Marie-Laure Betbeder
email: marie-laure.betbeder@femto-st.fr

Jean-Christophe Lapayre
email: jc.lapayre@femto-st.fr

A Tool for Telediagnosis of Cardiovascular Diseases in a Collaborative and Adaptive Approach

Keywords: Collaborative work, telediagnosis, adaptability, cardiovascular complications

In this paper, we present a new telediagnosis environment for the detection of cardiovascular problems.This tool, called VACODIS (VAscular COllaborative teleDIagnosiS), allows practitioners to semi-automatically identify and quantify a patient's potential cardiovascular complications. The system generates first-time automatic detection of cardiovascular abnormalities using Doppler ultrasound images.The system then provides remote collaborative sharing of this information among different doctors to allow distance telediagnostics. With this new system, different actors in the field of medicine (nurses, practitioners, etc.) will be able to contribute to a more reliable diagnosis in the cardiovascular domain.

Introduction

This investigation is the result of the combined efforts of UMSNH University (Morelia, Mexico) and the University of Franche-Comté (Besançon, France). The team from Morelia is specialized in algorithms for image processing and segmentation, particularly in ultrasonography images (echo-Doppler). The team from Franche-Comté is specialized in distributed algorithms for collaborative telediagnosis. The objective of this work is to provide the medical community with a distributed tool for semi-automatic diagnosis in order to identify and quantify potential cardiovascular complications in a patient. This diagnosis is composed of two steps. Firstly, cardiovascular anomalies are automatically detected from echo-Doppler images. Secondly, these images and results are adaptively and collaboratively shared to help practitioners during the collaborative diagnosis. Thus, multiple practitioners may use this tool to collaborate in a more reliable diagnosis, independently of their locations.

The second section of this article presents the medical aspect and pathology of the cardiovascular diseases. Then, in the third section, we present algorithms of automatic image processing which can detect three kinds of anomalies: atherosclerosis, endothelial dysfunction, and atherosclerosis plaque. In the fourth section we introduce a collaborative environment for telemedicine state of the art. Then, in the fifth section, we introduce a platform for automatic adaptation: WAVA. This platform ensures the sharing of information (ultrasonography images) and results of automatic detection of cardiovascular anomalies, adapting shared data in function of the available hardware capacity of each user. Finally, we end with a summary and some perspectives.

Medical Context

Cardiovascular disease (CVD) is still the leading cause of death worldwide. Although cardiovascular mortality rates have declined in many high-income countries over the last two decades, they have increased at an astonishingly fast rate in low-and middle-income countries. By the time heart problems are detected, the underlying cause, atherosclerosis, has progressed for decades and is usually quite advanced. In the CVD domain, practitioners use echography to obtain what is usually a reliable diagnosis. Echography provides images in real-time through an inexpensive and non-intrusive approach.

On the other hand, the processing of echographic images remains one of the most complex challenges in the area of image processing. Artery images present a broad dynamic range: useful details in the clinical zone may be found in a dark area as well as in the bright area of the image. Possible pathologies may be difficult to detect and remain unknown [Bishop 2000, Theodoridis et al. 2005].

Current Medical Problems

A report by the World Health Organization (WHO) 1 , published in September 2011, claims CVD and control states are already the principal causes of death and disability in the world. Although a large proportion of CDVs are avoidable, they continue to rise mainly because preventive measures are inadequate. The percentage of premature deaths from CVD range from 4% to 42% in highand low-income countries respectively. More than 17.3 million people died from CVDs in 2008, representing 30% of all global deaths. Over 80% of CVD deaths take place in low-and middle-income countries. The socio-economic impact is considerable: it has been predicted that approximately 23.6 million people will die from CVDs in 2030.

Pathology

Atherosclerosis is a condition in which the artery walls thicken as a result of the accumulation of fat residuals. It is a syndrome affecting arterial blood vessels, a chronic inflammatory response in the artery wall [START_REF] Brusseau | Fully Automatic Luminal Contour Segmentation in Intercoronary Ultrasound Imaging -A Statistical Approach[END_REF], Campbell et al.1997], caused by the formation of multiple plaques within the arteries.

Atherosclerosis typically begins in early adolescence and is found in major arteries, yet is asymptomatic and remains undetected by most diagnostic methods during most of the subjects lifetime [Fig. 1]. In addition, endothelial dysfunction is a systemic pathological state of the endothelium (the inner lining of blood vessels) and can be broadly defined as an imbalance between vasodilating and vasoconstricting substances produced by (or acting on) the endothelium [Campbell et al.1997].

Figure 1: The evolution of atherosclerosis

Endothelial dysfunction may result from and/or contribute to several disease processes. Endothelial function testing may have great potential prognostic value for the detection of CVD, but currently the available tests are too difficult, expensive, and/or variable for routine clinical use [START_REF] Rubin | Qualitative and Quantitative Effects of Harmonic Echocardiographic Imaging on Endo-cardial Edge Definition and Side-Lobe Artifact[END_REF]].

The Progression of Atherosclerosis

The existing methods for assessing endothelial dysfunction and atherosclerosis in humans are based on functional tests in the brachial [Fig. 2 There exists a noninvasive method for testing endothelial function, to find out whether abnormalities are present in symptom-free children and young adults at high risk of atherosclerosis. With high-resolution ultrasound, the diameter of the superficial brachial arteries is measured, while at rest, during reactive hyperaemia (with increased flow causing endothelium-dependent dilatation of 10%), and then assessed with Doppler ultrasonography [Fig. 3]: a well-tolerated, noninvasive and low-risk procedure. It is currently the most widely investigated method and shows the greatest promise for clinical application.

Carotid intima-media thickness (IMT) is considered a marker of early atheroscle- rosis and it has been demonstrated that it can predict future risk of cardiovascular disease in coronary patients [START_REF] Severo | Increased Atherothrombotic Markers and Endothelial Dysfunction in Steroid Users[END_REF]. Physicians use IMT testing to determine the age of the carotid arteries. Though patients may not be experiencing the symptoms of artherosclerosis, subtle changes in artery thickness may nevertheless be evidence of carotid atherosclerosis (carotid intima-media thickness > 0.8 mm) [Fig. 4]. The rate of change of progressive carotid atherosclerosis (determined by measurement of IMT, the distal common carotid arterial far wall) may exhibit evidence of coronary events. The structural and functional status of the vasculature are independent predictors of coronary events, as shown by noninvasive measurement of the endothelial function and carotid atheroma burden in patients. There is a risk of stratification in CVD development. Endothelial dysfunction is both an early marker of vascular disease and a facilitative process in the development of atherosclerosis. Some evidence indicates that the presence and degree of endothelial dysfunction are of prognostic importance. But these early results are tempered by the paucity of studies using noninvasive methods.

Methodology

Computerized methods are implemented with the intention to obtain results quickly and accurately. Calculating cardiovascular risk and detecting atherosclerosis, atheromatous plaques, and endothelial dysfunction. Digital image processing algorithms based on rules of prediction for pattern generation and acquisition systems, thereby complement detection methods using arterial images and pattern extraction to determine diameters and thicknesses of vascular layers. These algorithms indicate the presence and severity of vascular disorders, with the precision needed to favor clinical monitoring of patients.

Use of Ultrasonography Image with DICOM Format

Ultrasound is the most widely used technique for in vivo measurement of Intima-Media Thickness (IMT) of the carotid and brachial artery [Pianykh 2011] and has many advantages over other radiological or imaging methods: measurements of wall arteries, wall thickness, diameters and areas. In addition, ultrasound scanners are portable and available in hospitals and clinics. Research with ultrasonography is widespread because it covers a very large percentage of the population. Perhaps the major drawback of ultrasound is that it is highly operatordependent [START_REF] Theodoridis | Pattern Recognition[END_REF], meaning that a number of technical and methodological requirements must be met in order to perform a correct and effective computer-based endothelium measurement. To fully exploit the potential of ultrasound imaging in atherosclerosis research, image acquisition and analyses should meet DICOM standards [START_REF] Loizou | comparative evaluation of despeckle filtering in ultrasound imaging of the carotid artery[END_REF], Pham et al. 2007]. The DI-COM norm integrates additional medical data within the image itself. It makes it possible to measure both image acquisition and conformity with protocols.

