Steady-state double-angle method for rapid B1 mapping

White matter integrity and motor function: a link between cerebral myelination and longitudinal changes in gait speed in aging

Gait speed is a robust health biomarker in older adults, correlating with the risk of physical and cognitive impairments, including dementia. Myelination plays a crucial role in neurotransmission and consequently affects various functions, yet the connection between myelination and motor functions such as gait speed is not well understood. Understanding this link could offer insights into diagnosing and treating neurodegenerative diseases that impair mobility. This study analyzed 437 longitudinal observations from 138 cognitively unimpaired adults, aged 22 to 94 years, to investigate the relationship between myelin content and changes in gait speed over an average of 6.42 years. Myelin content was quantified using a novel multicomponent magnetic resonance relaxometry method, and both usual and rapid gait speeds (UGS, RGS) were measured following standard protocols. Adjusting for covariates, we found a significant fixed effect of myelin content on UGS and RGS. Longitudinally, lower myelin content was linked to a greater decline in UGS, particularly in brain regions associated with motor planning. These results suggest that changes in UGS may serve as a reliable marker of neurodegeneration, particularly in cognitively unimpaired adults. Interestingly, the relationship between myelin content and changes in RGS was only observed in a limited number of brain regions, although the reason for such local susceptibility remains unknown. These findings enhance our understanding of the critical role of myelination in gait performance in unimpaired adults and provide evidence of the interconnection between myelin content and motor function impairment.

Age-Related Differences in the Choroid Plexus Structural Integrity Are Associated with Changes in Cognition

The choroid plexus (CP) plays a critical role in maintaining central nervous system (CNS) homeostasis, producing cerebrospinal fluid, and regulating the entry of specific substances into the CNS from blood. CP dysfunction has been implicated in various neurological and psychiatric disorders, including Alzheimer's disease, Parkinson's disease, and multiple sclerosis. This study investigates the relationship between CP structural integrity and cognitive decline in normative aging, using structural and advanced magnetic resonance imaging techniques, including CP volume, diffusion tensor imaging indices (mean diffusivity, MD, and fractional anisotropy, FA) and relaxometry metrics (longitudinal, T1, and transverse, T2, relaxation times). Our results show that diminished CP microstructural integrity, as reflected by higher T1, T2, and MD values, or lower FA values, is associated with lower cognitive performance in processing speed and fluency. Notably, CP microstructural measures demonstrated greater sensitivity to cognitive decline than macrostructural measures, i.e. CP volume. Longitudinal analysis revealed that individuals with reduced CP structural integrity exhibit steeper cognitive decline over time. Furthermore, structural equation modeling revealed that a latent variable representing CP integrity predicts faster overall cognitive decline, with an effect size comparable to that of age. These findings highlight the importance of CP integrity in maintaining cognitive health and suggest that a holistic approach to assessing CP integrity could serve as a sensitive biomarker for early detection of cognitive decline. Further research is needed to elucidate the mechanisms underlying the relationship between CP structural integrity and cognitive decline and to explore the potential therapeutic implications of targeting CP function to prevent or treat age-related cognitive deficits.

Fast 3D 31P B 1 + mapping with a weighted stack of spiral trajectory at 7 T

PURPOSE

Phosphorus MRS (31P MRS) enables noninvasive assessment of energy metabolism, yet its application is hindered by sensitivity limitations. To overcome this, often high magnetic fields are used, leading to challenges such as spatialB 1 + $$ {\mathrm{B}}_1^{+} $$ inhomogeneity and therefore the need for accurate flip-angle determination in accelerated acquisitions with short TRs. In response to these challenges, we propose a novel short TR and look-up table-based double-angle method for fast 3D 31PB 1 + $$ {\mathrm{B}}_1^{+} $$ mapping (fDAM).

METHODS

Our method incorporates 3D weighted stack-of-spiral gradient-echo acquisitions and a frequency-selective pulse to enable efficientB 1 + $$ {\mathrm{B}}_1^{+} $$ mapping based on the phosphocreatine signal at 7 T. Protocols were optimized using simulations and validated through phantom experiments. The method was validated in the human brain using a 31P 1Ch-trasmit/32Ch-receive coil and skeletal muscle using a birdcage 1H/31P volume coil.

