Multiscale Computational Fluid Dynamics Modeling for Personalized Liver Cancer Radioembolization Dosimetry

Abstract

Yttrium-90 (⁹⁰Y) radioembolization is a minimally invasive procedure increasingly used for advanced liver cancer treatment. In this method, radioactive microspheres are injected into the hepatic arterial bloodstream to target, irradiate, and kill cancer cells. Accurate and precise treatment planning can lead to more efficient and safer treatment by delivering a higher radiation dose to the tumor while minimizing the exposure of the surrounding liver parenchyma. Treatment planning primarily relies on the estimated radiation dose delivered to tissue. However, current methods used to estimate the dose are based on simplified assumptions that make the dosimetry results unreliable. In this work, we present a computational model to predict the radiation dose from the ⁹⁰Y activity in different liver segments to provide a more realistic and personalized dosimetry. Computational fluid dynamics (CFD) simulations were performed in a 3D hepatic arterial tree model segmented from cone-beam CT angiographic data obtained from a patient with hepatocellular carcinoma (HCC). The microsphere trajectories were predicted from the velocity field. ⁹⁰Y dose distribution was then calculated from the volumetric distribution of the microspheres. Two injection locations were considered for the microsphere administration, a lobar and a selective injection. Results showed that 22% and 82% of the microspheres were delivered to the tumor, after each injection, respectively, and the combination of both injections ultimately delivered 49% of the total administered ⁹⁰Y microspheres to the tumor. Results also illustrated the nonhomogeneous distribution of microspheres between liver segments, indicating the importance of developing patient-specific dosimetry methods for effective radioembolization treatment.

1 Introduction

Understanding, the hepatic blood flow dynamics is important to study and optimize the delivery of therapeutic microspheres used in liver cancer embolization treatment. Liver cancer includes hepatocellular carcinoma (HCC) (85–90% of all primary liver cancers), which is the main cause of death in cirrhosis patients [1,2], and metastatic cancer such as intrahepatic bile duct cancer. The incidence rate and mortality of liver and colorectal cancer are continuously increasing in the United States [3]. Based on the cancer stage, localization, and other medical concerns different therapeutic strategies are preferred that include ablation, resection, transplantation, chemoembolization, and radioembolization.

Yttrium-90 (⁹⁰Y) radioembolization is a minimally invasive interventional radiology procedure performed under image guidance. It is increasingly used to treat unresectable liver tumors [4]. Although the hepatic artery provides only about 25% of the liver blood flow (the rest is provided by the portal vein), it plays an important role in therapeutic procedures because it supplies almost 100% of the tumor blood flow [5,6]. Therefore, in radioembolization, ⁹⁰Y microspheres (20–60 μm in diameter) injected in the hepatic artery can selectively deliver a specific amount of radioactivity to the malignant tissue. ⁹⁰Y microspheres emit high energy electrons that interact with tissue and deposit energy into them, which eventually can kill the tumor cells. To determine the optimal injection location as well as the ⁹⁰Y activity to inject, a multidisciplinary team consisting of interventional radiologists, hepatologists, nuclear medicine physicians, and physicists is required for the treatment planning. At this stage, an efficient and accurate ⁹⁰Y dosimetry is crucial to optimize the therapy such that the radiation dose delivered to the tumor is maximized while the exposure of surrounding liver parenchyma is minimized. The absorbed dose depends directly on the ⁹⁰Y activity as well as some of the characteristics of the isotope such as energy and emission type. For example, ⁹⁰Y is a beta emitter with a maximum energy of 2.28 MeV, and the activity of 3 GBq is typically injected to the patient. Current ⁹⁰Y dose calculation methods used in clinics follow the guidelines of the two microsphere manufacturers. These methods, the MIRD schema and the BSA model [7,8], are based on strong simplifications of the ⁹⁰Y physics and do not consider the microsphere transport or the anatomical variations of the hepatic arterial tree between patients. However, these well-known variations [9] affect the radionuclide distribution in the liver. Also, current methods assume that the ⁹⁰Y microspheres are homogeneously distributed in the liver, which is not realistic [7,10]. The partition model [11] improves upon MIRD by separating the tumor, liver, and lungs but its accuracy is also strongly limited by the manual CT segmentation and by the incorrect assumption of a uniform microsphere distribution. These three models are also not consistent in making recommendations for the ⁹⁰Y activity. This makes the current dosimetry unreliable and increases the risk of delivering an insufficient dose to the tumor while exposing the surrounding healthy parenchyma to a high radiation dose.

To develop a method that allows physicians to optimize the efficacy of radioembolization, we proposed a “patient-specific dosimetry” called CFDose that provides a volumetric estimation of the dose to the liver. It is based on three main steps [12] (Fig. 1):

Fig. 1.

Three steps of the CFD-based dosimetry for liver cancer radio embolization

1. Segmentation of the hepatic arterial tree from standard-of-care cone-beam computed tomography (CBCT) scans for each patient,

2. Simulation of hepatic artery hemodynamics and ⁹⁰Y microspheres transport using computational fluid dynamics (CFD),

3. Dose calculation using ⁹⁰Y decay physics simulation.

This work focuses on the patient-specific CFD simulation that can predict the microsphere transport in the blood flow inside the hepatic arterial tree, which is the key part of this three-step approach. The other two steps are briefly presented.

