Automated Segmentation of Brain Gray Matter Nuclei on Quantitative Susceptibility Mapping Using Deep Convolutional Neural Network
Tianjin First Central Hospital（天津市第一中心医院）；Tianjin Medical Imaging Institute（天津医学影像研究所）；Nankai University（南开大学）；Wayne State University（韦恩州立大学）；Tianjin Huanhu Hospital（天津市环湖医院）；Dalian University of Technology（大连理工大学）
Chao Chai, Pengchong Qiao, Bin Zhao, Huiying Wang, Guohua Liu, Hong Wu, E Mark Haacke, Wen Shen, Chen Cao, Xinchen Ye, Zhiyang Liu, Shuang Xia
Abnormal iron accumulation in the brain subcortical nuclei has been reported to be correlated to various neurodegenerative diseases, which can be measured through the magnetic susceptibility from the quantitative susceptibility mapping (QSM). To quantitively measure the magnetic susceptibility, the nuclei should be accurately segmented, which is a tedious task for clinicians. In this paper, we proposed a double-branch residual-structured U-Net (DB-ResUNet) based on 3D convolutional neural network (CNN) to automatically segment such brain gray matter nuclei. To better tradeoff between segmentation accuracy and the memory efficiency, the proposed DB-ResUNet fed image patches with high resolution and the patches with low resolution but larger field of view into the local and global branches, respectively. Experimental results revealed that by jointly using QSM and T$_\text{1}$ weighted imaging (T$_\text{1}$WI) as inputs, the proposed method was able to achieve better segmentation accuracy over its single-branch counterpart, as well as the conventional atlas-based method and the classical 3D-UNet structure. The susceptibility values and the volumes were also measured, which indicated that the measurements from the proposed DB-ResUNet are able to present high correlation with values from the manually annotated regions of interest.
University of Pennsylvania（宾夕法尼亚大学）；Lund University（隆德大学）
Ex vivo MRI of the brain provides remarkable advantages over in vivo MRI for visualizing and characterizing detailed neuroanatomy. However, automated cortical segmentation methods in ex vivo MRI are not well developed, primarily due to limited availability of labeled datasets, and heterogeneity in scanner hardware and acquisition protocols. In this work, we present a high resolution 7 Tesla dataset of 32 ex vivo human brain specimens. We benchmark the cortical mantle segmentation performance of nine neural network architectures, trained and evaluated using manually-segmented 3D patches sampled…
Introduction: Background field removal (BFR) is a critical step required for successful quantitative susceptibility mapping (QSM). However, eliminating the background field in brains containing significant susceptibility sources, such as intracranial hemorrhages, is challenging due to the relatively large scale of the field induced by these pathological susceptibility sources. Method: This study proposes a new deep learning-based method, BFRnet, to remove background field in healthy and hemorrhagic subjects. The network is built with the dual-frequency octave convolutions on the U-net architec…
Anatomy of the human brain constrains the formation of large-scale functional networks. Here, given measured brain activity in gray matter, we interpolate these functional signals into the white matter on a structurally-informed high-resolution voxel-level brain grid. The interpolated volumes reflect the underlying anatomical information, revealing white matter structures that mediate functional signal flow between temporally coherent gray matter regions. Functional connectivity analyses of the interpolated volumes reveal an enriched picture of the default mode network (DMN) and its subcompone…
Synthetic magnetic resonance images for domain adaptation: Application to fetal brain tissue segmentation
Lausanne University Hospital (CHUV)（洛桑大学医院（CHUV））；University of Lausanne (UNIL)（洛桑大学（UNIL））；CIBM Center for Biomedical Imaging（CIBM生物医学成像中心）；University Children’s Hospital Zurich（苏黎世大学儿童医院）；University of Zurich（苏黎世大学）；Neuroscience Center Zurich（苏黎世神经科学中心）
Shanghai Jiao Tong University（上海交通大学）；University of California, Berkeley（加州大学伯克利分校）；ShanghaiTech University（上海科技大学）；Zhejiang University School of Medicine（浙江大学医学院）；The Second Affiliated Hospital, Zhejiang University School of Medicine（浙江大学医学院附属第二医院）；Rui Jin Hospital, Shanghai Jiao Tong University, School of Medicine（上海交通大学医学院附属瑞金医院）；Stanford University（斯坦福大学）；Lucile Packard Children’s Hospital（露西尔·帕卡德儿童医院）；Helen Wills Neuroscience Institute（海伦·威尔斯神经科学研究所）
