# SenseCare: A Research Platform for Medical Image Informatics and Interactive
3D Visualization

###### Abstract

Clinical research on smart healthcare has an increasing demand for intelligent
and clinic-oriented medical image computing algorithms and platforms that
support various applications. To this end, we have developed SenseCare
research platform for smart healthcare, which is designed to boost
translational research on intelligent diagnosis and treatment planning in
various clinical scenarios. To facilitate clinical research with Artificial
Intelligence (AI), SenseCare provides a range of AI toolkits for different
tasks, including image segmentation, registration, lesion and landmark
detection from various image modalities ranging from radiology to pathology.
In addition, SenseCare is clinic-oriented and supports a wide range of
clinical applications such as diagnosis and surgical planning for lung cancer,
pelvic tumor, coronary artery disease, etc. SenseCare provides several
appealing functions and features such as advanced 3D visualization, concurrent
and efficient web-based access, fast data synchronization and high data
security, multi-center deployment, support for collaborative research, etc. In
this paper, we will present an overview of SenseCare as an efficient platform
providing comprehensive toolkits and high extensibility for intelligent image
analysis and clinical research in different application scenarios.

Index Terms—  Medical Imaging, Artificial Intelligence, Computer Aided
Diagnosis, Smart Health

##  1 Introduction

With the development of medical imaging techniques and computer science,
computer-aided systems for medical image analysis and downstream diagnosis and
treatment decision have been playing an increasing role in clinic practices.
In recent years, Artificial Intelligence (AI) has lead to a revolution of
image analysis and pattern recognition, and has a huge potential to be applied
for more efficient and intelligent medical image computing in a wide range of
medical departments towards smart healthcare. However, before AI is used in
clinic practice, extensive research studies are needed through the
collaboration between clinicians, radiologists, pathologists, surgeons, AI
scientists and engineers, which can validate the effectiveness, robustness,
reliability and security of AI systems. To this end, a research platform that
supports different medical image processing tasks and intelligent medical
image computing algorithms for various clinic applications are highly
desirable.

Despite the availability of several existing platforms for developing AI
algorithms for general image recognition or medical image computing, they are
not clinic-oriented and have limited support for clinical research. For
example, TensorFlow [1], Pytorch [2], TensorLayer [3] and Keras [4] are
general deep learning libraries that provide low-level functions to develop
complex deep learning models without specific functionality for medical image
computing. Some other libraries such as NiftyNet [5] and DLTK [6] are
developed for deep learning with medical images, but they are mainly designed
for AI algorithm developers and there is no graphic user interface, which is
difficult for clinicians and radiologists to use in specific clinic
applications. On the other hand, several medial image analysis solutions have
also been developed in the past decades. Tools that solve a specific medical
image processing task such as segmentation (e.g., NiftySeg [7]), registration
(e.g., NfityReg [8] and elastix [9]) and visualization (e.g., VTK [10]) can be
used for a part of a clinic application pipeline, but still not ready-to-use
for clinic researchers. Some research platforms such as MITK [11], NifTK [12]
and 3D Slicer [13] provide several traditional medical image analysis tools
and 3D visualization for image guided intervention. However, these platforms
have almost no AI models and are not suitable for clinical research on
intelligent medical image computing towards smart healthcare. In the era of
big data, a desired clinical research platform based on AI does not only need
to provide AI models, but also require data exchange with other image
management systems, functionality across diverse image modalities,
availability of sophisticated 3D visualization and high extensibility and
portability for different clinic scenarios.

Fig. 1: Architecture of SenseCare platform.

To solve these problems, we introduce the SenseCare Smart Health Platform
(SenseCare for short) that aims to take advantages of state-of-the-art AI
techniques to foster researchers from different clinical departments to
implement innovations for improvement through the whole process of clinical
diagnosis, treatment planning and rehabilitation management. SenseCare has the
following appearing functionalities for clinical research towards smart
healthcare:

  1. 1.

Equipped with powerful AI toolkits. SenseCare has integrated a wide range of
ready-to-use AI models for different medical image computing tasks such as
image segmentation, registration, lesion and landmark detection. It also
provides convenient tools for image annotation, model training and validation
for developing deep learning algorithms customized by users.

  2. 2.

Support for various applications in clinical research. SenseCare is clinic-
oriented and can be easily used by clinicians, radiologists and other clinical
researchers with several functional modules such as advanced 3D visualization,
data and research task management. Its use is not constrained to a specific
department or laboratory. SenseCare can be adapted to various clinic
applications including lung cancer diagnosis and treatment planning,
pathological image analysis, coronary artery disease diagnosis and modeling,
surgical planning and simulation for pelvic tumors, etc.

  3. 3.

Easy to access with high concurrency. Services of SenseCare can be accessed on
a browser and it does not rely on a specific device or operating system. In
addition, its high concurrency allows hundreds of users to use the intelligent
image computing services simultaneously. Its multi-center deployment makes it
much easier for researchers from multiple departments in different locations
to collaborate and perform multi-center studies.

  4. 4.

Support for various data modalities with high synchronization and security.
SenseCare can deal with various data modalities covering radiological and
pathological images, medical records and other types of data. It also provides
efficient synchronization of data from different information systems and
across multiple centers, and uses encryption algorithms to protect user
accounts and data from latent risk factors.

In this paper, we introduce favorable features distinguishing SenseCare from
other platforms and workstations targeting at AI-driven clinical research on
intelligent diagnosis and treatment towards smart healthcare. SenseCare has
increasingly assisted to achieve outputs in several research projects
including quantitative analysis of cardiac function [14], assessment of knee
articular cartilages [15], pathological image analysis [16, 17, 18], lung
cancer diagnosis [19, 20], quantitative brain tumor assessment [21, 22], spine
image analysis [23, 24] and radiotherapy planning for head and neck cancer
[25, 26], etc.

This paper is organized as follows: In Section 2, we give a brief summary of
the architecture of SenseCare, which is followed by detailed descriptions of
basic functional modules in Section 3. We then introduce SenseCare’s AI
toolkits in Section 4. In Section 5, we show several examples of clinical
applications powered by SenseCare. Finally, conclusions are given in Section
6.

##  2 Architecture Of SenseCare

As shown in Fig.1, SenseCare provides a wide range of artificial intelligence
algorithms based on deep learning for learning from and analyze different
kinds of medical big data. With these AI algorithms and some other supporting
modules such as data management, users can easily adapt the deep learning
framework to various clinical applications efficiently. SenseCare also
provides advanced visualization of medical images that enables users to
analyze complex anatomies and segmented structures. These modules are combined
with a browser/server architecture and multi-center deployment so that they
are accessible on different kinds of devices and at various locations.

The architecture of SenseCare follows a modular structure that consists of
three layers: 1) basic functional modules such as data management and
visualization, web-based access and multi-center deployment, 2) advanced AI
toolkits that include libraries for model training and a range of built-in
artificial intelligence algorithms for image segmentation, registration,
lesion detection, etc., and 3) application scenarios that adapt the basic
functional modules and AI algorithms to deal with different specific clinic
tasks such as computer assisted diagnosis of the lung and surgical planning
for bone tumors.

##  3 Basic Functional Modules

###  3.1 Data Support

Due to different imaging mechanisms, medical images are acquired in a variety
of modalities or protocols. Being able to deal with these images is important
for an intelligent image computing system to be useful in a clinical research
environment. Besides images, other medical data such as medical records are
also important for diagnosis of diseases and clinical follow-up. SenseCare
supports medical images in different modalities, and provides efficient and
convenient synchronization of data from different information systems in the
hospital or medical center.

####  3.1.1 Support of Various Imaging Modalities

SenseCare supports intelligent analysis of in images in various modalities
ranging from radiological images to pathological images. For radiological
images, it allows efficient import, query, retrieval, and storage of clinical
images using DICOM protocols and structures. Major radiological images
including Computed Tomography (CT), Magnetic Resonance Imaging (MRI), Digital
Radiology (DR) and Positron Emission Tomography (PET) are all well supported
by SenseCare in different clinical applications.

For pathological images, SenseCare supports several image formats including
SVS, TIFF, VMS, NPDI, KFB and others. It also provides a series of functions
of import, query, retrieval, storage, management, common measurements and
analysis to help pathologists perform diagnosis in a more efficient and
intuitive way.

####  3.1.2 Data Synchronization and Security

Medical data are commonly stored in different systems such as the Picture
Archiving and Communication System (PACS) and Radiology Information System
(RIS). Synchronization of data between these systems and the image computing
workstation is critical in a clinical research environment. To facilitate more
efficient and functional workflow for clinical researchers and medical
practitioners, SenseCare provides users with improved access by efficient data
synchronization. It is capable of synchronizing data from PACS, RIS and other
common information systems in hospitals without disturbing the original
clinical workflows. When newly acquired data are transmitted or changes in
status take place in these information systems, SenseCare will synchronize the
updated information and present the users with the latest information
automatically.

Besides real-time data synchronization, SenseCare also supports pulling data
directly from PACS/RIS based on user-defined rules. For example, users can
designate image modality and type or time range, and send queries to fetch the
data they want from a database. They can also synchronize all the follow-up
data after fetching the data of a patient based on patient ID. As for
pathological applications, SenseCare can be seamlessly connected to common
pathological scanners to obtain data, and users can retrieve digital slides by
giving their labels.