Computational analysis has been used to investigate particle transport in various medical applications [13,14]. Earlier studies have suggested the utility of CFD simulation to analyze the effect of blood flow dynamics on ⁹⁰Y microsphere transport inside simple models of the hepatic artery. Kennedy et al. carried out a computational study in a realistic generic hepatic artery model with four symmetric outlets in steady blood flow conditions and showed that the anatomy and downstream boundary conditions (BCs) affected the particle trajectories [6]. Using the same hepatic artery model, the ⁹⁰Y distribution at each outlet was predicted using a one-way coupled Eulerian–Lagrangian model of particle transport in a pulsatile flow [15,16]. Other studies [17,18] demonstrated that CFD simulations can be used to determine the optimal injection interval and location for tumor targeting using particle release maps (PRM) in simple patient-inspired hepatic artery configurations, limited to 2–4 outlets in the left and right hepatic arteries.

The imposed CFD BCs play an important role in the simulation results accuracy [19] and need to be carefully chosen. Lumped modeling of the vasculature is a method to estimate physiological BCs in a CFD simulation. A lumped parameter network (LPN) is an electrical representation of the circulatory system that can provide the inlet mass flowrate and stagnation pressure at different locations of the CFD model. The LPN components such as resistor, inductor, and capacitor simulate the hemodynamic parameters such as the blood viscous resistance, inertia, and the vascular wall elasticity. Recent studies have utilized LPN to simulate blood circulation in the liver and hepatic artery [20–22]. LPNs also have been used to provide an estimation of the BCs at the hepatic arterial tree outlets [23,24]. However, simplifications such as the symmetric distribution of blood flow inside the arterial tree might make them unrealistic.

This study aims at developing a computational hemodynamics pipeline that integrates with hepatic arterial tree segmentation to provide a base for the ⁹⁰Y decay physics simulation. In this study, a patient-specific hepatic arterial tree with 46 outlets, extracted from a clinical case, were used. The CFD BCs were tuned based on the patients' blood pressure and heart rate. The same injection locations that were used for the patient's radioembolization treatment were selected. After the CFD simulation, the administered dose to different liver segments was calculated from the blood flow to each segment. The pipeline described in this study can be eventually utilized to provide personalized dosimetry in future radioembolization treatment planning.

2 Materials and Methods

In this study, the hepatic arterial tree hemodynamics is investigated to predict the ⁹⁰Y microsphere distribution in the liver. The hepatic arterial tree 3D geometry was first segmented from CBCT scans. A CFD study was then conducted to calculate the microsphere delivery to different segments of the liver. Data acquisition, hepatic artery segmentation, and details of the CFD simulation are discussed in this section. This study was approved by the institutional review board of the University of California Davis.

2.1 Data Acquisition.

Image acquisition was done in the UC Davis Health hospital in Sacramento, CA, during treatment planning for a patient with HCC. CBCT scans were obtained under breath-hold in 6 s with a 198 deg coverage and 0.5 deg angular sampling using a Siemens Artis zeego angiography system. Omnipaque 300 was administered to visualize the arteries. Images were rendered using the Siemens Syngo DynaCT software as a 3D axial volume consisting of 185 images (1024 × 1024 pixels, ∼0.25 mm pixel size, and 1 mm slice thickness). Hepatic segments were labeled according to the Couinaud system [25] and will be referred to as such in the rest of this paper. Both, digitally subtracted angiograms and CBCT scans indicated the presence of a tumor in liver segments 7 and 8 (Figs. 2(a)–2(b)).

Fig. 2.

Patient's hepatic vasculature: (a) 2D digital subtraction angiogram of the right hepatic lobe, (b) coronal view of the 3D cone beam computed tomography showing the catheter tip and contrast inject in the right lobe, (c) CFD computational domain segmented from cone-beam CT scans in coronal, sagittal, and axial views, and (d) mesh features and prism layers used in this study

2.2 Hepatic Artery Segmentation.

The interior surface of the hepatic arterial wall was estimated using a marching cubes algorithm within the open-source vascular modeling toolkit, vmtk in the website link,1 using a gray level of 350 HU from CBCT scans. Extracted surfaces using this algorithm need to be preprocessed before they are used in CFD studies. The following procedure was developed to create the CFD computational domain from these surfaces. First, they were smoothed using Taubin's algorithm. The centerline and maximum inscribed radius of the arteries were calculated from the Voronoi diagram and smoothed using a moving average filter. The arterial wall surface was then reconstructed as a tubular surface around the smoothed centerline trajectory, such that the artery radius at each point along the centerline was equal to the local maximum inscribed radius (after smoothing). The solid model was further inspected and smoothed in open source software ParaView in the website link2 and OpenFlipper in the website link.3 More details are provided in Ref. [26].