Quantitative susceptibility mapping (QSM) estimates the underlying tissue magnetic susceptibility from MRI gradient-echo phase signal and typically requires several processing steps. These steps involve phase unwrapping, brain volume extraction, background phase removal and solving an ill-posed inverse problem. The resulting susceptibility map is known to suffer from inaccuracy near the edges of the brain tissues, in part due to imperfect brain extraction, edge erosion of the brain tissue and the lack of phase measurement outside the brain. This inaccuracy has thus hindered the application of …
Shanghai United Imaging Intelligence Co. Ltd（上海联影智能医疗科技有限公司）；Southern Medical University（南方医科大学）
In brain tumor diagnosis and surgical planning, segmentation of tumor regions and accurate analysis of surrounding normal tissues are necessary for physicians. Pathological variability often renders difficulty to register a well-labeled normal atlas to such images and to automatic segment/label surrounding normal brain tissues. In this paper, we propose a new registration approach that first segments brain tumor using a U-Net and then simulates missed normal tissues within the tumor region using a partial convolutional network. Then, a standard normal brain atlas image is registered onto such …
University Hospitals Leuven（鲁汶大学医院）；University Hospitals Tanta（坦塔大学医院）；University College London（伦敦大学学院）
Deep neural networks have increased the accuracy of automatic segmentation, however, their accuracy depends on the availability of a large number of fully segmented images. Methods to train deep neural networks using images for which some, but not all, regions of interest are segmented are necessary to make better use of partially annotated datasets. In this paper, we propose the first axiomatic definition of label-set loss functions that are the loss functions that can handle partially segmented images. We prove that there is one and only one method to convert a classical loss function for fu…
Concordia University（康考迪亚大学）；Concordia Institute for Information Systems Engineering（康考迪亚信息系统工程研究所）；London Health Sciences Centre（伦敦健康科学中心）；Western University（韦仕敦大学）；SpinTech MRI（SpinTech磁共振成像公司）；York University（约克大学）；Wayne State University（韦恩州立大学）
Brain iron deposition, in particular deep gray matter nuclei, increases with advancing age. Hereditary Hemochromatosis (HH) is the most common inherited disorder of systemic iron excess in Europeans and recent studies claimed high brain iron accumulation in patient with Hemochromatosis. In this study, we focus on Artificial Intelligence (AI)-based differential diagnosis of brain iron deposition in HH via Quantitative Susceptibility Mapping (QSM), which is an established Magnetic Resonance Imaging (MRI) technique to study the distribution of iron in the brain. Our main objective is investigatin…
Brain tumor segmentation intends to delineate tumor tissues from healthy brain tissues. The tumor tissues include necrosis, peritumoral edema, and active tumor. In contrast, healthy brain tissues include white matter, gray matter, and cerebrospinal fluid. The MRI based brain tumor segmentation research is gaining popularity as; 1. It does not irradiate ionized radiation like X-ray or computed tomography imaging. 2. It produces detailed pictures of internal body structures. The MRI scans are input to deep learning-based approaches which are useful for automatic brain tumor segmentation. The fea…
Purpose: This study demonstrated an MR signal multitask learning method for 3D simultaneous segmentation and relaxometry of human brain tissues. Materials and Methods: A 3D inversion-prepared balanced steady-state free precession sequence was used for acquiring in vivo multi-contrast brain images. The deep neural network contained 3 residual blocks, and each block had 8 fully connected layers with sigmoid activation, layer norm, and 256 neurons in each layer. Online synthesized MR signal evolutions and labels were used to train the neural network batch-by-batch. Empirically defined ranges of T…
Medical image segmentation is challenging especially in dealing with small dataset of 3D MR images. Encoding the variation of brain anatomical struc-tures from individual subjects cannot be easily achieved, which is further chal-lenged by only a limited number of well labeled subjects for training. In this study, we aim to address the issue of brain MR image segmentation in small da-taset. First, concerning the limited number of training images, we adopt adver-sarial defense to augment the training data and therefore increase the robustness of the network. Second, inspired by the prior knowled…
