Left Ventricular Fluid Mechanics: The Long Way from Theoretical
Models to Clinical Applications
G
IANNIP
EDRIZZETTI1and F
EDERICOD
OMENICHINI21
Dipartimento di Ingegneria e Architettura, University of Trieste, P.le Europa 1, 34127 Trieste, Italy; and2Dipartimento di Ingegneria Civile e Ambientale, University of Firenze, Florence, Italy
(Received 15 April 2014; accepted 25 August 2014)
Associate Editor Umberto Morbiducci oversaw the review of this article.
Abstract—The flow inside the left ventricle is characterized by the formation of vortices that smoothly accompany blood from the mitral inlet to the aortic outlet. Computational fluid dynamics permitted to shed some light on the fundamental processes involved with vortex motion. More recently, patient-specific numerical simulations are becoming an increasingly feasible tool that can be integrated with the developing imaging technologies. The existing computational methods are reviewed in the perspective of their potential role as a novel aid for advanced clinical analysis. The current results obtained by simulation methods either alone or in combination with medical imaging are summarized. Open problems are highlighted and perspective clinical applications are discussed.
Keywords—Cardiac mechanics, Fluid mechanics, Left ven-tricle, Mitral valve, Cardiac imaging.
INTRODUCTION
The left ventricle (LV) is the principal mechanical element of the human heart. It has the function of a volumetric pump that receives low pressure blood from the venous system through the left atrium and ejects it with higher pressure through the aortic valve into the arterial system. The LV chamber is surrounded by a muscular tissue, the myocardium, that operates in a sequence of mostly passive relaxations, when it receives the blood, and active contractions to push it into the circulatory system. Given the fundamental mechanical function of the heart, the myocardial tissue deforma-tion and the blood flow inside the LV represent a central issue of biomechanical research.
In the clinical setting, the evaluation of the LV function was mainly performed in terms of its volu-metric changes (heart as a voluvolu-metric pump), and only recently its characterization in terms of muscular deformation (heart as muscular pump) has become possible. On the other hand, the properties of the blood flowing inside the cardiac cavities received lim-ited attention.
The process of developing clinical indicators based on flow is different from that traced for tissue motion. In fact, while abnormalities of the wall motion, e.g., hypokinesia or asynchrony, directly testify the exis-tence of an overt clinical disease, abnormal flow pat-terns are rather a signal of a maladaptive function that may be present even before noticeable structural changes have arisen. The evaluation of blood flow is therefore a challenge that faces a new perspective of looking at cardiac mechanics. More about how the presence of a sub-optimal function may anticipate the progression of pathology than on the diagnosis of the pathology itself; fluid mechanics moves the focus from the diagnosis of an overt disease to the prediction of its development.75 In this context, unveiling the significance of LV fluid dynamics for clinical applica-tions requires a preliminary understanding of its fun-damental theoretical aspects and then the construction of firm interpretative schemes to be transferred into the clinical arena.
There are two different and complementary
approaches to this aim: computational fluid dynamics
and imaging technologies. Computational fluid
dynamics is the primary tool for the detailed compre-hension of the physical phenomena participating to the LV flow. Numerical simulations of the blood motion inside the LV, possibly including fluid–tissue interac-tion models, still represent a challenging task, and
Address correspondence to Gianni Pedrizzetti, Dipartimento di Ingegneria e Architettura, University of Trieste, P.le Europa 1, 34127 Trieste, Italy. Electronic mail: giannip@dica.units.it DOI: 10.1007/s10439-014-1101-x
different computational choices present own advanta-ges and drawbacks. Nevertheless, recent advances permit to start drawing a unitary picture about the LV fluid mechanics under normal conditions and its modifications in presence of pathologies.
Latest imaging technologies and their integration with computational methods permit, to a certain extent, in vivo visualization and quantification of LV fluid mechanics. Such imaging technology, supported by physics-based interpretative models, is now firmly entering in the clinical arena. First studies evidenced the intimate relationship between cardiac function and quality of the intra-ventricular flow, and built a growing consensus about the relevance of fluid dynamics on the cardiac performance.41,64,80,85,86
Here, we will review the journey that moved from computational fluid dynamics and the comprehension of the basic physical phenomena in LV blood flow and that, through medical imaging technology, led to the definition of aspects of patho-physiological relevance. This presentation is centered on the LV blood flow leaving aside the important topic of cardiac valves, unless strictly related. We will employ a synthetic approach with the focus on how scientific progresses and bioengineering research eventually fertilized the clinical field.
COMPUTATIONAL METHODS
The numerical approach to the study of the LV fluid dynamics has shown a rapid expansion in the last decades. This was supported by the continuous increase of the computational resources, the develop-ment of efficient numerical methods and, in addition, by the growing awareness of the relevance of the flow properties on cardiac performances. Numerically, the study of LV mechanics is a challenging problem because, in principle, one should write the correct equations for both blood and tissue elements (myo-cardium, valves etc.), the coupling relationships between these different media, and finally solve the complete system of equations. Difficulties arise due to the lack of a multiplicity of information: starting from the geometrical description of a rapidly time-varying domain, passing through the characterization of the mechanical properties of the tissues, to the coupling with the whole circulatory system or the definition of appropriate boundary-conditions.
Despite its general average characteristics, the LV geometry has a physiological variability, which depends by individual differences and by the presence of pathologies at different stages of evolution, in addition to natural modifications with ageing. More-over, the accuracy in the description of an individual
geometry is still limited by the space-temporal resolu-tion of the available technologies of clinical imaging.