Segmentation: Automatic Recognition of Artery

The automatic segmentation of anatomical structures in ultrasound imagery is a real challenge due to the acoustic interference (speckle) noise and artifacts which are inherent in these images. Speckle noise depends on the distribution of points within the tissue and appears to provide visual differentiation in terms of tissue texture. For our study of CVDs, speckle noise has not been a problem in automated recognition. The first issue in automatic image analysis is the segmentation of the area of interest. The artery must then be recognized in the image frame. Once recognized, the distal wall of the carotid and brachial artery can be segmented in order to extract the characteristic pattern of the artery.

The basic idea consists of assuming the artery as the dark region (lumen: space where blood circulates), comprised of two bright stripes (the near and far wall artery layers). Therefore, the artery is recognized when the boundaries of the near and far wall endothelium have been traced. This procedure combines feature extraction, fitting, and the classification approach. The image is processed by column to locate the maximum intensity of each generated column by a linear discriminant to detect the artery wall [Fig. 6]. These points are called seed points and they are then used to link all the sequences of segmentation lines. An intelligent procedure removes false positives inside the segmentation lines and joins the nearest aligned segments. This step avoids over-segmentation, reducing speckle noise using a Gaussian filter applied with a sliding kernel.

This technique is user independent. Its integrated approach exploits the morphologic characteristics of the arterial wall thus enabling segmentation (recognition) in more than 95% of cases.

Automatic Detection of Vascular Anomalies

Segmentation of IMT and lumen in arteries is perhaps the most challenging problem in our investigation. Three concentric layers constitute the arterial wall and are named, from inside out: tunica intima, tunica media and adventitia [Fig. 7]. Only the tunica intima and the tunica media are controlled.

Automatic detection of CVDs and features measuring of arteries should be reliable, this implies accuracy, repeatability, robustness to noise, applicability to diverse pathologies, independence from the human operator, independence from the ultrasound scanner, and measurement uncertainty characterization. For our study, we propose three new tools for assisting physicians in the detection

Measure of Lumen in Brachial Artery

Measurements of light diameter in the brachial artery must be taken between the two artery walls in the intima-lumen [Fig. 7]. In most cases the two leading edges defining the lumen diameter of the brachial artery can be easily recorded. The arterial lumen must be detected via the process of detecting the edges. It is therefore of high importance to accurately detect the borders of the artery. We use a cellular automaton introduced by Ullman and von Neumann [START_REF] Hunter | Operations on Images Using Quadtree[END_REF].

A cellular automaton is basically a computer algorithm that is discrete in space and time and operates on a lattice of sites (in our case, pixels of an image). This type of technique changes its state depending on the state of the neighborhood cells at a given time, and its own previous state. A set of triplet (i, j, k) (cells) characterizes each pixel, where (i, j) represents the position of the cell associated to k-level intensity. The ultrasound can be considered a specific cellular automaton where cellular space is a window N × N within the ultrasonographic image. In our process, we consider two kinds of cells: live cells (light) and dead cells (muscle) [Fig. 8(a)]. Measurement of the brachial artery lumen is determined for the internal area, a process known as region growing [Adams 1994]. It is also classified as a pixelbased image segmentation method since it involves the selection of initial seed points. This approach to segmentation examines neighboring pixels of initial seed points and determines whether the neighboring pixels should be added to the region. The process is iterated, in the same manner as general data clustering algorithms [Fig. 8(b)].

Measure of Endothelium in the Carotid Artery

The most important determinant in the diagnosis of atherosclerosis is the measurement of the carotid IMT that is defined on a side of the artery wall. The near-wall intima/lumen is defined by the leading edge. We use the ARC-Potential method [START_REF] Rivera | Adaptive Rest Condition Potentials: First and Second Order Edge-Preserving Regularization[END_REF] which quickly determines the IMT and preserves the edge. The IMT measure uses the edge and a collection of seed points determined with the gradient. The adventitia generally bright indicates precisely the end of IMT. Consequently, IMT measure eases to distinguish edges inside the area of common carotid of artery wall. This measure is done using the transition between the light and the artery wall (Lumen-Intima or LI) and also the transition between the media and the adventitia (MA) [Fig. 9]. The gradient satisfies the

Atherosclerosis Plaque

An atherosclerosis plaque in the ultrasound image may be defined as a focal thickening of the intima-media complex that is 50% thicker than that of the adjacent region, or as a focal thickening over 1.5 mm. For automatic edge detection, we used quad-trees by combining a nonparametric classifier [START_REF] Hunter | Operations on Images Using Quadtree[END_REF] based on a clustering algorithm, with a quad-tree representation of the image. The scheme performs well and is simple to implement, giving satisfactory results in spite of speckle noise interference [Fig. 11].

Carotid bulb IMT may be measured in the majority of cases with a low dropout rate. Areas with plaque should not be excluded during measurement.

Based on the ultrasonographic image process, the system has determined the Intima-Media thickness, the inner area of the artery, its diameter and the endothelium. Now the system determines the cardiovascular disease risk factor and a diagosis based on a set of rules. For example, an adult having an IMT greater than 1 milimeter has atherosclerosis. All this information is written in a text file. This text file with the original images of a patient have to be available to physicians in order to collaborate in the final diagnosis [Fig. 12]. The files have to be adapted to the different devices, networks and protocols.

In the following section, it is important to know how to adapt the diagnostic information (three images and a text) in function of devices, networks and protocols. The goal is to enable practitioners to work together and elaborate a final diagnosis in the best conditions.

In the next sections, we define (1) the real-time collaborative environment for telemedicine, and (2) our new adaptive platform Wava.

Collaborative Environments for Telemedicine

The CNOM (French National Council of Health Practitioners) defines Telemedicine as the practice that allows a patient to receive medical attention remotely from one or more practitioners through the use of information and communication technologies in order to deliver the required medical data.

Adaptation in Collaborative Environments

Collaborative environments consist solely of data exchange. Adaptation of data streams can be implemented to ensure that the data are adequate for user needs. In recent years, several studies have been undertaken in the area of adaptability and especially in the adaptation of multimedia flows [START_REF] Garcia | [END_REF], Dalmau et al. 2009]. Our study identifies research that ensures adaptation in distributed systems and especially in collaborative software. To understand the problems of adaptation data in these environments, it is necessary to study how these environments are used today and what constraints they face.

The ultimate goal of an adaptable collaborative system is to be able to adapt to new needs, considering the needs used in the initial analysis and evolution of the tools and technologies used [START_REF] Elmarzouqi | [END_REF], Mrak et al. 2009]. These systems have come about through the joint development of both designers and users.

A choice must always be made between the needs, skills and preferences, and technological constraints, but also using the budgetary skills and habits of designers as shown by the team Nallini Selvaraj, University of New York [Selvaraj et al. 2007]. At present, the adaptation is implemented through wider applications to adapt to new ways of using computer tools, such as mobility and access to data via various devices. Websites have adapted to needs and allow us to benefit from any information in just seconds. Whether this information is in the form of text, audio or video, new communication infrastructures and terminals can display it without using additional software or codecs, provided that these data are appropriate.

The term co-evolution, suggested by Jurgen Ziegler [START_REF] Ziegler | [END_REF], was chosen to reflect the fact that collaborative systems should be in continuous evolution, without, however, being independent or self-adaptive, because they are explicitly accountable for the changing needs, attitudes and skills of users, individually or collectively. Users must also change and adapt their practices and working methods to meet the changing needs of the organization. Following the development of specific applications and collaborative systems, a more industrial set up is needed in the form of platforms that integrate services and components designed for group work. New applications can be quickly developed using the building blocks that have been developed in several other studies [START_REF] Duque | [END_REF]] [Ziegler et al. 2010].

But, being able to adapt the data flow within the environment does not guarantee that the contents actually fit properly or even optimally. Adaptability can implement techniques when the user is not satisfied with the result. The system can be configured to adapt so as to improve the result, but the user must still know how to use these techniques and, especially, how to use them appropriately. Implementing a policy of transparent adaptability allows the user to completely forget the configuration and adaptation of the working environment. However, for this system to make the self-adaptation of context and data totally transparent, the system must take into account all relevant parameters and applications necessary for the operation. Different methods may be used as required:

-Adaptation: the system adapts data according to criteria or predefined sce-narios.

-Adaptability: the user is master of his system and makes decisions from the choices offered to him at the risk of rendering the system unstable.

-Adaptivity: The system selects the best options based on criteria selected by the user, proposing only modes compatible with the type of application, synchronous, asynchronous and system constraints (the need for bandwidth, computing capacity).