Susceptibility tensor imaging (STI) is an emerging magnetic resonance imaging technique that characterizes the anisotropic tissue magnetic susceptibility with a second-order tensor model. STI has the potential to provide information for both the reconstruction of white matter fiber pathways and detection of myelin changes in the brain at mm resolution or less, which would be of great value for understanding brain structure and function in healthy and diseased brain. However, the application of STI in vivo has been hindered by its cumbersome and time-consuming acquisition requirement of measuri…
University Medical Center Utrecht（乌得勒支大学医学中心）；Wilhelmina Children’s Hospital（威廉明娜儿童医院）；Utrecht University（乌得勒支大学）
MR images of fetuses allow clinicians to detect brain abnormalities in an early stage of development. The cornerstone of volumetric and morphologic analysis in fetal MRI is segmentation of the fetal brain into different tissue classes. Manual segmentation is cumbersome and time consuming, hence automatic segmentation could substantially simplify the procedure. However, automatic brain tissue segmentation in these scans is challenging owing to artifacts including intensity inhomogeneity, caused in particular by spontaneous fetal movements during the scan. Unlike methods that estimate the bias f…
Aix Marseille Univ（艾克斯-马赛大学）；CNRS（法国国家科学研究中心）；CRMBM（生物医学与医学影像学研究中心）；APHM（Assistance Publique - Hôpitaux de Marseille（马赛公立医院系统））；CHU Timone（蒂莫讷大学医院）；CEMEREM（马赛医学研究中心）；NeuroPoly Lab, Institute of Biomedical Engineering, Polytechnique Montréal（蒙特利尔理工大学生物医学工程系NeuroPoly实验室）；MILA - Québec AI Institute（魁北克人工智能研究所（MILA））
The spinal cord (SC), which conveys information between the brain and the peripheral nervous system, plays a key role in various neurological disorders such as multiple sclerosis (MS) and amyotrophic lateral sclerosis (ALS), in which both gray matter (GM) and white matter (WM) may be impaired. While automated methods for WM/GM segmentation are now largely available, these techniques, developed for conventional systems (3T or lower) do not necessarily perform well on 7T MRI data, which feature finer details, contrasts, but also different artifacts or signal dropout. The primary goal of this stu…
Detailed delineation of the fetal brain in diffusion MRI via multi-task learning
Harvard Medical School（哈佛医学院）；Elmhurst Hospital Center（埃尔姆赫斯特医院中心）；Icahn School of Medicine at Mount Sinai（西奈山伊坎医学院）；Massachusetts General Hospital（麻省总医院）；University of California Irvine（加州大学欧文分校）
ResNet-34 with Lightweight Decoder for Accurate and Efficient Segmentation of Fetal Brain MRI
Department of Biomedical Physics and Technology, University of Dhaka（达卡大学生物医学物理与技术系）；Department of Electrical Engineering, College of Engineering, Qatar University（卡塔尔大学工程学院电气工程系）；Department of Biomedical Engineering, Jashore University of Science and Technology（贾沙尔大学科学与技术学院生物医学工程系）
Automatic segmentation of brain MR images into white matter (WM), gray matter (GM), and cerebrospinal fluid (CSF) is critical for tissue volumetric analysis and cortical surface reconstruction. Due to dramatic structural and appearance changes associated with developmental and aging processes, existing brain tissue segmentation methods are only viable for specific age groups. Consequently, methods developed for one age group may fail for another. In this paper, we make the first attempt to segment brain tissues across the entire human lifespan (0-100 years of age) using a unified deep learning…
University of Science, Ho Chi Minh City（胡志明市自然科学大学）；John von Neumann Institute, Ho Chi Minh City（胡志明市约翰·冯·诺依曼研究所）；Vietnam National University, Ho Chi Minh City（胡志明市越南国家大学）；University of Arkansas, Fayetteville（阿肯色大学费耶特维尔分校）
While Magnetic Resonance Imaging (MRI) has played an essential role in infant brain analysis, segmenting MRI into a number of tissues such as gray matter (GM), white matter (WM), and cerebrospinal fluid (CSF) is crucial and complex due to the extremely low intensity contrast between tissues at around 6-9 months of age as well as amplified noise, myelination, and incomplete volume. In this paper, we tackle those limitations by developing a new deep learning model, named DAM-AL, which contains two main contributions, i.e., dilated attention mechanism and hard-case attention loss. Our DAM-AL netw…
