A dynamic blood flow model to compute absorbed dose to circulating blood and lymphocytes in liver external beam radiotherapy

Abstract

We have developed a novel 4D dynamic liver blood flow model, capable of accurate dose estimation to circulating blood cells during liver-directed external beam radiotherapy, accounting for blood recirculation and radiation delivery time structure. Adult male and adult female liver computational phantoms with detailed vascular trees were developed to include the hepatic arterial, hepatic portal venous, and hepatic venous trees. A discrete time Markov Chain approach was applied to determine the spatiotemporal distribution of 10⁵ blood particles (BP) in the human body based on reference values for cardiac output and organ blood volumes. For BPs entering the liver, an explicit Monte Carlo simulation was implemented to track their propagation along ~2000 distinct vascular pathways through the liver. The model tracks accumulated absorbed dose from time-dependent radiation fields with a 0.1s time resolution. The computational model was then evaluated for 3 male and 3 female patients receiving photon (VMAT and IMRT) and proton (passive SOBP and active PBS) treatments. The dosimetric impact of treatment modality, delivery time, and fractionation on circulating blood cells was investigated and quantified using the mean dose (𝜇_dose,,b), V_>0Gy, V_>0.125Gy, and D_2%. Average reductions in 𝜇_dose,b, V_>0Gy, V_>0.125Gy and D_2% of 45%, 6%, 53%, 19% respectively, were observed for proton treatments as compared to photon treatments. Our simulation also showed that V_>0Gy, V_>0.125Gy, and D_2% were highly sensitive to the beam-on time. Both V_>0Gy and V_>0.125Gy increased with beam-on time, whereas D_2% decreased with increasing beam-on time, demonstrating the tradeoff between low dose to a large fraction of blood cells and high dose to a small fraction of blood cells. Consequently, proton treatments are not necessarily advantageous in terms of dose to the blood simply based on integral dose considerations. Instead, both integral dose and beam-on time can substantially impact relevant dosimetric indices.

1 |. INTRODUCTION

Radiation causes lymphopenia, a rapid depletion of lymphocytes in circulating blood following the onset of radiotherapy. Radiation-induced lymphopenia has been associated with inferior clinical outcomes in patients with various cancer types including high-grade glioma (Grossman et al., 2015; Mendez et al., 2016; Rudra et al., 2018), esophageal cancer (Fang et al., 2017; Shiraishi et al., 2018), non-small cell lung cancer (Campian et al., 2013; Tang et al., 2014; Cho et al., 2016; Ellsworth et al., 2019) and pancreatic cancer (Wild et al., 2015, 2016). For patients with hepatocellular carcinoma (HCC), the most common primary liver malignancy (Balogh et al., 2016), recent studies have shown a decline of lymphocyte counts during radiation therapy, which correlates with inferior overall survival rate (Byun, Kim, Park, et al., 2019; Zhang et al., 2019). In addition, increased lymphocyte counts were reportedly associated with increased response and survival in immunotherapy (Pardee and Butterfield, 2012). As lymphocytes play a vital role in the effectiveness of immunotherapy in HCC patients, there is an emerging interest on preserving functioning lymphocytes in systemic circulation after radiation therapy (Byun, Kim, Yoon, et al., 2019). Despite the importance of lymphocytes, the dose received by circulating lymphocytes during radiotherapy is still poorly understood. Heylmann et al. found that apoptosis occurred in a significant fraction of T and B cells after only 0.125Gy in an in vitro study (Heylmann et al., 2021). Therefore, accurately quantifying dose to circulating lymphocytes during fractionated external beam radiation therapy (EBRT) is of clinical interest.

Aiming at immunity-sparing radiation therapy, previous studies have tried to estimate the dose to circulating lymphocytes. Jin et al. developed a framework based on differential equations to model radiation dose to circulating blood by dividing the total blood volume into 6 sub-volumes, including the upper body, aorta, non-abdomen and inferior vena cava, liver, kidney, bowels, other abdominal organs (Jin et al., 2020). While this work serves as an important first step in dose estimation to circulating blood, the model over-simplifies blood flow in the body and assumes an irradiation time equal to the blood circulation time, which then deviates from realistic treatment time course. Basler et al. improved the technique by applying a statistical approach to estimate liver dose-volume histograms (DVHs) and the dose to circulating blood for VMAT beam delivery (2018). The effects of dose rate, fractionation scheme and irradiation volume on dose to circulating blood cells were also investigated. Both studies considered the liver as a uniform entity without modeling its internal vasculature nor accounting for the three-dimensional dose distribution. Whole-body hemodynamics, re-entry of circulating blood, and the time structure of the radiation dose delivery were also not considered.

A recent simulation study by Hammi et al. has proposed an explicit four-dimensional (4D) blood flow model to estimated dose to the circulating blood during radiation therapy for intracranial targets (Hammi et al., 2020). The major cerebral vasculature was constructed based on segmented MRI data, whereas the finer blood vessels were abstracted to cover the entire brain volume. Individual blood particles (BP) representing a subset of the circulating blood cells were explicitly tracked using time-step simulations through the cerebral volume and into and out of the radiation fields. This study considered re-entry of blood to the brain and interactions between the blood particles and the radiation fields to deduce dose to circulating blood cells during brain EBRT.

The blood vasculature of the liver is vastly different from that in the brain. Compared to the brain, the liver contains 10 times more blood volume making accurate modeling of vascular elements even more important (ICRP, 2002). The aim of this work was the development of a novel 4D dynamic blood flow model for the liver, capable of estimating radiation absorbed dose to circulating blood cells during the course of fractionated EBRT.