Collaborative software for telemedicine furnishes practitioners with the collaborative tools necessary to perform online diagnostics [START_REF] Fuin | [END_REF]]. These tools allow them to exchange information such as medical images and previous diagnoses. More and more tools are available to enable real-time diagnostics including neurology. A camera (webcam or IP camera) is used to transmit video streams directly to the patient's computer specialist who no longer needs to travel. Thus, one hospital, which does not have a specialist on site, can, by means of these tools, call upon one or more practitioners for diagnosis in the shortest possible time, which in some cases may save the patient's life. To be sure the diagnosis takes place in good conditions we must implement flow control device videos. In recent years, the rapid advances in telecommunications and the development of more efficient terminals have allowed the practice of telemedicine to grow rapidly.

The New Challenges of Collaborative Environments: Mobility/Ubiquity

Collaborative environments are also called groupware and are used to assist the work of a group of people. These environments emerged in the early 90s. They have undergone many changes since their creation, through different approaches in the design and architecture that have supported collaborative learning in all its aspects.

Over the last twenty years, the concept of remote working (teleworking) has undergone great development, largely due to the combined growth of networks and the evolution of more efficient terminals, but also due to the improved security in these systems. Collaborative environments are used in various ways such as e-Learning, [START_REF] Watkins | [END_REF]], remote maintenance [START_REF] Sun | Consistency maintenance in realtime collaborative graphics editing systems[END_REF], collaborative design [START_REF] Duque | [END_REF]] and telemedicine [START_REF] Elmarzouqi | [END_REF].

Mobile computing can now be seen as a more holistic approach than ubiquitous computing. Also known as pervasive computing [Louberry et al. 2009], ubiquitous computing aims to render a multitude of services accessible from anywhere, while hiding the computer aspect. This desire to free the user from the constraints of the current use of a computer (sitting in front of a screen and using a keyboard) gives him his freedom of action and above all, freedom of movement. Ubiquity is built into an environment that integrates mobility. These new uses imply that new constraints must be taken into account, in addition to the constraints of working together.

Collaborative environments provide users with tools to be used in the field for which they were created, in addition to the traditional tools of chat, whiteboard and videoconferencing [START_REF] Duque | [END_REF]]. For medicine, the tools are specific and allow doctors to have discussions about the purpose of treating the cases diagnosed. Tools must allow all actors in the system access to the same information (images, previous diagnoses, patient history, etc.) so that they can collaborate in real time and arrive at a reliable diagnosis.

To ensure that practitioners can work in good conditions, information must be available at all times and data must be sent and presented to the various actors uniformly and in sufficient quality.

Real-Time Collaborative Environments for Telemedicine

Telemedicine can be defined in the following words: it is a form of collaboration in medical practice, by connecting remotely, through information and communication technologies, a patient (and / or medical data) with one or more physicians and health professionals for the purpose of medical diagnosis, decision support and treatment in accordance with the rules of medical ethics.

Telemedicine is used in the context of patient monitoring or education and has recently been introduced in the area of emergency care. Telemedicine can be defined as the use of new telecommunication technologies in providing information [START_REF] Mitra | Source separation on hyperspectral cube applied to dermatology[END_REF]] and medical services to patients and doctors through audio and visual communication. These new technologies are used to help practitioners located at remote sites to perform diagnostic procedures and consultations such as clinical examinations through the transfer of medical images [START_REF] Fuin | [END_REF]]. These images are often large and may take time to be routed according to the networks used.

A New Service of Data Flow Adaptation

We propose a new service, the adaptation of data provided by information from the environment capacity manager and the exchanged data type. This web service is called WAVA and allows us to change the properties of the exchanged data within a given environment, taking into account its constraints.

WAVA: Web Service for Automatic Adaptation

The WAVA adaptation determines marks in the function of properties used by specific devices. In its turn, the adaptation server determines and reports the best quality/capacity for data to meet the requirements of the devices. Quality is determined by adaptation of the encoding type used and also the resolution and compression rate. According to device performances (processors, graphic card, resolution) and the network used, the system provides a corresponding mark for each property. This information concerning the devices and their properties is stored in a database. A distinction must be made between two kinds of capacity evaluation. The first is the static mark that does not change during a session (Mark global (Mg)). This mark refers to CPU (Mcpu), resolution (Mres) and graphic card (Mgrph) that do not change during a session. If the devices are near each other then Mg is determined as the average of Mcpu, Mre and Mgrph, otherwise the lowest of them is used. The second kind of notation is a dynamic one. It is calculated with bandwidth (MBW) and the CPU charge (MC) that change during a session. To obtain the final mark (M), we re-evaluate the global mark with the dynamic one by taking the lower of the two marks to validate that the chosen adaptation is in concordance with the system. The use of WAVA is depicted [Fig. 13]. WAVA periodically receives information about the bandwidth and the CPU charge of the users computers/devices via information retrieved by the coordinator. The new mark is calculated in function of new information available.

WAVA Service Architecture

This prototype is also based on provided services capable of furnishing a data skeleton to applications that want to use this service. The algorithm for calculating associated marks is shown in Fig. 14. This algorithm marks the terminal that requires access to a resource from the data provided by the configuration manager. The mark ranges from zero to five, five being the highest performance. This phase is totally transparent to the user.

Performance

Medical research laboratories in Mexico provided 1,062 DICOM images from real patients to be used in the assessment of our automatic diagnosis system. The ROC curve obtained after the sensitivity analysis, performed to assess the diagnostic accuracy of Atherosclerosis in carotid artery ultrasound [Fig. 15(a)]. This curve was built varying the threshold from 0.1 mm to 1.3 mm with a step size of 0.01 mm; such a threshold is used by the system to emit a diagnosis. The ROC curve obtained after our tests for evaluation of the diagnostic accuracy in determining endothelial dysfunction in humeral artery ultrasound pre-and poststimulation [Fig. 15(b)]. This curve curve was built comparing the percentage of arterial dilation with a threshold that was varied from 5% to 40% with a step size of 0.1%.

In the VACODIS telediagnosis application, the system provides physicians with three images (lumen, endothelium, and plaque measurements) and a generated synthetic text report with an automatically generated diagnosis. The WAVA platform is used for the three images but is not required for the text report. In The web service converts images to the required format depending on the previous automatically determined mark. The lowest quality corresponds to a usable format on a Smartphone without loss of quality. Users don't relate any loss of precision on images becausedefintion of screen on smartphone is higher than on a computer screen. Test have been made in accordance with practitioners who consider the quality was sufficient to make the diagnosis.

To test the performance of this platform, different kinds of terminals have been used to connect practicioners to the platform for diagnoses. Terminal (A) an Android tablet connected to WiFi, (B) an iPad connected by HSDPA, (C) a Smartphone connected to 3G, and (D) a laptop with a WiFi connection. The results are shown in Fig. 15(c). The adapted images display faster in the termi-nals (less CPU usage) and use less space (less memory usage). These points are of great importance for mobile terminals.

Conclusion

We have proposed a new tool for detection of cardiovascular abnormalities in patients. This semi-automatic tool (so-called due to its professional validation) is in use among hospital practitioners in Mexico, but it remains a local and individual tool. It is important for small hospitals (for example, regional hospitals attached to a distant University Hospital Center), to be able to guarantee the same healthcare offer regardless of the patients location. It is important for hospital experts to be able to send their opinions and participate remotely and collaboratively in the telediagnosis.

In this study we have proposed combining the semi-automatic detection tool with a collaborative adaptive platform thus allowing practitioners to share data and to co-author their remote diagnosis. The first results show that practitioners benefit from a very high level of information availability, and performance results validate this use. The phase of clinical trials is now in progress.

Until now, we have worked on two directions: (1) discrete media (images and text), but healthcare experts use echo-Doppler video recordings to visualize some blood circulation problems. Our present investigations are oriented towards the adaptability of these continuous media and any new problems that emerge, (2) the use of ontology [START_REF] Hervas | [END_REF] for the traceability of telediagnosis and the representation of information.

  (a)] and carotid artery[Fig. 2(b)]. The role of the endothelium [Fig.2(c)] in human disease has recently become the focus of intense scientific investigation[Davignon et al. 2004]. Impaired endothelial function is associated with a number of disease states, including CVD and its major risk factors. Endothelial dysfunction precedes overt vascular disease by years and may itself be a potentially modifiable CVD risk factor.