RESULTS

The results of fDAM were compared with the classical DAM. A good correlation (r = 0.95) was obtained between the twoB 1 + $$ {\mathrm{B}}_1^{+} $$ maps. A 3D 31PB 1 + $$ {\mathrm{B}}_1^{+} $$ mapping in the human calf muscle was achieved in about 10:50 min using a birdcage volume coil, with a 20% extended coverage (number of voxels with SNR > 3) relative to that of the classical DAM (24 min). fDAM also enabled the first full-brain coverage 31P 3DB 1 + $$ {\mathrm{B}}_1^{+} $$ mapping in approximately 10:15 min using a 1Ch-transmit/32Ch-receive coil.

CONCLUSION

fDAM is an efficient method for 31P 3DB 1 + $$ {\mathrm{B}}_1^{+} $$ mapping, showing promise for future applications in rapid 31P MRSI.

Cerebral white matter myelination is associated with longitudinal changes in processing speed across the adult lifespan

Myelin's role in processing speed is pivotal, as it facilitates efficient neural conduction. Its decline could significantly affect cognitive efficiency during ageing. In this work, myelin content was quantified using our advanced MRI method of myelin water fraction mapping. We examined the relationship between myelin water fraction at the time of MRI and retrospective longitudinal change in processing speed among 121 cognitively unimpaired participants, aged 22-94 years, from the Baltimore Longitudinal Study of Aging and the Genetic and Epigenetic Signatures of Translational Aging Laboratory Testing (a mean follow-up duration of 4.3 ± 6.3 years) using linear mixed-effects models, adjusting for demographics. We found that higher myelin water fraction values correlated with longitudinally better-maintained processing speed, with particularly significant associations in several white matter regions. Detailed voxel-wise analysis provided further insight into the specific white matter tracts involved. This research underscores the essential role of myelin in preserving processing speed and highlights its potential as a sensitive biomarker for interventions targeting age-related cognitive decline, thereby offering a foundation for preventative strategies in neurological health.

Association of Plasma Markers of Alzheimer's Disease, Neurodegeneration, and Neuroinflammation with the Choroid Plexus Integrity in Aging

The choroid plexus (CP) is a vital brain structure essential for cerebrospinal fluid (CSF) production. Moreover, alterations in the CP's structure and function are implicated in molecular conditions and neuropathologies including multiple sclerosis, Alzheimer's disease, and stroke. Our goal is to provide the first characterization of the association between variation in the CP microstructure and macrostructure/volume using advanced magnetic resonance imaging (MRI) methodology, and blood-based biomarkers of Alzheimer's disease (Aß42/40 ratio; pTau181), neuroinflammation and neuronal injury (GFAP; NfL). We hypothesized that plasma biomarkers of brain pathology are associated with disordered CP structure. Moreover, since cerebral microstructural changes can precede macrostructural changes, we also conjecture that these differences would be evident in the CP microstructural integrity. Our cross-sectional study was conducted on a cohort of 108 well-characterized individuals, spanning 22-94 years of age, after excluding participants with cognitive impairments and non-exploitable MR imaging data. Established automated segmentation methods were used to identify the CP volume/macrostructure using structural MR images, while the microstructural integrity of the CP was assessed using our advanced quantitative high-resolution MR imaging of longitudinal and transverse relaxation times (T1 and T2). After adjusting for relevant covariates, positive associations were observed between pTau181, NfL and GFAP and all MRI metrics. These associations reached significance (p<0.05) except for CP volume vs. pTau181 (p=0.14), CP volume vs. NfL (p=0.35), and T2 vs. NFL (p=0.07). Further, negative associations between Aß42/40 and all MRI metrics were observed but reached significance only for Aß42/40 vs. T2 (p=0.04). These novel findings demonstrate that reduced CP macrostructural and microstructural integrity is positively associated with blood-based biomarkers of AD pathology, neurodegeneration/neuroinflammation and neurodegeneration. Degradation of the CP structure may co-occur with AD pathology and neuroinflammation ahead of clinically detectable cognitive impairment, making the CP a potential structure of interest for early disease detection or treatment monitoring.