Figure 2(c) shows the segmented hepatic arterial tree in coronal, axial, and sagittal views. This arterial tree included the right hepatic artery (RHA) and its daughter branches with 46 segmented outlets feeding liver segments 5–8. The inlet diameter of the RHA was 4.6 mm. The smallest detected outlet diameter was 0.45 mm. The tumor location, liver segments, and arteries feeding each segment were annotated by an interventional radiologist at the UC Davis Health hospital.

2.3 Computational Methods

2.3.1 Mesh Generation.

The computational domain was meshed using tetrahedral elements with regional refinement. Three prism layers were used near arterial walls to locally increase the mesh density and resolve the boundary layer. The thickness of the initial prism layer was half of the adjacent mesh elements and decreased with a ratio of 0.8 to the second and third layers. Figure 2(d) shows the main features of the computational mesh.

Mesh independency tests were carried out with three different mesh sizes detailed in Table 1. The uncertainty due to discretization, excluding the modeling error, was calculated based on the ASME recommendations for CFD studies [27]. Here, the maximum velocity at three different sections was examined while the mesh was progressively refined with a refinement factor of r = 2.29. Table 2 shows the approximate relative error (e_a) and fine-grid convergence (GCI_fine) error at these three locations. These results suggested that the percent difference of the axial velocities was within an acceptable level (i.e., within e_a = 0.63%) at geometric locations, independent of mesh element location, for meshes 1 and 2. Coarser mesh 3 resulted in a larger discretization error. This larger error was due to obtaining a different maximum velocity at a slightly different geometric location when mesh 3 was used compared to the other two mesh sizes. Therefore, mesh 2 was used in the rest of this study to save computational power compared to mesh 1.

Table 1.

Mesh information used for mesh independency study

Parameter	Mesh 1	Mesh 2	Mesh 3
Number of elements	20,168,069	8,784,761	3,838,909
Number of nodes	3,440,075	1,532,979	688,639
Max edge size	1.4 × 10−2 cm	1.8 × 10−2 cm	2.43 × 10−2 cm

Table 2.

Discretization error calculation

φ = maximum velocity in the cross-section	First injection inlet	Main branch that feeds tumor	Main branch that feeds segment 6
r21 = r32	2.29	2.29	2.29
φ1	74.40	29.41	24.04
φ2	73.93	29.27	24.10
φ3	73.88	38.82	28.14
P	2.57	5.07	5.10
φext21	74.46	29.42	24.04
φext32	73.94	29.13	24.04
ea21	0.63%	0.48%	0.24%
eext21	0.08%	0.01%	0.00%
GCIfine21	0.11%	0.01%	0.00%
ea32	0.07%	32.60%	16.78%
eext32	0.01%	0.50%	0.25%
GCIfine32	0.01%	0.62%	0.31%

Note: For parameter definitions, refer to Ref. [27].

2.3.2 Governing Equations.

It is widely accepted that blood can be modeled as a Newtonian fluid in large arteries [28] such as hepatic artery [29,30] (with a diameter of 4.3–12.4 mm [31]), Thus, blood was considered as a Newtonian incompressible fluid with a density and dynamics viscosity of 1.06 g cm⁻³ and 0.04 g cm⁻¹ s⁻¹, respectively. The maximum Reynolds number was below 900, which falls within the laminar flow regime. Navier–Stokes continuity and conservation of momentum equations were solved to calculate the 3D flow field

$$\frac{\partial u_i}{\partial x_i}=0$$ (1)

$$\rho \left(\frac{\partial u_i}{\partial t}+u_j\frac{\partial u_i}{\partial x_j}\right)=-\frac{\partial P}{\partial x_i}+\mu \frac{{\partial }^2{u}_i}{\partial x_j^2}+\rho F_B$$ (2)

where u, P, F_B, ρ, and μ are velocity, pressure, body forces, density, and viscosity, respectively. Subscripts i and j represent the Cartesian tensor notation where the repeated subscripts stand for summation over the three coordinates. The Navier–Stokes equations were solved using the finite element method and discretized by the Galerkin method. Finite element is proven to be a suitable method for cardiovascular simulations [32]. The streamwise upwind Petrov–Galerkin method was used to stabilize the solution [33]. The governing equations were advanced in time by a generalized-alpha method [34]. This is an implicit second-order accurate method which is unconditionally stable. The time steps were selected such that the Courant–Friedrichs–Lewy number was about 1. The simulation was run for six cardiac cycles to ensure a periodically stable state was reached. A residual criterion of 10⁻² was used for all the flow variables [34]. The open-source SimVascular in the website link4 was used to perform the meshing and CFD simulation. The solver has been validated in many peer-reviewed publications including validation of arterial blood flow modeling [35–37]. Data postprocessing was done in ParaView.

