Using Gaussian process for velocity reconstruction after coronary stenosis applicable in positron emission particle tracking: An in-silico study

Hamed Keramati; Adelaide de Vecchi; Ronak Rajani; Steven A. Niederer

doi:10.1371/journal.pone.0295789

Abstract

Accurate velocity reconstruction is essential for assessing coronary artery disease. We propose a Gaussian process method to reconstruct the velocity profile using the sparse data of the positron emission particle tracking (PEPT) in a biological environment, which allows the measurement of tracer particle velocity to infer fluid velocity fields. We investigated the influence of tracer particle quantity and detection time interval on flow reconstruction accuracy. Three models were used to represent different levels of stenosis and anatomical complexity: a narrowed straight tube, an idealized coronary bifurcation with stenosis, and patient-specific coronary arteries with a stenotic left circumflex artery. Computational fluid dynamics (CFD), particle tracking, and the Gaussian process of kriging were employed to simulate and reconstruct the pulsatile flow field. The study examined the error and uncertainty in velocity profile reconstruction after stenosis by comparing particle-derived flow velocity with the CFD solution. Using 600 particles (15 batches of 40 particles) released in the main coronary artery, the time-averaged error in velocity reconstruction ranged from 13.4% (no occlusion) to 161% (70% occlusion) in patient-specific anatomy. The error in maximum cross-sectional velocity at peak flow was consistently below 10% in all cases. PEPT and kriging tended to overestimate area-averaged velocity in higher occlusion cases but accurately predicted maximum cross-sectional velocity, particularly at peak flow. Kriging was shown to be useful to estimate the maximum velocity after the stenosis in the absence of negative near-wall velocity.

Citation: Keramati H, de Vecchi A, Rajani R, Niederer SA (2023) Using Gaussian process for velocity reconstruction after coronary stenosis applicable in positron emission particle tracking: An in-silico study. PLoS ONE 18(12): e0295789. https://doi.org/10.1371/journal.pone.0295789

Editor: Muhammad Shakaib, NED University of Engineering and Technology, PAKISTAN

Received: August 15, 2023; Accepted: November 28, 2023; Published: December 14, 2023

Copyright: © 2023 Keramati et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: The mesh for the anatomically accurate geometry and all the particle tracking data are available on Zenodo (doi: 10.5281/zenodo.10102125).

Funding: SAN is a Co-PI for the grant (EP/R045046/1) from Engineering and Physical Sciences Research Council (EPSRC - https://www.ukri.org/councils/epsrc/), UK. HK is a Research Associate under the same grant. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Blood flow reconstruction based on measured data in coronary arteries is crucial to calculate the measures used for estimating the severity of coronary artery disease. Using quantified velocity and pressure distributions guides clinicians to identify the exact location of the lesion and the suitable treatment procedure. Invasively, one can measure the pressure difference along a lesion during coronary angiography [1] to calculate the Fractional Flow Reserve (FFR), which links to the severity of the disease. To avoid invasive procedures, techniques such as coronary computed tomography (CT) [2,3], 4D MRI [4,5], and cardiac Positron Emission Tomography (PET) [6] have been used. However, all the current techniques are associated with various uncertainties and challenges.

Positron Emission Particle Tracking (PEPT) is a technique that enables us to track tracer particles in opaque environments [7] with novel medical applications [8–11]. PEPT is based on PET and benefits from high spatial and temporal resolution [12].

Tracer particles are labeled with positron emission radionuclides such as ¹⁸F, ⁶¹Cu, ⁸⁹Zr, and ⁶⁶Ga [12,13] and then the particles are injected into the system. When the particles are in the field of view, the positron-emitting traces are tracked, and the trajectories are reconstructed using the gamma rays detected by the gamma cameras and corresponding lines of response (LORs). PEPT particle tracking methods started with the Birmingham method [14,15], and have been expanded to include the line-density method [16], multiple location-allocation algorithm (MLAA) [17,18], K-Medoids [19], clustering methods [20], the feature point identification (FPI) method [21], Odo triangulation method [22], Voronoi-based multiple particle tracking (VMPT) [23], the time-of-flight PEPT (TOF-PEPT) algorithm to do motion correction in medical imaging [24,25] and recently-developed method of PEPT machine learning (PEPT-ML) which tracks multiple particles and does not require frame tracking [26].

PEPT was initially used in industrial applications with opaque and fast-moving environments such as industrial powder mixing systems [27] and complex deformation such as extrusion [28,29]. PEPT has also been used for the characterization and measurement of physical behaviors such as diffusion [30] and convection [31]. However, PEPTs ability to track fluid flow in opaque systems has yet to be widely exploited in medical applications.