In addition, encryption algorithms embedded in SenseCare reliably manage and
protect user accounts and data from potential risk factors. Although data and
images are utilized and transmitted over a shared database on which multiple
users manipulate, SenseCare strictly ensures data security, integrity and
consistency.

###  3.2 Advanced Visualization

Fig. 2: 3D reconstruction of the lung in SenseCare.

SenseCare provides advanced 3D reconstruction and visualization of medical
images to facilitate the analysis of complex anatomies and segmented
structures, which presents data and information in a more explicit way thus
improving the information interpretation. Comprehensive methods such as
Maximum Intensity Projection (MIP), Minimum Intensity Projection (MinIP),
Multi-Planar Reconstruction (MPR), Curved Planar Reformation (CPR) and 3D
volume rendering are available for users to perform 3D visualization and
enhance interactivity. These capabilities play an important role in clinical
diagnosis meanwhile contributing to surgical planning, simulation and
navigation, as well as radiotherapy planning, etc.

For instance, by providing 3D visualization of thoracic anatomies and
intervention path recommendation, SenseCare assists clinicians in performing
preoperative planning in a more intuitive and convenient way. Fig. 2 shows an
example of 3D visualization of lung structures performed by SenseCare. The
system provides automatic rotation of 3D models so that users can observe the
models from different views for a comprehensive understanding. Users can also
interactively zoom, pan and rotate the view and click to hide tracts or
vessels in real time.

###  3.3 Concurrent and Efficient Web-based Access

SenseCare is designed with high performance concurrency. The network
communication structure of SenseCare enables hundreds of users to concurrently
perform high-resolution image rendering and 3D post-progressing thus
satisfying multiple usage needs. With servers deployed, SenseCare platform
allows over 1,000 concurrent users to review and retrieve medical images while
its comprehensive toolkits for advanced image post-processing are designed to
support more than 160 users simultaneously.

SenseCare can be used in different devices and operating systems and no
plugins are needed. By adopting a browser/server architecture, it offers a
truly seamless user experience and eliminates the need for multiple logons.
Users can access not only MPR, MIP/MinIP, CPR tools, but also the full range
of three-dimensional capabilities through HTML5 websites. SenseCare grants
radiologists and physicians with efficient access and workflow-boosting
benefits even when working from iPad, smartphone or laptop.

###  3.4 Multi-center Deployment

Collecting data from multiple centers is important for developing robust
algorithms and large-scale clinic studies. To support such research that
requires a collaboration between researchers from different locations,
SenseCare can be deployed at multiple centers. This favorable feature
distinguishes SenseCare from traditional image computing workstations that are
located at a single institution.

In addition, SenseCare’s multi-center deployment facilitates the data
collection and access process. Various types of data from different centers
can be collected and cleaned under standardized rules. Data scientists,
clinical researchers, and pharmacists are all able to participate in the
process while respecting the original clinical process.

###  3.5 Support for Collaborative Research

To facilitate the collaboration between researchers, SenseCare provides a
specific Document Management System (DMS) to make it easier to organize,
secure, capture, digitize, tag, approve and complete tasks with research-
related files. It can handle a large amount of papers and images and cope with
different workflows by supporting advanced user permission management and task
management. This efficient tool enables researchers to manage multiple
sophisticated research projects easily, and helps a ground of researchers
collaborate with each other and exchange knowledge for efficient
accomplishment of comprehensive research projects.

For user permission management, SenseCare supports different levels of users.
Admin users can create normal users and set permissions for other normal users
to access, view, manipulate, share and remove folders and files. Nonetheless,
Admin users can change the permission if needed. For task management,
SenseCare’s DMS allows admin users to break down tasks and delegate tasks to
normal users. It also allows users to track task progress from the beginning
to the end and set small milestones to make sure the whole project will be
finished on time. Users can prioritize, organize and set deadlines for
themselves and are able to draw together the resources they need to achieve
their research goals.

##  4 Artificial Intelligence Toolkits

Recent years have seen a fast growing of novel deep learning algorithms for
medical image computing tasks [27], which play an important role for more
accurate and efficient diagnosis and treatment planning and assessment. To
boost research on deep learning algorithms for medical imaging, SenseCare has
integrated commonly used traditional image processing algorithms with advanced
state-of-the-art deep learning models. Based on these features, SenseCare can
largely improve the efficiency of development and validation of novel deep
learning algorithms by users. In this section, we introduce SenseCare’s AI
toolkits for medical image computing, including tools for users to develop new
algorithms, and built-in deep learning models for image segmentation,
registration and lesion detection, etc.

###  4.1 Deep Learning Framework

SenseCare’s AI toolkits are constructed on the basis of SenseParrots, a deep
learning framework independently developed by SenseTime. While matching
mainstream frameworks such as TensorFlow [1] and PyTorch [2], it also has
advantages in ultra-deep networks and ultra-large-scale data model training.
The integrated optimization from the underlying layer to the SenseParrots
framework enables the platform to outperform others under the same level of
configuration. Currently, the testing of a single training task uses more than
1,000 GPU card for parallel training. It takes less than 1.5 minutes to
complete the AlexNet neural network training, surpassing the previous fastest
4-minute record [28].

###  4.2 Annotation Tools for Model Training

Annotations of a large set of training images are critical for achieving high
performance of deep learning models. To facilitate the development of these
models, SenseCare provides a set of off-the-shelf tools for efficient image
annotation, such as contouring tumors and organs for segmentation tasks,
bounding box annotation for object and lesion detection tasks.

The annotation tools in SenseCare are available for both radiological images
and pathological images and they provide a variety of annotation types. For
example, users can choose from different interactive styles including mouse
click points, rectangles, circles, ellipses, polygons or hand-drawn shapes
based on the characteristics of the target. Since manual annotation is time-
consuming and annotation may vary from different annotators’ inputs, SenseCare
also supports semi-automatic annotation, which combines the efficiency of
automatic annotation and the accuracy of manual annotation. Annotators can
start from automatically generated annotations and only need to provide few
interactions to obtain refined annotations, which can largely reduce burdens
of annotators and improve the efficiency. As an example, Fig. 3 shows
efficient annotation of signet ring cell carcinoma in SenseCare, where
algorithms suggested some annotations in Fig. 3 (a), and the annotator only
provided manual refinement to obtain accurate annotations for a slide in Fig.
3 (b).

Fig. 3: Annotation of signet ring cell carcinoma.

###  4.3 Detection

Fig. 4: Lung nodule detection by SenseCare.

Automatic detection of objects of interest is a common task for computer
assisted diagnosis systems. SenseCare has several built-in deep learning
models for object detection tasks, such as landmark and lesion detection in 3D
radiological images and cancer cell detection in histopathological images.

For example, automatic localization of vertebrae in CT is important for image-
guided diagnosis, pre-operative planning and post-operative evaluation. Deep
learning models such as 3D Fully Convolutional Neural Networks (FCN) are
embedded in SenseCare for accurate vertebrae localization [24]. The model also
takes advantages of prior knowledge such as spatial and sequential constraints
to obtain high robustness in challenging cases. In addition, SenseCare is able
to predict tumor invasiveness and malignant in Ground Glass Opacity (GGO) on
the basis of its lung nodule detection model [19]. Fig. 4 shows an example of
lung nodule detection by deep learning models in SenseCare. For signet ring
cell carcinoma detection from Hematoxylin and Eosin (H&E) stained Whole Slide
Images (WSI), SenceCare is integrated with bottom-up approaches [16] that
obtain cell instance masks first and then derive bounding boxes for each
instance, which is more accurate than the general RCNN-based detection
methods.

###  4.4 Segmentation

Image segmentation is essential for most clinical applications such as
accurate modeling of anatomical structures, quantitative measurement of tumor
volumes, planning of radiotherapy and surgical treatment. Its output has a
large impact on the downstream workflows. However, due to the low contrast
between the target tissue and its surroundings, inhomogeneous appearance,
complex shape variation and image noise, accurate segmentation is extremely
challenging and traditional image segmentation algorithms are often faced with
large regions of over- and under-segmentation.

Supported by its deep learning-based image segmentation models, SenseCare can
overcome these challenges and has obtained state-of-the-art performance in a
range of segmentation tasks. For example, to segment the complex structures of
pulmonary vessels from CT images, SenseCare is equipped with a multi-view-
based 2.5D network with a low complexity, which outperformed other
contemporary networks by a large margin on the LIDC dataset [29]. To segment
intervertebral discs from MR images, a novel multi-resolution path network
with deep supervision is included in SenseCare and it achieved superior
performance on the MICCAI 2018 IVDM3Seg challenge dataset [23]. Several
specific CNN models are also developed for other applications, such as
multiple Organs-at-Risk (OAR) segmentation from CT for radiotherapy planning
[25], segmentation of optic disc and cup for glaucoma diagnosis [30], nuclei
instance segmentation in histopathological images [17, 18] and cartilage
segmentation from MR images for osteoarthritis assessment [15]. These
different models are ready-to-use and serve as strong baselines for the above
specific applications, and they can be easily adapted to new segmentation
tasks based on the user’s research interests.