2.3.3 Boundary Conditions.

The arterial wall was assumed to be rigid and a no-slip condition was used along them. A pulsatile flowrate waveform with a parabolic velocity profile was prescribed at the inlet (Fig. 3). This waveform was approximated based on the published data from Ref. [15] where the tumor presence in the right lobe increased the blood flow share of RHA by a factor of 1.45 compared to a healthy liver [24]. The inlet flow curve was smoothed by ten Fourier modes to ensure the periodicity. Determining the outlet boundary conditions in an arterial tree with multiple outlets is always challenging since the in vivo calculation of outflow static pressure and flowrate is not feasible. Here, the downstream vasculature between the computational domain and the distal arteries was modeled with an resistor-capacitor-resistor (RCR) circuit analogy LPN at the outlets (Fig. 4, coronal view). Such coupling of the LPN and CFD model saves significant computational time and power as well as accounts for the effect of peripheral flow. In the RCR circuit, R_d, R_p, C, and P_d are the distal and proximal resistances, proximal capacitance, and distal pressure, respectively. The distal pressure was obtained from a total-body LPN model in a closed network based on prior studies [38]. This network was tuned based on the patient's measured heart rate, systolic, and diastolic pressure. A distal pressure of 15 mmHg was used for the outlets feeding the right hepatic lobe, consistent with previous studies [21]. The resistance and capacitance values between the computational domain and the distal arterioles were adjusted to obtain a physiological pressure drop of ∼4.5–9 mmHg between the RHA and outlets throughout a cardiac cycle [39]. Since there was no information available about the relation between distal and proximal resistances in hepatic arteriole outlets, for simplicity, we assumed that R_d = R_p for all 46 outlets. The resistance and capacitance values of the arterial branches feeding each liver segment (Fig. 4, sagittal view) are listed in Table 3. They were then split between the outlets belonging to a given liver segment using the following equations [40]:

$$R_{ij}=\frac{\sum {\sqrt{A_i}}^k}{{\sqrt{A_i}}^k}{R}_j$$ (3)

$$C_{ij}=\frac{A_i}{\sum A_i}{C}_j$$ (4)

Fig. 3.

Flow rate imposed at the RHA inlet

Fig. 4.

Coronal and sagittal views of the CFD model. The outlets to different liver segments (S5-S8) and tumor (T) are shown in the sagittal view. A sample of the RCR circuits coupled with the CFD domain, and the first and second injection locations are shown in the coronal view. In the RCR circuit, R_d, R_p, C, and P_d are the distal and proximal resistances, capacitance, and distal pressure, respectively.

Table 3.

Proximal resistance (R_p), capacitance (C), and distal resistance (R_d) values for outlets feeding different liver segments and tumor

Liver segment	No. of outlets	(Rp + Rd) × 104 (dyne s/cm5)	C × 10−7 (cm5/dyne)
S5	6	9.1405	8.93
S6	10	14.5720	8.93
S7	17	6.8554	8.93
S8	13	4.4129	8.93

where R_i, C_i, and A_i are the resistance, capacitance, and cross-sectional area of outlet i in liver segment j. R_j and C_j are the total resistance and capacitance of the outlets supplying liver segment j. k is Murray's law coefficient. Murray's law coefficient of k = 2 was used in this study based on the diameter of the arterial tree branches [41].

2.4 Injection Location.

The ⁹⁰Y microspheres are injected into the hepatic artery during radioembolization. The injection location varies from one patient to another as determined by the interventional radiologist during the treatment planning. The tumor was located in the right lobe, segments 7 and 8. During treatment planning, the injection of contrast from the proper hepatic artery showed some reflux into the gastroduodenal artery. The injection locations selected by the physician specifically for this patient were identified from digitally subtracted angiogram images on which the catheter is visualized with the assistance of the interventional radiologist. The microspheres were administered through two slow injections (over ∼15 cardiac cycles); one into the RHA and the second one through an artery downstream of the RHA that fed the tumor directly intended to target the lesion more specifically for greater local dose delivery. The injection locations are shown in Fig. 4.

2.5 Dosimetry.

The ⁹⁰Y radioactive dose distribution in the liver is directly proportional to the number of injected microspheres and their volumetric distribution in the hepatic artery. Each ⁹⁰Y microsphere deposits a given amount of energy per tissue volume determined by the ⁹⁰Y decay physics and the microsphere unit activity (typically 2500 Bq for glass microspheres used to treat HCC). All simulations presented in this work assumed that the microspheres were transported with the blood flow, therefore the number of microspheres reaching each outlet was proportional to the cumulative blood flow at the outlet and was obtained from the CFD simulations. The dosimetry calculations were then done with custom code in Matlab (R2018b, The MathWorks, Inc., Natick, MA). The microsphere 3D distribution was first computed by arranging the microspheres at each outlet (typically 30,000 for the selective injection) in a cylinder. The diameter and axis of that cylinder were given by the vessel diameter and orientation [12]. Finally, the 3D microsphere distribution was histogrammed using cubic voxels of 3.6 mm and convolved with a dose point kernel [42] to calculate the dose distribution in Gy (energy deposited in tissue). The dose kernel for ⁹⁰Y was calculated with the same voxel size (corresponding to the voxel size of the positron emission tomography scanner used to image the patient, not shown here) with the open-source Monte Carlo simulation toolkit GATE [43] for a ⁹⁰Y point source in water. Because ⁹⁰Y emitted electrons can travel up to 11 mm, considering the actual spatial profile of the ⁹⁰Y dose (dose point kernel method) instead of assuming that all the dose is deposited at the location of the microsphere (local dose deposition) provides a more accurate 3D dose distribution.

