Descriptors of Flow in Computational Hemodynamics

Abstract

A considerable amount of scientific evidence demonstrates that the regime of magnitude, direction, and/or frequency of wall shear stress (WSS) modulates endothelial cell function and structure, influencing vascular biology in health and disease. Advances in computational fluid dynamics (CFD) and fluid–structure interaction (FSI) simulations in cardiovascular medicine have enabled accurate WSS quantification, correlating flow behavior and its interaction with the vessel wall with disease progression. To effectively analyze and interpret the results of numerical simulations, various descriptors of blood flow were defined. Such indicators allow researchers to quantify and characterize key aspects of blood flow, facilitating the study of healthy and pathological conditions, medical device design, and treatment planning. However, a very fast-growing collection of hemodynamic metrics were defined and used: whether called indicators, parameters, metrics, or indexes, they will be here referred to as hemodynamic descriptors. This narrative review was aimed at synthesizing scientific literature about the descriptors used to analyze blood flow in computational cardiovascular studies, highlighting their significance, applications, and advancements.

1. Introduction

Mechanical forces are important regulators of cell function and structure in all organ tissues, including the cardiovascular system. The pulsatile flow of blood through the arterial vasculature generates hydrostatic pressures, cyclic strains, and shear stresses, all of which control vessel wall biology. The endothelial cells (EC), lining the artery wall on the luminal side, sense a pressure pulse, i.e., the difference between systolic and diastolic blood pressure. The pressure pulse induces variation in diameter and cross-sectional area, resulting in radial and circumferential wall strain. The vascular endothelium is also directly exposed to wall shear stress (WSS), the tractive force exerted by the flowing blood acting tangential to the luminal surface. WSS plays an important role in maintaining the normal functionality of the vascular system, since blood vessels tend to remodel in response to long-term WSS variations [1,2]. Arteries attempt to re-establish a physiological level of WSS by remodeling to a larger diameter in the presence of increasing flow [3] or to a reduced diameter if the flow is decreasing [4,5]. This is different in cerebral arteries that dilate in response to reduced cerebral blood flow due to a mechanism called autoregulation, which helps maintain stable blood flow to the brain despite fluctuations in blood pressure [6]. A considerable amount of scientific evidence demonstrates that the regime of magnitude, direction, and/or frequency of WSS modulates endothelial cell function and structure [7] and vascular pathobiology in health and disease [2,8]. Depending on the location in the vascular system, the flow patterns and resulting hemodynamic forces may be either uniform or not. In the straight parts of the arterial tree there is laminar and forward pulsatile flow, and thus the resulting WSS is pulsatile with a definite direction throughout the whole cardiac cycle. Within branches or curvatures of the cardiovascular tree, blood flow features reverse or multi-directionality zones where WSS changes direction in space or with time, a flow regime commonly known as disturbed flow [9]. In contrast to the disturbed flow, the helical flow that develops in large non-planar arteries and veins is known to have protective effects against vascular diseases by reducing flow disturbances. Such helical blood flow may lessen the burden of arteries and protect them from atherosclerosis, thrombosis, and intimal hyperplasia [10]. Moreover, helical flow in the aorta is believed to be the result of natural optimization of fluid transport processes, aimed at obtaining efficient perfusion [11].

Blood flow stagnation implies a high residence time of blood particles in that specific region and is believed to trigger thrombus formation, a serious clinical issue. For example, formation of intralumenal thrombus in cerebral aneurysms adds an additional risk of thrombo-embolism beyond that posed by rupture. Although the pathways for platelet activation and clot formation are not yet fully understood, actual findings suggest that thrombosis risk is increased in the regions of flow stasis, with low WSS and high residence time [12,13]. In this context, computational hemodynamic studies may help to understand the relationship between blood flow patterns, stagnation, and residence time of blood elements.

Analyzing arterial pressure waveforms reveals information about heart function, vessel compliance, and the presence of wave reflections. The difference in pressure between two points in an artery, often used to assess the resistance to flow caused by a stenosis or other obstructions, and flow reserve may be indicative for patients at risk. To this aim, computational pressure-based descriptors may be obtained non-invasively from CT angiography imaging [14], helping clinicians decide on the appropriate treatment strategy.

These observations have driven researchers to study the connection between local hemodynamic forces and the onset of vascular diseases in vivo as well as in vitro models. By the mid-1980s image-based computational fluid dynamics (CFD) and, quite a decade later, fluid–structure interactions (FSIs) have emerged as powerful tools in the study of hemodynamics in silico, providing insights into the complex behavior of blood flow in large arteries [15]. Advances in CFD simulations in cardiovascular medicine [16] have enabled accurate blood flow characterization, correlating flow behavior and its interaction with the vessel wall with disease progression. To effectively analyze and interpret the results of CFD simulations, various descriptors of blood flow were defined. Such indicators allow researchers to quantify and characterize key aspects of blood flow, facilitating the study of healthy and pathological conditions, medical device design, and treatment planning. Nevertheless, descriptors became numerous and increasingly sophisticated, raising a concern that image-based CFD could also have the potential to “fuel a hemodynamic data explosion, without concomitant gains in our understanding of pathological mechanisms” [17].

In fact, throughout the entire computational modelling literature, a fast-growing collection of hemodynamic metrics were defined and used; whether called indicators, parameters, metrics, or indexes, they will be here referred to as hemodynamic descriptors. This narrative review is aimed at synthesizing scientific literature about the descriptors used to analyze blood flow in CFD cardiovascular studies, highlighting their significance, applications, and advancements.

2. Image-Based Computational Hemodynamics Studies

2.1. Computational Fluid Dynamics (CFD)

Blood is assumed to be an incompressible, isothermal fluid, which is justifiable for liquids under normal pressure levels. Conservation of mass for an incompressible fluid yields the following volume continuity equation for a fluid element:

$$\mathsf{\nabla }·\mathit{u}=0$$

(1)

The conservation of momentum yields the following Navier–Stokes equations:

$$\rho \left({\displaystyle \frac{\mathsf{\partial }\mathit{u}}{\mathsf{\partial }t}}+\mathit{u}·\mathsf{\nabla }\mathit{u}\right)=-\mathsf{\nabla }p+\mathsf{\mu }{\mathsf{\nabla }}^2\mathit{u}+\mathit{S}$$

(2)

where $\mathit{u}\left(\mathit{x},t\right)$ is the velocity vector, p(x,t) is the pressure field, $\mu$ is the dynamic viscosity of blood, and S is a source term accounting for body forces.

Equations (1) and (2) are partial differential equations (PDE) that can only be solved numerically. CFD programs solve the PDE by breaking up complex geometries into smaller but simpler shapes. A good description of the geometry of the flow domain is required to create a discrete model (i.e., mesh or grid) and proper initial and boundary conditions (BC), e.g., information about the flow entering and leaving the model boundaries. The growth of computational power and the development of numerical techniques like the finite element method (FEM) and finite volume method (FVM) enabled precise modeling of fluid mechanics problems. The difference between the two methods lies in the way they approximate the solutions to PDE. FEM uses a variational formulation and divides the domain into interconnected elements [18], while FVM discretizes the domain into control volumes (cells) and integrates the governing equations over each volume, using the divergence theorem to convert volume integrals to surface integrals [19]. Both methods converge towards the same numerical solution, providing the same computational domain and same boundary conditions are compared [20]. The simulation results are a detailed representation of velocity and pressure fields, from which a multitude of flow descriptors may be derived.