While the blood can be assumed to have a relatively simple rheological behavior, i.e., a Newtonian fluid at least in the heart chambers and in the main vessels, the cardiac tissues present nontrivial structural and mechanical properties. The myocardium is a living tissue with a fiber-oriented geometry, with evident anisotropy and inhomogeneity, and a mechanical behavior that rapidly changes from the almost passive relaxation during the LV diastolic filling to the systolic phase of active contraction (Nordsletten et al.68 for a recent perspective). The cardiac valves have a fibrous structure with nonlinear and non-homogeneous elastic properties.
A further critical point in the numerical modelling of LV fluid dynamics is represented by the definition of the boundary-conditions about the pressure field. From a pure fluid mechanical point of view, leaving aside the dynamic interaction with tissue and assuming it as a boundary with a prescribed motion, the value of the absolute pressure does not enter in the equations that depend by its spatial gradient only. This picture changes in case of Fluid Tissue Interaction (FTI), when the dynamical coupling requires the continuity of the stresses at the fluid–tissue interface. Details about this point will be discussed later.
As stated above, the mathematical definition of the LV fluid dynamics follows the widely accepted assumption of the Newtonian description of the blood, while the mechanical characterization of the sur-rounding tissues is still a matter of research. In order to give the largest as possible overview about the numerical approach, several schemes can be adopted in classifying the variety of methods used by different researchers.
From the pure perspective of the numerical solution of the Navier–Stokes equation (Direct Numerical Simulation, DNS), one can distinguish between the use of structured grids or unstructured computational mesh or, almost equivalently, between numerical schemes based on the finite-differences approximation, even coupled with spectral methods, and finite ele-ments (FE) or finite volumes (FV) methods. Very often, this distinction corresponds to the difference between studying idealized geometrical models of the LV, which can be more easily described in a mathe-matically well-defined system of coordinates,21 or simulating realistic geometries for patient-specific cases. The first option has the objective of providing a general physically based understanding of the main flow phenomena occurring in a model LV without entering into the details of the singular individual variability, while the second one is more projected to the application to specific cases with a clinically
oriented perspective.92 In both cases, the accuracy of the results is strictly dependent on the discretization of the flow domain.87
On top of this, the capability in solving the 3D fluid equation of finite-differences based methods was improved by the advent of efficient techniques based on the Immersed Boundary Method (IBM), where arbitrarily complex boundaries are placed into a reg-ular computational domain. IBM methods started with the pioneering works of Peskin and coauthors65–67,76 and have been widely extended to adapt to several different contexts.96
Theoretical Models
The first theoretical studies about intraventricular flow were produced with the aid of approximated models. The main simplification was the hypothesis of axial symmetry of the problem with the choice of a LV ellipsoidal shape.3,95 Vierendeels et al.95 used a FV method to solve the fluid equation, weakly coupling the algorithm with a nonlinear extension of the thin shell equations for the wall and imposing a blunt profile to the velocity at the valve-less mitral orifice. A finite-differences scheme was used for the solution of the Navier–Stokes equations written in the natural system of coordinates by Baccani et al.,3 forcing the system with the wall motion and leaving the trans-mitral flow free to develop. Both models aimed to provide some insights about the relationship between the formation of vortices and the space–time evolution of the axial (base to apex) velocity distribution com-monly used as clinical indicator in Doppler echocar-diography (color-Doppler M-mode). With a similar approach, modifications of the flow pattern due to the presence of different degrees of dilated cardiomyopa-thy (DCM) were analyzed, showing its correlation with a more stagnating flow in the apical region.5In another model, the presence of the mitral valve was accounted imposing different velocity profiles at the inlet orifice, suggesting that the valvular motion has a limited influence on the global flow properties because of its rapid opening.4 Early three-dimensional studies were based on the same simplified geometry. The diastolic phase of the left heart was studied using an IBM method, the ventricular wall was described as a system of connected elastic springs and the flow was forced by imposing a pressure variation at the pulmonary veins connection to the left atrium.59The diastolic filling of an ellipsoidal LV was studied coupling a FV solution of the flow equation with a FE description of a ven-tricular hyperelastic wall,12and forcing the system with physiological time histories of the intraventricular pressure and of the mean velocity at the mitral orifice. These studies successfully reproduced the vortex
formation process and its dynamics inside the LV chamber. Still within the same geometric approxima-tion, the three-dimensional diastolic filling in a model LV was studied using a mixed spectral/finite-differ-ences method in the prolate-spheroid coordinate sys-tem, and the presence of the mitral valve was mimed through a blunt velocity profile of the entering flow.21 The study analyzed the flow dependence on the dimensionless parameters governing the problem. It showed the well-defined vortex ring-like structure of the vorticity field, (see Fig. 1for an exemplary case), in qualitative agreement with a previous in vivo study51 and a following mixed numerical and experimental one.22 A further study analyzed the influence of the eccentric position of mitral orifice with respect to the LV geometry,73 suggesting that the physiological arrangement corresponds to the minimal kinetic en-ergy dissipation within the LV. The issue of enen-ergy dissipation was discussed later comparing the energetic balance of a ‘‘healthy natural’’ arrangement of the LV flow with that induced by a mitral valve replacement.97 Conclusions about the relevance of the viscous dissi-pation is still a question of debate.50,98 It was shown that, although the intra-LV energy losses are obviously small when compared with the total blood resistances in the entire circulation, modifications of the global forcing parameters of the LV motion lead to a drastic change of the flow properties that alters blood mixing and residence time in the ventricular chamber.87 Exemplary results are depicted in Fig.2, where the 3D pattern of the flow field is reported in comparison
FIGURE 1. Vortex formation in a young healthy left ventricle by DNS in an ellipsoidal-shape model at the end of the dias-tole. (a) Normal vorticity distribution in a 2D longitudinal slice (color-coded) and in-plane velocity vectors; (b) 3D vorticity field visualized by the k2method.