3 Results and Discussion

3.1 Velocity Distribution and Secondary Flow.

Understanding the hepatic arterial tree hemodynamics plays an important role in determining the ⁹⁰Y microsphere distribution in the liver. Axial velocity profiles were investigated in different branches of the hepatic arterial tree. Secondary flows (perpendicular to the artery's axis) were also calculated since they might affect the microspheres' trajectory as well. The general velocity field characteristics remained similar between the fifth and sixth cycles of the unsteady simulation. The fifth cycle results are thus presented in the rest of this study. The nonlinear residual was ∼10⁻⁵ in the last cycle of the simulation which was three order of magnitude smaller than the residual criteria (see Sec. 2.3.2). Figure 5 shows the velocity distribution and secondary flow streamlines at the RHA for three different time instants of t* = 0, 0.15 (i.e., max flowrate at the inlet), and 0.5 where t* is the dimensionless time vector normalized by the cardiac cycle duration. In the RHA, right after the inlet, the velocity distribution was changed from a parabolic profile such that the maximum velocity was moved from the artery's centerline to a location closer to the arterial wall.

Fig. 5.

The axial velocity distribution at the RHA inlet. The secondary flow streamlines are also plotted in each cross section. The velocity profiles are shown at t* = 0, 0.15, and 0.5 for 0 < u < 60 cm/s.

The general behavior of the flow field in liver segments 6–8 was also investigated (see Supplemental Material on the ASME Digital Collection). Results demonstrated different flow patterns in the arterial branches feeding each liver segment. The blood flow had a higher axial velocity in segment 8 branches compared to the other two segments. The variations in velocity profile and magnitude in different liver segments indicated the asymmetric distribution of blood flow in the liver.

There were two main daughter branches in segment 7 supplying blood to the tumor. The stems of these branches are labeled in Fig. 14 (S7) which is available in the Supplemental Materials on the ASME Digital Collection. The velocity distribution in three cross section in each of these daughter branches is shown in Fig. 6. In the first branch, section 11 bifurcates to sections 12 and 13 (Note that sections 1–10 are shown in Supplemental Material on the ASME Digital Collection.). The axial velocity had almost a parabolic profile in section 11 with a maximum velocity of >60 cm/s at t* = 0.15. After the bifurcation, the velocity profiles deviated from the parabolic shape and the mean axial velocity dropped from 29.5 cm/s in section 11 to 9.5 and 13.1 cm/s in sections 12 and 13, respectively. Similarly, the maximum velocity decreased from 67.2 cm/s to 61.5 and 40.4 cm/s. In the second daughter branch feeding the tumor, the blood flow was slower with less variation in the successive vessel generations (sections 14–16). In this branch, sections 15 and 16 are the major daughters of sections 14 and 15, respectively. The mean axial velocity changed from 5.4 cm/s in section 14 to 5.1 and 2.5 cm/s in sections 15 and 16 at t* = 0.15.

Fig. 6.

The axial velocity and secondary flow streamlines in two tumor-feeding branches of segment 7. The sagittal view of these branches is shown on the top left corner. The velocity profiles are shown at t* = 0, 0.15, and 0.5 for 0 < u < 60 cm/s.

The axial velocity changes in the tumor-feeding branches can also be seen in Fig. 7 where the velocity isosurfaces are plotted at the maximum inlet flowrate (t* = 0.15). These isosurfaces show that the velocity is lowered significantly (∼70–80 cm/s) from the RHA to the tumor outlets. However, some regions with high axial velocity (50–80 cm/s) can be still seen close to the tumor-feeding branches. Higher axial velocity in a branch results in higher blood flow and consequently delivery of more ⁹⁰Y microspheres to that region. Therefore, the isosurfaces illustrate how the right region of the tumor received more microspheres than the left region due to a higher blood flowrate.

Fig. 7.

(a) Axial view of the hepatic arterial tree and tumor-feeding branches (highlighted in light brown). (b) Velocity magnitude isosurfaces (10–50 cm/s) in the tumor-feeding branches.

The secondary flows appeared as two nonsymmetric counter-rotating vortices in some branches, such as cross section 6, 10, and 12, similar to Dean flow patterns for laminar flow in a curved pipe. These vortices were not symmetric, as expected in an ideal Dean flow because of the irregular curvatures of the hepatic arterial tree anatomy.

The velocity distribution in most branches showed a different pattern than Poiseuille flow. For example, the velocity profiles were asymmetric with a maximum value occurring close to the arterial walls rather than on the artery's center. This might be due to factors such as the lateral distance between the centerline of the mother and daughter branches as well as the filet structure of the bifurcation and downstream flow field conditions.