Evidence of association between higher cardiorespiratory fitness and higher cerebral myelination in aging

Emerging evidence suggests that altered myelination is an important pathophysiologic correlate of several neurodegenerative diseases, including Alzheimer and Parkinson's diseases. Thus, improving myelin integrity may be an effective intervention to prevent and treat age-associated neurodegenerative pathologies. It has been suggested that cardiorespiratory fitness (CRF) may preserve and enhance cerebral myelination throughout the adult lifespan, but this hypothesis has not been fully tested. Among cognitively normal participants from two well-characterized studies spanning a wide age range, we assessed CRF operationalized as the maximum rate of oxygen consumption (VO2max) and myelin content defined by myelin water fraction (MWF) estimated through our advanced multicomponent relaxometry MRI method. We found significant positive correlations between VO2max and MWF across several white matter regions. Interestingly, the effect size of this association was higher in brain regions susceptible to early degeneration, including the frontal lobes and major white matter fiber tracts. Further, the interaction between age and VO2max exhibited i) a steeper positive slope in the older age group, suggesting that the association of VO2max with MWF is stronger at middle and older ages and ii) a steeper negative slope in the lower VO2max group, indicating that lower VO2max levels are associated with lower myelination with increasing age. Finally, the nonlinear pattern of myelin maturation and decline is VO2max-dependent with the higher VO2max group reaching the MWF peak at later ages. This study provides evidence of an interconnection between CRF and cerebral myelination and suggests therapeutic strategies for promoting brain health and attenuating white matter degeneration.

Fast 3D 31P B1 + mapping with a weighted stack of spiral trajectory at 7 Tesla

Purpose

Phosphorus Magnetic Resonance Spectroscopy (31P MRS) enables non-invasive assessment of energy metabolism, yet its application is hindered by sensitivity limitations. To overcome this, often high magnetic fields are used, leading to challenges such as spatialB 1 + inhomogeneity and therefore the need for accurate flip angle determination in accelerated acquisitions with short repetition timesT R ) . In response to these challenges, we propose a novel shortT R and look-up table-based Double-Angle Method for fast 3D 31PB 1 + mapping (fDAM).

Methods

Our method incorporates 3D weighted stack of spiral gradient echo acquisitions and a frequency-selective pulse to enable efficientB 1 + mapping based on the phosphocreatine signal at 7T. Protocols were optimised using simulations and validated through phantom experiments. The method was validated in phantom experiments and skeletal muscle applications using a birdcage 1H/31P volume coil.

Results

The results of fDAM were compared to the classical DAM (cDAM). A good correlation (r=0.94) was obtained between the twoB 1 + maps. A 3D 31PB 1 + mapping in the human calf muscle was achieved in about 10 min using a birdcage volume coil, with a 20% extended coverage relative to that of the cDAM (24 min). fDAM also enabled the first full brain coverage 31P 3DB 1 + mapping in approx. 10 min using a 1 Tx/ 32 Rx coil.

Conclusion

fDAM is an efficient method for 31P 3DB 1 + mapping, showing promise for future applications in rapid 31P MRSI.

A preparation pulse for fast steady state approach in Actual Flip angle Imaging

BACKGROUND

Actual Flip angle Imaging (AFI) is a sequence used for B1 mapping, also embedded in the Variable flip angle with AFI for simultaneous estimation of T1 , B1 and equilibrium magnetization.

PURPOSE

To investigate the design of a preparation module for AFI to allow a fast approach to steady state (SS) without requiring the use of dummy acquisitions.

METHODS

The features of a preparation module with a B1 insensitive adiabatic pulse, spoiler gradients, and a recovery timeT r e c $T_{rec}$ were studied with simulations and validated via experiments and acquired with different k-space traveling strategies. The robustness of the flip angle of the preparation pulse on the acquired signal is studied.

RESULTS

When a 90° adiabatic pulse is used, the forthcomingT r e c $T_{rec}$ can be expressed as a function of repetition times and AFI flip angle only asTR 1 ( n + cos α ) / ( 1 - cos 2 α ) $\mathrm{TR_1}(n+\cos \alpha )/(1-\cos ^2\alpha )$ , where n represents the ratio between the two repetition times of AFI. The robustness of the method is demonstrated by showing that using the values further away from 90° still allows for a faster approach to SS than the use of dummy pulses.

CONCLUSIONS

The preparation module is particularly advantageous for low flip angles, as well as for AFI sequences that sample the center of the k-space early in the sequence, such as centric ordering acquisitions, and for ultrafast EPI-based AFI methods, thus allowing to reduce scanner overhead time.

