Calibration of medical imaging systems that provide quantitative measures relating to complex physiological flows is challenging. Physical test objects available for the purpose either offer a known simple flow far removed from the complexity of pathology (e.g. parabolic flow in a straight pipe) or complex relevant flows in which the details of the flow behaviour are unknown. This paper presents the ring vortex as a candidate for a complex flow phantom, since it is marked by inherently complex flow features that are controllable, predictable, reproducible and stable. These characteristics are demonstrated by a combination of analytical, numerical (CFD) and experimental methods. Together they provide a consistent perspective on ring vortex behaviour and highlight qualities relevant to phantom design. Discussion of the results indicates that a liquid phantom based on the ring vortex may have merit as a complex flow phantom for multimodal imaging. Furthermore, availability of such a flow reference may also serve as a benchmark for quality assurance of simulation methodologies.
This paper argues for a novel concept in medical imaging flow phantom design based around the flow phenomenon known as the ring vortex. Relevant medical imaging technologies include Doppler ultrasound, and (contrast) angiography techniques that involve planar X-ray, MRI and CT [
Flow related quantities obtained from imaging systems vary according to scanner technology and performance. Assessment of the latter is recommended through approved quality assurance and control processes1 [
We propose a new reference for complex flow imaging in the form of the ring vortex. This benchmark flow can be studied as a physical phantom for imaging performance assessment, and may also be useful as a reference for quality assurance of computational approaches. It offers the capacity to produce known complex flow features (velocity, acceleration, pulsatility, vorticity, etc.) that are predictable, reproducible, controllable and stable. Consequently, practical experience of the ring vortex is described below and its suitability as a candidate for an innovative complex flow phantom is discussed and presented.
The ring vortex is a fundamental component of flow complexity and consists of an annular vortex core that propagates perpendicular to the plane of the ring. It is a natural phenomenon that is variously known as a “ring vortex”, “smoke ring” or “toroidal vortex”. Ring vortices represent one of the most fundamental phenomena in fluid dynamics. As described by Akhmetov “…a vortex ring is a toroidal volume of vortical fluid moving in a surrounding medium at an approximately constant translational speed perpendicular to the ring plane. The fluid motion is axisymmetric, and the vector of vorticity in the torus is directed along the circles concentric with the circular axis of the torus. A certain volume of the fluid which embraces the ring and looks like an ellipsoid flattened along the direction of motion is moving together with the toroidal vortex ring. This enclosed volume of fluid is called vortex atmosphere. Inside the vortex atmosphere the fluid is circulating along the closed streamlines encompassing the toroidal core of the vortex. Motion of the fluid surrounding the vortex atmosphere resembles a pattern of flow without separation past a corresponding solid body.” [
The vortex ring clearly offers vorticity, but because it is stable and can be controlled it also offers reproducibility, pulsatility, and can be produced in various sizes, travelling at distinct velocities. These characteristics make it particularly suitable as a candidate for a complex flow phantom. As an illustration of its predictability, reproducibility, controllability and stability, the following sections present analytical and computational analyses of the ring vortex, as well as our own experimental experiences with this phenomenon. Together they provide a body of evidence, considered in the discussion section, which is used to evaluate the suitability of this flow for calibration and performance evaluation of medical imaging systems and benchmarking of computational simulations.
The behaviour of the ring vortex can be generally described by solution of the Navier-Stokes equations [
Model (class) | Thin (Lamb) [ | Thick (Kaplanski-Rudi) [ | Spherical (Hill) [ |
---|---|---|---|
Schematic Stream Function | |||
Circulation | Const | ||
Impulse | Const I | ||
Energy | |||
Velocity | Const |
Symbols:
the frame of reference so that the ring is static as the free field passes at velocity -v. Broadly speaking the analytical solutions can be characterised according to three particular idealizations, namely:
・ spherical vortex ring―Hill [
・ thick vortex ring―such solutions are identified by a larger core radius to ring radius ratio (Limit as a/R ®
・ thin vortex ring―in contrast to the above, the thin ring approximation is characterised by a smaller core radius to ring radius ratio (Limit as a/R ® 0) (Lamb [
Key characteristics are expressed analytically in
and ring radius (R) is related to the impulsive volume of fluid used to generate the vortex (characterised by a length parameter L)2:
These are important relationships that offer insights to ring vortex behaviour, with implications for physical phantom design.
The analytical descriptions presented do not include any concept of ring vortex creation―the mathematics describes only the fluid dynamics once the ring has been generated. Experimental studies require a mechanism to create the vortex and this involves propelling a slug of fluid through an orifice.
Amplifier MAX9744 (USA)] to the displacing membrane of a loud speaker [Monacor SP-45/8 (Germany)] to propel a slug of fluid through an orifice with subsequent ring vortex creation and propagation.
For proof of concept purposes the experiment was performed in air, with the smoke filled generator chamber producing a visible smoke ring that was captured by video camera at 30 frames per second. With an orifice diameter of 1 centimetre, rings were generated at Reynolds numbers of 500, 1000 and 2000 in the orifice throat. Post processing of the digital video stream enabled salient features of the ring to be measured. Ring size and velocity were used to create plots of vortex behaviour as a function of time and distance.