###  4.5 Image Registration

In clinical applications, images acquired in different modalities are often
need to be fused to provide sufficient information for diagnosis and treatment
decision. In addition, a patient may be scanned several times at different
stages of a disease to obtain a better understanding of the evolution of the
disease. Therefore, it is necessary to register two or more images into a
common spatial coordinate system for better interpretation of anatomical and
pathological characteristics thus improving diagnosis and treatment for
patients.

Fig. 5: An example of non-rigid lung image registration in SenseCare.

SenseCare provides several registration algorithms for different tasks. For
example, the combination of MR and CT is quite useful since the former is
better suited for delineation of tumor regions while the latter is needed for
accurate computation of the radiation dose. Registration between MR and CT
images supported by SenseCare makes it more efficient and accurate to obtain
radiotherapy planning. Registration algorithms embedded in SenseCare include
both rigid and non-rigid registration between images from a single modality or
multiple modalities. Rigid methods are useful for the the registration in the
presence of rigid bodies such as bones. Non-rigid registration is used for
applications such as correcting soft-tissue deformation during imaging and
modeling the dynamic heart. Fig. 5 shows an example of non-rigid lung image
registration in SenseCare.

###  4.6 Docker Integration

Users of SenseCare can employ the built-in deep learning models mentioned
above for several image computing tasks, and they can also develop their own
models with the help of SenseCare’s training and testing pipelines. When the
user creates a new algorithm, SenseCare provides a dockerized version of the
algorithm, and it can package the algorithm with all its dependencies together
in the form of containers, so that the algorithm can work seamlessly in any
new environment. Therefore, researchers can focus on the development of AI
algorithms without worrying about the testing and production environment.
Since dockers are lightweight, SenseCare makes it more convenient and
efficient for users to develop, test and deploy algorithms and deep learning
models for various clinical applications.

##  5 Clinical Applications

Based on the above basic functional modules and AI toolkits with high
extensibility, the most important high-level goal of SenseCare is to serve as
a platform for clinical research in various applications. In this section, we
give several examples of user scenarios where SenseCare is adapted for
different specific clinical applications.

###  5.1 Lung-oriented Application

Fig. 6: Intelligent analysis of lung nodules and preoperative planning in
SenseCare.

Lung cancer is a malignant lung tumor and becomes the leading cause of death
in China. It has a five-year survival rate of 16%-17%, which is the lowest of
all types of cancers. Researches have shown that early diagnosis and
intervention will improve the five-year survival rate up to 54%. Although
computer-assisted image analysis have been adopted to facilitate the lung
cancer detection and diagnosis in an early stage, the various sizes, shapes
and types of pulmonary nodules are placing great burdens on clinical
physicians, thus causing misdiagnosis and missed diagnosis due to their
fatigue and overwork. For lung cancer treatment, current clinic practice has a
lack of effective preoperative planning tools. As a result, respiratory
physicians find it difficult to personalize treatment plans in the surgical
practices.

Based on its leading algorithms, SenseCare supports a thorough research and
analysis of pulmonary nodules by automatic detection, segmentation, and
quantitative analysis. It can automatically detect and locate the nodules [19]
and then provide further quantitative information of each nodule such as its
volume and density, in addition to qualitative estimation of its type and
malignancy. As shown in Fig. 4, four kinds of nodules are automatically
detected and distinguished by SenseCare [20]. In addition, it offers
intelligent three-dimensional surgical planning for clinicians. By providing
3D visualization of thoracic anatomies including nodules, respiratory tracts
and pulmonary vessels, as well as bronchoscopy intervention path
recommendation, SenseCare facilitates clinicians performing preoperative
planning in a more intuitive and convenient way. Fig. 6 displays the user
interface of a lung image analysis and preoperative planning application in
SenseCare.

###  5.2 Pathology-related Application

Pathological diagnosis is regarded as the most reliable criteria for diagnosis
of cancers such as Gastrointestinal cancer. An early diagnosis is critical for
good prognosis. However, it is labor-insensitive and time-consuming to
manually detect lesion areas and pathological cells from
100,000$\times$100,000-pixel whole slide pathological images, which easily
leads to fatigue of human and cannot satisfy the increasing demand for
pathological diagnosis due to a lack of well trained experts in developing
countries.

SenseCare-Pathology is designed to support pathological diagnosis covering
gastrointestinal and cervical diseases at both tissue and cell levels. It
enables pathology department of hospitals and the third-party pathological
diagnosis centers to perform efficient and intelligent analysis of
pathological images by providing AI-based functional modules such as lesion
and cell detection [16], nuclei/gland segmentation [18] and cell segmentation
[31], etc. Especially, SenseCare supports efficient retrieval of pathological
images from a large-scale database [31, 32, 33]. Users can employ the
retrieval module to search for similar images for a given input image. These
modules help clinical researchers build capability to conduct large-scale
cancer screening projects. Fig. 7 shows an example of lesion localization from
pathological images.

Fig. 7: Pathological image analysis in SenseCare.

###  5.3 Pelvic Tumor Surgical Planning

Fig. 8: Pelvic tumor surgical planning system in SenseCare.

Pelvic tumor is one of highly malignant tumors, and surgical treatment is the
most effective treatment for pelvis tumors, where an accurate preoperative
simulation and planning based on segmentation and modeling of the tumor is
critical for the success of surgery.

SenseCare provides an intelligent preoperative surgical planning for for limb
salvage surgery of malignant pelvic tumors, where the accuracy and efficiency
is improved by our deep learning-based algorithms. The surgical planning
workflow mainly consists of three parts. First, the pelvic tumor is segmented
from MR scans with a U-Net like model [34]. Then the pelvic bone is segmented
from CT scans with a CNN combined with self-attention blocks [35]. Finally, a
robust rigid/affine inverse-consistency registration method that is an
extension of SymMirorr [36] is conducted to align the corresponding MR-CT
pair. Based on the registered CT-MR pair and the corresponding segmentation
results, surgeons and radiologists could make accurate preoperative surgical
planning rapidly. Fig. 8 shows the user interface of pelvic tumor surgical
planning system in SenseCare. The system efficiently cuts down the multi-party
communication costs between radiology, orthopedics and 3D printing centers,
and ultimately reduces doctor’s workload and facilitates patient-tailored
treatment.

###  5.4 Automatic Diagnosis of Coronary Artery Disease

Fig. 9: Coronary artery application in SenseCare.

Cardiovascular diseases (CVDs) are the most common causes of death throughout
the world. Non-invasive morphological and functional assessment of
cardiovascular structures plays an important role in diagnosis and treatment
of CVDs. However, accurate and robust reconstruction of cardiovascular system
from Computed Tomography Angiography (CTA) is laborious and time-consuming due
to the large variation of heart and low contrast between arteries and other
tissues. SenseCare provides a full-stack solution towards automatic diagnosis
of coronary artery disease, including fully automatic segmentation of 3D whole
heart and coronary arteries, extraction of coronary artery centerlines,
labeling of important artery branches, reconstruction of MPR and CPR, real-
time volumetric rendering, detection of plaques and quantification of
stenosis, and automatic generation of diagnose reports.

More specifically, we propose a novel cascaded method for automatic
segmentation of 3D whole heart and coronary arteries, which is integrated into
a semi-supervised framework [37] and has achieved state-of-the-art performance
using very few manual annotations. A unified deep reinforcement learning [38]
framework is proposed to automatically traverse tree-structure centerlines of
coronary arteries. Pixel-level segmentation followed by 3D classification and
segmentation of point sets [39] is proposed for coronary artery labeling. We
utilize a recurrent CNN to automatic detect and classify the type of coronary
artery plaque [40], and the degree of coronary artery stenosis is quantified
by a multi-class segmentation of plaque and vessel lumen from the
reconstructed probe images. Finally, a structured diagnose report is
summarized based on the results obtained from all steps mentioned above, and
the whole procedure is finished within one minute. Fig. 9 shows a snapshot of
the coronary artery application in SenseCare.

###  5.5 Lower Limb-oriented Application

Fig. 10: Lower limb application in SenseCare. (a) shows limb parameters
measured on (b). (b) and (c) show the detected landmarks and HTO planning,
respectively.

Knee Osteoarthritis (OA) is an important public health issue that causes
chronic disability. Lower limb osteotomies is a well-established and commonly
utilized technique in medial knee osteoarthritis secondary to different forms
of knee joint malalignments, trying to establish a better alignment by passing
the load bearing leg axis through hip, knee and ankle joint. The measurement
of lower limb axial alignment with radiographs is a critical step in the
preoperative planning to exactly define the characteristic of the osteotomy.

SenseCare’s lower limb surgical planning system provides advanced AI
algorithms for automatic radiograph analysis such as segmentation of femur and
tibia and detection of lower limb landmarks for accurate measurement of the
critical angles and distances of lower limbs. The results are used for
planning of surgical treatments such as High Tibial Osteotomy (HTO), as shown
in Fig. 10. The segmentation of femur and tibia is achieved by a U-Net [34]
followed by post-processing based on connected components to discriminate the
left and right instances. Localization of landmarks is based on a coarse-to-
fine recurrent network [41] combined with Gaussian heatmap regression [42].
The heatmap corresponding to a landmark location is a sum of multiple Gaussian
functions centered at that landmark. At inference time, the largest values in
the heatmap are taken as the detected landmark positions. Such a strategy
helps to achieve stable and accurate limb landmark detection results that
ensure the reliability of downstream surgical planning.