Quantitative susceptibility mapping (QSM) is a valuable magnetic resonance imaging (MRI) contrast mechanism that has demonstrated broad clinical applications. However, the image reconstruction of QSM is challenging due to its ill-posed dipole inversion process. In this study, a new deep learning method for QSM reconstruction, namely xQSM, was designed by introducing modified state-of-the-art octave convolutional layers into the U-net backbone. The xQSM method was compared with recentlyproposed U-net-based and conventional regularizationbased methods, using peak signal to noise ratio (PSNR), st…
University of Iceland（冰岛大学）；Landspitali - University Hospital（兰斯皮塔利大学医院）；The Icelandic Heart Association（冰岛心脏协会）；The Johns Hopkins University（约翰斯·霍普金斯大学）
Lesions that appear hyperintense in both Fluid Attenuated Inversion Recovery (FLAIR) and T2-weighted magnetic resonance images (MRIs) of the human brain are common in the brains of the elderly population and may be caused by ischemia or demyelination. Lesions are biomarkers for various neurodegenerative diseases, making accurate quantification of them important for both disease diagnosis and progression. Automatic lesion detection using supervised learning requires manually annotated images, which can often be impractical to acquire. Unsupervised lesion detection, on the other hand, does not r…
DeepMultiConnectome: Deep Multi-Task Prediction of Structural Connectomes Directly from Diffusion MRI Tractography
Eindhoven University of Technology（埃因霍温理工大学）；Brigham and Women’s Hospital（布莱尔妇女医院）；Harvard Medical School（哈佛医学院）；The University of Sydney（悉尼大学）；University of Electronic Science and Technology of China（电子科技大学）；Harvard-MIT Health Sciences and Technology（哈佛-麻省理工健康科学与技术）
Center for Imaging Research, Medical College of Wisconsin（威斯康星医学院成像研究中心）；Department of Biomedical Engineering, Marquette University and Medical College of Wisconsin（马凯特大学与威斯康星医学院生物医学工程系）；Department of Radiology, Medical College of Wisconsin（威斯康星医学院放射学系）
Quantitative susceptibility mapping (QSM) utilizes MRI phase information to estimate tissue magnetic susceptibility. The generation of QSM requires solving ill-posed background field removal (BFR) and field-to-source inversion problems. Because current QSM techniques struggle to generate reliable QSM in clinical contexts, QSM clinical translation is greatly hindered. Recently, deep learning (DL) approaches for QSM reconstruction have shown impressive performance. Due to inherent non-existent ground-truth, these DL techniques use either calculation of susceptibility through multiple orientation…
Image Sciences Institute, University Medical Center Utrecht（乌得勒支大学医学中心图像科学研究所）；Wilhelmina Children’s Hospital, University Medical Center Utrecht（乌得勒支大学医学中心威廉明娜儿童医院）；Brain Center Rudolf Magnus, University Medical Center Utrecht（乌得勒支大学医学中心鲁道夫·马格努斯脑中心）
Automatic neonatal brain tissue segmentation in preterm born infants is a prerequisite for evaluation of brain development. However, automatic segmentation is often hampered by motion artifacts caused by infant head movements during image acquisition. Methods have been developed to remove or minimize these artifacts during image reconstruction using frequency domain data. However, frequency domain data might not always be available. Hence, in this study we propose a method for removing motion artifacts from the already reconstructed MR scans. The method employs a generative adversarial network…
Automated brain structure segmentation is important to many clinical quantitative analysis and diagnoses. In this work, we introduce MixNet, a 2D semantic-wise deep convolutional neural network to segment brain structure in multi-modality MRI images. The network is composed of our modified deep residual learning units. In the unit, we replace the traditional convolution layer with the dilated convolutional layer, which avoids the use of pooling layers and deconvolutional layers, reducing the number of network parameters. Final predictions are made by aggregating information from multiple scale…
To rapidly obtain high resolution T2, T2* and quantitative susceptibility mapping (QSM) source separation maps with whole-brain coverage and high geometric fidelity. We propose Blip Up-Down Acquisition for Spin And Gradient Echo imaging (BUDA-SAGE), an efficient echo-planar imaging (EPI) sequence for quantitative mapping. The acquisition includes multiple T2*-, T2'- and T2-weighted contrasts. We alternate the phase-encoding polarities across the interleaved shots in this multi-shot navigator-free acquisition. A field map estimated from interim reconstructions was incorporated into the joint mu…
Center for Data Science, New York University（纽约大学数据科学中心）；New York University School of Medicine（纽约大学医学院）