2.2. Fluid–Structure Interaction (FSI)

The coupling of fluid and solid models is referred to as fluid–structure interaction or FSI. The governing equation for the solid can be described by the equation of motion:

$${\rho }_s{\displaystyle \frac{{\mathsf{\partial }}^2{\mathit{d}}_{\mathit{s}}}{\mathsf{\partial }{t}^2}}=\mathsf{\nabla }·{\mathsf{\sigma }}_s+{\mathit{f}}_{\mathit{s}}$$

(3)

where subscript s denotes solid, ρ_s is the wall density, d_s is the displacement vector, σ_s is the Cauchy stress tensor, and f_s is an externally applied body force vector. The BC on the FSI interface must guarantee that the velocity of the fluid and solid is equal and that the tractions at the boundaries are in equilibrium:

$${\mathit{u}}_{\mathit{s}}={\mathit{u}}_{\mathit{f}}$$

(4)

$${\mathsf{\sigma }}_{\mathit{s}}·{\mathit{n}}_{\mathit{s}}={\mathsf{\sigma }}_{\mathit{f}}·{\mathit{n}}_{\mathit{f}}$$

(5)

where u are velocities, σ are stress tensors and n surface normals. Together with a constitutive relationship linking stress and strain in the solid material, the system of equations can be solved. Solving the FSI equations can be accomplished either in a monolithic or in a decoupled, iterative way. In the monolithic solution, both the fluid and solid equations are solved in a single matrix, while in a decoupled approach, forces and displacements are passed between two solvers through a common interface.

Early FSI studies likely date back to the late 1990s [21], as researchers began exploring the interaction between blood flow and vessel wall mechanics. FSI techniques have significantly evolved over the years, driven by the advancements in computational power, numerical methods, and interdisciplinary research. Initial FSI studies were limited by computational resources and relied on simplified models, the early approaches often using a decoupled approach. With the advent of computational power, fully coupled simulations became feasible, and such methods allowed for a more accurate representation of the dynamic interplay between blood and vessel wall structure.

Commonly used CFD and FSI programs in cardiovascular modeling are either commercially available packages (proprietary license) or open-source projects (GPL, LGPL, or BSD licenses) as well as in-house developed and validated numerical packages [21]. It is worth taking into account that CFD solvers are sensible to initial settings [22] and therefore should be thoroughly tuned to obtain accurate results, especially for capturing high-frequency laminar flow instabilities or even transitional flows [23]. Moreover, solver numerics must be considered as important as mesh quality and time-step resolution in determining the quality of CFD simulations [24,25].

Most image-based CFD studies use the International System of Units (SI) for dimensions and fields, hence [m] for meshes, [m/s] for velocity, and Pascals [Pa] for pressure and stresses. However, since in vitro studies historically used the [dynes/cm²] for the WSS acting on EC and physiological values of WSS are similarly expressed (i.e., 10–70 dynes/cm² for normal arteries [2]), the centimetre–gram–second (CGS) system of units is often preferred in computational hemodynamics studies, for instance [cm] for grid dimensions, [cm³/s] for flow rates and [cm/s] for velocity, resulting in [dynes/cm²] for pressure and WSS. Additionally, since in clinical settings the millimeter of mercury is used for blood pressure readings, the pressure BC, or that resulting from solver numerics is often converted to [mmHg].

2.3. Post-Processing

All proprietary computational packages contain a dedicated post-processing module. Non-commercial and in-house developed frameworks often provide the post-processing phase via paraview (www.paraview.org (accessed on 16 July 2025)), an open-source, multi-platform data analysis and visualization application. Processing results are contour maps for hemodynamic wall descriptors, velocity iso-contours for high-speed jets or regions of stagnation flow, and flow helicity iso-contours or simulated dye or particle visualizations.

Post-processing is an important task in computational hemodynamics and becomes even more influential when it is applied to the calculation of blood flow descriptors. While most of the blood descriptors can be directly computed from the flow field, particle residence time and Lagrangian coherent structures (LCS) calculations require additional post-processing simulations. Lagrangian-based approaches to study cardiovascular flows were proposed since 2008 by Shadden and Taylor [26]. They introduced the computation of finite-time Lyapunov exponent (FTLE) fields, which provide a scalar accounting for the unsteady, nonlinear behavior of blood but can be visualized as an Eulerian field. Furthermore, FTLE fields reveal flow features known as LCS used to better characterize blood flow kinematics and to quantify transport topology in computational models of large vessels. Such approaches have been applied to identify LCS in intravascular flows, refer to the review of Shadden and Arzani [27] for a detailed overview of Lagrangian post-processing in cardiovascular flows.

3. Blood Flow Descriptors

The number of flow descriptors that have been used to characterize hemodynamics from the velocity field, or its derivatives, are too numerous to discuss them all. There are, however, descriptors based on the pressure field. In this narrative review only the blood flow descriptors highly used for characterizing the hemodynamics are treated. Blood flow descriptors are therefore parameters or indexes used to characterize and analyze the movement of blood and its interaction with the surrounding walls to localize zones of blood stagnation, blood recirculation, pressure losses, etc., through the cardiovascular system. Based on their properties these descriptors can be divided into the following categories.

3.1. Core Flow Descriptors

3.1.1. Blood Volumetric Flow Rate (Q(t))

Blood volumetric flow rate is mainly used as BC in numerical analyses, under the strong assumption that such a waveform is periodic with the heartbeat, which is not the case in vivo. It may be derived from echo-color Doppler or 4D (time) magnetic resonance imaging (MRI) examinations of the vascular tracts, but care should be taken in the acquisition phase because the blood flow rate (or the velocity resulting from it) may be inaccurate in poorly resolved imaging data.

Q = f(t)

(6)

3.1.2. Velocity Field (u)

Three-dimensional velocity vectors are the direct result of CFD numerical analyses in the whole blood vessel domain, representing the foundational element from which any number of hemodynamic descriptors may be derived.

u = f(x,t)

(7)

3.1.3. Pressure Distribution (p)

Similar to velocity, pressure is the result of the numerical CFD analyses, and it is a scalar describing the amount of force per unit surface area acting normal to the surface.

p = f(x,t)

(8)

3.1.4. Reynolds Number (Re)

The Reynolds number is the ratio of inertial forces to viscous forces and is widely employed in recognizing laminar, transitional, or turbulent flows [28], although its broad application in cardiovascular flows and clinical applications is still needing validation. It has the formula:

$$Re ={\displaystyle \frac{\rho vD}{\mu }}$$

(9)

where $v$ is blood mean velocity and $D$ is the diameter of the vessel.