##  6 Conclusion and Future Work

With the development of medical imaging techniques and artificial
intelligence, AI-based medical image computing systems have an increasing
potential for more intelligent diagnosis and treatment in clinic practice.
However, existing deep learning platforms rarely support specific clinical
applications while most current medical image analysis platforms are not well
supported by advanced AI algorithms. To boost clinical research towards smart
healthcare, SenseCare is developed as a generic research platform for
intelligent medical image computing that can support various research needs
across different medical disciplines.

SenseCare provides a large set of AI algorithms for medical image
segmentation, registration, detection and other tasks that can help clinicians
and radiologists conduct various clinic-oriented translational research
programs, such as lung cancer diagnosis and surgical planning, efficient
pathological image analysis, pelvic tumor and limb surgical planning, coronary
artery disease diagnosis and modeling, etc. In addition to the built-in AI
algorithms, SenseCare also provides several tools for users to develop and
deploy customized AI models efficiently. The AI toolkits and other appearing
functional modules such as advanced visualization, web-based access and multi-
center deployment in SenseCare can efficiently boost clinical research
programs and applications towards smart healthcare.

In the future, we will continue working on improving the reliability and
stability of SenseCare while enlarging its capability and functionality for
wider use scenarios. It should be emphasized that SenseCare is temporarily
only used for research purposes and we are conducting clinical studies under
regulations, to translate AI techniques into a revolution in clinic practice
with more efficient, accurate and intelligent healthcare.

##  7 Acknowledgements

We would like to express our gratitude to our research and product team and
clinical collaborators, without whom we cannot develop our algorithms and
user-interface so efficiently and rapidly.

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# Effect of dynamical traps on chaotic transport in a meandering jet flow

M.Yu. Uleysky, M.V. Budyansky, and S.V. Prants

###### Abstract

We continue our study of chaotic mixing and transport of passive particles in
a simple model of a meandering jet flow [Prants, et al, Chaos 16, 033117
(2006)]. In the present paper we study and explain phenomenologically a
connection between dynamical, topological, and statistical properties of
chaotic mixing and transport in the model flow in terms of dynamical traps,
singular zones in the phase space where particles may spend arbitrary long but
finite time [Zaslavsky, Phys. D 168–169, 292 (2002)]. The transport of passive
particles is described in terms of lengths and durations of zonal flights
which are events between two successive changes of sign of zonal velocity.
Some peculiarities of the respective probability density functions for short
flights are proven to be caused by the so-called rotational-islands traps
connected with the boundaries of resonant islands (including the vortex cores)
filled with the particles moving in the same frame and the saddle traps
connected with periodic saddle trajectories. Whereas, the statistics of long
flights can be explained by the influence of the so-called ballistic-islands
traps filled with the particles moving from a frame to frame.

##  1 Introduction

Chaotic advection of water masses with their physical and biochemical
characteristics in quasi two-dimensional geophysical flows in the ocean and
atmosphere can be studied within the framework of Hamiltonian dynamics. In a
recent paper [1] we have studied chaotic mixing and transport of passive
particles in a simple kinematic model of a meandering jet flow motivated by
the problem of lateral mixing in the western boundary currents in the ocean.
We found all the possible bifurcations of advection equations, described the
structure of the phase space (which is the physical space for advected
particles), and computed some statistical characteristics of chaotic
transport. In the present paper we establish a phenomenological connection
between dynamical, topological, and statistical properties of chaotic
transport and mixing in the same flow. Specific singular zones in the phase
space where particles may spend arbitrary long but finite time (dynamical
traps in terminology by Zaslavsky [2]), are responsible for anomalous
statistical properties. The dynamical traps, connected with rotational islands
and saddle trajectories, are responsible, mainly, for anomalous mixing,
whereas those ones, connected with ballistic islands — for anomalous
transport. These dynamical traps may have strong impact on transport and
mixing in real geophysical jet flows.

Methods of the theory of dynamical systems are actively used to describe
advection of water (air) masses and their properties in the ocean and
atmosphere [1, 14, 13, 15, 16, 26, 3, 4, 5, 6, 7, 17, 18, 19, 20, 21, 32]. The
geophysical jet currents, like the Gulf Stream, the Kuroshio, the Antarctic
circumpolar current, and others in the ocean and the polar night Antarctic jet
in the atmosphere, are robust structures whose form typically changes in space
and time in a meander-like way. If advected particles rapidly adjust their own
velocity to that of background flow and do not affect the flow properties,
then the particles are called passive (scalars, tracers, or Lagrangian
particles) and their equation of motion is very simple

$\frac{d{\mathbf{r}}}{dt}={\mathbf{v}}({\mathbf{r}},t)$ (1)

where ${\mathbf{r}}=(x,y,z)$ and ${\mathbf{v}}=(u,v,w)$ are the position
vector and the particle velocity at a point $(x,y,z)$. If the corresponding
Eulerian velocity field is supposed to be regular, the vector Eq. (1) in
nontrivial cases is a set of three nonlinear deterministic differential
equations whose phase space is a physical space of advected particles. It is
well known from dynamical systems theory that solutions of this kind of
equations can be chaotic in the sense of exponential sensitivity to small
variations in initial conditions and/or control parameters. As to advection
equations, it was Arnold [8] who firstly suggested chaos in the field lines
(and, therefore, in trajectories) for a special class of three-dimensional
stationary flows (so-called ABC flows), and this suggestion has been confirmed
numerically by Hénon [9]. In the approximation of incompressible planar flows,
the velocity components can be expressed in terms of a stream function [10]:
$u=-\partial\Psi/\partial y$ and $v=\partial\Psi/\partial x$. The equations of
motion (1) in a two-dimensional incompressible flow have now a Hamiltonian
form with the streamfunction $\Psi$ playing the role of a Hamiltonian and the
coordinates $(x,y)$ of a particle playing the role of canonically conjugated
variables.

Advection of passive particles has been shown to be chaotic in a number of
theoretical [4, 5, 6, 7, 1, 14, 13, 15, 16, 17, 26, 32] and laboratory [18,
19, 20, 21] models of geophysical jet currents. In our recent paper [1] (Paper
I) we have studied mixing, transport, and chaotic advection in a simple
kinematic model of a meandering two-dimensional jet flow with a Bickley zonal
velocity profile $u\sim\operatorname{sech}^{2}y$ being motivated by the
problem of lateral mixing of water masses (together with salinity, heat,
nutrients, pollutants, and other passive scalars) in the western boundary
currents in the ocean. We derived advection equations in the frame moving with
the phase velocity of a running wave imposed on the Bickley jet, found the
stationary points, conditions of their stability, and all the possible
bifurcations of these equations which were shown to be autonomous in the co-
moving frame. Under a periodic perturbation of the wave amplitude, the phase
plane of the chosen model flow has been shown to consist of a central eastward
jet, peripheral westward currents, and chains of northern and southern
circulations (vortex cores) immersed in a chaotic sea which, in turn, contains
islands of regular motion. Statistical properties of chaotic transport of
advected particles have been characterized in terms of particle’s zonal
flights (any event between two successive changes of the sign of the
particle’s zonal velocity $u$). Probability density functions (PDFs) of
durations and lengths of flights, computed with a number of very long chaotic
trajectories, have been found to be complicated functions with local maxima
and fragments with exponential and power-law decays.

The aim of this paper is to study a phenomenological connection between
dynamical, topological, and statistical properties of chaotic mixing and
transport in the meandering jet flow considered in Paper I and to explain
transport properties by phenomenon of the so-called dynamical traps. Following
to Zaslavsky [2], the dynamical trap is a domain in the phase space of a
Hamiltonian system where a particle (or, its trajectory) can spend arbitrary
long finite time, performing almost regular motion, despite the fact that the
full trajectory is chaotic in any appropriate sense. In fact, it is the
definition of a quasi-trap. Absolute traps, where particles could spend an
infinite time, are possible in Hamiltonian systems only with a zero measure
set. The dynamical traps are due to a stickiness of trajectories to some
singular domains in the phase space, largely, to the boundaries of resonant
islands, saddle trajectories, and cantori. There are no classification and
description of the dynamical traps. Zaslavsky described two types of dynamical
traps in Hamiltonian systems: hierarchical-islands traps around chains of
resonant islands [27, 28, 12] and stochastic-layer traps which are stochastic
jets inside a stochastic sea where trajectories can spend a very long time
[29, 2, 12]. It is expected that classification and description of the most
typical dynamical traps would help us to construct kinetic equations which
will be able to describe transport properties of chaotic systems including
anomalous ones [2, 11, 12].