To accelerate the development of flow reconstruction methods and optimize PEPT experimental designs, Computational Fluid Dynamics (CFD) is advantageous. CFD can encode inherent physical constraints to reconstruct flow fields to distinguish the effects of various sources of uncertainties in this novel technology. Previously, CFD-PEPT studies have focused on industrial applications. The location of tracers in turbulent flow was predicted in a 3D tube with an obstacle using PEPT and Large Eddy Simulation (LES) models [32]. Experimental investigations were also performed to trace a single particle moving through a bend in a pneumatic system [33], in a hydro-cyclone with a turbulent flow for industrial applications [34], and in a tube with a pulsatile flow [35]. PEPT was also used as an experimental validation for a granular dynamics model coupled with CFD [36].

Recently, PEPT has been adopted for biomedical research and applications. Such applications include the improvement of radiation therapy [37] and tracking cells in the body [8,11].

For the clinical application of PEPT, the main advantage is that it enables us to track particles in vivo, i.e., inside the patient’s body [9]. In contrast to conventional particle tracking imaging techniques, e.g., PIV, PEPT does not require a transparent environment. The second advantage is that PEPT uses a PET scanner to collect the particle data. By improving the PEPT technology, one can use the existing and standard PET scanner to image and diagnose new conditions.

By injecting the tracer in the blood flow upstream of the location of interest, using PEPT one can relate the blood velocity to the measured tracing particles and can estimate the blood velocity in different locations in vessels. Using PEPT enables researchers and clinicians to reconstruct the velocity profile. This can be used to quantify other metrics of diagnostic relevance, such as the wall shear stress [38], which is related to the risk of thrombus formation [39,40]. Moreover, the velocity profile after the stenosis can be used to estimate the pressure drop across the blockage without catheterization [41].

Assuming the surface forces between the particles, the effect of the blood rheology, and the interaction between the particles and the blood cells are negligible, the low Stokes number reported by [33] implies the path of the particles in the flow is only governed by the velocity field of the flow, not the particle inertia, meaning the effect of the particles on the flow is negligible. Bruggemann and co-workers [42] used the Monte Carlo technique implemented in the package GEANT4 Applications for Tomographic Emission (GATE) [11] to reconstruct the flow downstream of an orifice to find the required acquisition duration to resolve the flow features of interest for a given activity and the Reynolds number for simulated PET data. GATE was also used to assess the ability of PEPT to track particles in turbulent flows [43] and to model a PET scanner to for the for the development of new PEPT algorithms [44]. Clinical CFD-PET simulations have been reported in [45] to predict the radioisotope distribution in the hepatic artery during a dosimetry procedure to optimize the injection site for tumor targeting. Although they did not perform particle tracking analysis, they observed that the number of microspheres was proportional to the cumulative blood flow.

One of the sources of uncertainties in reconstructing the velocity profiles is the sparsity of the particle data, especially in a time-dependent flow. To overcome this challenge, we proposed and evaluated the kriging Gaussian process to calculate the blood flow velocity based on the velocities of the particles and to quantify the uncertainty based on the sampled data. Kriging, an advanced geostatistical technique, has emerged as a powerful tool in various fields, including geology [46], design optimization [47–49], health-related data analysis [50–54], and medical image processing [55–57]. Kriging models complex spatial patterns and incorporates spatial dependency, offering an innovative solution for capturing and predicting data in areas where traditional interpolation methods fail [40]. The model is trained using the measured or sampled data, and the kernel hyperparameters are optimized. By utilizing a Gaussian process and a set of predefined parameters, kriging not only estimates unknown values at unsampled locations but also provides measures of uncertainty, enabling researchers to make informed decisions based on the reliability of predictions.

So far, PEPT has mainly been used to record the position of a single particle. However, to reconstruct the velocity profile of blood flow with the accuracy required for diagnostic purposes, more data points are required. In addition to the number of particles, other factors like the number of seeding instances in a cardiac cycle play a central role in the accuracy of the velocity reconstruction. Therefore, we investigated the effect of the number of particles and the measurement sampling rate per cardiac cycle on the uncertainty associated with the velocity reconstruction and quantified the error in velocity field estimates in a set of models with increasing levels of complexity.

In this paper, we present a fully computational methodology for blood flow reconstruction from multiple particles to assess the potential of clinical applications of PEPT for coronary disease diagnosis and estimation of the associated uncertainty. Specifically, we apply Gaussian process to estimate blood velocity and the associated uncertainty, first in an idealized straight vessel with stenosis, then in a bifurcation with a stenotic branch, and finally in a patient-specific model of coronary obstruction.