3.1.5. Dean Number (De)

The Dean number is specific to curved vessels and utilized to understand curvature-induced hemodynamic effects in arteries. If the vessel has diameter $D$ and has a curvature with a radius $R_c$, the Dean number is linked to the Reynolds number by the formula:

$$De = Re\sqrt{{\displaystyle \frac{D}{2R_c}}}$$

(10)

3.1.6. Turbulent Kinetic Energy (TKE)

Turbulent kinetic energy provides the mean energy per unit mass generated by turbulent or transitional flows, i.e., the kinetic energy per unit mass from velocity fluctuations. It is calculated from the mean of the squares of the fluctuating velocity component in all directions:

$$TKE = {\displaystyle \frac{1}{2}} \left(\overline{{u\left(t\right)}^2}+ \overline{{v\left(t\right)}^2}+ \overline{{w\left(t\right)}^2}\right)$$

(11)

where $u\left(t\right), v\left(t\right), w\left(t\right)$ are the turbulent velocity fluctuations about the mean, the overbar denoting averaging of the variable. Fully developed turbulence is rarely seen in healthy humans [29]. However, the flow in the aortic arch can be in the transitional regime between laminar and turbulent flow, especially during the deceleration phase where flow instabilities can occur [30], or in a diseased aorta [31,32]. Therefore, turbulence modelling in such specific regions must be carefully applied.

3.1.7. Reynolds Shear Stress (RSS)

The Reynolds shear stress (RSS) with units of stress, e.g., [dynes/cm²] or [Pa], is:

$$RSS = \rho \overline{u\left(t\right)v\left(t\right)}$$

(12)

where v(t) is the root mean square of the turbulent velocity fluctuations perpendicular to u(t).

3.2. Helicity Descriptors

Blood swirling flows develop due to the non-planar curvature and branching of arteries [33] and are believed to have protective effects against atherosclerosis by reducing flow disturbances. Helicity-based descriptors are used to analyze the swirling blood flow within arteries and are derived from the vorticity vector ($\mathsf{\omega }$), which is the curl of flow velocity field:

$$\mathsf{\omega }=\nabla \times \mathit{u}$$

(13)

3.2.1. Helicity (H(t))

The helicity in a fluid flow confined into a volumetric domain D can be expressed as [34]:

$$H\left(t\right)={\int }_D\mathit{u}\left(\mathit{x},\mathit{t}\right)· \mathsf{\omega }\left(\mathit{x},\mathit{t}\right)\mathrm{d}\mathrm{V}={\int }_D{H}_k\left(\mathit{x},\mathit{t}\right)\mathrm{d}\mathrm{V}$$

(14)

where $\mathit{u}(\mathit{x},t)$ is the velocity vector, $\mathsf{\omega }(\mathit{x},t)$ is the vorticity vector, and $H_k$, which is the inner product of the two, is called kinetic helicity density.

If T is the period of the cardiac cycle, based on kinetic helicity, the helical content of the flow can be further characterized in terms of strength, size, and relative rotational direction using the following additional descriptors.

3.2.2. Cycle-Average Helicity (h₁)

The cycle average helicity can be calculated as:

$$h_1={\displaystyle \frac{1}{T}}{\int }_0^T{H}_k dt$$

(15)

3.2.3. Helicity Intensity (h₂)

The helicity intensity is calculated as:

$$h_2={\displaystyle \frac{1}{T}}{\int }_0^T\left|H_k\right| dt$$

(16)

3.2.4. Balance Between Counter-Rotating Helical Structures (h₃)

The balance between counter-rotating helical structures is calculated as:

$$h_3={\displaystyle \frac{h_1}{h_2}}$$

(17)

3.2.5. Absolute Value (h₄)

The absolute value of helicity is simply the modulus of balance:

$$h_4=\left|h_3\right|$$

(18)

3.2.6. Localized Normalized Helicity (LNH)

A useful descriptor for visualizing complex flow patterns is the localized normalized helicity (LNH), which is obtained through the normalization of the product of the velocity and vorticity vectors:

$$LNH={\displaystyle \frac{\mathit{u}·\mathsf{\omega }}{\left|\mathit{u}\right|\left|\mathsf{\omega }\right|}}$$

(19)

In this formulation, LNH represents the cosine of the angle between the velocity and the vorticity vectors and thus can take values from −1 to 1, with the sign indicating the direction of rotation of the helical structures.

Of note, many other helicity descriptors were defined and used in computational hemodynamics, like, for example, the helicity flow index (HFI), refer to [11,35,36] or the vorticity components ratio (VRI) [37].

3.3. Hemodynamic Wall Descriptors

The role of hemodynamics in the near-wall region is of highest importance since at this level blood flow exerts stresses on the vessel wall, regulating the local transport of circulating molecules and their interaction with the surrounding tissue. A growing collection of metrics were introduced over the years to localize and quantify disturbed flow. These descriptors are commonly determined by integrating WSS over a single cardiac cycle, under the strong assumption that blood flow rate is periodic in the modeled vascular tract. It is worth noting that WSS calculation is strongly influenced by the mesh quality, therefore performing a mesh independence test [38] is indispensable in hemodynamic simulations.

  

Wall shear stress vector

The WSS is defined as the tangential component of traction on the wall [39]. The traction on the wall can be computed as:

$$\mathit{t}= \mathsf{\sigma } · {\mathit{e}}_{\mathit{n}}$$

(20)

where $\mathsf{\sigma }$ is the stress tensor and ${\mathit{e}}_n$ is the unit normal vector, both evaluated on the wall. Accordingly, the WSS vector can be calculated as the tangential component of t:

$$\mathit{W}\mathit{S}\mathit{S}= \mathit{t}-\left(\mathit{t} · {\mathit{e}}_n\right) {\mathit{e}}_{\mathit{n}}$$

(21)

3.3.1. Time-Averaged WSS (TAWSS)

The time-averaged WSS is calculated by integrating each wall surface WSS magnitude over the cardiac cycle as:

$$TAWSS ={\displaystyle \frac{1}{T}}{\int }_0^T\left|\mathit{W}\mathit{S}\mathit{S}\left(t\right)\right|dt$$

(22)

3.3.2. Oscillatory Shear Index (OSI)

OSI was first introduced by Ku et al. in a model of human carotid bifurcation [40] and later generalized by He and Ku for the three-dimensional flow in [41]:

$$OSI ={\displaystyle \frac{1}{2}}·\left[1-{\displaystyle \frac{\left|{\int }_0^T\mathit{W}\mathit{S}\mathit{S}\left(t\right)dt\right|}{{\int }_0^T\left|\mathit{W}\mathit{S}\mathit{S}\left(t\right)\right|dt}}\right]$$

(23)

3.3.3. Relative Residence Time (RRT)

RRT was introduced by Himburg et al. [42] who showed that the residence time of particles near the wall is proportional to a combination of OSI and TAWSS:

$$RRT~{\displaystyle \frac{1}{\left(1 -2 OSI\right) TAWSS}}$$

(24)

3.3.4. Aneurysm Formation Indicator (AFI)

This descriptor was introduced by Mantha et al. in a study of cerebral aneurysms [43] aimed at capturing the rotational effect of the WSS vectors on EC. It is simply the cosine of the angle between the instantaneous WSS vector and the time-averaged WSS vector:

$$AFI =\mathrm{c}\mathrm{o}\mathrm{s}\left(\theta \right) = {\displaystyle \frac{\mathit{W}\mathit{S}\mathit{S}\left(\mathit{t}\right)·{\int }_0^T\mathit{W}\mathit{S}\mathit{S}\left(t\right)dt}{\left|\mathit{W}\mathit{S}\mathit{S}\left(t\right)\right|\left|{\int }_0^T\mathit{W}\mathit{S}\mathit{S}\left(t\right)dt\right|}}$$

(25)

where $\mathit{W}\mathit{S}\mathit{S}\left(t\right)$ is the instantaneous WSS vector.