Quantitative, volumetric analysis of Magnetic Resonance Imaging (MRI) is a fundamental way researchers study the brain in a host of neurological conditions including normal maturation and aging. Despite the availability of open-source brain segmentation software, widespread clinical adoption of volumetric analysis has been hindered due to processing times and reliance on manual corrections. Here, we extend the use of deep learning models from proof-of-concept, as previously reported, to present a comprehensive segmentation of cortical and deep gray matter brain structures matching the standard…
Technical University of Denmark（丹麦技术大学）；Copenhagen University Hospital Hvidovre（哥本哈根大学医院维兹奥勒医院）；Medical Image Analysis Center (MIAC AG)（医学图像分析中心（MIAC AG））；Technical University of Munich（慕尼黑工业大学）；Copenhagen University Hospital Bispebjerg（哥本哈根大学医院比斯佩比约医院）；University of Copenhagen（哥本哈根大学）；Massachusetts General Hospital（麻省总医院）；Harvard Medical School（哈佛医学院）
Here we present a method for the simultaneous segmentation of white matter lesions and normal-appearing neuroanatomical structures from multi-contrast brain MRI scans of multiple sclerosis patients. The method integrates a novel model for white matter lesions into a previously validated generative model for whole-brain segmentation. By using separate models for the shape of anatomical structures and their appearance in MRI, the algorithm can adapt to data acquired with different scanners and imaging protocols without retraining. We validate the method using four disparate datasets, showing rob…
Mayo Clinic（梅奥诊所）；Moffitt Cancer Center and Research Institute（莫菲特癌症中心与研究所）
Whole brain extraction, also known as skull stripping, is a process in neuroimaging in which non-brain tissue such as skull, eyeballs, skin, etc. are removed from neuroimages. Skull striping is a preliminary step in presurgical planning, cortical reconstruction, and automatic tumor segmentation. Despite a plethora of skull stripping approaches in the literature, few are sufficiently accurate for processing pathology-presenting MRIs, especially MRIs with brain tumors. In this work we propose a deep learning approach for skull striping common MRI sequences in oncology such as T1-weighted with ga…
Igor Sikorsky Kyiv Polytechnic Institute（伊戈尔·西科尔斯基基辅理工学院）
Images with abnormal brain anatomy produce problems for automatic segmentation techniques, and as a result poor ROI detection affects both quantitative measurements and visual assessment of perfusion data. This paper presents a new approach for fully automated and relatively accurate ROI detection from dynamic susceptibility contrast perfusion magnetic resonance and can therefore be applied excellently in the perfusion analysis. In the proposed approach the segmentation output is a binary mask of perfusion ROI that has zero values for air pixels, pixels that represent non-brain tissues, and ce…
Undoubtedly, textural property of an image is one of the most important features in object recognition task in both human and computer vision applications. Here, we investigated the neural signatures of four well-known statistical texture features including contrast, homogeneity, energy, and correlation computed from the gray level co-occurrence matrix (GLCM) of the images viewed by the participants in the process of magnetoencephalography (MEG) data collection. To trace these features in the human visual system, we used multivariate pattern analysis (MVPA) and trained a linear support vector …
Effective, robust, and automatic tools for brain tumor segmentation are needed for the extraction of information useful in treatment planning from magnetic resonance (MR) images. Context-aware artificial intelligence is an emerging concept for the development of deep learning applications for computer-aided medical image analysis. In this work, it is investigated whether the addition of contextual information from the brain anatomy in the form of white matter, gray matter, and cerebrospinal fluid masks and probability maps improves U-Net-based brain tumor segmentation. The BraTS2020 dataset wa…
Full-Head Segmentation of MRI with Abnormal Brain Anatomy: Model and Data Release
The City College of New York（纽约市立学院）；New York University（纽约大学）；Georgetown University（乔治城大学）；University of North Carolina School of Medicine（北卡罗来纳大学医学院）；Soterix Medical
Photoacoustic tomography (PAT) is a newly developed medical imaging modality, which combines the advantages of pure optical imaging and ultrasound imaging, owning both high optical contrast and deep penetration depth. Very recently, PAT is studied in human brain imaging. Nevertheless, while ultrasound waves are passing through the human skull tissues, the strong acoustic attenuation and aberration will happen, which causes photoacoustic signals' distortion. In this work, we use 10 magnetic resonance angiography (MRA) human brain volumes, and manually segment them to obtain the 2D human brain n…