43
The diastolic jet is natu-rally deviated along the lateral wall and the ring displaced on that side leaving room to the blood redirection toward the aortic outlet (on the top-left side in the pictures). Adapted from the numerical results previously published.20,21,74
between a normal case and one with an ideal diastolic dysfunction.87
Simplified models of the LV were also aimed to discriminate the flow between healthy and abnormal hearts. Reduction of the vortex-induced flow proper-ties were evidenced in the case of diastolic dysfunc-tion.60 More recently, the modification of the flow pattern due to the presence of a modeled infarcted region showed how the reduced mobility of the myo-cardium directly reflects into a more stagnating flow close to the ventricular apex, which is a consequence of the reduced penetration of the mitral jet.20With similar numerical schemes, based on the solution of the flow equations on a Cartesian grid coupled with different versions of the IBM method, the flow changes due to impaired relaxation and systolic dysfunction associ-ated to hypertrophic cardiomyopathy were studied in terms of ‘‘virtual’’ M-Mode analysis of the propaga-tion velocity, showing how different pathologies
modify the normal pattern.101 The influence of a mechanical prosthetic mitral valve was simulated modifying the velocity profile at the mitral orifice,74 demonstrating that this may lead to the inversion of the intra-LV rotating pattern and to an increase of energy dissipation.97
Patient-Specific Models
Moving from the foundations built on the studies in ideal geometries, a substantial effort was devoted to the development of more clinically oriented approaches to the problem, integrating information recorded in vivo with numerical simulations. Cardiac Magnetic Reso-nance (CMR) data, after interpolation, was used as kinematical boundary condition for the LV flow, opening the perspective to the application of the numerical approach to the specific-patient study of the intraventricular fluid mechanics.81,82 In these works,
FIGURE 2. Three-dimensional vortex formation during diastole visualized by the k2method,43as computed by direct numerical
simulations in a model left ventricle. The isosurfaces are colored by the value of the axial (z-direction) velocity. Comparison between a healthy case (a) characterized by the ratio E/A 5 1.2 and a case with an virtual diastolic dysfunction with a reduced E/ A 5 0.5 (b). The dimensionless parameter E/A is the ratio between the peaks of the mitral flow rates during the E and A-waves of the diastole. Adapted from the numerical results previously published.87Courtesy of J. H. Seo and R. Mittal.
the flow was forced by prescribed pressure values at the mitral and aortic sections, where the valves were modelled as simple open/closed orifices, with uniform velocity profiles imposed as boundary condition. A hybrid pressure–velocity boundary conditions was used in a subsequent study, mainly aimed to analyze the dependence of the LV flow on the direction of the mitral jet.61 The CMR-derived description of the LV wall in a healthy subject was integrated by a simplified representation of the valves in terms of temporally and spatially varying pressure drops.83 In this case, modi-fication of the inlet conditions resulted in the inversion of the sense of rotation of the leading vorticity struc-ture. Developments along this line, where CMR and Echocardiographic data are used to reconstruct the human (four chambers) heart, reported a study in a subject with antero-apical myocoardial infarction before and after surgical ventricular remodeling, in comparison with a healthy volunteer.19The heart was coupled with a model of the circulatory system, in order to gain information about the pressure field, and the valves were modelled as 2D planar orifices, on the basis of Echocardiographic recordings.70
More recently, the three-dimensional arrangement of the early diastolic flow was investigated simulating the heart wall motion with a cell-based activation method, using an unstructured description of the endocardial wall immersed in a stationary curvilinear grid. The focus was on the birth and development of coherent vorticity structures and their possible insta-bility due to the interaction with the wall boundary layers.57,58
The DNS study of the LV flow applied to the evaluation of patient-specific mechanical function could be considered, in perspective, as an additional tool in the clinical evaluation. To this aim DNS should be carefully integrated with 3D Echocardiographic/ CMR clinical imaging techniques, from which tissue-blood interfaces must be segmented, to provide an evaluation of LV flow with high spatial resolution. A high-technology clinical trial was performed by inter-facing numerical simulation with tissue tracking methods of 3D echocardiography in a group of 20 healthy subjects and 8 patients with dilated cardio-myopathy.62Results revealed the basic physical chan-ges associated with the pathology, one case of which is shown Fig.3. The same imaging integration approach was previously applied to explore the flow in the right ventricle.63
Numerical patient-specific studies have the advan-tage of a high resolution, although prone to the rela-tively low frame rate (e.g., Cardiac tomography CT) or spatial accuracy (e.g., Echocardiography) due to the 3D imaging from which boundaries are obtained. It must be reminded, however, that such flow results are
reconstructed and not observed; several details are missing in the geometrical definition and some ele-ments (like the mitral valve) are modeled, therefore care must be taken to ensure the validity of the com-puted flow.