3.3.5. Transverse WSS (transWSS)

The transverse WSS was ideated by Peiffer et al. [44] to take into account the multidirectional wall shear, by quantifying the WSS component orthogonal to the cycle-averaged WSS direction:

$$transWSS ={\displaystyle \frac{1}{T}}{\int }_0^T\left|\mathit{W}\mathit{S}\mathit{S}\left(t\right)· \left(\mathit{n} \times {\displaystyle \frac{{\int }_0^T\mathit{W}\mathit{S}\mathit{S}\left(t\right)dt}{\left|{\int }_0^T\mathit{W}\mathit{S}\mathit{S}\left(t\right)dt\right|}}\right)\right|\mathrm{d}\mathrm{t}$$

(26)

where $\mathit{n}$ is the normal vector to the vessel wall. In this formulation, transWSS averages over the cardiac cycle the magnitude of those components of the instantaneous WSS vector that are perpendicular to the TAWSS vector in the plane of the endothelium [45].

3.3.6. Cross Flow Index (CFI)

The CFI was introduced by Mohamied et al. [46] to account for the directionality of the WSS vector but not its magnitude. It is the sine of the angle between the temporal mean WSS vector and the instantaneous WSS vector, the CFI being the time average of these instantaneous components:

$$CFI ={\displaystyle \frac{1}{T}}{\int }_0^T{\displaystyle \frac{\mathit{W}\mathit{S}\mathit{S}\left(\mathit{t}\right)}{\left|\mathit{W}\mathit{S}\mathit{S}\left(\mathit{t}\right)\right|}} · \left(\mathit{n} \times {\displaystyle \frac{{\int }_0^T\mathit{W}\mathit{S}\mathit{S}\left(t\right)dt}{\left|{\int }_0^T\mathit{W}\mathit{S}\mathit{S}\left(t\right)dt\right|}}\right)\mathrm{d}\mathrm{t}$$

(27)

In this formulation, CFI differs from the AFI, which is the cosine of the above angle.

3.3.7. The Axial WSS (WSS_ax)

The axial WSS was introduced by Morbiducci et al. [47] in the attempt to create a robust framework for calculating WSS vector components based on vessel centerline. In particular, the axial component aligned along the main flow direction is:

$${\mathit{W}\mathit{S}\mathit{S}}_{ax}={\displaystyle \frac{\mathit{W}\mathit{S}\mathit{S}·{\mathit{C}}^{\mathbf{\prime }}}{\left|{\mathit{C}}^{\mathbf{\prime }}\right|}}·{\displaystyle \frac{{\mathit{C}}^{\mathbf{\prime }}}{\left|{\mathit{C}}^{\mathbf{\prime }}\right|}}$$

(28)

where C’ is the tangent vector of the vessel centerline C. Of note, a similar framework for characterization of WSS field, based on centerline, tangent, and binormal WSS components, was precedingly introduced by Arzani and Shadden [39].

3.3.8. WSS Gradient (WSSG)

This descriptor captures the spatial variation of shear stress along the vessel wall. It was ideated by Lei et al. [48] in a model of rabbit aorto–celiac junction, assuming that WSSG is the kernel of a wall permeability function, playing a key role for simulating enhanced low-density lipoprotein (LDL) transport into the arterial wall. It was further refined in CFD simulations for arterio-venous grafts by Longest and Kleinstreuer [49] as:

$$WSSG={\displaystyle \frac{1}{T}}{\int }_0^T\sqrt{{\left({\displaystyle \frac{{\mathsf{\partial }\mathit{W}\mathit{S}\mathit{S}\left(t\right)}_{\mathit{m}}}{\mathsf{\partial }m}}\right)}^2+{\left({\displaystyle \frac{{\mathsf{\partial }\mathit{W}\mathit{S}\mathit{S}\left(t\right)}_{\mathit{n}}}{\mathsf{\partial }n}}\right)}^2}\mathrm{d}\mathrm{t}$$

(29)

where m is the direction of TAWSS vector, and n is perpendicular on m on the wall’s surface.

3.3.9. WSS Angle Gradient (WSSAG)

Longest and Kleinstreuer [49] also proposed the WSSAG to highlight regions exposed to large changes in WSS direction, irrespective of magnitude. WSSAG is calculated as the time-averaged magnitude of the angle-scalar gradient:

$$WSSAG={\displaystyle \frac{1}{T}}{\int }_0^T\left|{\displaystyle \frac{1}{A_i}} {\int }_S\mathsf{\nabla }{\varnothing }_{j}d{A}_i\right|\mathrm{d}\mathrm{t}$$

(30)

where ${\varnothing }_j$ is the relative shear stress angle deviation that can be expressed as:

$${\varnothing }_j=arccos\left({\displaystyle \frac{{\mathit{W}\mathit{S}\mathit{S}}_i·{\mathit{W}\mathit{S}\mathit{S}}_j}{\left|{\mathit{W}\mathit{S}\mathit{S}}_i\right|·\left|{\mathit{W}\mathit{S}\mathit{S}}_j\right|}}\right)$$

(31)

where the stress vector at the location of interest is ${\mathit{W}\mathit{S}\mathit{S}}_i$ and ${\mathit{W}\mathit{S}\mathit{S}}_j$ represents the surrounding stress vectors and $A_i$ is the area control of surface area.

3.3.10. Dominant Harmonic (DH)

Himburg and Friedman [50] suggested the harmonic content of the WSS waveform as a possible metric of disturbed flow, subsequently linking it to the frequency-dependent responses of EC [51]. Following these authors, the time-varying WSS magnitude is Fourier-decomposed, with the DH defined as the harmonic with the highest amplitude.

$$DH=max\left[F_w\left(n{\omega }_0\right),F_w=FFT\left(\left|\mathit{W}\mathit{S}\mathit{S}\left(t\right)\right|\right)\right], {\omega }_0= {\displaystyle \frac{2\pi }{T}}$$

(32)

where ${\omega }_0$ is the fundamental frequency of the WSS signal.

3.3.11. Harmonic Index (HI)

The HI was defined by Gelfand et al. [52] as the relative fraction of the harmonic amplitude spectrum arising from the pulsatile flow components:

$$HI ={\displaystyle \frac{\sum _{n=1}^{\infty }{F}_w\left(n{\omega }_0\right)}{\sum _{n=0}^{\infty }{F}_w\left(n{\omega }_0\right)}}$$

(33)

It ranges from zero, in the case of a steady nonzero shear stress signal, to one, in the case of a purely oscillatory signal.