Figure 2 :

 2 Figure 2: (a) Brachial artery, (b) Carotid artery, and (c) Diseased artery

Figure 3 :Figure 4 :

 34 Figure 3: Flow causing endothelium-dependent dilatation

Figure 5 :

 5 Figure 5: Atherosclerosis plaque

Figure 6 :

 6 Figure 6: Artery recognition

Figure 8 :

 8 Figure 8: (a) Edge detection using cellular automata, (b) Measurement of lumen in brachial artery

Figure 9 :

 9 Figure 9: Carotid artery wall segmentation based on an intensity criterion and a gradient approach

Figure 10 :

 10 Figure 10: Measure of IMT carotid by segmentation

Figure 11 :

 11 Figure 11: Plaque segmentation using the quad-tree technique

Figure 13 :

 13 Figure 13: WAVA Principle

Figure 14 :

 14 Figure 14: WAVA Algorithm

Figure 15 :

 15 Figure 15: ROC curve (a)Atherosclerosis diagnosis, (b)Endothelium dysfunction diagnosis, (c)WAVA Performances

http://www.who.int/cardiovascular_diseases/en/


Z Wang

Y Sui

P D M Spelt

W Wang

Three-dimensional dynamics of oblate and prolate capsules in shear flow

We study computationally the dynamics of oblate and prolate spheroidal capsules in simple shear flow with small inertia for a range of dimensionless shear rates. The capsule is modelled as a liquid droplet enclosed by a hyperelastic membrane, and its equatorial plane is initially tilted out of the plane of shear. We find at low shear rates, the well accepted tumbling motion is not always stable for both oblate and prolate capsules. For an oblate capsule, the dominant stable modes for increasing dimensionless shear rate are: rolling with the equatorial plane staying in the plane of shear, precessing around the vorticity direction, and tumbling. Interestingly, the order of modes is reversed for a prolate capsule: tumbling, precessing and rolling with increasing dimensionless shear rate. At transitional regimes, we find the stable motion of a capsule can depend on its initial titled angle even at the same shear rate. At high dimensionless shear rates, a spheroidal capsule undergoes a complicated oscillating-swinging motion: its major axis oscillates about the plane of shear in addition to the swinging about a mean angle with flow direction found previously, and the amplitudes of both oscillations decrease when increasing the dimensionless shear rate, towards a steady tank treading motion asymptotically. We summarize the results in phase diagrams and discuss the reorientation of both oblate and prolate capsules in a wide range of dimensionless shear rates.

I. INTRODUCTION

A capsule is a small liquid droplet enclosed by a thin membrane which has shear elasticity. The dynamic motion of a capsule subject to an external flow has been drawing much attention for many years, because it is important for both fundamental research and biomedical or industrial applications (e.g., in micro-encapsulation and drug delivery [1,2]). Furthermore, it forms the first step for considering more complicated situations such as capsule suspensions [3,4].

Various types of motion have been identified for a capsule freely suspended in simple shear flow. An initially spherical capsule has always been found to exhibit a steady tank-treading motion, wherein the capsule has deformed into a stationary configuration with a finite inclination with the flow direction and the membrane rotates around the liquid interior (e.g., in Refs [5][6][7][8] and a recent review by Barthès-Biesel [9]). The motion of non-spherical capsules is richer in that it is known to yield three modes: a tumbling mode, wherein a capsule flips continuously, a swinging mode, corresponding to periodic shape deformation and small amplitude inclination oscillation with a non-zero mean angle with flow direction when the membrane is rotating around the liquid interior, and a vacillating-breathing mode, in which the capsule's major axis oscillates around the flow direction and the shape shows ample deformation (breathing) [10,11]. The vacillating-breathing mode lies between tumbling and swinging. As to which class of motion a non-spherical capsule adopts is known to be affected by the viscosity ratio between the internal and exter-nal liquids [12][13][14], the membrane viscosity [15][16][17][18], the shear rate [11,[19][20][21][22][23][24][25][26][27] and the membrane bending stiffness [28,29]. In general, the transition from swinging to tumbling can be induced by increasing the viscosity ratio or membrane viscosity, reducing the shear rate, or increasing the membrane bending stiffness when the capsule has a non-spherical reference shape.

All of these modes (tank-treading, swinging, tumbling and vacillating-breathing) are symmetric about the plane of shear, and this symmetry has been imposed (at least on initial conditions) in most previous theoretical and numerical studies on capsules in shear flows. Early experiments by Goldsmith and Marlow [12] and a very recent study by Dupire et al. [START_REF] Dupire | Proc. Natl. Acad. Sci[END_REF] on red blood cells have suggested the existence of an asymmetrical mode, however. In other systems, such a mode is well documented: a rigid ellipsoidal particle can exhibit a precessing orbit wherein the trajectory of its axis of revolution forms a cone [START_REF] Jeffery | [END_REF], and the angle of the cone depends on the initial inclination angle of the rigid particle. A similar off-theshear-plane motion has been found recently for vesicles in shear flow both theoretically [32] and numerically [33][34][35] and it was named as a 'kayaking' motion [33], which appears when the viscosity ratio and the shear rate are both high. However, a vesicle is different from a capsule in that its membrane is incompressible but does not have any shear elasticity, and such differences in membrane mechanical property lead to very different dynamics in shear flows (e.g., in Refs [10,34,36,37]). For example, the behaviour of a vesicle is critically determined by the viscosity ratio. At low viscosity ratios a vesicle always undergoes a steady tanking treading motion, indepen-dent of the shear rate; however, a non-spherical capsule could have a swinging or a tumbling motion depending on the shear rate at such low viscosity ratios. Without the constraint of membrane shear elasticity, the dynamics of vesicles appears to be richer and recently novel modes such as squaring and parity breaking have been discovered in numerical simulations [35]. So far, the existence of an asymmetrical precessing mode for capsules is still largely an open issue. The only related study is by Dupont et al. [38] on prolate capsules. Clearly there is a need for a numerical study of this possibility for oblate capsules, which forms one motivation of the present paper. In a larger context, it is well accepted that a red blood cell can be modelled mechanically as a liquid-filled oblate capsule (e.g., in Refs [26,[39][40][41]), so it would be of interest to see to what extent an asymmetrical mode suggested by experimental work cited above would indeed be predicted by this model.

In experiments, yet another mode has been found for red blood cells besides the suggestion of precessing motion discussed above (e.g., in Refs [12,[START_REF] Dupire | Proc. Natl. Acad. Sci[END_REF]42]): a cell spins around its axis of revolution with its equatorial planes staying in the plane of shear, like a rolling wheel. These experiments show that this rolling motion can exist under the same conditions as tumbling. It is in general accepted that a red blood cell shares many features in common with an oblate spheroidal capsule, however, such rolling motion has not been reported for oblate capsules with numerical simulation or theoretical approaches, to the best of our knowledge. Only recently with a boundary integral method, Dupont et al. [38] considered the dynamics of prolate spheroidal capsules, whose revolution axis are initially not in the plane of shear. They found that tumbling motion is not stable at low shear rates, and that a prolate capsule will gradually adopt a log-rolling motion with its revolution axis aligned with the vorticity axis.

In summary, for non-spherical capsules in shear flow, with the shear rate increasing, well documented modes are tumbling, vacillating-breathing and swinging; all are symmetric about the plane of shear. Experiments of red blood cells in shear flow have identified an asymmetric off-plane precessing mode and a rolling mode, but these have not been recovered in numerical simulations using the oblate-capsule model, and the relations of these two motions with other modes remain unclear. In this paper, we therefore study the effects of initially tilting a capsule's equatorial plane out of the plane of shear, as an off-plane perturbation, on its subsequent dynamics, seeking specifically to establish the sequence of events when increasing the dimensionless shear rate, which would provide a platform for future theoretical work, as well as future work on suspensions of capsules; also, the work establishes the possibility of coexistence of modes, which has consequences for experimental observations. We use an immersed boundary lattice Boltzmann method [43], which has been well validated for simulating flow-induced deformation of capsules. Both oblate and y z x ry rx r FIG. 1. A spheroidal capsule in shear flow. r represents the revolution axis. Vorticity axis is the y-axis. θry and θrx are angles that the capsule's revolution axis makes with the vorticity axis and x-axis respectively. φ is the phase angle defined as the angle that the revolution axis makes with xaxis in the plane of shear.

prolate spheroidal capsules are considered. The capsule is modelled as a Newtonian liquid droplet enclosed by a hyperelastic membrane, in the limit wherein the membrane is very thin, hence bending stiffness is neglected. A unity viscosity ratio is used; also, the capsule is assumed to be unstressed at its initial shape. Although real situations are usually more complicated, for example a capsule membrane could be pre-stressed due to osmotic effects, these assumptions form an important limit that has been studied extensively for initial conditions wherein the capsule is symmetric about the plane of shear (e.g., in Refs [14,24,26]). Results for a range of initial tilting angles and dimensionless shear rates are categorized (in particular, new regimes for capsules are identified and analysed further) and discussed in light of the above summary of prior work. The sensitivity of the results to the membrane constitutive law used is assessed by comparing results obtained with two different models.