On the other side, this approach has the peculiarity of allowing the reproduction of the flow that would realize under hypothetical conditions, even if these do not exist yet. In other terms, different options in an individual surgical procedure, for example, can be simulated and thus evaluated before being effectively performed. The potential of developing such a virtual surgery approach can refine patient-specific solutions and improve outcomes.92
As a final perspective warning, however, the appli-cation of the numerical approach to the study of the cardiac flow should not be separated by a validation analysis to ensure reliability and reproducibility. A careful validation protocol of computational results vs. in vivo data could accelerate the transfer of LV flow simulation into the clinical setting. In a recent work, starting from the numerical results, virtual Echocar-diographic and PhonocarEchocar-diographic signals were syn-thetized, showing the feasibility to reproduce in silico the data normally acquired in the clinical practice.88
Fluid–Tissue Interaction
The large part of the above mentioned works was based on the kinematical coupling between the blood and the surrounding tissue. The wall motion was pre-scribed as a boundary condition forcing the system on the basis of a model or extraction of medical imaging data. The actual mechanics of the heart, however, comprises the interaction between the blood and the surrounding tissues. The latter is composed by the myocardial walls, able to actively work during the systolic contraction, and the valves’ leaflets, which are mainly moved by the flow itself. In both cases, the mechanical properties of the tissue are non-uniform and anisotropic, showing marked differences in differ-ent regions and along differdiffer-ent direction relatively to the underlying fibers’ structure.
Several hypotheses were introduced in modelling the tissue mechanics and solving the coupled fluid–tissue dynamical equations. The fluid and structure dynamics were solved simultaneously using FE methods and adopting a non-linear elastic12 or a hyperelastic97 representation of the wall properties, while the pre-sence of the cardiac valves was typically accounted imposing simplified boundary conditions at the ori-fices. In the computational framework of the FV method, the diastolic filling was studied in a model LV, with a fibrous elastic description of a thin ventricular wall based on the IBM approach and the use of an
iterative procedure to solve simultaneously the whole system of equations.59 More recently, a similar approach was used with additional schemes to asses both the passive diastolic and active systolic phases, although still neglecting the presence of valvular structures.68
A special remark must be acknowledged to the IBM method, that starting from the seminal work of Peskin76 has received a growing attention in cardiac fluid dynamics. Basically, this is a finite-differences method for simulating the flow interacting with boundary elements immersed in the flow field. The fluid equations are written on a regular fixed (Eulerian) grid, while a moving (Lagrangian) grid is employed to account for the boundaries. The form of the dynamical equation depends on their mechanical properties, e.g., if the boundary is assumed to be elastic, a generalized Hooke’s law must be satisfied. The fluid and solid systems are coupled with a proper strategy that ensures the continuity of momentum and stresses between the two media. Without entering into the mathematical and numerical details,65,66 the heart chambers were modelled as a number of one-dimensional fibers, using elastic and contractile models incorporating many anatomical features of the heart, sometime including the valvular structures. Developments of this method incorporated non-linear elastic models of the fibers, also time dependent in the case of the heart muscles, with different time functions for ventricles and atria.55 Actually, the heart walls and the valvular leaflets were commonly considered as neutrally buoyant in the blood, e.g., it was assumed that the tissue has the same density of the blood. A more complex model of the mitral valve was implemented, taking into account the almost passive behavior of the leaflets coupled with a forcing-dependent elastic model of the papillary mus-cles.96 A methodology able to reproduce the LV flow interacting with cardiac tissue was tested to study the diastolic function in hypoplastic patients, with the aim to support planning of future intervention.18
Despite the large effort in improving the computa-tional tools for the study of the LV fluid dynamics, and despite the relevance for LV flow, the modelling of mitral valves has received limited attention. This delay is partly imputable to the difficulties in modelling the correct interaction. In fact, fluid–tissue interaction is markedly influenced by the details of the nearby flow, where the birth and development of unsteady bound-ary-layers occur at very small scales. Here, the wall vorticity distribution directly affects the local, sharply varying, pressure field, which is the mover for the valvular leaflets; in turn, the details of the differential motion between fluid and leaflets immediately affect such a bound vorticity distribution. This makes the flow-leaflets a strongly interacting system that chal-lenges accurate numerical strategies.
Some studies introduced structural models to understand the dynamical response of different types of valves to imposed pressure drops,38,91,96but a sat-isfactory description of the reciprocal influence between pressure distribution and the rapidly (inertial dominated) developing motion of the valvular leaflets is not yet available.
The integration of flow simulations with tissue dynamics, and/or with multi-scale models of the entire circulatory system, opens the possibility of studying the influence of cardiac mechanics onto the systemic function and vice versa. In a patient-specific perspec-tive, such a comprehensive approach should permit to embed the numerical analysis into subject’s unique individuality. On the other hand, this requires the estimation of several parameters that cannot be mea-sured directly, like the mechanical properties of tissues and the parameterizations of the circulation. In some cases these can best be estimated indirectly and pro-spectively used for intra-individual modifications. Nevertheless, researchers must be aware of the numerous approximations involved, experience and care must be taken to ensure the reliability of the eventual outcomes.
FIGURE 3. Three-dimensional vortex structures visualized by the k2method,43as computed by direct numerical simulations in a
patient with DCM (EF 5 18). (a) Peak of the E-wave. (b) Diastasis. (c) Peak of the A-wave. (d) End of the diastole. (e) Beginning of the systole. Adapted from the numerical results previously published.62The support by J. O. Mangual is kindly acknowledged.
