we get

$-\partial_{\mu}\partial^{\mu}\phi-m^{2}\phi-2\sum_{n=1}^{\infty}\frac{n(q-1)^{n}}{(n+1)!}x^{n-1}\phi^{(n+1)}-2\sum_{n=1}^{\infty}\frac{(q-1)^{n}}{(n+1)!}x^{n}\phi^{(n+2)}$

$+\sum_{k=2}^{\infty}(-1)^{k}\frac{2(q-1)^{k-1}x^{2k-1}}{k!}\sum_{n=0}^{\infty}\sum_{m=0}^{k}\binom{k}{m}\frac{(q-1)^{n}(n+k-1)!}{(n+1)!(n+2k-m-1)!}x^{n-m}\phi^{(n+k+1)}$
$+(x\rightarrow t)=0.$ (5)

This is a partial differential equation of infinite order with variable
coefficients.

If we consider only small deformations i.e. $q\approx 1$, then we can only
keep terms up to the linear order in $q-1$, the first order equation will be

$-\partial_{\mu}\partial^{\mu}\phi-m^{2}\phi-(q-1)[\phi^{(2)}+x\phi^{(3)}-\frac{x^{3}}{6}\phi^{(3)}-x^{2}\phi^{(3)}-x\phi^{3}+(x\rightarrow
t)]=0.$

This equation is a stiff equation i.e. it is numerically unstable, this may
indicate an instability in the theory due to the linear approximation used,
but as seen from the full equation of motion the full theory is stable.

The solution is $\phi=F(t)G(x)$ where

$F(t)=c_{1}e^{iAt/\sqrt{q}}+c_{2}e^{-iAt/\sqrt{q}}+(q-1)\frac{e^{iAt/\sqrt{q}}}{2iA\sqrt{q}}[\frac{iA}{24q}t^{4}+(\frac{iA^{3}}{3}-\frac{1}{12\sqrt{q}}-\frac{i}{8A})t^{3}$

$+(\frac{A+q}{2q}-\frac{A\sqrt{q}}{2}-\frac{i}{8A})t^{2}+(\frac{i(A+q)}{2A\sqrt{q}}-\frac{iqA}{4}+\frac{\sqrt{q}}{8A^{2}})t$
$+(\frac{A+q}{4A^{2}}+\frac{q\sqrt{A}}{4}+\frac{iq}{16A^{3}})]+O((q-1)^{2}),$

$G(x)=c_{3}e^{ikx/\sqrt{q}}+c_{4}e^{-ikx/\sqrt{q}}+(q-1)\frac{e^{ikx/\sqrt{q}}}{2ik\sqrt{q}}[\frac{ik}{24q}x^{4}+(\frac{ik^{3}}{3}-\frac{1}{12\sqrt{q}}-\frac{i}{8A})x^{3}$

$+(\frac{k+q}{2q}-\frac{k\sqrt{q}}{2}-\frac{i}{8k})x^{2}+(\frac{i(k+q)}{2k\sqrt{q}}-\frac{iqk}{4}+\frac{\sqrt{q}}{8k^{2}})x$
$+(\frac{k+q}{4k^{2}}+\frac{q\sqrt{k}}{4}+\frac{iq}{16k^{3}})]+O((q-1)^{2}),$

where $c_{1},c_{2},c_{3},c_{4},A$ are normalisation constants and
$k=\pm\sqrt{A^{2}+m^{2}}$. When $q=1$, it reduces to the solution to the Klein
Gordon equation as expected.

##  4 Numerical results

Here, we present numerical solutions to the equation of motion to the first
order in $q-1$, we focus on $G(x)$ only since the remaining part is similar.
The solutions are exponentially growing in time establishing that the equation
of motion was stiff.

We set $c_{3}=c_{4}=k=1$, $A^{2}=\frac{1}{2}$ and we plot the solution for
different values of the parameter $q$.

Figure 1: At $q-1=0.1$ the solution grows exponentially with $|x|$. This is a
feature of a stiff equation with unstable numerical solution. In the vicinity
of $x=0$ it is close to the usual Klein-Gordon solution but as we go further
it becomes more and more distant  Figure 2: At $q-1=0.001$ the solution still
grows exponentially but slower. Figure 3: At $q-1=10^{-6}$ the solution
resembles the Klein-Gordon solution up to $|x|=50$ then decays for a bit but
eventually blows up.  Figure 4: At $q-1=10^{-9}$ the solution has the same
behaviour as the previous graph but the decay happens at larger $|x|$, all
smaller $q-1$ values follow this pattern.

The above results shows an instability in the theory leading to divergent
solutions to the equations of motion as $x\rightarrow\infty$. To remove the
instability we must add infinite terms corresponding to an infinite series of
higher derivatives i.e. we have to consider the full theory. However, this
approximation gives us an intuition on how the q-deformation affects the
space, small q-deformations beside leading to non local effects appear to
affect the space irregularly with only small effects locally.

##  5 Conclusion and outlook

In conclusion, we showed that defining a field theory on a q-deformed space
leads to an infinite series of higher derivatives in the Lagrangian even with
static background. In the case presented the algebra was commutative so no new
product of functions is needed. We also demonstrated that any approximation or
truncation to the theory will lead to stiff equations of motion resulting from
instabilities in the theory.
While we made a progress in the field, much more is to be studied, future
research in this direction should focus on defining more complicated theories
on q-deformed spaces with non-commutative function algebras and with dynamical
spacetimes, also to define higher spin fields on such space and study the new
symmetries of the theories as well as the types of instabilities arise if the
Lagrangian is truncated.

## Acknowledgments

We would like to thank Dr.Ivan Kolar for the useful discussions on the topic

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