3.3.12. Spectral Power Index (SPI)

SPI was introduced by Khan et al. [25] to account for harmonic signals arriving from flows developing transitional instabilities, and it is calculated as:

$${SPI}_Y={\displaystyle \frac{{\sum _{n=n_c}^{\infty }\left|Y\left[n{\omega }_0\right]\right|}^2}{{\sum _{n=1}^{\infty }\left|Y\left[n{\omega }_0\right]\right|}^2}}$$

(34)

where $\left|Y\left[n{\omega }_0\right]\right|$ is the magnitude of the Fourier-transformed signal, ${\omega }_0$ is the fundamental frequency of the periodic signal, and n_c is the harmonic corresponding to the cut-off frequency. It was exemplified for WSS (i.e., SPI_WSS) in cranial aneurysms [25], but it might be applied to any harmonic signal derived from flows developing internal instabilities. In this form, SPI is a normalized quantity in the interval [0–1], with zero meaning no flow instabilities and 1 indicating a completely unstable flow.

3.4. WSS Topological Skeleton Descriptors

  

WSS Topological Skeleton Analysis

Stimulated by the need to improve the localization of zones of EC under contraction or expansion, topological skeleton analysis of the WSS vector field has received increased interest. Regarding such analysis, it is worth noting that the WSS could be interpreted as either the frictional force acting on EC or the near-wall velocity vector. Inspired by the dynamical system theory, the WSS topological skeleton is composed of a collection of fixed points, i.e., focal points where WSS locally vanishes, and unstable/stable manifolds, consisting of contraction/expansion regions linking fixed points. The fixed points of the time-averaged WSS vector field and their associated manifolds govern the near-wall transport in cardiovascular flows. Blood flow features such as flow separation and impingement affect these fixed points. Physiologically, the instantaneous WSS fixed points can potentially affect the EC, as these points have no preferred direction (zero magnitude), and the WSS vector direction and magnitude change dramatically around them [53]. An explanatory sketch of the WSS topological skeleton is presented in Figure 1.

Based on such Lagrangian approach, studies of WSS topological skeleton analysis in cardiovascular mechanics have been performed starting from a decade ago [21,22,23,24].

Though LCS are typically preferred to reveal inherently transient hemodynamic conditions, their extensive use has practical limitations, specifically the computational cost. More recently, starting from the WSS vector distribution at the luminal surface, for the WSS topological skeleton analysis, an Eulerian approach was proposed by Mazzi et al. [55]. Their method was further rationalized [56], and currently linked to the vorticity topology and revised as a unified theory [57].

  

Lagrangian descriptors of WSS topological skeleton

3.4.1. WSS Exposure Time (WSSET)

WSS exposure time was introduced by Arzani et al. [58] as a measure computed from Lagrangian tracking of surface-born tracers, which can account for stagnation (low flow) and species redistribution. It has the advantage of being a Lagrangian-based descriptor accounting for transport but has less computational cost compared to explicitly solving a full transport problem. Stable and unstable manifolds of fixed points can be computed from the TAWSS vector field, without having to actually perform the Lagrangian surface transport calculations required to compute WSSET. Namely, unstable manifolds determine the regions that tend to attract nearby trajectories in time, and stable manifolds mark areas of repulsion of nearby trajectories [58], e.g., the regions where near-wall species become attracted to particular TAWSS fixed points or TAWSS unstable manifolds, as exemplified in Figure 2.

Since WSS provides an approximation of the near-wall blood flow velocity, WSSET is computed for each surface element as the accumulated amount of time that all the WSS trajectories spend inside that element, with proper normalizations:

$$\begin{gathered} WSSET = {\displaystyle \frac{1}{\mathcal{T}}} \sqrt{{\displaystyle \frac{A_m}{A_e}}} \sum _{p=1}^{N_t}{\int }_0^{\mathcal{T}}{H}_e\left(p, t\right)dt \\ {H}_e = \left\{\begin{array}{c}1 if {\mathit{x}}_p\left(t\right) \in e\\ 0 if {\mathit{x}}_p\left(t\right) \notin e\end{array}\right. \end{gathered}$$

(35)

where A_e is the area of the surface element, A_m is the average area of all surface elements, x_p(t) is the position of the WSS trajectory, H_e is the indicator function for element e, N_t is the total number of trajectories released, and $\mathcal{T}$ is the integration time [58].

  

Eulerian descriptors of WSS topological skeleton

3.4.2. Topological Shear Variation Index (TSVI)

The TSVI measures the amount of variation in the WSS contraction/expansion exerted at the vessel luminal surface during a cardiac cycle. Based on a WSS topological skeleton analysis carried out applying the Eulerian method [55], the TSVI is defined as the root mean square deviation of the divergence of normalized WSS with respect to its average over the cardiac cycle [54]:

$$TSVI={\left\{{\displaystyle \frac{1}{T}}·{\int }_0^T{\left[\mathsf{\nabla }·{\mathit{W}\mathit{S}\mathit{S}}_u\left(t\right)-\overline{\mathsf{\nabla }·{\mathit{W}\mathit{S}\mathit{S}}_u\left(t\right)}\right]}^{2}dt\right\}}^{{\displaystyle \frac{1}{2}}}$$

(36)

where ${\mathit{W}\mathit{S}\mathit{S}}_u$ is the normalized WSS vector (calculated as ${\displaystyle \frac{\mathit{W}\mathit{S}\mathit{S}}{\left|\mathit{W}\mathit{S}\mathit{S}\right|}}$) and the overbar denotes a cycle-average operation.

3.5. Residence Time Descriptors

Blood flow stagnation and high residence time are critical factors in understanding vascular health, particularly in the context of aneurysm development and progression. Residence time descriptors can be calculated either with Lagrangian or Eulerian approach.

  

Lagrangian approach

3.5.1. Particle Residence Time (PRT)

In this approach, a number of tracers (massless particles) are seeded inside the region of interest transported by the flow, then the minimum time a particle takes to leave the region of interest is traced back to the initial position of the released particle. This defines the PRT field as introduced by Shadden and Arzani [27]:

$$PRT\left({\mathit{x}}_0, t_0;\Gamma \right)=\min \left(\mathrm{t}\right)\in \left(0,\mathrm{\infty }\right)\mathrm{s}.\mathrm{t}.\mathit{x}\left({\mathit{x}}_0, t_0+t\right)\notin \Gamma$$

(37)

where ${\mathit{x}}_{\mathbf{0}}$ is the initial position of a particle, $\Gamma$ is the region of interest, and $\mathit{x}\left(t\right)={\mathit{x}}_0,\left(t_0\right)+{\int }_{t_0}^t\mathit{u}\left(\mathit{x}\left(s\right),s\right)ds$ governs the tracer position.

3.5.2. Mean Exposure Time (MET)

Similar to WSSET, to calculate this descriptor, particles are seeded inside the fluid domain region of interest and released during ten equally spaced intracardiac time points [27]. After ten cardiac cycles of integration, the accumulated average time that all of the particles spend inside each computational element is measured and normalized by element volume to obtain the MET for each element:

$$\begin{gathered} MET = {\displaystyle \frac{1}{N_e\sqrt[3]{V_e}}} \sum _{p=1}^{N_t}{\int }_0^{\infty }{H}_e\left(p, t\right)dt \\ {H}_e = \left\{\begin{array}{c}1 if {\mathit{x}}_p\left(t\right) \in e\\ 0 if {\mathit{x}}_p\left(t\right) \notin e\end{array}\right. \end{gathered}$$

(38)

where N_e is the number of times a particle passes through element e, V_e is the volume of the element, x_p(t) is the position of the tracer, H_e is the indicator function for element e, and N_t is the total number of particles released.