In most branches, the axial blood velocity decreased with the arterial generations. For example, the maximum blood velocity at t* = 0.15 dropped from > 60 cm/s at the RHA cross section (Fig. 5) to ∼35 and 10 cm/s in cross section 2 and 4 (Fig. 14 available in the Supplemental Materials on the ASME Digital Collection), respectively. However, there were a few exceptions where the maximum axial velocity was > 60 cm/s in the downstream branches such as cross section 11 and 12 in the tumor-feeding branches (Fig. 6).

3.2 Velocity Streamlines and Microsphere Trajectories.

With the size of blood vessels segmented from the CBCT images, and the microspheres and blood properties, the Stokes number St was calculated using the following equation:

$$\mathrm{St}=\frac{{\rho }_p{d}_p^{2}u}{18\mu d}$$ (5)

where the subscript p stands for microsphere particles, and d, u, ρ, and μ are diameter, velocity, density, and viscosity, respectively. Here, the diameter and density of the microspheres assumed to be 20 μm and 3600 kg m⁻³, respectively [19]. St was calculated at the RHA inlet (d = 4.6 mm) for a flow velocity of 50 cm/s. St was ∼ 0.02, which indicated that the microspheres tended to closely follow the blood flow streamlines in the absence of collision forces. At higher generations of the arterial tree with smaller d, the flow velocity was slower than the inlet. St thus remained far less than unity with the same order of magnitude as the one calculated at the RHA inlet. Therefore, we assumed that the particle deposition along the wall and drift from the blood velocity were negligible at small St numbers as both microspheres and blood density have the same order of magnitude. Therefore, the velocity streamlines were used to estimate the microsphere trajectories and volumetric distribution along the hepatic arterial tree. In the absence of catheter modeling in this study, we assumed that the microspheres are spatially injected homogeneously into the entire cross section of the artery at each injection location. Therefore, all the streamlines crossing this location were selected as the microsphere trajectories. Figure 8(a) shows the flow streamlines at the time of maximum inlet flowrate (t* = 0.15), with a color scale based on the axial velocity magnitude. Results showed no flow reversal in the entire arterial tree. The streamlines in the tumor feeding branches are shown in Fig. 8(b). While the flow streamlines were smooth in some branches (inset 2), insets 1 and 3 demonstrate the formation of recirculation regions after other bifurcations similar to the ones reported in previous work in a simplified hepatic artery [15]. Recirculation mostly occurred in the regions with higher vorticity shown with dotted circles in Fig. 8(b).

Fig. 8.

Velocity streamlines in the (a) hepatic arterial tree and (b) tumor-feeding branches as shown in Fig. 8(a). The streamlines are colored based on the velocity magnitude. Regions with high vorticity are shown with dotted circles (please see the online version for colorful figure).

3.3 Flow Rate at Arterial Outlets.

As described in Sec. 3.2, the microspheres follow the blood flow streamlines. This was used to estimate the number of ⁹⁰Y microspheres delivered to each arterial outlet from the blood flow into that outlet. Figures 9(a) and 9(b) show the flowrate to the tumor and other liver segments as a percentage of the inlet flowrate during the first and second injections, respectively. Based on the known activity used in each injection, the number of microspheres for the first (i.e., lobar) and second (i.e., selective) injections was 654,800 and 547,200, respectively. The selective injection targeted a specific liver segment containing the tumor. Therefore, charts in Fig. 9(b) only included segment 7 and tumor. During the second injection, the flowrate to segment 7 and tumor outlets were 18% and 82% of the inlet flowrate (i.e., the flowrate at the injection cross section), respectively, and was not affected by the blood velocity variations through the cardiac cycle. On the other hand, although the flowrate to tumor outlets remained constant during the first injection as shown in Fig. 9(a) (∼22%), the blood supply to other segments slightly changed over the cardiac cycle due to changes in the inlet flowrate with time (Fig. 3). This reveals the effect of downstream conditions such as the location and arrangement of the bifurcations on the blood flow distribution. In this study, we assumed that the downstream vascular resistance did not change from the first to the second injection because the ⁹⁰Y microspheres do not typically have a strong embolic effect due to their size. However, future work can investigate the effect of injected microspheres during the first injection on the partial embolization of the tumor-feeding branches and downstream resistance alteration in the subsequent injections.

Fig. 9.

Flow rate to the tumor (T) and other liver segments (S5–S8) as a percentage of the inlet flowrate during the (a) first and (b) second injections. The first injection was a general lobar injection, while the second injection was more selective. (c) Microspheres delivered to different liver segments after both injections.

The number of ⁹⁰Y microspheres delivered to each liver segment was then calculated from the cumulative blood flow to the group of outlets feeding that liver segment. Figure 9(c) demonstrates that ∼49% of the injected microspheres were delivered to the tumor after the two injections. Other liver segments received from 7% to 22% of microspheres mostly through the first injection. The results indicate that a specific injection would be more efficient to target the tumors located in a particular part of the hepatic right lobe. In this clinical case, the reason for a lobar injection was to shrink the right lobe for subsequent resection.