METHODS FORIN VIVO FLOW ASSESSMENT Diagnostic imaging technologies represent a fun-damental support to cardiologists for the assessment of cardiac function. Here, echocardiography and CMR play a leading role for their temporal resolution and non-invasiveness. Imaging underwent rapid techno-logical advances during the last years and real-time 3D imaging has become a widely available clinical tool. At the same time, these achievements have wildly increased the amount of produced data, which are difficult to be included in the normal clinical workflow. As a result, the improvements in imaging technology produced a pressing demand for computational tech-niques and interpretative models able to rearrange such a large amount of information into synthetic clinically valuable representations. Visualization tools are being progressively integrated with post-processing techniques capable of quantifying properties related to the hearth function. The principal imaging technolo-gies currently available for flow assessment are briefly recalled here.
CMR
Phase-contrast CMR is capable to provide details of the 3D flow field and represents the gold standard for in vivo flow measurements.39 The methodological process and the fundamental applications to cardio-vascular flow were extensively reviewed in the past and are not repeated here.64
This technology is the most complete and reliable currently available for multidirectional 3D flow eval-uation, it presents some limitations in the space–time resolution, but it is characterized by excellent ana-tomical details and reliability due to high signal-to-noise ratio. Resolution is directly related to the acquisition time; in fact, the velocity values are pro-gressively decoded from the signal, and acquisitions are increasingly refined during the time of recording. This means that the increase in terms of accuracy has the drawback that flow results are averaged over a large number of heartbeats, and therefore that the small scale fluctuations responsible for blood mixing or intermittent acute phenomena may be hidden. How-ever, advanced methods that allows decoding the fluctuating part of the signal, in addition to its average, are under development23,24 and mitigate this limita-tion.
CMR has the drawback of a certain complexity and relevant costs; however its diffusion is largely increased and costs are being substantially reduced in many countries. CMR analysis must also be combined with an extensive post-processing for organ segmentation and quantification of flow information of clinical
relevance. Solutions for automation of the quantifica-tion process are under development.27Although CMR for 3D flow field cannot be forecasted for a short-term usage in routine clinical practice, it certainly represents the reference standard for in vivo assessment of intra-cardiac fluid dynamics, as discussed later in the clinical applications.
Doppler-Based Echocardiography
Echocardiography is the most diffused imaging technique in the medical practice due to its easy application. Moreover, the blood velocity can be obtained with ultrasound equipment that includes Doppler acquisition. The Color Doppler, where the velocity is displayed color-coded on the echocardio-graphic images, is largely applied in two-dimensional echography and, since more recently, in 3D echo-cardiography as well. Doppler echography is widely used and represents a valuable diagnostic tool. Its fundamental limitation stays in being sensitive to the velocity component along the direction of a scan-line only, being able to measure the velocity at which blood is moving towards or away from the trans-ducer, but being blind to the transversal flow com-ponents.
Some methods to reconstruct the 2D velocity field from the single component recorded have been intro-duced under some assumptions. An original approach was first employed by a team working in collaboration with industry,71,94 the same method was methodolog-ically improved more recently.32All of them are based on the mass conservation law that, written in echo-cardiographic polar coordinates (r, h), gives the esti-mation of the transversal velocity
vhðr; hÞ ¼ vhðr; h0Þ
Zh h0
@rvr
@r dh; ð1Þ
where the radial velocity vr(r, h) is known by the
Doppler measurement and the integration can start from any (initial) position, here indicated by h0, where
the value of the transversal velocity vh(r, h0) is
avail-able. The difference between the various approaches depends on the way boundary conditions are applied. This method has the advantage of an extreme ease of usage, although it may suffer of the limited Doppler sensitivity to the low velocities that characterize the LV vortex. The main drawback is that the transversal velocity is reconstructed on the basis of hypothetical assumptions and, in some cases, using boundary con-ditions that may not be accurate. Further advances are necessary for its extension to 3D vector reconstruction. In addition, this approach is intrinsically anisotropic
because velocity is measured along one direction (the scan-line along which Doppler velocity component is measured) which is different from the others along which velocity is reconstructed; this fact may become critical for the evaluation of directional indicators which could be affected by their relative orientation with the direction of scanning.
Echocardiographic PIV
A recent echocardiographic technique for the evaluation of the instantaneous LV blood flow uti-lizes B-mode harmonic imaging seeded with ultra-sound micro bubbles contrast agent. Such bubbles visually evidence the swirling intracardiac motion, and the images can be processed by Echo-PIV, an echocardiographic adaptation of the Particle Image Velocimetry (PIV) technology long used in fluid dynamics laboratory.2 The Echo-PIV concept has been proven in vascular measurements and validated in accurate in vitro settings.52,77,100 The technique underwent a careful validation utilizing an in vitro model of the LV and comparing the results with
those coming from laser-based digital Particle
Imaging Velocimetry.46,99
Echo-PIV has the advantage of permitting
recording at very high frame rate (up to over 200 Hz) and allowing beat-to-beat variability. On the other hand, the high frame rate is also a requirement of the method otherwise it presents a cut-off in the velocity magnitude: in fact the time resolution, not to mention image quality, is lower than that usually required for laser-based PIV in fluid dynamics lab-oratories. This technique presents a high sensitivity to low velocity while it is prone to underestimate high ones, in this sense it can be considered as complementary to Doppler-based techniques. Echo-PIV has the practical disadvantage of requiring
intra-venous injection of contrast agents; however,
preliminary tests suggested that a dedicated pro-cessing of the echocardiographic data may reveal a spontaneous signal from the moving blood itself.