Eulerian approach

3.5.3. Eulerian Residence Time (ERT)

ERT is computed by solving an advection-diffusion-reaction equation where the reaction term represents the time integrated in an Eulerian framework. The equation is written as:

$$\begin{gathered} {\displaystyle \frac{\mathsf{\partial }c}{\mathsf{\partial }t}}+\mathit{u}·\nabla c = \nabla ·\left(D\nabla c\right)+H \\ H= \left\{\begin{array}{c}1 if \mathit{x} \in \Gamma \\ 0 if \mathit{x} \notin \Gamma \end{array}\right. \end{gathered}$$

(39)

where c represents the ERT, D is the mass diffusivity and H is the source term, which is equal to 1 inside the region of interest and 0 outside.

3.5.4. Virtual Ink Residence Time (RT_VI)

This residence time was defined by Rayz et al. [59] by virtually injecting a passive scalar at the inlet and monitoring the transport of the scalar throughout the flow domain by solving an advection–diffusion equation. The injected scalar represents a virtual ink.

$${\displaystyle \frac{\mathsf{\partial }C}{\mathsf{\partial }t}}+\mathit{u}·\mathsf{\nabla }\mathrm{C}=D∆\mathrm{C}$$

(40)

where D is the diffusion coefficient and C is the local concentration of the virtual ink. In this study they set the diffusion coefficient D to zero, which ensures a pure advection of the virtual ink.

3.5.5. Point-Wise Residence Time (RT_x)

To define a point-wise measure of residence time that is independent from the volume, Esmaily-Moghadam et al. [60] established the residence time per unit length, RT_x(x), as the time required for a fluid particle to travel a unit length from any given point in the direction of flow:

$${RT}_x\left(\mathit{x}\right)={\left[{\displaystyle \frac{1}{T}} {\int }_{\left(n-1\right)T}^{nT}‖u\left(\mathit{x},t\right)‖dt\right]}^{-1}, \mathit{x}\in \mathsf{\Omega }-{\mathsf{\Gamma }}_w$$

(41)

in which Γ_w is the wall boundary, where u = 0 and hence RT_x is not defined. In this formulation the RT_x reduces to the inverse of the velocity magnitude at that point.

3.6. Pressure-Based Descriptors

Pressure-based descriptors are essential to cardiovascular mechanics, providing insights into heart function, arterial properties, and overall circulatory health. These metrics are essential for both clinical applications and research, aiding in diagnosis, monitoring, and treatment of cardiovascular diseases. The pressure-based descriptors are not so numerous as the parameters drawn from the velocity field and its derivatives, like WSS or vorticity.

Fractional Flow Reserve (FFR)

The fractional flow reserve (FFR) was proposed in the early 1990s as an index of the physiological significance of coronary stenosis, defined as the ratio of the maximum blood flow through a stenosis to the maximum blood flow in the same vessel in the hypothetical absence of the stenosis [61,62]. It is clinically measured at coronary angiography during induced hyperemia, with a pressure guidewire, as the ratio of distal coronary pressure to the aortic pressure, a value < 0.80 indicating a hemodynamically significative stenosis that should be treated. The FAME trial demonstrated that routine measurement of FFR in patients with multivessel coronary artery disease (CAD) undergoing percutaneous coronary intervention (PCI) with drug-eluting stents significantly reduced mortality and myocardial infarction at 2 years when compared with standard angiography-guided PCI [63].

Based on CT angiography imaging of coronaries, CFD proved to be an accurate and non-invasive tool to calculate FFR [64]. Such virtual FFR_CT can be calculated from the patient-specific model, providing that patient-specific heart, hyperemic and downstream coronary vascular bed BC are correctly imposed [65,66]. The FFR_CT is therefore calculated as the ratio between the pressure distal to the stenosis and the pressure in the aorta:

$${FFR}_{CT}={\displaystyle \frac{p_d}{p_a}}$$

(42)

4. Use of Descriptors in Cardiovascular Mechanics

The number of articles employing some of the main blood flow descriptors in the cardiovascular mechanics literature was roughly estimated by interrogating the Scopus database (https://www.scopus.com (accessed on 4 June 2025)). The bibliographic search query contained the names of descriptors in the title, abstract and key; the publication year equal to that of the defining article; and excluded subject areas not pertinent to the topic (e.g., “Earth and Planetary Sciences” or “Agricultural Sciences”). The number of articles employing specified descriptors are summarized in

Table 1. Number of articles using hemodynamic descriptors since their definition.

Descriptor	Defined in	Year	N of Articles	Humans
OSI	[40]	1985	865	575
RRT	[42]	2004	289	176
transWSS	[44]	2013	24	15
TAWSS	[48]	1996	403	253
WSSG	[49]	2000	204	97
Axial WSS	[47]	2015	9	6
SPI	[25]	2017	13	10
WSSET	[58]	2017	2	2
TSVI	[67]	2021	9	7
FFR	[61]	1993	8946	8160

.

As shown in Table 1, the highest number of scientific publications is on the fractional flow reserve, owing to the fact that FFR is most used in the clinical setting, too. Of note, if limited to CFD, the number of FFR studies becomes 484, highlighting the importance of computational hemodynamics studies in CAD. Development and localization of atherosclerotic plaques, which narrow the arteries impairing blood flow, have been correlated with regions of low and oscillating WSS since the early 1990s [1]. Since then, OSI, TAWSS, and RRT (and to a lesser extent, WSSG), being the oldest and relatively the easiest to compute, have been by far the most used descriptors. Since OSI does not account for WSS vector direction, Arzani and Shadden proposed an algorithm to calculate a descriptor (${∆}_{90}$) as the number of instances that the WSS angle changes more than 90 degrees, although any cut-off angle may be chosen [39]. Although less used in literature, the SPI is an example of a descriptor that is reduced to a single number, allowing for mapping of hotspots of turbulent WSS. The analysis of cycle averaged WSS fixed points and manifolds has been applied to analyze cardiovascular flows. Arzani et al. [58] computed WSS LCS from stable and unstable manifolds of cycle-averaged WSS and exemplified WSSET on six patient-specific models of abdominal aortic aneurysms (AAA), two carotid arteries, one cerebral aneurysm, and one coronary aneurysm, all models taken from previous studies, characterizing near-wall flow topology and transport of species in blood. For a systematic review of image-based CFD and FSI studies in cardiovascular diseases, refer to He et al. [21].

Abnormal hemodynamic descriptors are associated with several cardiovascular diseases, which often arise from, or lead to, disturbances in the flow field within blood vessels. The following are some key diseases linked to abnormal hemodynamic descriptors.