Charts in Fig. 9 are shown for R_d = R_p. Similar flow distribution with minimal changes was observed when the ratio R_d/R_p varies from 1 to 3. Our results showed that R_tot had a larger effect on the flow distribution than the ratio R_d/R_p.

The blood flow in different vessel generations was plotted as a percentage of total inlet flow in Fig. 10. Arterial trees and airways trees share similar traits. Thus, the generation number was determined similar to a previous study of airways trees [44]. Here, generation zero was assigned to the RHA. A schematic of the generation numbering method is shown in Fig. 10. The flow ratio dropped by 39% on average from generation 1 to generation 5 in the right hepatic lobe (Fig. 10). The flow ratio then remained constant at 2.89 ± 0.06% (mean ± SEM) of the inlet flow, without significant changes after generation 9. Results also showed the flow ratio was different in different liver segments, indicating the dependency of the flow ratio to the downstream conditions including distal resistance, number of branches, and bifurcation characteristics. For example, the flow ratio was 1.5–3.1 times higher in S8 than tumor-feeding branches. These flow ratio variations are the main cause of the nonhomogeneous distribution of the microspheres in different liver segments.

Fig. 10.

Right lobe blood flow ratio variation with the vessel generation. The markers and error bars show the mean and standard error of mean (SEM), respectively.

All mother branches split into two daughter branches at each bifurcation in this geometry, no trifurcations were observed. The bar plots in Fig. 11 show the flow distribution among the daughter branches at each bifurcation (Q₁ and Q₂) in the tumor-feeding tree, i.e., Q₁/(Q₁ + Q₂) and Q₂/(Q₁ + Q₂) where Q₁ and Q₂ are the flow in daughter 1 and 2, respectively. Daughter branches were numbered as daughters 1 and 2 such that Q₁/(Q₁ + Q₂) increases by the arbitrary bifurcation number shown in Fig. 11. Results indicated that the flow did not split equally between the daughter branches, which suggested the limitation of the assumption of symmetric and homogeneous flow distribution in the hepatic arterial tree made in previous studies [24]. These results were also compared with Poiseuille's law. In Poiseuille's flow, the volumetric flowrate is proportional to the second power of the pipe cross-sectional area in the following equation:

$$Q=\frac{{\Delta PA}^2}{8\pi \mu L}$$ (6)

Fig. 11.

Blood flow distribution among two daughter branches of bifurcations in the tumor-feeding tree. The red and gray bars represent the flow ratios Q₁/(Q₁ + Q₂) and Q₂/(Q₁ + Q₂), respectively. The area ratio A₁²/(A₁² + A₂²) from Poiseuille's law shown in green does not present any correlation with the flow ratios.

where ΔP, L, and A are the pressure drop, length, and cross-sectional area of the pipe, respectively. The green line in Fig. 11 shows the ratio between the area squared of the daughter branches, i.e., A₁²/(A₁² + A₂²) where A₁ and A₂ are the cross-sectional areas of daughter branches 1 and 2, respectively. The spikes in A₁²/(A₁² + A₂²) were due to the fact that daughter branch numbering was not such that daughter 1 has always a smaller cross section than daughter 2. Results did not illustrate any meaningful relation between Q₁/(Q₁ + Q₂) and A₁²/(A₁² + A₂²) as derived from Poiseuille's law. Although employing Poiseuille's law can reduce the high computational cost of blood flow in the hepatic arterial tree, nonrealistic assumptions such as constant cross-sectional area throughout each daughter branch make the blood flow estimations inaccurate. This indicates the importance of carefully considering the anatomy of every patient and carrying out accurate CFD simulations rather than simplifications such as blood flow distribution based on Poiseuille's law assumed in previous studies [45]. This also justifies our choice of using a multiscale model that combines CFD in an image-segmented 3D domain with LPNs for smaller downstream blood vessels rather than a fully LPN model from the RHA inlet to the outlets. A fully LPN model cannot also properly account for local flow recirculation (see Fig. 8) and some geometry variables such as varying bifurcation angles which are likely to exist in a complex geometry such as the one included in this study.

3.4 Application to ⁹⁰Y Microsphere Dose Prediction.

Figure 12(a) shows the heterogeneous 3D distribution of the microspheres in the vasculature. Figure 12(b) shows the dose point kernel and illustrates the broad distribution of the ⁹⁰Y dose deposition in space due to the average and maximum travel lengths of 2.5 mm and 11 mm, respectively. Figure 12(c) shows a projection of the 3D dose distribution, illustrating the heterogeneous dose distribution and the spread of the dose around the microsphere clusters due to the ⁹⁰Y dose kernel distribution. This spread is also increased by the downstream travel of the microspheres farther than what can be identified on the segmented structure and modeled in the CFD domain. In future work, we will devise methods to model a realistic microsphere distribution in smaller arterioles [12]. By accounting for these heterogeneities, this 3D dosimetry can ultimately be used to predict the dose delivered to different parts of the liver with greater accuracy and precision than global models such as the partition model.

Fig. 12.