The current paper presents a fully in-silico study, which is a necessary step to inform our next steps for in-vitro and finally in-vivo experiments. These controlled in-silico datasets form a testbed to understand the feasibility of PEPT and Gaussian process in in-vitro phantoms of arterial flow. Full in-silico approaches improve one’s understanding of the strength and weakness of the reconstruction algorithm and are beneficial for designing in-vitro and in-vivo experiments.

Methods

In our test scenario, pulses of biocompatible tracer particles [10,58] are injected into the patient upstream of the region of interest. Sufficient particles in the plane of interest in the vessel are needed to reconstruct the velocity field. If only one particle in the region is used, it is not possible to calculate the mean velocity as the true velocity profile is not known, and because of this, we have increased the number of tracers.

Fig 1 shows the workflow for the present analysis. CFD simulations are first performed to generate a physiological flow field (a), from which random points are iteratively sampled at a specific location to reconstruct the profile in space and time (a1). An error analysis is then performed to establish the optimal number of sampling points necessary to reconstruct the velocity field (a2). We then determine the optimal number of simulated PEPT particles that need to be released based on the optimal number of sampling points from the error analysis in the CFD model. At the next step, we simulate particle tracking to approximate the PEPT procedure (c1). The velocity profile is then reconstructed using kriging and the particles’ velocity as they pass through a plane of interest (c2). Finally, the reconstructed velocity field is quantitatively compared with the velocity field from CFD (c3).

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Fig 1. Schematic of the research methodology.

The velocity field is generated using CFD (a). By sampling from the CFD velocity field, the number of required particles for particle tracking is estimated (b). The particle tracking is performed as a post-processing procedure and the velocity profiles are reconstructed (c).

https://doi.org/10.1371/journal.pone.0295789.g001

CFD simulation

We propose a workbench of CFD problems for evaluating PEPT spanning from a straight tube to a bifurcation and finally a patient specific coronary obstruction model.

Computational domains.

For the simplest model we considered a straight tube with a 50% blockage (Fig 2a), where the occlusion profile was generated using a sinusoidal shape. Both inlet and outlet boundaries were extended by 60 mm.

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Fig 2. Geometry and boundary conditions.

Tube stenosis geometry, the stenosis geometry, velocity (solid line) and the pressure (dashed line) regenerated from [59] (a), idealized bifurcation with 50% stenosis at LCx (b), segmented coronary artery on the CT scan images (c) and anatomical bifurcation with an artificially generated 50% occlusion at LCx and the equivalent electrical circuit for the outlet boundaries (d).

https://doi.org/10.1371/journal.pone.0295789.g002

For the idealized bifurcation model (Fig 2b), the diameters and lengths of the vessels were based on previous studies on coronary flow [60,61]. Murray’s law was used to relate the radii of the branches to the radius of the main inlet, as well as the angles between each branch and the centerline, which are 35° and 40° for the left anterior descending artery (LAD) and the left circumflex (LCx), respectively [62]. In each case, models with a narrowing of 30%, 50% and 70% in the cross-sectional diameter were created. Finally, the anatomy of a patient-specific coronary tree (Fig 2c) was reconstructed from anonymized cardiac CT images using 3D Slicer 4 [63] and MeshMixer (Autodesk, California, USA). The computational domain is presented in Fig 2d.

CFD simulation and boundary conditions.

The Navier-Stokes and continuity equations for incompressible flows were solved using Ansys^® FLUENT^®, Release 18.1 (ANSYS, Canonsburg, PA, USA). Regarding the Reynolds number for the average velocity (Re<500) the flow was assumed to be laminar [64]. Even though Re at the peak flow in a stenosis might reach transition values, the pulsatility effect means that turbulence will not develop at the scale that requires turbulent modeling. The blood was assumed to be a Newtonian flow with a density of 1050 kg/m3 and a viscosity of 0.0036 Pa∙s. According to [65], assuming the blood to be Newtonian is acceptable in coronary artery, which leads to more computationally efficient simulations. The velocity waveform recorded in 20 patients with relatively low probability of coronary disease [59] (Fig 2a) was prescribed as the inlet boundary condition (BC). For the straight tube, the outlet pressure presented in Fig 2a [59] was used as the outlet BC. For the idealized bifurcation geometry and the patient-specific anatomy, a 0D model was assigned to the outlet boundaries based on the model described by [66] as shown in Fig 2d. The equation for the 0D model is: (1) where P is the pressure and Q is the flow rate. The rest of the parameters are as follow: where P_LV is the pressure of the left ventricle and the lumped parameters for the resistances and capacitances R_a, R_v, R_a-m, C_im, and C_a are illustrated in Fig 2d. The lumped parameters were then separately calibrated in both geometries based on physiological flow curves [59]. The parameter optimization was performed using Simulink^® (MathWorks, Natick, MA, USA) by minimizing the error between the physiological inlet pressure [59] and the calculated inlet pressure. The final parameter values are listed in S1 and S2 Tables for idealized bifurcation and anatomically-accurate geometry, respectively. The CFD simulation was performed for three cardiac cycles until the solution reached a periodic state.