REUSED: A deep neural network method for rapid whole-brain high-resolution myelin water fraction mapping from extremely under-sampled MRI

Changes in myelination are a cardinal feature of brain development and the pathophysiology of several central nervous system diseases, including multiple sclerosis and dementias. Advanced magnetic resonance imaging (MRI) methods have been developed to probe myelin content through the measurement of myelin water fraction (MWF). However, the prolonged data acquisition and post-processing times of current MWF mapping methods pose substantial hurdles to their clinical implementation. Recently, fast steady-state MRI sequences have been implemented to produce high-spatial resolution whole-brain MWF mapping within ∼20 min. Despite the subsequent significant advances in the inversion algorithm to derive MWF maps from steady-state MRI, the high-dimensional nature of such inversion does not permit further reduction of the acquisition time by data under-sampling. In this work, we present an unprecedented reduction in the computation (∼30 s) and the acquisition time (∼7 min) required for whole-brain high-resolution MWF mapping through a new Neural Network (NN)-based approach, named NN-Relaxometry of Extremely Under-SamplEd Data (NN-REUSED). Our analyses demonstrate virtually similar accuracy and precision in derived MWF values using NN-REUSED compared to results derived from the fully sampled reference method. The reduction in the acquisition and computation times represents a breakthrough toward clinically practical MWF mapping.

Lower Myelin Content Is Associated With Lower Gait Speed in Cognitively Unimpaired Adults

Mounting evidence indicates that abnormal gait speed predicts the progression of neurodegenerative diseases, including Alzheimer's disease. Understanding the relationship between white matter integrity, especially myelination, and motor function is crucial to the diagnosis and treatment of neurodegenerative diseases. We recruited 118 cognitively unimpaired adults across an extended age range of 22-94 years to examine associations between rapid or usual gait speeds and cerebral myelin content. Using our advanced multicomponent magnetic resonance relaxometry method, we measured myelin water fraction (MWF), a direct measure of myelin content, as well as longitudinal and transverse relaxation rates (R1 and R2), sensitive but nonspecific magnetic resonance imaging measures of myelin content. After adjusting for covariates and excluding 22 data sets due to cognitive impairments or artifacts, our results indicate that participants with higher rapid gait speed exhibited higher MWF, R1, and R2 values, that is, higher myelin content. These associations were statistically significant within several white matter brain regions, particularly the frontal and parietal lobes, splenium, anterior corona radiata, and superior fronto-occipital and longitudinal fasciculus. In contrast, we did not find any significant associations between usual gait speed and MWF, R1, or R2, which suggests that rapid gait speed may be a more sensitive marker of demyelination than usual gait speed. These findings advance our understanding on the implication of myelination in gait impairment among cognitively unimpaired adults, providing further evidence of the interconnection between white matter integrity and motor function.

Hypertensive Adults Exhibit Lower Myelin Content: A Multicomponent Relaxometry and Diffusion Magnetic Resonance Imaging Study

BACKGROUND

It is unknown whether hypertension plays any role in cerebral myelination. To fill this knowledge gap, we studied 90 cognitively unimpaired adults, age range 40 to 94 years, who are participants in the Baltimore Longitudinal Study of Aging and the Genetic and Epigenetic Signatures of Translational Aging Laboratory Testing to look for potential associations between hypertension and cerebral myelin content across 14 white matter brain regions.

METHODS

Myelin content was probed using our advanced multicomponent magnetic resonance relaxometry method of myelin water fraction, a direct and specific magnetic resonance imaging measure of myelin content, and longitudinal and transverse relaxation rates (R1 and R2), 2 highly sensitive magnetic resonance imaging metrics of myelin content. We also applied diffusion tensor imaging magnetic resonance imaging to measure fractional anisotropy, mean diffusivity, radial diffusivity, and axial diffusivity values, which are metrics of cerebral microstructural tissue integrity, to provide context with previous magnetic resonance imaging findings.

RESULTS

After adjustment of age, sex, systolic blood pressure, smoking status, diabetes status, and cholesterol level, our results indicated that participants with hypertension exhibited lower myelin water fraction, fractional anisotropy, R1 and R2 values and higher mean diffusivity, radial diffusivity, and axial diffusivity values, indicating lower myelin content and higher impairment to the brain microstructure. These associations were significant across several white matter regions, particularly in the corpus callosum, fronto-occipital fasciculus, temporal lobes, internal capsules, and corona radiata.