A further major limitation of Echo-PIV is that it furnishes blood velocities only on 2D slices (scan planes) of the actual 3D flow. Although 3D Echo-PIV is theoretically feasible by the direct processing of three-dimensional echography, this modality presents a largely insufficient space–time resolution. Recently, extensions of this technique to multi-planar acquisi-tions have permitted to gain an understanding of the three-dimensional flow structure in the LV.86A careful analysis of multi-plane flow velocity data, that includes the constraint of incompressibility, may permit a reli-able reconstruction of the 3D blood flow motion.29
APPLICATIONS IN CLINICAL RESEARCH First Studies on Vortex Formation
Studies with emphasis on LV vortex formation were initiated by Gharib et al.,33 based on the previous theoretical discovery about the existence of a limiting, in some sense optimal, timing of this process.16,17,34 The concept of vortex formation time (VFT) appeared as a potentially relevant indicator in the assessment of LV functionality, being related to the dynamical properties of the intraventricular flow and valvular motion.45,48,49 Subsequent studies on VFT evidenced its alteration in patients with diastolic dysfunction as well in those with dyskinetic regions.35,44,47,69,78 How-ever, those clinical applications could only rely on global echocardiographic measurements, namely LV volumes and mitral flow parameters, with limited insight on the actual intraventricular flow pattern. The VFT was computed by the LV Ejection Fraction (EF) with two proportionality coefficients (one given by the percentage contribution of the early filling, E-wave, to the total diastolic flow; the second as a shape factor given by the ratio between the end diastolic volume and that of a sphere with diameter equal to that of the mitral valve). Computed that way, the VFT is essen-tially a physically based combination of previously employed indicators (mainly EF and diastolic E/A ra-tio), therefore the added value that it can bring is limited by the fact that additional information from the intraventricular vortex flow are not involved.
A first seminal study about modifications of the LV vortex formation process in presence of pathologies was performed with ultrasound and published in 1988,6 where authors described the distortion of the normal flow patterns in dogs with ischemic heart dis-ease. The videos were systematically reviewed frame by frame and the resulting flows were examined to testify the reduction of vortex penetration in the LV with the increase of infarct size. In vivo visualization of the actual LV vortex pattern in normal subjects were first recorded by CMR,54and the asymmetric sinuous flow paths around the vortex in the human heart were ele-gantly described shortly after.51 It was suggested that the asymmetric vortical arrangement is the fluid func-tional counterpart of the looped heart structure, which enhances the atrio-ventricular mechanical synergy and supports the conservation of momentum from the entry jet to the ejected flow. Vortex flow visualization were performed by Echo-PIV in the animal work done by Sengupta et al.84 during impaired electrical activa-tion and by Hong et al.42on patients with DCM where modifications in vortex characteristics were described. More recently, using the same method, quantitative assessment of the vortex alteration due to hypertrophic
nonobstructive cardiomyopathy and pharmacological inotropic modulation was given.53,79 These works demonstrated the possibility of appreciating in the clinical settings the theoretically predicted LV vortical motion as well as its variations with diseases. From this basis it was possible to advance from qualitative visualization into quantitative clinical indicators as described below.
Kinematic Aspects: Flow Transit
Major progresses were achieved by processing 3D CMR flow measurements to evaluate the properties of the flow transport across the LV. To that purpose the flow transit was evaluated, using integration of parti-cles trajectories originating from the end-diastolic volume, VED, both backward to the beginning of
diastole and forward to the end of the following sys-tole. In this way the VED was divided into 4
sub-volumes depending on the eventual origin/destination of such trajectories.8 The direct flow, Vdirect, is the
volume of blood that entered during diastole and transits directly to the aortic outlet during systole; the retained volume, Vretained, is the part that entered
during diastole but not ejected during the following systole; the delayed volume, Vdelayed, was already
present in the LV at the beginning of diastole and then ejected during the following systole; and the residual volume, Vresidual, that was present in the LV and yet
not ejected in the next systole. In formula, the VEDis
divided into four contributions
VED¼ Vdirectþ Vretainedþ Vdelayedþ Vresidual; ð2Þ
two were present in the LV before the beginning of
diastole, in the previous end systolic volume,
VES= Vdelayed+ Vresidual, and two are the ejected
stroke volume VED2 VES= Vdirect+ Vdelayed.
This approach, which can be partly automated,28 extends the standard LV volumetric measures intro-ducing a flow-based division that provides a measure of the quality of the blood transit inside the LV. It was successfully applied to differentiate blood flow prop-erties in heart failure patients from normal sub-jects,10,26 with an evident reduction of the direct flow component in dilated LV. An example of the blood visualization can be depicted in Fig.4 where the four sub-volumes fractions are differentiated by colors for three-different cases at the peak of the diastolic E-wave.26
In order to evidence measures about the blood flow, it is worth to remind that the four sub-volumes are related by three linear relationships (see Eq. (2) and lines below). Therefore three of them can be recovered from the knowledge of a single one (for exampleVdirect,
but any other could be preferred) that represents a single independent measure of the LV blood transit. From the knowledge of this and of the LV volumes, VEDand VES, all other sub-volumes can be computed
by their linear combinations.