The experimental system detailed above was simulated using a computational description. ANSYS Fluent [ANSYS Fluent 16.1 (Canonsburg, PA, USA)] was used to investigate the behaviour of the vortex ring, avoiding the idealisation constraints associated with the analytical formulation (see
The geometry modelled was a 2D axisymmetric domain from the vortex generator extending 50 cm in the direction of propagation of the vortex and 10 cm from the symmetry axis in the radial direction to represent the free field region. These axial and radial extents of the computational domain were chosen to be large enough to observe the propagation of the vortex ring over an extended distance (i.e. protracted time period) and to prevent the lateral and downstream pressure boundary conditions from affecting the dynamics of the propagating ring. A structured, mapped mesh of quadrilateral elements was employed, biased in the radial direction to obtain a high density of elements close to the symmetry axis, where the greatest velocity gradients would be located. Preliminary sensitivity tests were performed to determine appropriate mesh and time step size to provide acceptable numerical accuracy. Optimal values were 6.25 × 10−4 m for the mean size of the mesh elements and 1 × 10−3 s for the time step size. Residuals of 0.1% offered a satisfactory threshold for effective computation.
Boundary conditions included simulation of the ring vortex generator speaker
membrane, with the aim of reproducing the experimental conditions of the speaker displacement. The time dependent uniform inlet velocity (normal to the boundary) was of the form
where t represents time and w refers to the angular frequency of membrane oscillation with period
The analytical solutions suggest simple relationships between ring size, velocity, time and position as follows;
・ ring velocity varies with the reciprocal of time
・ ring velocity varies exponentially with distance
・ ring size can be expected to grow linearly with propagation time
The results reported in this study encourage use of the ring vortex as a complex flow benchmark for imaging. Notably the analytical, experimental and computational analyses of ring vortex behaviour are all in agreement. Key features of the flow which are well reproduced include:
・ Ring diameter (2R) is dependent on swept volume of the piston (i.e. volume of the slug of displaced fluid)
・ Ring core diameter (2a) is dependent on the velocity/energy of the slug of displaced fluid
・ Ring velocity is dependent on slug velocity
・ Ring velocity decays approximately exponentially with distance (R2 ≈ 0.99)
The numerical simulations are particularly helpful in confirming that most of the fluid volume displaced by the piston (
Experimental results have been presented for Re = 2000 because the fluid behaviour in this regime (transitional within the throat of the orifice) can be expected to demonstrate greater instabilities/complexities than at Re = 1000 or Re = 500. Results at Re = 2000 tend to improve at lower Reynolds number, also confirmed by the CFD data shown in
As an additional assessment of the validity of our computational approach, an investigation involving reproduction and comparison with the work of Danaila and Hélie [
It is important to recognise that flow phantoms set the standards against which scanner performance is judged, which by implication influences the diagnostic thresholds that are used to manage the patient treatment pathway. Consequently, poorly calibrated imaging systems adversely impact patient management, although there are no accepted, general imaging, independent standards for complex flow.
The ring vortex represents a reference flow that is both well understood yet complex, with direct physiological significance (ring vortices are reported to be a feature of ventricular action and are also associated with valve function [
・ Predictability (within specified tolerances, the flow can be known at every point in space and time)
・ Reproducibility (the flow is repeatable, it does the same thing every time)
・ Controllability (the flow characteristics can be varied in a controlled fashion)
・ Stability (the flow is robust and easy to produce; resistant to disturbances)
Critique of the vortex analyses presented here identifies that the analytical solutions are highly idealised, with assumptions that limit the relevance of their solutions to real world problems (e.g. phantoms). Nonetheless they do offer insight into ring vortex behaviour and are invaluable as a validation tool for numerical analyses (CFD). When the CFD is applied to real experimental geometries, it becomes a predictive tool suited to phantom design and its subsequent operation. Of course the assumptions associated with CFD modelling may not translate to effective description of actual ring vortex behaviour, but our experiments show otherwise―despite the proof of concept nature of these experiments undertaken in air―indicating correlation between theory and experimental behaviours. The vortex in air was chosen since construction of an air based system was cheap and effective. However, this does not come without compromises― neutral buoyancy of the visible smoke was lacking; the propagating ring was sensitive to air currents within its environment despite being enclosed by a tunnel; smoke ring visibility was limited etc. Nonetheless encouraging controllability, predictability, reproducibility and stability of this complex flow was obtained, and there is good reason to anticipate much improved performance in a better controlled and more refined liquid environment. This opens numerous opportunities for flow phantom development, including the potential for a liquid-based programmable unit capable of delivering repeatable, precomputed complex flows to MRI, ultrasound and CT. A test object such as this could be used to generate cyclic flow features, produce accelerating behaviour consistent with pulsatility, or to deliver precise repeatable eddy formations by virtue of vorticity. All of this can be achieved in a reproducible manner consistent with previously established characterisation (from both theory and experiment). An aspirational goal would be the creation of a system that would present full field, dynamic and complex known flows at known tolerances to aid diagnostic interpretation of flow imaging data from existing medical imaging systems and to support novel imaging hardware and software design.
In the wider context of CFD benchmarking, the flexibility of such a flow phantom also has the potential to exercise and quantify the predictive capabilities (to within defined tolerances) of existing and novel computational approaches, particularly for flows at higher Reynolds numbers where laminar approximations may be insufficient to successfully capture flow behaviour.
This paper has reported analytical, experimental and computational behaviour of the ring vortex. The work demonstrates that this inherently complex flow has features that are sufficiently predictable, reproducible, controllable and stable to warrant its consideration as a candidate for a complex flow phantom in medical imaging. A flow benchmark such as this also has implications for quality assurance of numerical simulation methodologies.
This work is funded by the European Commission through the H2020 Marie Sklodowska-Curie European Training Network H2020-MSCA-ITN-2014 VPH- CaSE, www.vph-case.eu, GA No. 642612.
Ferrari, S., Ambrogio, S., Walker, A., Verma, P., Narracott, A.J., Wilkinson, I. and Fenner, J.W. (2017) The Ring Vortex: Concepts for a Novel Complex Flow Phantom for Medical Imaging. Open Journal of Medical Imaging, 7, 28-41. https://doi.org/10.4236/ojmi.2017.71004