We carried out a grid independence analysis to ensure the results were in the asymptotic region. The used mesh for all stenotic cases had 800k to 831k cells. The error between the area-pressures was less than 1.5% and the maximum velocity in the field had 0.7% error when the mesh was refined about 50%.

We ensured that the pressure is within a physiological range (70–140 mmHg), allowing us to compare simulated PEPT particle distributions within under physiologically plausible conditions [67].

Flow measurement.

To estimate the flow across an occlusion, particles were released randomly in a cross section at the inlet of the vessel. The location at which each particle experienced its maximum velocity was then used to estimate the position of the occlusion. The particles’ velocity through a plane of interest (POI) perpendicular to the vessel centerline 10 mm downstream of the minimum lumen area (MLA) was then calculated. This location was chosen based on the guideline for the Fractional Flow Reserve (FFR) measurements [68,69]. The reconstructed profile, performed using kriging, was compared with the results from CFD.

Sampling

The velocity vector from the CFD solution was sampled at a varying number of points uniformly distributed in the POI (i.e., 10, 20, 40, 60, 80 and 100). T_c is the cardiac cycle time (T_c = 0.96 s), t_c ∈ [0,T_c] is the time in a cycle, and Δt_c is the difference between the t_c values. The data were sampled at Δt_c values of 96 ms, 64 ms and 48 ms, equivalent to sampling 10, 15 and 20 times per cardiac cycle. Δt_c was considered as a control parameter in this study.

Sampling allowed us to find the optimal number of particles and Δt_c for the particle tracking algorithm (Fig 1b). In the particle tracking phase, Δt_c can be interpreted as the time points at which the particles are injected within the cardiac cycle.

Particle tracking

The Stokes number of a particle in a flow represents the time it needs to follow the changes in the flow field normalized by the characteristic time of the flow, i.e., the ratio of the particle relaxation time, τ_p, and the flow typical time scale, τ_f [70]: (2)

The characteristic time of a particle is defined [71] as (3) where ρ_p, d_p and μ_f are the density of the particle material, the particle diameter and the surrounding fluid viscosity. For a particle in range of 1–100 μm [12] and with density similar to that of blood, τ_p is between 1.62 × 10⁻⁸ s and 1.62 × 10⁻⁴ s. Therefore, for τ_f of about a second in the coronary artery, the Stokes number will be less than 10⁻³, which means that the particles are driven by the flow without disturbing it. The velocity of a particle at a point can therefore be approximated by the velocity of the carrier fluid at the same point and the particle tracking analysis can be de-coupled from the flow simulation. Based on these considerations, particles were released in the computed flow field before the stenosis and tracked by calculating their location at the next particle tracking time step (Δt_PT) using in-house software. The updated location of the particles at the time k+1 was calculated as: (4) where v and a are the velocity and acceleration vectors from the CFD simulation. Both v and a depend on space and time and should be updated at each time step. Where historical information was lacking (for k = 0, 1), the acceleration was assumed to be zero. Algorithm 1 presents the step by stem algorithm.

Algorithm 1 Post processing particle tracking

Load CFD results for all time steps

Initialize the location x of the tracer particles

Initial the time t = 0

while t < t_end do

Check all elements of the particle location to be in the domain

if no element located in the domain

end procedure

Read the velocities at x and t

Calculate the acceleration

if t = 0 or Δt_PT then

Acceleration is zero a = 0

else

Linear acceleration based on the previous velocities

Update t to t + Δt_PT

Update x using Eq 4

end while

Similar to the sampling phase, we grouped the data points with similar Δt_c. Therefore, although the particles could be injected in different cardiac cycles, we grouped the information of the particles that have identical t_c values. As for the CFD sampling, the quantity Δt_c was also considered as a control parameter for the particle tracking. It should be emphasized that Δt_c is a property of seeding/grouping the particles during the analysis of the results.

The assumption of multiple particles was used because the velocity depends on both space and time, therefore using multiple particles is necessary for generating enough data to train the Gaussian process.