CONCLUSIONS

These original findings suggest a direct association between myelin content and hypertension and form the basis for further investigations including longitudinal assessments of this relationship.

Investigation of the association between central arterial stiffness and aggregate g-ratio in cognitively unimpaired adults

Stiffness of the large arteries has been shown to impact cerebral white matter (WM) microstructure in both younger and older adults. However, no study has yet demonstrated an association between arterial stiffness and aggregate g-ratio, a specific magnetic resonance imaging (MRI) measure of axonal myelination that is highly correlated with neuronal signal conduction speed. In a cohort of 38 well-documented cognitively unimpaired adults spanning a wide age range, we investigated the association between central arterial stiffness, measured using pulse wave velocity (PWV), and aggregate g-ratio, measured using our recent advanced quantitative MRI methodology, in several cerebral WM structures. After adjusting for age, sex, smoking status, and systolic blood pressure, our results indicate that higher PWV values, that is, elevated arterial stiffness, were associated with lower aggregate g-ratio values, that is, lower microstructural integrity of WM. Compared to other brain regions, these associations were stronger and highly significant in the splenium of the corpus callosum and the internal capsules, which have been consistently documented as very sensitive to elevated arterial stiffness. Moreover, our detailed analysis indicates that these associations were mainly driven by differences in myelination, measured using myelin volume fraction, rather than axonal density, measured using axonal volume fraction. Our findings suggest that arterial stiffness is associated with myelin degeneration, and encourages further longitudinal studies in larger study cohorts. Controlling arterial stiffness may represent a therapeutic target in maintaining the health of WM tissue in cerebral normative aging.

Fast B1 Mapping Based on Double-Angle Method with T1 Correction Using Standard Pulse Sequence

Radiofrequency (RF) field (B1) mapping by combining the double-angle method (DAM) and T1 correction was investigated. The signal intensities S1 and S2 acquired by flip angle (FA) α and double FA 2α at short repetition time (TR) were converted to a signal intensity at TR=∞ by T1 correction. Then, these were used for DAM calculation. The T1 values are measured from two different images acquired with different TRs based on the saturation recovery (SR) method preliminarily. The effects of imaging parameters for T1 estimation and measured FA were investigated using CuSO₄-doped water phantoms. A two-dimensional gradient echo type echo planar imaging pulse sequence was used. T1 values obtained by the 2-SR method were underestimated compared to the multipoint inversion recovery method. FA error was less than 5% when the appropriate imaging parameters were used. The acquisition time could be shortened to under 25 s by the use of T1-corrected DAM.

A 16-Channel 13C Array Coil for Magnetic Resonance Spectroscopy of the Breast at 7T

OBJECTIVE

Considering the reported elevation of ω-6/ω-3 fatty acid ratios in breast neoplasms, one particularly important application of ¹³C MRS could be in more fully understanding the breast lipidome's relationship to breast cancer incidence. However, the low natural abundance and gyromagnetic ratio of the ¹³C isotope lead to detection sensitivity challenges. Previous ¹³C MRS studies have relied on the use of small surface coils with limited field-of-view and shallow penetration depths to achieve adequate signal-to-noise ratio (SNR), and the use of receive array coils is still mostly unexplored.

METHODS

This work presents a unilateral breast 16-channel ¹³C array coil and interfacing hardware designed to retain the surface sensitivity of a single small loop coil while improving penetration depth and extending the field-of-view over the entire breast at 7T. The coil was characterized through bench measurements and phantom ¹³C spectroscopy experiments.

RESULTS

Bench measurements showed receive coil matching better than -17 dB and average preamplifier decoupling of 16.2 dB with no evident peak splitting. Phantom MRS studies show better than a three-fold increase in average SNR over the entirety of the breast region compared to volume coil reception alone as well as an ability for individual array elements to be used for coarse metabolite localization without the use of single-voxel or spectroscopic imaging methods.

CONCLUSION

Our current study has shown the benefits of the array. Future in vivo lipidomics studies can be pursued.

SIGNIFICANCE

Development of the 16-channel breast array coil opens possibilities of in vivo lipidomics studies to elucidate the link between breast cancer incidence and lipid metabolics.