This paper is organized as follows. We start with advection equations derived
in Paper I in the frame of reference moving with the phase velocity of a
meander whose amplitude changes in time periodically. We compute in Sec. II
PDFs for lengths $x_{f}$ and durations $T_{f}$ of zonal flights for a number
of chaotic trajectories and show analytically that all the flights start and
finish only inside a strip confined by two curves whose form is defined by the
condition $u=0$. Some prominent peaks in statistics of short flights
($|x_{f}|<2\pi$) are proved to be caused by stickiness of trajectories to the
boundaries of rotational resonant islands filled with regular particles
rotating in the same frame (Sec. III). We call this type of dynamical traps as
a rotational-island trap (RIT). In Sec. IV we study dynamical traps connected
with periodic saddle trajectories, emerged from saddle points of the
unperturbed system (4) under the perturbation (5), and prove numerically that
the saddle traps (STs) contribute to the statistics of short flights as well.
Another type of islands, ballistic islands (filled with regular particles
moving from frame to frame), is proved to contribute to the statistics of long
flights ($|x_{f}|\gg 2\pi$) in Sec. V. Both the ballistic-islands trap (BITs)
and RITs belong to the class of hierarchical-island traps by the Zaslavsky’s
classification.

##  2 Description of chaotic transport in terms of flights

###  2.1 Basic features of the flow

We take the following specific stream function as a kinematic model of a
meandering jet flow in the laboratory frame of reference:

$\Psi^{\prime}(x^{\prime},y^{\prime},t^{\prime})=\\\
=-\Psi^{\prime}_{0}\tanh{\left(\frac{y^{\prime}-a\cos{k(x^{\prime}-ct^{\prime})}}{\lambda\sqrt{1+k^{2}a^{2}\sin^{2}{k(x^{\prime}-ct^{\prime})}}}\right)},$
(2)

where the width of the jet is $\lambda$. Meandering is provided by a running
wave with the amplitude $a$, the wave-number $k$, and the phase velocity $c$.
The normalized streamfunction in the frame moving with the phase velocity is

$\Psi=-\tanh{\left(\frac{y-A\cos x}{L\sqrt{1+A^{2}\sin^{2}x}}\right)}+Cy,$ (3)

where $x=k(x^{\prime}-ct^{\prime})$ and $y=ky^{\prime}$ are new scaled
coordinates, and $A=ak$, $L=\lambda k$, and $C=c/\Psi^{\prime}_{0}k$ are the
control parameters. Equations, governing advection of passive particles (1) in
the co-moving frame, are the following:

$\begin{gathered}\begin{aligned}
\dot{x}&=\frac{1}{L\sqrt{1+A^{2}\sin^{2}x}\cosh^{2}\theta}-C,\\\
\dot{y}&=-\frac{A\sin x(1+A^{2}-Ay\cos
x)}{L\left(1+A^{2}\sin^{2}x\right)^{3/2}\cosh^{2}\theta},\end{aligned}\\\
\theta=\frac{y-A\cos x}{L\sqrt{1+A^{2}\sin^{2}x}},\end{gathered}$ (4)

where dot denotes differentiation with respect to dimensionless time
$t=\Psi^{\prime}_{0}k^{2}t^{\prime}$.

In Paper I (for more details see [17]) we have found and analyzed all the
stationary points, their stability, and the bifurcations of the equations of
motion (4). Being motivated by the problem of mixing and transport of water
masses and their properties in oceanic western boundary currents like the Gulf
Stream and the Kuroshio, we chose the phase portrait shown in Fig. 1a among
all the possible flow regimes. Passive particles can move along stationary (in
the co-moving frame) streamlines in a different manner. They can move to the
east in the jet ($J$) and to the west in northern and southern (with respect
to the jet) peripheral currents ($P$). There are also particles rotating in
the northern and southern circulation cells (C) in a periodic way. The
northern separatrix connects the saddle points at $x_{s}^{(n)}=2\pi n$ and
$y_{s}^{(n)}=L\operatorname{Arcosh}\sqrt{1/LC}+A$ and the southern one
connects the saddle points at $x_{s}^{(s)}=(2n+1)\pi$ and
$y_{s}^{(s)}=-L\operatorname{Arcosh}\sqrt{1/LC}-A$, where $n=0,\pm 1,\dots$.

Figure 1: (a) Stationary streamfunction of a meandering jet in the co-moving
frame (3). The flow is divided into three different regimes: circulations
($C$), jet ($J$), and peripheral currents ($P$). (b) Poincaré section of the
perturbed meandering jet in the co-moving frame. The parameters of the steady
flow are: the jet’s width $L=0.628$, the meander’s amplitude $A_{0}=0.785$ and
its phase velocity $C=0.1168$. The perturbation amplitude and frequency are:
$\varepsilon=0.0785$ and $\omega=0.2536.$ (c) Turning points of a single
chaotic trajectory on the cylinder $0\leq x\leq 2\pi$ are in a strip confined
by two curves (6) with $A=A_{0}\pm\varepsilon$.

As a perturbation, we took in Paper I the simple periodic modulation of the
meander’s amplitude

$A=A_{0}+\varepsilon\cos(\omega t+\phi).$ (5)

Under the perturbation, there arise resonances between the perturbation
frequency $\omega$ and the frequencies $f$ of the particle’s rotation in the
circulations $C$. A frequency map $f(x_{0},y_{0})$, computed in Paper I (see
Fig. 2 in that paper), shows the values of $f$ for particles with initial
positions $(x_{0},y_{0})$ in the unperturbed flow. With a given value of the
perturbation frequency and fixed values of the other control parameters, the
vortex cores in the circulations survive, stochastic layers appear along the
unperturbed separatrix, and the central jet $J$ is a barrier to transport of
particles across the jet. In Paper I we fixed the scaled values of the
parameters of the unperturbed flow, the jet’s width $L=0.628$, the meander’s
amplitude $A_{0}=0.785$ and its phase velocity $C=0.1168$, that are in the
range of the realistic values for the Gulf Stream [22, 23], and took the
initial phase to be $\phi=\pi/2$. The perturbation frequency $\omega=0.2536$
chosen in Paper I is close to the values of the rotation frequency $f$ of the
particles circulating in the inner core of the regions $C$ (see Fig. 2 in
Paper I). In Fig. 1 b we show the Poincaré section (for a large number of
trajectories) of the meandering jet whose amplitude is modulated with the
frequency $w=0.2536$ and the strength $\varepsilon=0.0785$. The vortex cores
survive under this perturbation, the stochastic layers appear along the
unperturbed separatrix, and a central jet $J$ is a barrier to transport of
particles across the jet.

The equations of motion (4) with the perturbation (5) are symmetric under the
following transformations: (1) $t\to t$, $x\to\pi+x$, $y\to-y$ and (2)
$t\to-t$, $x\to\ -x$, $y\to y$. It implies that the meridional transport
(north-south and south-north) is symmetric but the zonal transport (west-east
and east-west) is symmetric under a time reversal. Due to these symmetries
motion can be considered on the cylinder with $0\leq x\leq 2\pi$ and $y\geq
0$. The part of the phase space with $2\pi n\leq x\leq 2\pi(n+1)$, $n=0,\pm
1,\dots$, is called a frame.

It is convenient to characterize chaotic mixing and transport in terms of
zonal flights. A zonal flight is a motion of a particle between two successive
changes of signs of its zonal velocity, i. e. the motion between two
successive events $\dot{x}=u=0$. Particles (and corresponding trajectories) in
chaotic jet flows can be classified in terms of the lengths of flights $x_{f}$
as follows. The trajectories with $|x_{f}|<2\pi$ correspond to the particles
moving in the same frame or in neighbor frames. In the global stochastic layer
there are particles moving chaotically forever in the same frame but they are
of a zero measure. Among the particles with inter-frame motion, there are
regular and chaotic ballistic ones. Regular ballistic trajectories can be
defined as those which cannot have two flights with $|x_{f}|>2\pi$ in
succession. They correspond to particles moving in regular regions of the
phase space persisting under the perturbation, (eastward motion in the jet and
western motion in the peripheral current) and those moving in the stochastic
layer (trajectories belonging to ballistic islands). Typical chaotic
trajectories have complicated distributions over the lengths and durations of
flights.

In the laboratory frame of reference, all the fluid particles move to the east
together with the jet flow and a flight is a motion between two successive
events when the particle’s zonal velocity $U$ is equal to the meander’s phase
velocity $c$. If $U<c$, the corresponding particle is left behind the meander
(it is a western flight in the co-moving frame), if $U>c$, it passes the
meander (an eastern flight in the co-moving frame). Short flights with
$|x_{f}|<2\pi$ (motion in the same spatial frame in the co-moving frame of
reference) correspond to the motion in the laboratory frame when two
successive events $U=c$ occur on the space interval less than the meander’s
spatial period $2\pi/k$. Ballistic flights between the spatial frames in the
co-moving frame with $|x_{f}|>2\pi$ correspond to the motion in the laboratory
frame when the particles move through more than one meander’s crest between
two successive events $U=c$.