University of California, San Francisco（加州大学旧金山分校）
Quantitative susceptibility mapping (QSM) is a powerful MRI technique that has shown great potential in quantifying tissue susceptibility in numerous neurological disorders. However, the intrinsic ill-posed dipole inversion problem greatly affects the accuracy of the susceptibility map. We propose QSMGAN: a 3D deep convolutional neural network approach based on a 3D U-Net architecture with increased receptive field of the input phase compared to the output and further refined the network using the WGAN with gradient penalty training strategy. Our method generates accurate QSM maps from single …
University of Sydney（悉尼大学）；University of New South Wales（新南威尔士大学）；Sydney Neuroimaging Analysis Centre（悉尼神经影像分析中心）
Multiple sclerosis (MS) is a chronic inflammatory and degenerative disease of the central nervous system, characterized by the appearance of focal lesions in the white and gray matter that topographically correlate with an individual patient's neurological symptoms and signs. Magnetic resonance imaging (MRI) provides detailed in-vivo structural information, permitting the quantification and categorization of MS lesions that critically inform disease management. Traditionally, MS lesions have been manually annotated on 2D MRI slices, a process that is inefficient and prone to inter-/intra-obser…
Connectivity information derived from diffusion-weighted magnetic resonance images~(DW-MRIs) plays an important role in studying human subcortical gray matter structures. However, due to the $O(N^2)$ complexity of computing the connectivity of each voxel to every other voxel (or multiple ROIs), the current practice of extracting connectivity information is highly inefficient. This makes the processing of high-resolution images and population-level analyses very computationally demanding. To address this issue, we propose a more efficient way to extract connectivity information; briefly, we con…
GE Research（通用电气研究院）；GE Healthcare（通用电气医疗）；Mayo Clinic（妙佑医疗国际）；Walter Reed National Military Medical Center（沃尔特里德国家军事医疗中心）；Uniformed Services University（美国三军卫生科学大学）
Purpose: To accelerate brain 3D MRI scans by using a deep learning method for reconstructing images from highly-undersampled multi-coil k-space data Methods: DL-Speed, an unrolled optimization architecture with dense skip-layer connections, was trained on 3D T1-weighted brain scan data to reconstruct complex-valued images from highly-undersampled k-space data. The trained model was evaluated on 3D MPRAGE brain scan data retrospectively-undersampled with a 10-fold acceleration, compared to a conventional parallel imaging method with a 2-fold acceleration. Scores of SNR, artifacts, gray/white ma…
Parkinson's Disease(PD) is one of the major nervous system disorders that affect people over 60. PD can cause cognitive impairments. In this work, we explore various approaches to identify Parkinson's using Magnetic Resonance (MR) T1 images of the brain. We experiment with ensemble architectures combining some winning Convolutional Neural Network models of ImageNet Large Scale Visual Recognition Challenge (ILSVRC) and propose two architectures. We find that detection accuracy increases drastically when we focus on the Gray Matter (GM) and White Matter (WM) regions from the MR images instead of…
Deep Generative Model-Based Generation of Synthetic Individual-Specific Brain MRI Segmentations
Department of Informatics, University of Zurich（信息学院，苏黎世大学）；School of Computing, Dublin City University（计算学院，都柏林城市大学）；UZH Healthy Longevity Center, University of Zurich（UZH健康长寿中心，苏黎世大学）；Department of Psychology, University of Zurich（心理学系，苏黎世大学）
Purpose: Arterial Spin Labeling (ASL) is a quantitative, non-invasive alternative to perfusion imaging with contrast agents. Fixing values of certain model parameters in traditional ASL, which actually vary from region to region, may introduce bias in perfusion estimates. Adopting Magnetic Resonance Fingerprinting (MRF) for ASL is an alternative where these parameters are estimated alongside perfusion, but multiparametric estimation can degrade precision. We aim to improve the sensitivity of ASL-MRF signals to underlying parameters to counter this problem, and provide precise estimates. We als…