A practical method for post-acquisition reduction of bias in fast, whole-brain B1-maps

Large consistent differences have been observed between maps of the flip angle correction factor (commonly called "B₁-maps") produced with different fast methods in the human brain. We present an empirical procedure for first-order multiplicative bias correction that can be applied when more than one B₁-mapping method is available. We use a B₁-map measurement in a calibration phantom as a reference and the voxel-wise histogram mode between ratios of B₁-maps produced from different methods to calculate determine the bias as a multiplicative correcting scale factor. Institutional implementations of four common methods of B₁-mapping were assessed: Method of Slopes, FSE and EPI double angle methods (DAM), and Bloch-Siegert. In human subjects, the multiplicative bias used to correct for each of the four methods was: Method of Slopes = 1.005, FSE-DAM = 0.956, EPI-DAM = 1.080, and Bloch-Siegert = 1.128. Scaling to remove this bias between methods produces more consistent B₁-maps which enable more consistent values for any computations requiring flip angle correction. In addition, we present evidence that the corrected B₁ maps, using our calibration method, are also more accurate.

Four-angle method for practical ultra-high-resolution magnetic resonance mapping of brain longitudinal relaxation time and apparent proton density

Changes in longitudinal relaxation time (T₁) and proton density (PD) are sensitive indicators of microstructural alterations associated with various central nervous system diseases as well as brain maturation and aging. In this work, we introduce a new approach for rapid and accurate high-resolution (HR) or ultra HR (UHR) mapping of T₁ and apparent PD (APD) of the brain with correction of radiofrequency field, B₁, inhomogeneities. The four-angle method (FAM) uses four spoiled-gradient recalled-echo (SPGR) images acquired at different flip angles (FA) and short repetition times (TRs). The first two SPGR images are acquired at low-spatial resolution and used to accurately map the active B₁⁺ field with the recently introduced steady-state double angle method (SS-DAM). The estimated B₁⁺ map is used in conjunction with the two other SPGR images, acquired at HR or UHR, to map T₁ and APD. The method is evaluated with numerical, phantom, and in-vivo imaging measurements. Furthermore, we investigated imaging acceleration methods to further shorten the acquisition time. Our results indicate that FAM provides an accurate method for simultaneous HR or UHR mapping of T₁ and APD in human brain in clinical high-field MRI. Derived parameter maps without B₁⁺correction suffer from large inaccuracies, but this issue is well-corrected through use of the SS-DAM. Furthermore, the use of SPGR imaging with short TR and phased-array coil acquisition permits substantial imaging acceleration and enables robust HR or UHR T₁ and APD mapping in a clinically acceptable time frame, with whole brain coverage obtained in less than 2 min or 5 min, respectively. The method exhibits high reproducibility and benefits from the use of the conventional SPGR sequence, available in all preclinical and clinical MRI machines, and very simple modeling to address a critical outstanding issue in neuroimaging.

Channel-combination method for phase-based |B1+| mapping techniques

PURPOSE

The aim of this study was to propose a channel combination method for |B₁⁺| mapping methods using phase difference to reconstruct |B₁⁺| map.

THEORY AND METHODS

Phase-based |B₁⁺| mapping methods commonly consider the phase difference of two scans to measure |B₁⁺|. Multiple receiver coils acquire a number of images and the phase difference at each channel is theoretically the same in the absence of noise. Affected by noise, phase difference is approximately governed by Gaussian distribution. Considering data from all channels as samples, estimation can be achieved by maximum likelihood method. With this method, all phase differences at each channel are combined into one. In this study, the proposed method is applied with Bloch-Siegert shift |B₁⁺| mapping method. Simulations are performed to illustrate the phase difference distribution and demonstrate the feasibility and facility of the proposed method. Phantom and vivo experiments are carried out at 1.5 T scanner equipped with 8-channel receiver coil. In all experiments, the proposed method is compared with weighted averaging (WA) method.

RESULTS

Simulations revealed appropriateness of approximating the distribution of phase difference to Gaussian distribution. Compared with WA method, the proposed method reduces errors of |B₁⁺| calculation. Phantom and vivo experiments provide further validation.

CONCLUSION

Considering phase noise distribution, the proposed method achieves channel combination by finding the estimation from data acquired by multiple receivers coil. The proposed method reduces |B₁⁺| reconstruction errors caused by noise.