The analysis of flow transit and residence time is relevant for recognizing stagnating regions and helping to stratify the risk of thrombus formation. A drastic reduction of flow exchange in dilated cardiomyopathy patients was confirmed in a DNS-based clinical study.62A ‘‘blood transit curve’’ was introduced there, whose time profile shows the course of LV blood wash-out beat after beat, and whose first value corresponds to Vdirect. Assessment of LV stagnation regions in vivo
was also performed on trajectory-based computation of Lagrangian coherent structures from CMR data,93 and on velocity fields reconstructed by the color-Doppler approach.40 It was demonstrated how these methods are capable of revealing the underlying topological structure of blood mixing and providing qualitative and quantitative assessments of blood transport patterns.
Local stagnating regions has been observedin vivo close to a dyskinetic wall, after ischemia or myocardial infarction, in particular when it is localized in the median-apical regions of the LV.6It was demonstrated numerically that this occurs either because the flow may stagnate adjacent to walls with reduced mobility or, in presence a reduction of EF, because the trans-mitral jet is shorter and the weaker diastolic vortex
does not translate deep toward the LV apex.20
Recently, a clinical study, performed by echo-PIV, analyzed the risk of thrombus formation in patients with myocardial infarction on the anterior wall.89 In agreement with previous results, they demonstrated that the vortex remains in a more basal location within the LV, as measured from the steady streaming flow, and this vortex characteristic is strongly associated with LV thrombus formation.
Dynamic Aspects: Energy and Momentum Balance The volumetric subdivision introduced in CMR for flow transit was further extended associating the single sub-volumes with their specific properties (like kinetic energy, KE, or momentum) in order to find out whe-ther such a division could be more remarkable when weighted in terms of the transported dynamics quan-tities.28 Further advances in CMR technology permit to evaluate the variance of the fluid velocity compo-nents, in addition to their average value, whose sum is usually referred to a turbulent KE.23 Although prob-ably this is not strictly a turbulent KE, because the entire spectrum of different frequency and wavenum-ber fluctuations is not acquired, beside that the flow is
also not fully turbulent, this represents a degree of sub-voxel variability that may be useful to describe the level of high-frequency disturbance of the mean flow. Methodological developments are in progress24; potential clinical applications are envisaged especially in the description of aortic transvalvular flow, although clinical verifications are at a preliminary stage.
The possible physiological significance of energetic optimization is still a matter of debate. It was initially argued that the LV asymmetry gives rise to the rotating flow that in turn conserves momentum,51 and it was demonstrated numerically that viscous dissipation is effectively minimized in correspondence of a physio-logical LV asymmetry.73 Another numerical study found that this energetic saving is not related to the flow paths,97however the spatial resolution was far too low to effectively resolve the dissipation field (2 9 104 elements compared to well above 106of most compu-tational studies) and results were not accurate. Dif-ferently, it is obvious that the amount of energetic losses occurring inside the LV is a very small per-centage of those occurring in the entire systemic cir-culation and could be negligible in a global balance, as shown in a recent high resolution study.82 This does not mean, however, that a reduced mechanical effi-ciency of the LV does not affect the heart activity. In other words, an abnormal, sub-optimal LV function is expected to be associated with a cardiac suffering that may give rise to feedback mechanisms that participate in adaptation and LV remodeling.9,72,75
A reduction in energetic efficiency was documented by CMR and numerical methods in patients with
DCM.10,28,62 However, numerous clinical indicators are available for this disease and the incremental contribution of fluid dynamics is not decisive for this specific application.
Advances in the characterization of pathologies were also sought in terms of LV vortex properties. Along this line, the persistence of the vorticity struc-tures from early to late diastole characterizes a normal hemodynamic coupling between diastole and systole that appears to be lost in non-dilated patients with failing heart.1An alteration of LV flow was also doc-umented in Fontan patients,56 where the main vortex has been found to be shorter and wider than normal ones. Other investigators used clinical analysis and theoretical modelling to define modification of the LV vortices after mitral valve replacement,30,74 showing that in most cases the vortex reverts its direction of rotation, a phenomenon that leads to the increase of energy dissipation and a deviation of intraventricular pressure gradients.
In any case cases, whether the modified vortex characteristics are related with clinical outcomes at follow-up is a great challenge that requires further investigation.
Intraventricular Pressure Gradients
The intraventricular pressure gradient field (IVPG), Ñp, plays an important role in LV filling and may be abolished, reduced or abnormally oriented in presence of either systolic or diastolic dysfunction. Despite its
recognized importance in ventricular diastolic
FIGURE 4. Blood flow visualization: path line visualization of the four flow components defined in Eq. (2). Direct flow Vdirect
(green), retained inflow Vretained(yellow), delayed ejection flow Vdelayed(blue), and residual volume Vresidual(red). Results
corre-spond to the diastolic E-peak for three subjects. (a) A healthy 50 years old female with normal diastolic function. (b) A 62 years old male with DCM and relaxation abnormality. (c) A 61 years old female with DCM and restrictive LV filling. Semi-transparent three-chamber images provide anatomical orientation. Ao, aorta; LA, left atrium; LV, left ventricle. Adapted from the previously published results.26Courtesy of J. Eriksson and T. Ebbers.