###  2.2 Turning points

As in Paper I, we will characterize statistical properties of chaotic
transport by probability density functions (PDFs) of lengths of flights
$P(x_{f})$ and durations of flights $P(T_{f})$ for a number of very long
chaotic trajectory. Both regular and chaotic particles may change many times
the sign of their zonal velocity $\dot{x}=u$. From the condition $\dot{x}=0$
in Eq. (4), it is easy to find the equations for the curves which are loci of
turning points

$Y_{\pm}(x,A)=\pm L\sqrt{1+A^{2}\sin^{2}{x}}\times\\\
\times\operatorname{Arsech}{\sqrt{{LC\sqrt{1+A^{2}\sin^{2}{x}}}}}+A\cos x.$
(6)

We consider the northern curve, i. e., Eq. (6) with the positive sign. Taking
into account that the perturbation has the form (5), we realize that all the
northern turning points are inside a strip confined by two curves of the form
(6) with $A=A_{0}\pm\varepsilon$. Let us analyze the derivative over the
varying parameter A

$\frac{\partial Y}{\partial A}=\cos x+\\\
+\frac{ACL^{2}\sin^{2}x}{2D}\biggl{(}2\operatorname{Arsech}{\sqrt{D}}-\frac{1}{\sqrt{1-D}}\biggr{)},$
(7)

where $D=LC\sqrt{1+A^{2}\sin^{2}x}$. If the derivative at a fixed value of $x$
does not change its sign on the interval $A_{0}-\varepsilon\leq A\leq
A_{0}+\varepsilon$, then $Y$ varies from $Y(x,A_{0}-\varepsilon)$ to
$Y(x,A_{0}+\varepsilon)$, and for each value of $y$ we have a single value of
the perturbation parameter $A$. However, there may exist such values of $x$
for which the equation $\partial Y/\partial A=0$ has a solution on the
interval mentioned above. In this case one may have more than one values of
$A$ for a single value of $y$. Thus, the width of the strip, containing
turning points, is defined by the values of $Y$ at the extremum points and at
the end points of the interval of the values of $A$. In Fig. 1 c we show the
turning points of a single chaotic trajectory on the cylinder $0\leq x\leq
2\pi$ confined between two corresponding curves.

In the numerical simulation throughout the paper we use the Runge-Kutta
integration scheme of the fourth order with the constant time step $\Delta
t0.0247$. To study chaotic transport we have carried out numerical experiments
with tracers initially placed in the stochastic layer. It was found that
statistical properties of chaotic transport practically do not depend on the
number of tracers provided that the corresponding trajectories are
sufficiently long ($t\simeq 10^{8}$). The PDFs for the lengths $x_{f}$ and
durations $T_{f}$ of flights for five tracers with the computation time
$t=5\cdot 10^{8}$ for each tracer are shown in Figs. 2 a and b, respectively,
both for the eastward (e) and westward (w) motion. Both $P(x_{f})$ and
$P(T_{f})$ are complicated functions with local extrema decaying in a
different manner for different ranges of $x_{f}$ and $T_{f}$. The main aim of
our study of chaotic transport is to figure out the basic peculiarities of the
statistics and attribute them to specific zones in the phase space, namely, to
dynamical traps strongly influencing the transport.

Figure 2: Probability density functions of (a) lengths $x_{f}$ and (b)
durations $T_{f}$ of the westward (w) and eastward (e) flights. The PDFs
$P_{\rm w}(T_{f})$ and $P_{\rm e}(T_{f})$ are normalized to the number of
westward ($4.23\cdot 10^{7}$) and eastward ($4\cdot 10^{7}$) flights,
respectively. Statistics for five tracers with the computation time $t=5\cdot
10^{8}$ for each one.

##  3 Rotational-islands traps

It is well known, that in nonlinear Hamiltonian systems a complicated
structure of the phase space with islands, stochastic layers, and chains of
islands, immersed in a stochastic sea, arises under a perturbation due to a
variety of nonlinear resonances and their overlapping [24]. The motion is
quasiperiodic and stable in the islands. The boundaries of the islands are
absolute barriers to transport: particles can not go through them neither from
inside nor from outside. Invariant curves of the unperturbed system (see Fig.
1 a) are destroyed under the perturbation (5) (see Fig. 1 b). As the
perturbation strength $\varepsilon$ increases, a closed invariant curve with
frequency $f$ is destroyed at some critical value of $\varepsilon$. If the
$f/\omega$ is a rational number, the corresponding curve is replaces by an
island chain, while the curves with irrational frequencies are replaced by
cantori (for a review see [25]). There are uncountably many cantori forming a
complicated hierarchy. Numerical experiments with a variety of Hamiltonian
systems with different number of degrees of freedom provide an evidence for
the presence of strong partial barriers to transport around the island’s
boundaries (for review, see [12]) which manifest themselves on Poincaré
sections as domains with increased density of points.

In Paper I we have found that with chosen values of the control parameters
there exist in each frame a vortex core (which is an island of the primary
resonance $\omega=f$) immersed into a stochastic sea, where there are six
islands of a secondary resonance emerged from three islands of the primary
resonance $3f=2\omega$ (see Fig. 3 in Paper I). Chains of smaller size islands
are present around the vortex core and the secondary-resonance islands.
Particles belonging to all of these islands (including the vortex core) rotate
in the same frame performing short flights with the lengths $|x_{f}|<2\pi$. So
we will call them rotational islands and distinguish from the so-called
ballistic islands to be considered below.

Stickiness of particles to boundaries of the rotational islands has been
demonstrated in Paper I. It means that real fluid particles can be trapped for
a long time in a singular zone nearby the borders of the rotational islands
which we will call rotational-islands traps (RITs). To illustrate the effect
of the RITs we demonstrate in Figs. 3 and 4 the Poincaré sections of a chaotic
trajectory in the frame $0\leq x\leq 2\pi$ sticking to the vortex core and to
the secondary-resonance islands, respectively. The contour of the vortex core
is shown in Fig. 4 by the thick line. The small points are tracks of the
particle’s position at the moments of time $t_{n}=2\pi n/\omega$ (where
$n=1,2,\dots$) and the thin curves are fragments of the corresponding
trajectory on the phase plane. Increased density of points indicates the
presence of dynamical traps near the boundaries of the rotational islands.
Contribution of the vortex-core RIT (Fig. 3) to chaotic transport is expected
to be much more significant than the one of the RITs of the other islands
(Fig. 4).

Figure 3: The vortex-core trap. Poincaré section of a chaotic trajectory in
the frame $0\leq x\leq 2\pi$ with a fragment of a trajectory.

Figure 4: The secondary resonance islands trap. A fragment of a chaotic
trajectory sticking to the islands is shown.

Figure 5: The PDFs for the eastward (e) and westward (w) flights with the
length shorter than $2\pi$. The PDFs $P_{\rm w}(T_{f})$ and $P_{\rm e}(T_{f})$
are normalized to the number of westward ($4.19\cdot 10^{7}$) and eastward
($3.7\cdot 10^{7}$) flights, respectively. Statistics for five tracers with
the computation time $t=5\cdot 10^{8}$ for each one.

Figure 6: The vortex-core trap PDFs of durations $T_{f}$ of the eastward (e)
and westward (w) flights. (a) Regular quasiperiodic trajectory with the
duration $t=2\cdot 10^{5}$ inside the vortex core close to its boundary. Both
the PDFs are normalized to the number $8\cdot 10^{3}$ of corresponding
flights. (b) Chaotic trajectory with the duration $t=2\cdot 10^{5}$ sticking
to the boundary of the vortex core from the outside. Both the PDFs are
normalized to the number $4\cdot 10^{3}$ of corresponding flights.

It is reasonable to suppose that RITs contribute to the statistics of short
flights. By short flights we mean the flights with the length shorter than
$2\pi$. In Fig. 5 we show the part of the full PDF $P(T_{f})$ (Fig. 2 b) for
the eastward (e) and westward (w) short flights separately. There are a
comparatively small number of the eastward flights with $T_{f}<11$. Let us
note the prominent peak of the corresponding PDF at $T_{f}\simeq 11$ followed
by an exponential decay. As to the westward short flights, there are two small
local peaks around $T_{f}\simeq 17$ and $21$.

To estimate the contribution of the vortex-core RIT to the statistics of short
flights, we compute and compare the statistics of the durations of flights
$T_{f}$ for two trajectories: a regular quasiperiodic one with the initial
position close to the inner border of the vortex core (Fig. 6 a) and a chaotic
one with the initial position close to the vortex-core border from the outside
(Fig. 6 b). Each full rotation of a particle in a frame consists of two
flights, eastward and westward, with different values of $T_{f}$ because of
the zonal asymmetry of the flow. The statistics for the chaotic trajectory,
sticking to the vortex core (Fig. 6 b), may be considered as a distribution of
the durations of flights in the vortex-core RIT. The minimal flight duration
in this RIT is $T_{f}\simeq 11$ (the flights with smaller values of $T_{f}$
are rare and they occur outside the trap). Positions of the local maxima of
the PDF for the sticking trajectory in Fig. 6 b correlate approximately with
the corresponding local maxima of the PDF for the regular trajectory inside
the core in Fig. 6 a. The similar correlations have been found (but not shown
here) between the local maxima of the PDFs for the lengths of flights
$P(x_{f})$ for the interior regular and sticking chaotic trajectories. These
correlations and positions of the peaks prove numerically that short flights
with $|x_{f}|<2\pi$ and $11\lesssim T_{f}\lesssim 21$ may be caused by the
effect of vortex-core RIT. We conclude from Fig. 6 b that the vortex-core RIT
contributes to the statistics of the short flights in the range $11\lesssim
T_{f}\lesssim 20$ for the eastward flights with the prominent peak at
$T_{f}\simeq 11$ and in the range $15\lesssim T_{f}\lesssim 21$ for the
westward flights with small peaks at $T_{f}\simeq 17$ and $21$.