II. MEMBRANE MODEL AND NUMERICAL METHOD

We consider an initially spheroidal capsule of various aspect ratios in an incident linear shear flow, u = (kz, 0, 0) where k is the shear rate (Fig. 1). The aspect ratio of the capsule is defined as the length of its revolution axis over that of the equatorial axis, and is thus larger than unity for a prolate capsule and smaller than unity for an oblate capsule. The axis of revolution of the capsule is initially in y-z plane and the capsule's equatorial plane is tilted out of shear plane under an angle β; an equivalent capsule radius is denoted by a = (3V /4π) 1/3 , where V is the volume of the capsule. As stated in the Introduction, some simplifying assumptions are made: the fluids inside and outside the capsule are both taken to be Newtonian and to have the same density ρ and viscosity µ, and the capsule membrane is assumed to be very thin, hence bending stiffness is neglected. This leads also to the formation of wrinkles in the simulations that would otherwise be suppressed by bending, but we have found these to remain at acceptable levels (as in related work, e.g., [7,26]); also, results are consistent with analytical work, as discussed in Sec. III-IV. Barthès-Biesel and co-workers studied the effect of membrane bending and found that a moderate bending rigidity would alleviate wrinkles but this does not have significant effects on the global deformation and the dynamic motions of a capsule [START_REF] Dupont | 11e Colloque National en Calcul des Structures[END_REF].

Results have been obtained for two different membrane constitutive laws in the present study. One is the neo-Hookean (NH) law, assuming linear dependence of strain energy density with the first invariant of the left Cauchy-Green deformation tensor λ 1 2 + λ 2 2 + λ 3 2 . The terms λ 1 , λ 2 and λ 3 are the principle extension ratios in the plane of the membrane, and in the direction perpendicular to the plane, respectively. The NH law corresponds to thin membranes made of three-dimensional isotropic and volume-incompressible polymerized materials. The area dilation is unrestricted and is compensated by the thinning of the membrane: λ 3 = 1/λ 1 λ 2 . The strain energy function of NH law for a two-dimensional membrane is given by

W = 1 6 E ( I 1 -1 + 1 I 2 + 1 ) , (1)
where E is the surface shear elasticity modulus, and I 1 and I 2 are the first and second strain invariants in two-dimensional, with

I 1 = λ 1 2 + λ 2 2 -2 and I 2 = (λ 1 λ 2 ) 2 -1 = (dA/dA 0 ) 2 -1.
The parameters dA 0 and dA are initial and final areas of a membrane element. The principal membrane elastic tensions τ 1 and τ 2 , result from NH law are

τ 1 = E 3λ 1 λ 2 (λ 2 1 - 1 λ 2 1 λ 2 2 ) (likewise for τ 2 ). ( 2
)
The second constitutive equation used is Skalak's (SK) law [START_REF] Skalak | [END_REF] to model the membrane of a red blood cell:

W = 1 6 E ( 1 2 I 2 1 + I 1 -I 2 ) + 1 12 CEI 2 2 . (3)
Postulated in such a way, SK law can capture the special feature of biological membranes that deform easily under shear while almost keep a constant area. For such membranes, on the right hand side of Eq. ( 3) the factor C (in the second term) must be quite large to ensure negligible area dilation (I 2 → 0), and the first term mainly accounts for shape deformation. These are clearer when looking at the principle membrane tensions

τ 1 = Eλ 1 3λ 2 [(λ 2 1 -1 + Cλ 2 2 I 2 ] (likewise for τ 2 ). ( 4
)
We have divided the original strain energy function in [START_REF] Skalak | [END_REF] by a factor of 1.5 so that Eq. ( 3) leads to the same small deformation behavior as NH law when C = 1.

The 3D capsule membrane is discretized into flat triangular elements, and a finite-element membrane model is employed to obtain the forces acting at the discrete nodes of the membrane. The present simulation method is based on the immersed boundary lattice Boltzmann method of Sui et al. [43,[46][47][48]. The fluid-capsule interaction is solved by the immersed boundary method of Peskin [49], the flow field is solved by the lattice Boltzmann method with a multi-block technique to refine the mesh around the deformable capsule. The method has been validated extensively against results of boundary element simulations and small deformation theory for three-dimensional capsules in shear flow [22,43]. The simulation set-up (e.g. domain and mesh sizes) in the present study is the same as that in our earlier work, where detailed convergence tests were carried out. For most cases in the present study, the global error in mass conservation of the capsule is within 1.5%.

The following dimensionless groups are used: the Reynolds number is based on the dimension of the capsule, and is defined as Re = 4ρka 2 /µ; the ratio of viscous and elastic forces is represented as G = µka/E, which can be considered to be a dimensionless shear rate or a capillary number. In the present numerical simulations, the shear rate is kept constant and the membrane shear elasticity is varied, so that the Reynolds number does not change but a range of values of dimensionless shear rates can be achieved. We investigate the effect of the Reynolds number on the results towards the end of this paper. The same approach has been used in many previous studies and in such a way the effect of the dimensionless shear rate can be isolated from that of the Reynolds number. In the presentation of the results, the time variable has been made dimensionless with k -1 .

III. MOTIONS OF OBLATE CAPSULES

Results are first presented for an oblate spheroidal capsule with an initial aspect ratio of 2:3 and an NH membrane, for increasing the dimensionless shear rate G from O(10 -4 ) to O(10 -1 ) whilst keeping the Reynolds number fixed at 0.2. Effects of Re and the initial capsule aspect ratio are discussed further below.

At low dimensionless shear rates, G ≤ 0.01, it is in general accepted that the capsule will undergo a tumbling motion. In the present study, we find tumbling is not stable to off-plane perturbations; instead, the stable modes of an initially oblate capsule for increasing dimensionless shear rates are: rolling, precessing and tumbling.

At very low dimensionless shear rates, G ≤ 10 -3 , the capsule's global deformation is not visible, and it is found that independent of the value of the initial angle β, the capsule gradually adopts a rolling motion (shown in Fig. 2 and discussed later on). This can be best characterized by the time evolution of the angles that the capsule's revolution axis makes with the y-axis (direction of vorticity) θ ry (illustrated in Fig. 1) and x-axis (flow direction) θ rx , as shown in Fig. 3a for a capsule whose equatorial plane is initially perpendicular to the plane of shear. It is seen that the angle θ ry decreases in an oscillatory fashion towards zero and that θ rx exhibits a dampened oscillation around 90 • . These results indicate that the motion of the capsule loses its symmetry about the shear plane quickly; a capsule precesses with its equatorial plane gradually becomes aligned with the plane of shear, and then carries out a rolling motion as illustrated in Fig. 2a, in which the capsule spins about its revolution axis like a wheel. From Fig. 2b, it is seen that the capsule membrane in the plane of shear resembles a closed streamline which also suggests a steady configuration. It should be noted that a similar transition to rolling has been observed recently in numerical simulations of rigid oblate spheroidal particles in shear flow with small inertia [50,51], which we will discuss in Sec. V. The effect of dimensionless shear rate on the time evolution of θ ry is presented in Fig. 3b. It is seen that at the same initial tilted angle, the dimensionless time it takes for the capsule to change to rolling motion increases monotonically with the shear rate. This transient time also depends on the initial tilted angle, and it is longest for β = 90 • for the same dimensionless shear rate. We have also carried out simulations of capsules with an SK membrane and have found a similar transition to rolling motion at low shear rates. An example has been included in Fig. 3b for C = 10 at G = 0.001.

When the dimensionless shear rate is increased, a new mode of motion dominates a large range of initial tilted angle values. The oblate spheroidal capsule still largely keeps its initial shape; it carries out a precessing motion and seems to achieve a periodic state after a long transient stage as shown in Fig. 4a. At G = 0.004, an initial tilted angle of β = 22.5 • is not enough to trigger the precessing motion. The equatorial plane of the capsule returns to the plane of shear shortly and the capsule finally carries out a rolling motion. For sufficiently large tilted angles, such as β = 45 • or 60 • , θ ry finally oscillates around a constant value which suggests a periodic state. When G is increased slightly to 0.005, an initial tilted angle of β = 22.5 • is sufficient to lead to a precessing motion (Fig. 4b andc). Figure 4b shows the trajectory of the capsule's revolution axis projected onto a unit sphere. After the transient stage, the revolution axis forms a cone, and its intersection point with the surface of the unit sphere falls into a limit circle.