function, IVPGs have never been utilized in clinical cardiology due to the complexity of their acquisition. Therefore, the possibility of computing IVPG non-invasively opens new perspectives for clinical applica-tions of fluid dynamics. The Ñp field is the dynamic counterpart of the intraventricular velocity space–time pattern, v(x,t). The Navier–Stokes equation can be rearranged as
rp ¼ q @v
@tþ v rv
þ lr2v; ð3Þ
to show that the pressure gradient is balanced by the change of fluid momentum (right hand side) net to viscous losses (last term) that are usually nearly neg-ligible along the short intra-chamber paths. IVPG is therefore essentially made of the sum of the inertial (local) acceleration and the convective term. The pressure gradient vector field can be computed directly from the right side of Eq. (3). Operatively, it is some-time preferable to directly compute the relative pres-sure fieldp by solving the Poisson’s equation
r2p¼ qr v rvð Þ ð4Þ
obtained by taking the divergence of Eq. (3), although care must be taken in imposing appropriate boundary conditions.13,25,37 Because Eq. (4) is second order and the average pressure gradient (tri-linear terms) follows from the boundary conditions only.
The physiological non-synchronous sequence of activation–relaxation of the myocardial segments determines the natural distribution of IVPGs. Any condition that interferes with the normal sequence of regional contraction might be expected to alter the
physiological IVPG pattern and, consequently, the LV diastolic vortex dynamics.7Several studies have shown marked changes in the intraventricular filling pattern and in the diastolic IVPGs in patients with myocardial ischemia or congestive heart failure.90In this perspec-tive, the seminal papers by Courtois and co-workers14,15 demonstrated a relationship between the decrease in IVPG and the presence of LV dysfunction. The analysis of intraventricular pressure gradients was possible by processing the Color-Doppler M-mode recordings and applying the Navier–Stokes equation along the scan-line. Within the approximation of this method, because the M-mode scan-line is not a streamline and cross-flow terms, that account for the vorticity therein, are neglected, it could be demonstrated that improvements in LV systolic and diastolic function correlate directly with the recovery of the diastolic IVPG.31Using two-dimensional CMR it was also demonstrated that the vortex formation facilitates the LV filling process with relatively low pressure gradients.11
On these premises, a method able to depict vortex dynamics and quantify IVPGs could be of great value in understanding the effects that the complex heterogeneity in timing of regional loading (or unloading) conditions could exert in triggering and maintaining the spiral of events leading to LV remodeling and heart failure. Moreover, it was shown by Echo-PIV that the LV vortex formation process is immediately modified, within few heartbeats, by de-activation of pacing therapy, despite changes in the tissue dynamics could not be evidenced.36 The differences in the spatial distribution of IVPG are shown in Fig.5in a subject under cardiac resynchroni-zation therapy (CRT, bi-ventricular pace-maker) showing that the normal base-to-apex orientation is lost
FIGURE 5. Flow in the LV at the end of diastole—early systole computed by Echo-PIV in a patient who responded to resyn-chronization therapy with pace maker (CRT). Pictures report the instantaneous velocity vectors superimposed to the color-map of intraventricular pressure, the polar histogram represents the frequency of pressured gradient in the entire heartbeat. (a) Flow under normal therapy shows the vortex redirecting blood toward the outlet (bottom-right in the pictures), the longitudinal arrangement orientation of pressure gradient is compliant with the filling-emptying mechanism. (b) When pace-maker is tempo-rarily de-activated this alignment is immediately lost, in that situation the patient is expected to undergo to ventricular remodeling.
when pacing is temporarily discontinued. This demon-strated that LV flow is extremely sensible to minor changes in cardiac function, therefore it may represent a natural early indicator of sub-clinical physiological changes and a predictor of cardiovascular diseases.75
CONCLUSIONS AND PERSPECTIVES Direct numerical simulations have permitted to shed
lights on the fundamental physical phenomena
involved in left ventricular fluid dynamics. The appli-cations to patient specific cases are at an early stage and present promising developments either in the integration with medical imaging or in the individual parameterization. Nevertheless several methodological challenges are still ahead to overcome modeling approximations and to move toward a full integration in clinical applications.
The significance of fluid dynamics to the cardiac patho-physiology has still to be fully uncovered. Although there is evidence that several pathologies alter the LV fluid dynamics, it is by no mean evident whether such flow alterations may facilitate the diag-nosis of a pathology when compared with existing clinical indicators. The distinctive feature of LV flow stands in its immediate response to modification in the surrounding (boundary) conditions, thus proposing it as the natural early indicator of pre-clinical physio-logical changes after acute events, well before that the surrounding tissue have undergone to irreversible changes. Clinical studies are thus progressively ori-ented to verify the degree whether maladaptive fluid mechanics is involved in cardiac inefficiency and how it participates in the modulation of phenotypic patterns of heart failure. Computational support to medical imaging can provide physically based advancements synergic with the parallel improving quantification of in vivoobservations. In perspective, the potential of LV fluid dynamics will stand in its ability to predict out-comes of an altered cardiovascular state. In this case, numerical simulations of intraventricular flow permit to investigate the details of different options in an individual therapy before any of these is effectively performed. This virtual therapy approach can poten-tially define patient-specific procedures and improve follow-up management.
ACKNOWLEDGMENTS
We wish to thank J. O. Mangual, J. H. Seo, R. Mittal, J. Eriksson, and T. Ebbers for helping us in the preparation of the manuscript. They kindly prepared
original figures based on their previously published results. We are deeply indebted to G. Tonti for his invaluable competence and enthusiastic support about clinical cardiology. We are also indebted to all the colleagues that shared their knowledge, capability, and enthusiasm in exploring the charming, winding world of the cardiac fluid dynamics.
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