The effect of the RIT of the secondary-resonance islands is illustrated in
Fig. 4. To find the characteristic times of this RIT we compute two
trajectories: a regular quasiperiodic one with the initial position inside one
of these islands and a chaotic one with the initial position close to the
outer border of the island. The respective PDFs $P(T_{f})$, shown in Figs. 7 a
and b, demonstrate strong correlations between the corresponding peaks at
$T_{f}\simeq 12,23$, and $27$. Computed (but not shown here) PDFs $P(x_{f})$
for these trajectories confirm the effect of the islands RIT on the statistics
of short flights.

Figure 7: The secondary-resonance islands trap. The PDFs of durations $T_{f}$
of the eastward (e) and westward (w) flights. (a) Regular quasiperiodic
trajectory inside the islands with the duration $t=5\cdot 10^{5}$. Both the
PDFs are normalized to the number $1.5\cdot 10^{4}$ of corresponding flights.
(b) Chaotic trajectory sticking to the island’s boundary from the outside with
the duration $t=5\cdot 10^{5}$. $P_{\rm w}(T_{f})$ and $P_{\rm e}(T_{f})$ are
normalized to the number of westward ($1.1\cdot 10^{4}$) and eastward ($9\cdot
10^{3}$) flights, respectively.

##  4 Saddle traps

As a result of the periodic perturbation (5), the saddle points of the
unperturbed system (4) at $x_{s}^{(n)}=2\pi n$,
$y_{s}^{(n)}=L\operatorname{Arcosh}\sqrt{1/LC}+A$ and at
$x_{s}^{(s)}=(2n+1)\pi$, $y_{s}^{(s)}=-L\operatorname{Arcosh}\sqrt{1/LC}-A$
($n=0,\pm 1,\dots$) become periodic saddle trajectories. These hyperbolic
trajectories have their own stable and unstable manifolds and play a role of
specific dynamical traps which we call saddle traps (ST). In this section we
demonstrate that the STs influence strongly on chaotic mixing and transport of
passive particles and contribute, mainly, in the short-time statistics of
flights.

Tracers with initial positions close to a stable manifold of a saddle
trajectory are trapped for a while performing a large number of revolutions
along it. To illustrate the effect of the STs we show in Fig. 8 a and b
fragments of two chaotic trajectories sticking to the saddle trajectory and
performing about 20 full revolutions before escaping to the east (Fig. 8 a)
and to the west (Fig. 8 b). We have managed to detect and locate the
corresponding periodic unstable saddle trajectory which is situated in Figs. 8
a and b in the domain where a few fragments of the chaotic trajectory imposed
on each other. Because of the flow asymmetry, the duration of eastern flights
of a particle along the saddle trajectory $T_{\rm e}\simeq 11.9$ is shorter
than the duration of western flights $T_{\rm w}\simeq 12.9$. The black points
are the tracks of the particle’s positions on the flow plane at the moments of
time $t_{n}=2\pi n/\omega\simeq 24.8\,n$ (where $n=1,2,\dots$). They belong to
smooth curves which are fragments of the stable and unstable manifolds of the
saddle trajectory at the chosen initial phase $\phi=\pi/2$.

Figure 8: The saddle trap. Fragments of two chaotic trajectories sticking to
the periodic saddle trajectory one of which escapes to the east (a) and
another one to the west (b). (c) The number of the eastward $(N_{\rm e})$ and
westward $(N_{\rm w})$ short flights with duration $T_{f}$ for those two
trajectories. Statistics with two trajectories with the duration $t=10^{3}$
and the total number of western $N_{\rm w}=55$ and eastern $N_{\rm e}=51$
flights.

To estimate the contribution of the STs to the statistics of short flights
shown in Fig. 5, we compute and plot in Fig. 8 c the number of the eastward
$(N_{\rm e})$ and westward $(N_{\rm w})$ short flights with a given duration
$T_{f}$ for those two chaotic trajectories sticking to the saddle trajectory
arising from the saddle point with the position $x_{s}=0,y_{s}\simeq 2.02878$.
Each full rotation of the particles consists of an eastward flight with the
duration $T_{\rm e}\simeq 11.9$ and an westward flight with the duration
$T_{\rm w}\simeq 12.9$. The flights with $T_{\rm e}\simeq 11.9$ contribute to
the main peak in Fig. 5 and the flights with $T_{\rm w}\simeq 12.9$ to ‘‘the
wesward’’ plateau in that figure.

The mechanism of operation of the STs can be described as follows. Each saddle
trajectory $\gamma(t)$ possesses time-dependent stable $W_{s}(\gamma(t))$ and
unstable $W_{u}(\gamma(t))$ material manifolds composed of a continuous sets
of points through which pass at time $t$ trajectories of fluid particles that
are asymptotic to $\gamma(t)$ as $t\to\infty$ and $t\to-\infty$, respectively.
Under a periodic perturbation, the stable and unstable manifolds oscillate
with the period of the perturbation. It was firstly proved by Poincaré that
$W_{s}$ and $W_{u}$ may intersect each other transversally at an infinite
number of homoclinic points through which pass doubly asymptotic trajectories.
To give an image of a fragment of the stable manifold of the periodic saddle
trajectory, we distribute homogeneously $2.5\cdot 10^{5}$ particles in the
rectangular $[-0.4\leq x\leq 0.45;2\leq y\leq 2.1]$ and compute the time the
particles need to escape the rectangular. The color in Fig. 9 modulates the
time $T$ when particles with given initial positions $(x_{0},y_{0})$ reach the
western line at $x=-1$ or the eastern line at $x=1$. The particles with
initial positions marked by the black and white colors move close to the
stable manifold of the saddle trajectory and spend a maximal time near it
before escaping. The black and white diagonal curve in Fig. 9 is an image of a
fragment of the corresponding stable manifold. The particles with initial
positions to the north from the curve escape to the west along the unstable
manifold of the saddle trajectory whereas those with initial positions to the
south from the curve escape to the east along its another unstable manifold.

Figure 9: The saddle-trap map. Color modulates the time $T$ which $2.5\cdot
10^{5}$ particles with given initial positions ($x_{0},y_{0}$) need to reach
the lines at $x=-1$ or $x=1$ escaping to the west (w) and to the east (e),
respectively. The black and white diagonal curve is an image of a fragment of
the stable manifold of the saddle trajectory. The cross is a position of a
particle on that trajectory at the initial time moment. The integration time
is $t=500$.

We have found that particles quit the ST along the unstable manifolds in
accordance with specific laws. We distribute a large number of particles along
the segment with $x_{0}=0$ and $y_{0}=[2.02;2.06]$, crossing the stable
manifold $W_{s}$, and compute the time $T$ particles with given initial
latitude positions $y_{0}$ need to quit the ST. More precisely, $T(y_{0})$ is
a time moment when a particle with the initial position $y_{0}$ reaches the
lines with $x=-1$ or $x=1$. The ‘‘experimental’’ points in Fig. 10 a fit the
law $T_{\rm e}=(-85.81\pm 0.04)-(31.216\pm 0.007)\ln(y_{0s}-y_{0})$ for the
particles which quit the trap moving to the east and the law $T_{\rm
w}=(-60.61\pm 0.03)-(28.933\pm 0.006)\ln(y_{0}-y_{0s})$ for those particles
which move to the west when quitting the trap, where $y_{0s}=2.0405755472$ is
a crossing point of $W_{s}$ with the segment of initial positions.

Figure 10: (a) Time $T$ a particle with an initial latitude position $y_{0}$
needs to quit the saddle trap. (b) The number of short flights $n$ such a
particle performs before quitting the saddle trap. The ranges of $y_{0}$ from
which particles quit the trap moving to the west and east are denoted by ‘‘w’’
and ‘‘e’’, respectively.

The ST attracts particles and force them to rotate in its zone of influence
performing short flights, the number of which $n$ depends on particle’s
initial positions $y_{0}$. The $n(y_{0})$ is a steplike function (see Fig. 10
b) with the lengths of the steps decreasing in a geometric progression in the
direction to the singular point, $l_{j}=l_{0}\,q^{-j}$, where $l_{j}$ is the
length of the $j$-th step and $q\simeq 2.27$ for the western exits and
$q\simeq 2.20$ for the eastern ones. The seeming deviation from this law in
the range $y_{0}=[2.045;2.046]$ (see a small western segment between two
larger ones in Fig. 10 b) is explained by crossing the initial line
$y_{0}=[2.02;2.06]$ by the curve of zero zonal velocity $u$. To have the
correct law for the western exits, it is necessary to add the two segments of
that cut step. The asymmetry of the functions $T(y_{0})$ and $n(y_{0})$ is
caused by the asymmetry of the flow.

##  5 Ballistic-islands traps

Besides the rotational islands with particles moving around the corresponding
elliptic points in the same frame, we have found in Paper I ballistic islands
situated both in the stochastic layer and in the peripheral currents. Regular
ballistic modes [30] correspond to stable quasiperiodic inter-frame motion of
particles. Only the ballistic islands in the stochastic layer are important
for chaotic transport. Mapping positions of the regular ballistic trajectories
at the moments of time $t_{n}=2\pi n/\omega$ $(n=1,2,\dots)$ onto the first
frame, we obtain chains of ballistic islands both in the northern and southern
stochastic layers, i. e., between the borders of the northern (southern)
peripheral currents and of the corresponding vortex cores. A chain with three
large ballistic islands is situated in those stochastic layers. The particles,
belonging to these islands, move to the west, and their mean zonal velocity
can be easily calculated to be
$\left<u_{f}\right>=-2\pi/3T=-\omega/3\simeq-0.0845$. There are also chains of
smaller-size ballistic islands along the very border with the peripheral
currents.