Heart valves and left ventricle (LV)

Charonko et al. [68] quantified the vortex ring volume by computing LCS from in vivo LV phase contrast MRI data of healthy (n = 4, 75% males, mean age 28 years) and diseased (n = 10, 30% males, mean age 58 years) patients. Similarly, Töger et al. [69] extracted LCS from in vivo LV phase contrast MRI data to measure vortex ring volume during the rapid filling of the LV in healthy volunteers (n = 9, 66% males, mean age 30 years) and patients with dilated ischemic cardiomyopathy (n = 4, 75% males, mean age 65 years). Hendabadi et al. [70] identified attracting and repelling LCS from LV conventional transthoracic Doppler-echocardiogram data, a criterion adopted to discriminate between healthy (n = 6, 17% males, mean age 60 years) and diseased (n = 6, 66% males, mean age 61 years) patients. Other studies applied LCS to characterize the flow field through heart valves. In particular, LCS were extracted in an idealized 3D model of the aortic root, to delineate the boundaries of the outflow jet downstream of aortic valves, used as a measure of the severity of the valve’s stenosis [71]. Soltany Sadrabadi et al. [72] employed FTLE-based LCS detection on FSI models of a 2D bicuspid aortic valve and a 3D mechanical heart valve to study mass transport processes related to valve disease. Ge and Sotiropulos [73] hypothesized a link between WSS fixed points observed on the aortic side of the aortic valve leaflet and the focal distribution of calcification lesions in the valve.

Coronary artery disease (CAD)

Low TAWSS and multi-directionality of WSS (i.e., high transWSS and/or CFI) have emerged as possible predictors of atherosclerosis in twenty patients (13% males, mean age 54 years) with coronary artery disease at baseline and followed up for six months [74], as well as in a model of hypercholesterolemic pigs (n = 10) [75]. In a similar animal model (n = 5), De Nisco et al. [76] demonstrated the physiological significance of helical flow, revealing its protective role against atherosclerotic growth in coronary arteries. WSS topological skeleton analysis was applied by Mazzi et al. in coronary arteries in minipigs (n = 3) [67], showing that early atherosclerotic changes are associated with endothelium shear stress contraction/expansion variability, namely zones of high TSVI. Mahmoudi et al. studied the near-wall transport of species contributing to the initiation and progression of atherosclerosis in ten coronary artery models of diseased and healthy subjects, highlighting the strength of cycle-average WSS LCS as a template of luminal surface concentration and flux patterns of biochemicals transported with blood [77].

In the clinical setting, computational modelling of CAD has the great advantage of being a non-invasive tool for clinical risk assessment and treatment planning [66]. Estimation of FFR_CT to detect hemodynamically significant stenoses in patient-specific computational models has effectively predicted disease severity while reducing unnecessary invasive procedures in clinical trials [78] (n = 103, 72% males, mean age 63 years) [79] (n = 252, 71% males, mean age 63 years). Such trials have led to the approval of the first simulation-based risk assessment tool (FFR_CT, www.heartflow.com (accessed on 4 June 2025)) for clinical use by the FDA. Because of the relatively high computational demand of CFD, machine-learning approaches are being developed as alternatives with potentially faster evaluation of FFR_CT [64,80,81]. Gosling et al. developed a novel tool for virtual coronary intervention based on the CT angiography that predicts the physiological response to stenting with a high degree of accuracy [82], which is part of the VIRTUheart^TM framework (https://www.virtuheart.com (accessed on 4 June 2025)). For a contemporary review of computational studies for the assessment of disturbed coronary flow, refer to Ekmejian et al. [83].

Cerebrovascular disease

Numerous image-based computational studies were performed with the aim to identify hemodynamic predictors of cerebral aneurysm progression [84]. Cerebral computational hemodynamics studies are challenging in terms of model parameters and BC [85,86]. The size, strength, and location of the jet emanating from the neck and impinging on the aneurysm wall were correlated to the risk of aneurysm rupture in a pilot clinical study of 57 subjects (33% males, mean age 51 years) diagnosed with one or more cerebral aneurysms and followed by transfemoral catheterizations of the cerebral vessels and rotational digital subtraction imaging [87]. High OSI and morphological size ratio of intracranial aneurysms discriminated the rupture status in a pilot clinical study on 119 intracranial aneurysms (38 ruptured, 81 unruptured) [88]. Rakesh et al. [89] calculated hemodynamic wall descriptors in one patient-specific healthy and one stenosed human carotid bifurcation, finding higher OSI and RRT in the diseased one. Analysis techniques focused on topological features of the flow field itself, including LCS and the topology of the WSS field, have recently been shown to provide promising clinical indicators for clinical risk and disease progression in two carotid arteries and one cerebral aneurysm [58]. Similarly, in a pilot study aimed at identifying direct associations between the WSS topological skeleton and markers of long-term restenosis after carotid endarterectomy (n = 13, 46% males, mean age 73 years) [54]. Farghadan et al. [90] used WSS topology and magnitude analysis to predict surface concentration patterns in cardiovascular transport problems by computing WSS LCS from manifolds of cycle-average WSS in a coronary artery stenosis and a carotid artery model. In addition, FTLE-based analysis was adopted to highlight the hemodynamic impact of flow diverter stents in the treatment of intracranial aneurysms, in an aneurysm sac of a 45-year-old woman [91]. Fixed points were considered in patient-specific computational models of cerebral aneurysm [92,93,94,95].

High TAWSS was found to be present at thinning sites, while low TAWSS and high RRT were observed at thickening sites, in a pilot computational study in 4 subjects (3 males, mean age 60 years) [96]. Of note, AFI was defined and applied in a pilot computational study in three cases of paraclinoid aneurysms, reconstructed from data acquired by 3D-DSA imaging [43]. Performing a pilot study on patients (n = 3) with fusiform basilar intracranial aneurysms, Rayz et al. [97] found a strong similarity between the intra-aneurysmal regions with CFD-predicted slow and recirculating flows and the regions of thrombus deposition observed in vivo in the follow-up MRI acquisitions. Subsequently, the same group [59] simulated passive scalar advection by CFD in the same cases, finding a co-localization between thrombotic regions and prolonged residence time estimated by the virtual ink (RT_VI) method.

Aortic disease

Cilla et al. proposed a parametric study of aortic hemodynamics, with models informed from a data set of five patient-specific aortas acquired by means of CT, evaluating the effect of geometric parameters on TAWSS and OSI [98]. LCS have been used by Arzani and Shadden to capture vortex transport in six patients with small AAA, reconstructed from MRI data [99]. They proved how LCS can be used to clearly reveal flow stagnation, separation, and partitioning of fluid to downstream vasculature. Abdallah et al. [100] applied Lagrangian descriptors to extract three-dimensional coherent structures from patient-specific 4D flow MRI data in healthy subjects and patients with aortic regurgitation. The study cohort included eight healthy subjects and three patients with severe aortic regurgitation. Regions of strong near-wall turbulence were collocated with regions of elevated transWSS in a patient-specific model of aortic coarctation [32]. LCS was applied for WSS topological skeleton analysis in aortic aneurysmal flows [39,101]. A clinical study on 125 patients (81% males, mean age 61 years) with ascending aortic aneurysms found that high TAWSS is associated with high levels of circulating biomarkers [102]. For a systematic review of computational studies in human aortic disease, refer to Ong et al. [103], and to Zhu et al. [104] for a comprehensive review on the application of image-based CFD simulations and 4D-flow MRI for risk prediction in aortic dissection.

5. Discussion

The study of hemodynamic descriptors has significantly advanced our understanding of cardiovascular physiology and pathology. By analyzing these descriptors, researchers and clinicians may assess various aspects of blood circulation, including velocity profiles, WSS distributions, and transitional flow patterns, which were suggested as “mechanical irritation” of the arterial wall [105], revealing sites of initiation and progression of cardiovascular diseases. As highlighted in this narrative review, while the conventional descriptors, such as the hemodynamic wall parameters, have proven useful in identifying abnormal flow conditions linked to atherosclerosis and thrombosis, emerging metrics, including Lagrangian coherent structures and WSS topological skeleton, particle residence time, or helicity-based descriptors, have further refined our ability to characterize complex flow phenomena.