Dosimetry from microsphere distribution. The microsphere 3D distribution directly gives the activity distribution (left), which in turn is convolved with the dose kernel (middle) to compute the dose distribution (right).

3.5 Application to Optimal Injection Location Selection.

The PRMs were estimated for the first lobar injection. A total number of 107,000 streamlines were tracked from the inlet to the arterial outlets to determine the regions in the injection cross section that supply blood to each liver segment (Fig. 13). These transient PRMs can guide targeting the tumor while avoiding the delivery of ⁹⁰Y microspheres to other healthy tissues including healthy liver segments. For example, the injection region corresponding to the catheter tip location in the cross section of the artery can be selected such that the number of microspheres delivered to segments 5 and 6 is minimized (see the example catheter tip illustrated by the crosshairs in Fig. 13). It is worth noting that these PRMs were actually “estimated” particle release maps since they were estimated from the flow streamlines and were not directly calculated from a particle transport model. From here on, they were however called PRMs for simplicity.

Fig. 13.

Transient ⁹⁰Y microspheres PRM for different liver segments (S5-S8) and tumor (T), showing different patterns throughout the cardiac cycle. The PRMs calculated assuming that the microspheres were injected homogeneously into the arterial cross section. The black crosshairs show an example of the catheter tip location.

The PRMs show an organized pattern at different time instants. However, they were not symmetric as observed in other studies carried out in simplified hepatic arterial trees [18]. This might be due to the irregular and asymmetric anatomy of the patient hepatic arterial trees. The pattern of the particle maps remained rather constant for 0.45 < t* < 1 (Fig. 13). However, the particle maps varied more drastically for 0 < t* < 0.3, corresponding to a higher inlet flowrate regime. Drastic changes in the PRMs make it more difficult to target specific segments by placing the catheter tip in a specific region. For instance, the catheter tip position shown in Fig. 13, would spare liver segments 5 and 6 at t* = 0.3 while delivering some microspheres to segment 6 at other time instants. Similar analyses can help determine the optimal injection region for each patient based on their specific anatomy and tumor location. Despite, the PRMs shown in this study were calculated in the absence of a catheter in the hepatic artery, they can inform on the initial injection region and time as suggested by previous studies [17]. With current catheter designs, it is hard to precisely position the catheter in clinical practice. However, there are recent efforts [17,46] to design new catheters capable of implementing these ideas (e.g., controlling the injection time and location). The catheter presence can also cause local perturbations in the flow field and affect the flow streamlines. In addition, the effect of additional fluid mass injected and injection fluid velocity was not considered in this study. Due to the slow injection (usually during 12–15 cardiac cycles), the injection velocity might have a smaller effect than the catheter presence on the flow field. While this study provides a first look at how microspheres travel in the hepatic arterial tree under these conditions, the PRMs should be further affected by accounting for the catheter presence and additional fluid mass injected in future CFD simulations.

A limitation of this study was the assumption that blood behaves as a Newtonian fluid inside the hepatic artery and that all the arterial walls are rigid. They might affect the CFD simulation results and consequently the final dosimetry predictions. Although the literature suggests the validity of these assumptions in large arteries, they need to be verified for the hepatic arterial tree in future in vivo experiments. Furthermore, in this study, we employed an inlet flowrate profile based on previous studies. In future work, the flowrate and velocity profile shape can be determined using ultrasound imaging for each patient to achieve more personalized dosimetry. While CFD studies provide a first estimation of ⁹⁰Y microspheres trajectory inside the hepatic arterial tree, a fluid–structure interaction study might give more insight on interactions between the microspheres in motion, the surrounding blood flow, and hepatic arterial wall, to study the potential effect on the final ⁹⁰Y microsphere distribution and dosimetry. An accurate patient-specific fluid–structure interaction simulation would however require the material properties of the arterial walls for each patient to be known. We provided a proof of concept of our CFD based dosimetry in this study. The next step is to validate our workflow as well as our assumptions in a relevant animal model.

4 Conclusion

The unsteady flow in a patient-specific hepatic arterial tree was numerically studied using CFD, as a part of a new dosimetry method for liver cancer radioembolization. The number of microspheres delivered to different liver segments was calculated for different injection locations based on clinical data. Results showed the heterogeneous distribution of ⁹⁰Y microspheres in liver segments, indicating the importance of patient-specific CFD simulations for personalized dosimetry. Compared to current dosimetry methods, personalized dosimetry has the potential to maximize the dose delivered to the tumor while minimizing the number of microspheres deposited in other healthy tissues.

Supplementary Material

Supplementary Material

Supplementary PDF

Funding Data

• The National Institutes of Health (NIH) grants NCI (CCSG) (Grant Nos. P30 CA093373, R35 CA197608, and R21 CA237686; Funder ID: 10.13039/100000002).

Nomenclature

BC =

boundary condition

CBCT =

cone beam computed tomography

HCC =

hepatocellular carcinoma

LPN =

lumped parameter network

PRM =

particle release map

Re =

Reynolds number

RHA =

right hepatic artery

S =

liver segment

T =

tumor

⁹⁰Y =

Yttrium-90