Apparently similar off-the-shear-plane motion has been described by Jeffery [START_REF] Jeffery | [END_REF] for rigid particles in shear flow. For precessing motion of an ellipsoid of aspect ratio χ in shear flow without inertia, this corresponds to [START_REF] Jeffery | [END_REF] tan φ = χ tan

( kt χ + χ -1 ) , (5)
tan

θ ry = Bχ √ χ 2 cos 2 φ + sin 2 φ , (6)
where the phase angle φ is defined as the angle that the revolution axis of the particle makes with the flow direction in the plane of shear (see Fig. 1); B is the so-called orbit constant which only depends on the initial position of release of a rigid particle; its value determines the maximum and minimum values of θ ry during the periodic precessing motion through

tan θ min ry = Bχ, tan θ max ry = B. (7)
Eq. ( 7) can be used to fit the value of B to the results of the numerical simulation. In Fig. 4c we show an example comparison of the precessing state with Jeffery's orbit [START_REF] Jeffery | [END_REF] for χ = 2/3 and a fitted value of B. Similar good agreement has been observed for other cases in precessing regime.

We have extracted the time-averaged value and the oscillation amplitude of θ ry for capsules in the precessing motion; one example is shown in Fig. 5. The results suggest that transition from rolling to precessing happens via a Hopf bifurcation. It is also seen that the oscillation amplitude of θ ry arrives at a maximum value when the time averaged value is between 30 • and 60 • . The time averaged value of θ ry increases with the dimensionless shear rate in precessing regime, and for sufficiently high dimensionless shear rates and large initial tilted angles, the average value of θ ry reaches 90 • , which corresponds to the axis of revolution of the capsule arriving in the plane of shear and thus a tumbling motion which is symmetric about the plane of shear. Our simulation results show that when G is between 0.01 and 0.02, the capsule finally carries out the tumbling motion for a wide range of β. (see the inset). Interestingly, for β = 5 • , which can be considered as a small off-the-shear-plane perturbation to a capsule whose equatorial plane is in the plane of shear, the capsule turns back to rolling motion, which suggests rolling motion is strongly stable for an oblate capsule.

At low shear rates (G ≤ 0.02), the shape of the capsule largely resembles an oblate spheroid. We can find the length of the revolution axis c and the equivalent diameter of the capsule in its equatorial plane with area S, b = 2(S/π) 0.5 . A Taylor shape parameter D = (bc)/(b + c) has been computed and the results are presented in Fig. 6b. It is seen that the oscillation amplitude of the shape parameter increases with G and the minimum value approaches zero. When D becomes close to zero, the projection of the capsule onto the plane of shear is close to a circle and it is difficult to identify a major axis.

The limit D min → 0 has been proposed recently in [26] as a criterion for a transition from tumbling to swinging. A swinging regime is indeed achieved when G is beyond 0.03, where the major axis of the capsule can be identified from the principle axis of the capsule membrane's moment of inertia tensor. In previous studies where the capsule is symmetric about shear plane during deformation (e.g., in Refs [21,22,26]), it has been found that the angle between the major axis and the flow direction oscillates around a mean value with a small amplitude in a swinging motion. Interestingly, in the present study, we find that initially tilting the capsule out of the shear plane can lead to very complicated swinging states, in which the major axis oscillates about the plane of shear in addition to the previously found swinging about a mean angle with the flow direction. Such an off-plane swinging motion has also been found for prolate capsules in shear flow, and was named as oscillating-swinging (OS) motion [38]; here we use the same name. Figure 7a shows the trajectory of the long axis projected to a unit sphere for such a oscillatingswinging state, and it is seen to form a lemniscate with size decreases when G is increased. Figure 7b shows the variations of the angles that the major axis of the capsule makes with the shear plane and xy plane. When increasing the dimensionless shear rate, both oscillation amplitudes decrease, towards zero which corresponds to a steady tank treading mode. However, there are two exceptions. One corresponds to β = 90 • , where the capsule carries out a swinging motion whilst adhering to symmetry about the shear plane; this has been well studied before. The other exception is for β = 0 • , i.e., when the capsule's equatorial plane is initially in the plane of shear, where we find the capsule to always be symmetric about the shear plane and this is very stable to small perturbations at β ≤ 5 • . The capsule achieves a stationary shape with a constant inclination angle with the xy plane and the membrane rotates around the liquid inside, just like a spherical capsule in shear flow.

Recently, Bagchi and co-workers identified a novel vacillating-breathing mode for non-spherical capsules in shear flow [11], notably when the fluid inside the capsule is much more viscous. This motion happens between tumbling and swinging when increasing the shear rate. Vacillating-breathing mode was first predicted theoret- ically by Misbah [10] for vesicles. It is different from swinging in that the major axis of the capsule/vesicle oscillates around the flow direction and thus has a zero mean inclination angle. In the present study assuming a unity viscosity ratio, we do find swinging motion with the major axis of the capsule periodically crossing the plane of zero velocity, but the time averaged value of the inclination angle with the flow direction is always considerably larger than zero. In a recent work by Omori et al. [52] on the dynamics of an oblate spheroidal capsule in shear flow, an interesting reorientation phenomenon was discovered wherein the capsule is in the swinging regime at high shear rates. Defining a unit vector which extends from the mass center of the capsule to the material point of the membrane that initially intersects with the revolution axis of the capsule, it was found that this vector would gradually approach the plane of shear at relatively low shear rates (similar to that in the tumbling motion and here we use OS(T) to represent this mode), and the unit vector would approach the vorticity axis of the shear flow at relatively high shear rates (similar to that in the Rolling motion and here we use OS(R) to represent this mode). In the present study we observed a similar reorientation for oblate capsules. For example in Fig. 7, G = 0.05 belongs to the OS(T) regime and G = 0.1 and 0.2 belongs to the OS(R) regime. Detailed features of the reorientation can be found in [52] and are thus not repeated here; such a reorientation will be further discussed in Sec. V, covering both oblate and prolate capsules in a much wider range of dimensionless shear rates.

To summarize the various regimes as a function of the dimensionless shear rate and the initial tilted angle, a phase diagram has been constructed and is shown in Fig. 8, noting that the viscosity ratio is unity and bending has been neglected. Several novel features can be observed from the phase diagram: first, at low dimensionless shear rates (G ≤ 0.02 for the present oblate capsule), previous studies assuming an initial symmetry about the shear plane have reported only one tumbling mode, while the present results indicate that when increasing the dimensionless shear rate, the dominant stable motion exhibits the following modes: rolling, precessing and tumbling; second, it is seen that there can be different modes at the same G, for example at G = 0.01, which would explain the coexistence of different modes (e.g., tumbling and rolling) observed in experiments on red blood cells [12]; third, at high dimensionless shear rates, a sufficient initial titled angle could lead to a more complicated oscillating-swinging motion.

IV. MOTIONS OF PROLATE CAPSULES

In previous studies of prolate capsules in shear flow (e.g., in [26]), the revolution axis of the capsule was initially in the plane of shear. It was found that the capsule undergoes a tumbling motion at low shear rates, and transits to a swinging motion with the shear rate increasing. Recently, Dupont et al. [38] studied for the first time the motions of prolate capsules in shear flow, with the revolution axis being initially titled out of the plane of shear. They found at low dimensionless shear rates, the stable mode is a log-rolling motion, where the long axis of the capsule aligns with the vorticity axis, as illustrated in Fig. 9. They also discovered that with the shear rate increasing the rolling motion becomes unstable, and the capsule precesses around the vorticity axis, which was named as wobbling (illustrated in Fig. 10). At even higher shear rates, the long axis of the prolate capsule tends to approach and to oscillate about the plane of shear, in an oscillating-swinging motion similar to that of an oblate capsule as described in Sec. III. This similarity is possibly because at high shear rates both prolate and oblate capsules deform into elongated ellipsoidal shapes. In the present study, we have also considered the dynamics of prolate spheroidal capsules in shear flow, for situations similar to that of Dupont et al. [38]. The difference between our study and that of Dupont et al. is that we include small inertia, and the lowest dimensionless shear rate considered has been reduced to G = 0.0001.