The implications of blood flow descriptors for clinical practice are significant, as they may enhance diagnostic precision and improve treatment strategies for cardiovascular diseases. By providing a detailed characterization of blood flow patterns, hemodynamic descriptors may help clinicians to identify early signs of vascular abnormalities, such as regions of disturbed flow that may contribute to atherosclerosis or thrombus formation. One major advantage might be a personalized risk assessment; for example, patients with abnormal descriptors may be at higher risk for some vascular diseases. This would also allow for early interventions, such as lifestyle modifications or targeted medical therapies. Additionally, blood flow descriptors, like for example FFR_CT, contribute to surgical planning in CAD, particularly in procedures like catheterization, stent placement [82] or bypass grafting, where the understanding of local hemodynamics can optimize outcomes.

There are a number of considerations researchers must be aware of.

• For several hemodynamic descriptors, existing studies report conflicting findings. The transWSS was conceived because of new discoveries on EC alignment under flow that existing hemodynamic wall descriptors could not justify. In fact, the whole theory of low [106,107] and oscillatory WSS [40] was challenged by evidence underlining how the EC are exposed to a complex hemodynamic environment that cannot be described by the low/oscillatory WSS [44,108]. In a systematic review by Peiffer et al. [109], the various descriptors proposed for quantifying low and oscillatory WSS (low TAWSS, OSI) resulted in moderately weak predictors of vascular wall dysfunction in specific arteries. A prospective study of human coronary arteries [75] found that transWSS was not significantly correlated with the change in plaque over time. On the other hand, Pedrigi et al. [110] found a spatial correlation between transWSS and advanced lesions induced by a shear-modifying stent in hypercholesterolemic minipigs. Similarly, the role of WSS in plaque initiation and progression remains debated. While metrics like FFR_CT are suitable for guiding mild-to-severe CAD treatment, there is less consensus on assessing de novo plaque growth and early-stage risk. A serial MRI study of carotid artery plaques found that plaques tend to progress in regions exposed to low WSS [111]. On the other hand, a serial MRI study demonstrated an association between the site of plaque ulceration and high WSS [112], a finding confirmed also by an IVUS-based study of plaques that found an association with elevated WSS [113].
• It would be a difficult task deciding which descriptor is most useful for a specific vascular disease. There are many reasons for this: vascular pathologies are diverse and have different initiation mechanisms; cardiovascular disease depends on the vascular territory, on age, on gender, and on species; and there are no guidelines based on actual literature knowledge. We can, however, rely on existing clinical computational studies. OSI, for example, was originally proposed to describe a positive correlation between plaque location and low oscillating shear stress in the carotid sinus [40], where the flow instabilities are low; therefore, its applicability is uncertain when the flow is unstable. RRT also was recommended as a robust single metric of low and oscillating shear in a cross-sectional study of 50 normal carotid bifurcations Lee et al. [114]. The group of Weinberg found that atherosclerosis at aortic branch sites in immature and mature rabbits correlates three-fold better with transverse transWSS than with other hemodynamic wall descriptors [45]. In a cross-sectional study of human (n = 10) non-stenosed right coronary arteries [115], the same group found that RRT can account for the anatomical variation in fatty streak prevalence. Regarding residence time descriptors, a critical comparison of the different methods in an AAA and a cerebral aneurysm by Reza and Arzani [116] highlighted that most RT methods have a conceptually distinct definition and therefore should be utilized depending on the specific application of interest.
• In spite of achieving such complexity in characterizing the blood flow in arteries, we must reflect on the fact that, except for FFR_CT, blood descriptors are not integrated into the clinical workflows. The reasons for this are three-fold: creating an image-based computational model remains difficult and, in a clinical setting, would raise reproducibility problems; the numerical simulations have high computational costs; and the multiple CFD datasets are difficult to interpret into clinically relevant criteria.

Despite the advances highlighted in the previous sections, major challenges remain towards integrating blood flow descriptors into routine clinical practice. Firstly, inaccuracies may arise from the variability in imaging modalities or use of wrong, approximated, non-patient-specific BC in the computational techniques used to derive these metrics. Discrepancies in measurement approaches can lead to inconsistencies and to errors in three-dimensional geometry reconstructions, as warned by Perinajová et al. [99] about the geometrically induced WSS variability in CFD-MRI coupled simulations in the thoracic aortas. Secondly, these descriptors are generally determined by integrating WSS over a single cardiac cycle (i.e., the majority of computational studies simulate three pulsatile cycles, the first one is discarded, and final results are obtained by post-processing the final cycle). Nevertheless, it was proven that the common assumption of a periodic cardiac cycle may determine important errors in the calculation of hemodynamic wall descriptors [117]. Thirdly, the hemodynamic wall descriptors themselves are not perfect, and there is a certain level of redundancy among them. In fact, several studies were aimed at finding correlations among the hemodynamic wall descriptors [114,118]. OSI is frequently claimed to be a descriptor of disturbed flow, even if incomplete descriptor [118]. In fact, it cannot discriminate slow unidirectional oscillatory flow from fast multidirectional flow, as discussed in previous studies [23,24,109]. Moreover, blood is quite always assumed to be a single phase, while, for example, Buradi and Mahalingam [119], using multiphasic blood flow in an ideal model of carotid artery stenosis and several descriptors (TAWSS, WSSG, and OSI), found more disturbed fluid dynamics downstream of stenosis. Finally, while descriptors provide valuable insights, their clinical interpretation requires a nuanced understanding of physiological context, patient-specific factors, and disease progression with age [109]. Therefore, the medical community may have difficulties in understanding some of the hemodynamic descriptors, as well as remaining up to date with this increasing collection of descriptors.

Future research should focus on standardizing descriptor definitions and improving computational methodologies to enhance their reproducibility and clinical applicability. Efforts should be made to enhance existing hemodynamic descriptors, by preserving only those that really correlate or predict the future sites of vascular wall diseases. At the same time, new descriptors should arise from the unmet needs in current clinical practice, while the validation of these descriptors may be obtained in in vitro studies on EC, in in vivo studies in animal models, or, even better, in controlled clinical trials, towards acceptance by the FDA and EMEA for clinical use.

6. Concluding Remarks

The hemodynamic descriptors discussed in this review illustrate the evolving capabilities of image-based CFD and FSI in cardiovascular research. From the foundational metrics to the emerging descriptors, scientific literature underscores their importance in understanding physiological and pathological states, as well as in optimizing medical device designs. Moreover, blood flow descriptors will continue to evolve, offering deeper insights into cardiovascular health and disease. Their refinement and integration into clinical workflows will be essential for advancing precision medicine and improving patient outcomes. Future studies are expected to validate such descriptors and patient-specific data, further advancing computational studies’ role in cardiovascular research.

Machine learning and artificial intelligence may aid in synthesizing novel hemodynamic descriptors to create predictive models, potentially transforming cardiovascular diagnostics and treatment planning.