We have demonstrated in Paper I a stickiness of chaotic trajectories to the
borders of those three large ballistic islands (see Figs. 6 and 7 in Paper I).
The Poincaré section with fragments of two chaotic trajectories in the
northern stochastic layer is shown in Fig. 11 a. One particle performs a long
flight sticking to the very border with the regular westward current, and
another one moves to the west sticking to the very boundaries of three large
ballistic islands. A magnification of a fragment of the border and tracks of a
sticking trajectory around a smaller-size ballistic island are demonstrated in
Fig. 11 b. Fig. 11 c demonstrates the effective size of the trap of the large
ballistic islands with tracks of a sticking trajectory around them.

Figure 11: (a) Poincaré section of the northern stochastic layer where
stickiness to the very border with the regular westward current and to three
large ballistic islands are shown. Increased density of points along the
border with the peripheral current is caused by the traps of the border
ballistic islands one of which is shown in (b). (c) The trap of the large
ballistic islands.

Figure 12: The distribution of a number of long westward flights with
$T_{f}\geq 10^{3}$ over their mean zonal velocities $\left<u_{f}\right>$. The
sharp peak corresponds to the trap connected with the very boundaries of the
large ballistic islands, the left wing — to a number of traps of families of
the border ballistic islands, and the right wing — to the trap situated around
the large ballistic islands. Statistics for five tracers with the total number
of long westward flights $N_{f}=5\cdot 10^{4}$ and the computation time
$t=5\cdot 10^{8}$ for each tracer.

It is reasonable to suppose that the ballistic-islands traps (BIT) contribute,
largely, to the statistics of long flights with $|x_{f}|\gg 2\pi$. All the
ballistic particles, moving both to the west and to the east, can finish a
flight and make a turn only in the strip shown in Fig. 1 c. The loci of the
corresponding turning points have a complicated fractal-like structure. We
consider further only long westward flights, taking place in the northern
stochastic layer, because it is much wider than the stochastic layer between
the regular central jet and the southern parts of the vortex cores where
eastward flights take place.

To distinguish between contributions of the traps of different ballistic
islands (and, maybe, other zones in the phase space) to the statistics of long
flights, we compute for five long chaotic trajectories (up to $t=5\cdot
10^{8}$) the distribution of a number of westward flights with $T_{f}\geq
10^{3}$ over the mean zonal velocities $\left<u_{f}\right>=x_{f}/T_{f}$ of the
particles performing such flights. The distribution in Fig. 12 has a prominent
peak centered at the mean zonal velocity $\left<u_{f}\right>\simeq-0.0845$
which corresponds to a large number of long flights of those particles (and
their trajectories) which stick to the very boundaries of the large ballistic
islands (see Fig. 11 a) moving with the mean velocity
$\left<u_{f}\right>\simeq-0.0845$. The flat left wing of the distribution
$N(\left<u_{f}\right>)$ corresponds to the traps of smaller-size ballistic
islands nearby the border with the peripheral current. There are different
families of these islands (see one of them in Fig. 11 b) with their own values
of the mean zonal velocity which are in the range
$-0.092\lesssim\left<u_{f}\right>\lesssim-0.0845$. Stickiness to the
boundaries of the border islands is weaker because they are smaller than the
large islands and their contribution to the statistics of long flights is
comparatively small.

The right wing of the distribution $N(\left<u_{f}\right>)$ with
$-0.084\lesssim\left<u_{f}\right>\lesssim-0.075$ deserves further
investigation. The value $\left<u_{f}\right>\simeq-0.075$ is a minimal value
of the zonal velocity for long westward flights possible in the northern
stochastic layer. Increasing the minimal duration of a flight from
$T_{f}=10^{3}$ to $T_{f}=(2\div 5)\cdot 10^{3}$, we have found splitting of
the broad distribution with $-0.084\lesssim\left<u_{f}\right>\lesssim-0.08$
into a number of small distinct peaks. Comparing trajectories with the values
of $\left<u_{f}\right>$ corresponding to these peaks, we have found that all
they move around the large ballistic islands. The particles with smaller
values of $\left<u_{f}\right>$ used to penetrate further to the south from the
islands more frequently than those with larger values of $\left<u_{f}\right>$
which prefer to spend more time in the northern part of the dynamical trap
connected with those islands. Thus, we attribute the right wing of the
distribution $N(\left<u_{f}\right>)$ to an effect of the trap situated around
the large ballistic islands.

To estimate the contribution of different BITs to the statistics of long
westward flights in Fig. 2 we have computed the PDFs $P(x_{f})$ and $P(T_{f})$
for particles performing westward flights with $x_{f}\geq 100$ and $T_{f}\geq
1000$ and with the mean zonal velocity $\left<u_{f}\right>$ to be chosen in
three different ranges shown in Fig. 12:
$-0.092\lesssim\left<u_{f}\right>\lesssim-0.085$ (particles sticking to the
border islands) $-0.085<\left<u_{f}\right>\lesssim-0.084$ (particles sticking
to the very boundary of three large islands), and
$-0.084<\left<u_{f}\right>\lesssim-0.075$ (the trap of the three large
islands). All the PDFs $P(x_{f})$ decay exponentially but with different
values of the exponents equal to $\nu\simeq-0.005$ and
$\nu\simeq-0.0018\div-0.0014$ for the traps of border and the large ballistic
islands, respectively. The tail of the PDF $P(x_{f})$ for westward flights,
shown in Fig. 2, decays exponentially with $\nu\simeq-0.0014$. Thus, the
contribution of the large island’s BIT to the statistics of long westward
flights is dominant. As to temporal PDFs $P(T_{f})$ for westward long flights,
they are neither exponential nor power-law like with strong oscillations at
the very tails. The slope for the border BITs is again smaller than for the
large ballistic islands trap.

##  6 Conclusion

A meandering jet is a fundamental structure in oceanic and atmospheric flows.
We described statistical properties of chaotic mixing and transport of passive
particles in a kinematic model of a meandering jet flow in terms of dynamical
traps in the phase (physical) space. The boundaries of rotational islands
(including the vortex cores) in circulation zones are dynamical traps (RITs)
contributing, mainly, to the statistics of short flights with $|x_{f}|<2\pi$.
Characteristic times and spatial scales of the RITs have been shown to
correlate with the PDFs for the lengths $x_{f}$ and durations $T_{f}$ of short
flights. The stable manifolds of periodic saddle trajectories play a role of
saddle traps (STs) with the specific values of the lengths and durations of
short flights of the particles sticking to the saddle trajectories. The
boundaries of ballistic islands in the stochastic layers (including those
situated along the border with the peripheral current) are dynamical traps
(BITs) contributing, mainly, to the statistics of very long flights with
$|x_{f}|\gg 2\pi$.

Dynamical traps are robust structures in the phase space of dynamical systems
in the sense that they present at practically all values of the corresponding
control parameters. We never know exact values of the parameters in real
flows, especially, in geophysical ones. We don’t know exactly the structure of
the corresponding phase space, however, we know that typical features, like
islands of regular motion, vortices, and jets, exist in real flows (see their
images in some laboratory flows in Ref. [31]). In this paper we chose specific
values of the control parameters for which specific PDFs have been computed
and explained by the effect of those dynamical traps that exist under the
chosen parameters. We have carried out computer experiments with different
values of the control parameters and found that the phase space structure has
been changed, of course, with changing the values of the parameters, but the
corresponding RITs, STs, and BITs with specific temporal and spatial
characteristics have been found to contribute to the corresponding statistics.

After finishing our work, we were acquainted with Ref. [32] where meridional
chaotic transport, associated with a similar kinematic model of a meandering
jet, has been studied by the method of lobe dynamics [33]. In difference from
our study of zonal chaotic transport, a geometric structure of cross jet
transport has been considered in Ref. [32] where values of the control
parameters have been chosen to be sufficiently large to break up the central
jet as a barrier to transport of particles across the jet. The mechanisms for
particles to cross the jet have been described in terms of lobe dynamics and
the mean time to cross the jet for particles entering the jet and the mean
residence time for particles in the jet have been estimated in Ref. [32]. We
have studied a more realistic situation (at least, for surface oceanic jet
currents) when the jet is an absolute barrier to cross jet transport and we
explained statistical properties of transport in terms of dynamical traps of
saddle trajectories, rotational and ballistic islands. The method of lobe
dynamics is hardly applicable for study zonal chaotic transport since it is
practically impossible to trace out lobe evolution for a large number of
frames.

## Acknowledgments

The work was supported by the Russian Foundation for Basic Research (Grant no.
06-05-96032), by the Program ‘‘Mathematical Methods in Nonlinear Dynamics’’ of
the Russian Academy of Sciences, and by the Program for Basic Research of the
Far Eastern Division of the Russian Academy of Sciences.

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