In this section, results are presented for a prolate capsule with an initial aspect ratio of 2:1 enclosed by an SK membrane with C = 1. The dimensionless shear rate considered ranges from G = 10 -4 to 0.4 and the Reynolds number is kept at 0.2. We summarize the results in a phase diagram as shown in Fig. 11. At relatively high dimensionless shear rates, G ≥ 0.01, our results are consistent with those of Dupont [38]: with the dimensionless shear rate increasing, we find rolling, wobbling (or precessing) and oscillating-swinging motions as the stable modes. However, at lower dimensionless shear rates, where the deformation of the capsule is very small, we have observed fundamentally new features. At very low dimensionless shear rates, G ≤ 0.0005, we find independent of the initial tilted angle a capsule finally adopts a tumbling motion. The capsule places its long axis in the shear plane and rotates about its minor axis. Figure 12a shows the evolution of θ ry at low dimensionless shear rates with an initial tilted angel of 45 • , it is seen that the transient dimensionless time towards tumbling motion (θ ry = 90 • ) increases with the dimensionless shear rate. When the dimensionless shear rate is increased to around 0.002, a precessing motion dominates a wide range of initial tilted angles. In this mode, the long axis of a capsule precesses around the vorticity axis with a nutation and forms a cone. The time evolution of θ ry of a prolate capsules in a precessing motion is presented in Fig. 12a, and a comparison with Jeffery's orbit [START_REF] Jeffery | [END_REF] is presented in Fig. 12b, where very good agreement is observed. From Fig. 12a, we also infer a dependency of the final stable motion on the initial titled angle, similar to that of an oblate capsule. At G = 0.001, an initial titled angle of β = 45 • leads to a tumbling motion, and β = 30 • results in a precessing motion.

In the precessing regime, the time-averaged value of θ ry decreases with the dimensionless shear rate, towards 0 • which corresponds to a log-rolling motion. For the dimensionless shear rate G ≥ 0.004, rolling motion is a dominant stable mode where the prolate capsule finally rotates around its long axis which is aligning with the vorticity axis. The time evolutions of θ ry for prolate capsules in the rolling regime are presented in Fig. 13 for a range of dimensionless shear rates. It is interesting to find that upon increasing the dimensionless shear rate, the transient dimensionless time towards steady rolling first decreases, until approximately G = 0.01, beyond which it increases until the next stable mode of motion, which is wobbling. We also conclude from Fig. 13 that the tumbling motion of the prolate capsule is strongly stable: initially tilting the revolution axis out of the pane of shear by 15 • is not sufficient to trigger the rolling motion at G = 0.004, when rolling is the dominant mode. The qualitative difference between the present results and that of Dupont [38] may originate from the inertial effect, which we will discuss in Sec. V.

V. DISCUSSION

A. Particle shear stress and effect of Reynolds number

We investigate the particle shear stress for both oblate and prolate capsules at low shear rates, in order to test the relevance of an hypothesis of Jeffery [START_REF] Jeffery | [END_REF] that nonspherical rigid particles in Stokes shear flow finally adopt a motion with minimum energy dissipation. The normalized particle shear stress is [6]: (8) where f x is the membrane force in the direction of x and λ is the viscosity ratio. The particle shear stress can be used to indicate the shear viscosity of the suspension of capsules.

σ xz = 1 µkV ∫ A [f x x z + µ(λ -1)(u x n z + u z n x ))]dA
For an oblate capsule with an NH membrane and an initial aspect ratio of 2 : 3, at G = 0.001 the value of σ xz averaged over one period equals 3.15 for tumbling motion and 3.87 for rolling motion, so it appears the capsule tends to adopt a mode of motion which makes larger contribution to the shear stress. The same phenomenon is observed for prolate capsules, thereby also negating the applicability of Jeffery's hypothesis when extended to flows with finite inertia. However, the solution of the Stokes equations are known to minimize energy dissipation only for given boundary velocities (e.g., in [START_REF] Kim | Microhydrodynamics[END_REF]), and the presence of the membrane around the capsule makes the applicability of this theorem here unclear. Furthermore, numerical studies on rigid particles in shear flow [50,51] have shown that when the Reynolds number is within [0.1, 100], rigid oblate spheroids tend to carry out a rolling motion and prolate particles tend to tumble, which are very similar to the present findings.

In early experiments of Karnis et al. [START_REF] Karnis | [END_REF] about axial migration of particles in Poiseuille flow, it was found that inertia becomes sufficient to break the reversibility of the flow when the particle Reynolds number (defined in a similar way to the present study) is of O(10 -4 ), above which both rigid rods and disks adopt motions that contribute to increasing the suspension viscosity. Drawing an analogy between rigid particles and capsules, we would expect there could be a transition in stable modes of motions for capsules when the Reynolds number is at a similar order. However, reducing Re from the present 0.2 to 10 -4 will increase the computational time by three orders of magnitude, which is currently not accessible. An attempt has been made to assess the significance of inertia here, by carrying out a simulation with reduced Re from 0.2 to 0.02, for a capsule with an NH membrane with β = 45 • and G = 0.001. The results as shown in Fig. 14, indicate that the capsule transits to rolling motion in a slightly shorter dimensionless time with decreasing Reynolds number. We therefore leave this issue to future study, when new algorithms that can reduce the computation time to an acceptable level are adopted.

B. Flow decomposition

In order to understand the mechanism whereby a spheroidal capsule adopts a motion that increases the suspension shear viscosity, we have decomposed the simple shear flow into an elongational flow and a rotational flow for the case presented in Fig. 14 at Re = 0.2, and studied the dynamics of the capsule in each of these basic flows separately. We have found that the capsule finally undergoes a tumbling motion in the rotational flow; in the elongational flow the capsule achieves a steady state with the revolution axis aligned with the direction of compression and the equatorial plane being perpendicular to the original plane of shear. In both flows, the revolution axis of the capsule is within the original plane of shear, and thus combining these two modes together would naturally lead to a tumbling motion for the oblate capsule. However, the capsule adopts a rolling motion in the combined flow (simple shear flow). This analysis, although does not completely explain the mechanism, it suggests that the fluid-capsule system with small inertia is complicated and highly nonlinear.

C. Effect of initial tilted angle

For both oblate and prolate spheroidal capsules, it is interesting to find that the long-term dynamics of a capsule can depend on the initial inclination angle in transitional regimes in the phase diagrams: at the same dimensionless shear rate, a capsule can adopt different modes of motion. For example in Fig. 4a, an oblate capsule adopts either a rolling or a precessing motion; even when β falls into the range where a capsule undergoes a precessing motion, the average value of θ ry (the cone angle of the trajectory of the capsule's revolution axis) still depends on the initial tilted angle β. We have observed a similar phenomenon for prolate capsules, which suggests the fluid-capsule system to be a high-dimensional dynamical system. In fact, the dependence of the final stable motion on the initial condition has been observed in other systems. Zhao and Shaqfeh [34] considered vesicles in shear flow with zero inertia, Huang et al. [51] studied prolate and oblate rigid particles in shear flow with finite inertia, and both reported off-the-shear-plane modes that are sensitive to the initial inclination angle of the vesicle or particle. These studies may shed some light on the intriguing phenomenon of red blood cells subjected to shear flow where individual cells adopt different types of motion even under the same flow condition [12].

D. Reorientation

We generalize the reorientation phenomenon discussed in Sec. III to both oblate and prolate capsules and a much wider range of dimensionless shear rates covering all modes of motion observed in the present study. Interestingly, for an oblate capsule, upon increasing the dimensionless shear rates, the unit vector (extending from the mass center of the capsule and pointing to the membrane material point that initially intersects with the revolution axis) changes from being aligned with the vorticity axis (in rolling motion) to being within the plane of shear (in tumbling motion), and finally turns back to being aligned with the vorticity axis (in the OS(R) mode). In between, the capsule adopts transitional off-the-shearplane motions (i.e., precessing and oscillating-swinging modes). It is also interesting to find the reorientation of the unit vector for a prolate capsule with increasing dimensionless shear rate is in an opposite order: the unit vector changes from being within the shear plane (in tumbling motion), to being aligned with the vorticity axis (in rolling motion), and finally changes back to being within the plane of shear. Similar to that of an oblate capsule, off-the-shear-plane modes are observed in between. Such a reorientation phenomenon and the difference between the reorientation of prolate and oblate spheroidal capsules can be utilized in applications such as particle alignment techniques.