It is worth noting that we initially assumed perfect tracking, i.e., the location of the location of the particles were exactly measured and the positron range was assumed to be zero. However, the positron range in a blood-like environment is in submillimeter scales for ¹⁸F [72]. This range affects the spatial resolution of PEPT. We initially assume the limiting case, where we decay events occur at a sufficient rate that allow an accurate partial position to be determined through averaging. To estimate the impact of uncertainty on our analysis we consider the effect of noise. We have introduced random noise to the location of particles before feeding them to the kriging process. Moreover, this approach provides an estimate of what can be achieved at given levels of uncertainty and provides an estimate of the accuracy upper bound that could be achieved with improved sensors that approached optimal data acquisition.

Kriging

To reconstruct the velocity using the data from a limited number of particles, we used the kriging technique. In kriging, the points closer together have more similar values than points further apart. The values corresponding to the points are related through the semivariogram. For the current study, we used an anisotropic exponential semivariogram with a higher impact on the spatial correlation than the temporal correlation. In kriging, one uses the data value at some special points and calculates the continuous distribution of the parameter. During the training phase, the semivariogram (i.e., the kernel) was calculated based on the sampled data and the values for the hyper-parameters, i.e., sill, range and nugget were calculated. The nugget, i.e., the y-intercept for the semivarogram, was consistently zero for all cases in our analysis. Therefore, the kriged field behaved as an exact interpolator.

Regarding the pulsatile nature of the flow, we used universal kriging that does not assume the constant mean value for the data. The data for training the kriging process was prepared based on the locations and velocities of the sampled points (for the sampling) and the particle locations and the corresponding velocity (for the particle tracking). We transform the 3D coordinate system to local 2D coordinates at the POI. By doing so we transformed (x,y,z,t) into (X,Y,t) to use a 3D kriging process, where x, y and z were the global 3D coordinates for the data and X and Y were the local 2D corresponding coordinates on the POI. The time was not changed in the transformation. Zero velocity was set at the boundary of the POI for all time steps to apply the no-slip condition of the flow at the vessel wall. As the rigid wall was assumed for the CFD domain and the fluid was modelled as a continuous material, no-slip boundary condition was chosen for the study. The S1 Text presents the universal kriging process which was used for reconstructing the normal velocity profile.

Error analysis.

In this study, we employed two concepts of error and uncertainty. The error refers to the difference between the CFD (ground-truth) and the reconstructed velocity profile. The reconstructed profile at each point is space and time is the mean of the value of the values predicted from the Gaussian process. On the other hand, the range that the Gaussian process predicts for a point with no particle or sample data is the quantified uncertainty. The uncertainty of the prediction is calculated during the Gaussian process and reflected in the kriging variance. The uncertainty is not compared to the true values from the CFD simulations.

We used two error measures to evaluate the accuracy of the kriging velocity reconstruction based on the sparse data from particles: the root-mean-square (RMS) error between the reconstructed profile and the true velocity profile from the CFD simulations, and the difference of the peak velocity value from the reconstructed profile and that from the CFD simulation. For the first error, the reconstructed velocities were compared to the CFD results and the RMS effective surface error, e_RMS, was used to quantify accuracy [73]: (5) where e is the absolute error between the reconstructed profile and the CFD profile and A is the area of the POI. The error was normalized with the average velocity at the corresponding timepoint.

For the maximum velocity error, e_max(v), the relative error of the maximum reconstructed velocities was calculated as: (6) where max(v_R) is the maximum value of the reconstructed velocity normal to the POI at a given timepoint and max(v_CFD) is the maximum value of the normal velocity from the CFD simulation at the same timepoint.

For more clarification, Table 1 presents all the errors used in this paper with their explanations.

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Table 1. The errors and their explanations.

https://doi.org/10.1371/journal.pone.0295789.t001

Results

Stenosis detection

Fig 3 shows the maximum velocity and the corresponding position for each particle in the three geometries, with and without stenosis (red and black symbols, respectively).

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Fig 3. The maximum velocity of particles and the coordinates in which the maximum velocity occurred.

Results in a straight tube (a), a bifurcation (b) and an anatomically accurate left coronary artery (c). Each point represents a particle. Points presented with black × correspond to the case without stenosis and the red dots ● correspond to the case with a 50% stenosis. The black triangle shows the location of the stenosis based on particle data in different geometries.

https://doi.org/10.1371/journal.pone.0295789.g003

The cycle time difference between the particle releases was Δt_c = 64 ms and the particles were released in batches of 40.

The axial location of the stenosis was identified by the accumulation of particles with elevated maximum velocity. An illustration of the estimated Cartesian location of the stenosis based on the maximum velocity is presented in the embedded panel for each case.