E. Effect of membrane constitutive laws and capsule aspect ratio

Different membrane constitutive laws have been used in the present study: the strain-softening neo-Hookean (NH) law and strain-hardening Skalaks (SK) law. When C = 1 in Eq. ( 3), both constitutive laws lead to the same results at low dimensionless shear rates. We carried out simulations for an oblate capsule with an SK membrane with C = 1, aspect ratio 2 : 3 and β = 45 • . The phase diagram, corresponding to a horizontal line in Fig. 8, is the same to that of a capsule with an NH membrane. In the oscillating-swinging regime, we observe larger oscillating amplitudes of θ lxy and θ lxz for the capsule with an SK membrane. This is mainly because an SK membrane is strain-hardening and leads to less deformation at high shear rates when compared with an NH membrane. The effect of membrane constitutive laws has also been discussed by Dupont et al. [38]; they found that the main factor that determines the stable mode of motion and the transition is the deformation of the capsule instead of the membrane constitutive law. On these bases, we argue that the phase diagrams obtained in the present study are robust to membrane constitutive laws of the capsules.

We have also carried out simulations of capsules with higher sphericity. For an oblate capsule with an aspect ratio of 9 : 10, an NH membrane and an initial angle of β = 45 • , we observe all types of motion and in the same order as in Fig. 8 when increasing the dimensionless shear rate. However, the upper boundary of G for rolling motion is increased from 0.003 to 0.006 when compared with an oblate spheroidal capsule with aspect ratio of 2 : 3. When G is further increased to 0.007, the capsule is already in the oscillating-swinging regime. This suggests that the window of dimensionless shear rate for precessing and tumbling motions would be reduced when increasing the sphericity of an oblate spheroidal capsule. A similar phenomenon has been found for prolate capsules, the range of rolling motion for a prolate capsule with an aspect ratio of 3 : 2 and an SK membrane is 0.005 ≤ G ≤ 0.1, slightly smaller than that of a capsule with aspect ratio of 2 : 1. In precessing motion at low dimensionless shear rates, we find the dependence of the precessing cone angle of the revolution axis on initial tilted angle is weaker for capsules with lower sphericity. For a prolate capsule with aspect ratio of 3 : 2, we observed that different values of β can lead to precessing with different cone angles at the same dimensionless shear rate. However, for a capsule with an aspect ratio of 2 : 1, the cone angle seems only to be a function of the dimensionless shear rate for all cases considered. Finally, at high dimensionless shear rates where the capsule is undergoing the oscillating-swinging motion, we find the oscillation amplitudes of the angles θ lxy and θ lxz are smaller for a capsule of higher sphericity.

VI. CONCLUSION

We have studied the dynamics of oblate and prolate spheroidal capsules in simple shear flow with small inertia, for the case of equal internal and external fluid viscosity and without bending stiffness, using an immersed boundary lattice Boltzmann method. The effect of the initial tilted angle that the capsule's equatorial plane makes with the plane of shear on the motion the capsule would carry out after an initial transient stage has been investigated. Having identified the sequence of events when increasing the dimensionless shear rate, and characterized the various modes observed, form a necessary step to further experimental and theoretical work.

For oblate capsules, it is found here that tumbling, a previously well accepted mode of motion at low shear rates, is unstable. Instead, at very low dimensionless shear rates, independent of the initial angle, the capsule is found at late times to be in a rolling motion, wherein the capsule's equatorial plane tends to stay in the plane of shear. At larger dimensionless shear rates, a precessing motion is observed for most initial inclination angles over a long computational time. It is shown that this motion can be represented by Jeffery's theory [START_REF] Jeffery | [END_REF] for a rigid spheroid in shear flow. Further increasing the shear rate leads to the well-known tumbling motion as the stable mode for a large range of inclination angles, in which the capsule flips with its revolution axis staying in the plane of shear. We find in transitional regimes, the capsule's motion could depend on the initial inclination angle. At high dimensionless shear rates, the capsule carries out a complicated oscillating-swinging motion: while the membrane is rotating around the liquid inside, the major axis of the capsule oscillates about the shear plane in addition to the previously found swinging about a mean angle with the flow direction. Both oscillations become weaker with the dimensionless shear rate, towards a steady tank treading motion asymptotically. The motion of a prolate spheroidal capsule is found to be different from that of an oblate one. When increasing the dimensionless shear rate, the stable modes are tumbling with the revolution axis staying in the plane of shear, precessing following Jeffery's orbit, log-rolling with revolution axis aligned with the vorticity axis, precessing with large deformation (wobbling), and finally oscillating-swinging at high dimensionless shear rates.

Both the rolling motion of an oblate capsule and the tumbling motion of a prolate capsule at very low dimensionless shear rates correspond to modes that contribute to increasing the particle shear stress. These are similar to the recent findings for rigid oblate and prolate solid particles in shear flow with small inertia [51], suggesting that the hypothesis of Jeffery [START_REF] Jeffery | [END_REF] that non-spherical particles in Stokes shear flow finally adopt a motion with minimum energy dissipation does not apply to situations where inertial effects are significant. Finally, we have also reported here the interesting reorientation phenomenon, wherein modes are observed in opposite order for oblate and prolate capsule upon increasing the dimensionless shear rate, which can be used in particle alignment techniques.

FIG. 2 .

 2 FIG. 2. Illustration of the rolling motion of an oblate capsule: (a) instantaneous profiles, black dots represent the same membrane node on the capsule's cross section in the plane of shear; (b) flow field around the cross section of the capsule in the plane of shear.

  FIG. 3. Rolling motion of oblate capsules: temporal evolutions of (a) angles that the revolution axis of a capsule with an NH membrane makes with x-axis θrx, and y-axis θry for β = 90 • and G = 0.0005; (b) θry for capsules with NH or SK membranes at different dimensionless shear rates and β = 45 • .

FIG. 4 .FIG. 5 .

 45 FIG. 4. (Color online) Precessing motion of an oblate capsule with an NH membrane: (a) temporal evolution of θry of a capsule with various initial tilted angles at G = 0.004; (b) the stable trajectory of the revolution axis projected to a unit sphere for a capsule at β = 22.5 • and G = 0.005; (c) comparison between numerical simulation (symbols) and Jeffery's theory [31] (curves) in angles φ and θry for β = 22.5 • , G = 0.005.

FIG. 6 .

 6 FIG. 6. (Color online) Tumbling motion of an oblate capsule with an NH membrane: (a) temporal evolution of θry of a capsule at G = 0.01. The inset shows the stable trajectory of the revolution axis projected to a unit sphere for a capsule with β = 22.5 • ; (b) variations of the Taylor shape parameter at different dimensionless shear rates for a capsule with β = 45 • .

FIG. 7 .

 7 FIG. 7. (Color online) Oscillating-swinging motion of an oblate capsule with an NH membrane. (a) Trajectories of the long axis of a capsule projected to a unit sphere. The values of G are 0.05, 0.1 and 0.2 respectively with the sizes of the lemniscates decreasing. β = 67.5 • . (b) Variations of the angle that the long axis of a capsule makes with the plane of shear θ lxz (blue lines) and the angle that the long axis makes with xy plane θ lxy (red bold lines) for increasing dimensionless shear rate.

FIG. 8 .

 8 FIG.8. Phase diagram for an oblate capsule with an initial aspect ratio of 2 : 3 and an NH membrane. R represents rolling, P represents precessing, T is for tumbling, S for swinging, OS(T) and OS(R) are for two types of oscillatingswinging motion as explained in the text, and TT is for steady tank-treading.

FIG. 9 .

 9 FIG. 9. Illustration of the rolling motion of a prolate capsule. Black dots represent the same membrane node on the capsule's cross section in the plane of shear.

FIG. 10 .FIG. 11 .

 1011 FIG.10. Snapshots of a prolate capsule during a wobbling motion. Time sequence is from (a) to (h). Two small spheres are attached to the surface of the capsule as tracer particles.

FIG. 12 .

 12 FIG.12. (a)Temporal evolution of θry for prolate capsules in tumbling and precessing motions; (b) comparison between numerical simulation (symbols) and Jeffery's theory[START_REF] Jeffery | [END_REF] (curves) in time evolution of angles φ and θry for β = 30 • and G = 0.001 when a capsule is in a precessing motion.

1 FIG. 13 .FIG. 14 .

 11314 FIG. 13. (Color online) Temporal evolution of θry for prolate capsules that are finally in tumbling and rolling motions.

ACKNOWLEDGMENTS

Z.W. is supported by a PhD studentship from Queen Mary University of London.