2 |. METHODS AND MATERIAL

2.1. Liver vasculature model

Polygon-mesh models of the livers from within the whole-body ICRP reference adult male (AM) and reference adult female (AF) computational phantoms were extracted (Figures 1A and 1B). As noted in the source document – ICRP Publication 145 – the total liver volumes were 2228 ml for the AM model, and 1710 ml in the AF model, both inclusive of in-situ blood content (Kim et al., 2020). As the vasculature in each liver segment is independent in humans, AF/AM reference livers were further partitioned into eight independent functional segments based on the Couinaud classification (Figure 1C) (Strasberg et al., 2000). Reported values of the percentage of total liver volume based on CT studies of a cohort of healthy liver patients were then used as the target parameters to perform the AM/AF liver segmentation. Major blood vessels including the hepatic proper artery (HA), portal vein (PV), and the right, middle, and left hepatic veins (HV) were constructed based on the surgical anatomy 3D model of the liver as developed at Emory University¹. An in-house vessel generation algorithm using the Constrained Constructive Optimization method was developed to create three detailed vascular trees, including the HA tree (Figure 1D), PV tree (Figure 1E) and HV tree (Figure 1F) (Correa-Alfonso et al., 2021).

Figure 1.

Development of the vasculature model inside the liver of the adult female ICRP mesh-type phantom (A). The adult female liver (B) is extracted and divided into 8 Couinaud segments (C), represented by different colors. The major vessel vasculature (cylinders) along with the generated hepatic arterial vessel (D), hepatic portal venous (E) and hepatic venous (F) trees are added to the adult female liver (G).

Hemodynamic and geometric parameters of the major vessels were used as inputs for hepatic vascular tree construction. Conservation of blood is considered in the algorithm; Poiseuille’s and Murray’s laws (Murray, 1926) are thus applied at each bifurcation to obtain the hemodynamic and geometrical parameters that characterize each vessel within the tree. Pressure, blood flow, and radius of vessels in the detailed vascular trees are updated each time a new vessel is created and connected to the optimal bifurcation site. The vessel generation algorithm was developed using Grasshopper, a graphical algorithm editor integrated into the computational modeling software Rhinoceros 6.0 (Robert McNeel & Associates, Seattle, WA). More details on the vascular tree model are documented in Correa-Alfonso et al. (2021)

Detailed vasculature trees were developed separately for the ICRP AM and AF reference livers, as they differ in both shape and size (Figure 2). Each vessel tree contains 1987 vessel branches to guarantee even overall blood supply across the entire liver. The modeled blood vessels cover 15% and 13% of the blood volume in the AF and AM liver respectively. Blood vessels were modeled as cylinders of various radii, with a radius of 0.1 mm for the smallest blood vessel and a radius of 13 mm for the inferior vena cava.

Figure 2.

The adult male and female livers both contain three vascular trees: hepatic arterial (red), portal venous (green) and hepatic venous (blue) trees. The thicker lines represent the centerlines of the main vessels; finer lines represent the centerlines of vessel branches.

2.2. Blood flow simulation

2.2.1. Whole body blood flow

A whole-body blood flow network based on the cardiac output, blood volume, and the flow rates from ICRP Publication 89 was created to produce the spatiotemporal distribution of BPs in 28 organs across the entire body (ICRP, 2002). BPs are defined as finite elements of blood cells and the smallest unit volume of blood considered in the simulation. Using a discrete-time Markov process, a sequence of organ indices (R_i,0, R_i,1, R_i,2, …,R_{i, K}) is simulated for each BP, where i ∈ {0, 1, 2, …, N−1} for N organs and K time-steps. Each sequence represents the circulation of a BP in the human body. The organ index of the first step R_i,0 is randomly chosen proportional to the ICRP reference organ volume. The spatiotemporal information of a BP depends on two variables - the organ index R and the resident time τ in organ R. When a BP transitions to the next organ, R is updated and τ is set to zero. If a BP stays in the same organ, the resident time is increased by one time-step. A detailed description of this simulation model was recently published by Shin et al. (Shin et al., 2021). For this study, spatiotemporal BP distributions of 10⁵ BPs with a 1s time resolution were generated for both the ICRP adult AM and AF for a 10-minute period. The blood volumes for the AM and AF were set to 5.3 L and 3.9 L with cardiac outputs of 6.5L/min and 5.9L/min, respectively. The resident time for BPs in the liver was set to 19.2 s for the AM liver and 14.4 s for the AF liver to match the reference hemodynamical values given in ICRP Publication 89 (ICRP, 2002).

2.2.2. Explicit tracking of blood flow in the liver vasculature

Using the simulated spatiotemporal distribution of 10⁵ BPs, each individual BP entering or re-entering the liver was explicitly tracked. For each BP entering the liver at time point t_i, the location of the BP at t_{i − 1} was assessed. Incoming BPs from the aorta were sent through the HA tree, whereas BPs from the pancreas, spleen, stomach, esophagus, or small intestine were sent through the PV tree. All BPs exit the liver through the HV tree (see Figure 3). This method mimics the physiological blood supply to the liver, where the HV supplies ~ 75% of the blood to the liver and carries venous blood drained from the spleen, gastrointestinal tract, and its associated organs. The HA supplies arterial blood and accounts for the remainder of the liver blood flow. A simulation was implemented to track the propagation of individual BPs through the vascular trees. The trajectory of each BP was generated in real time. At each vessel branching point of the HA and PV trees, BPs chose a vessel branch probabilistically, with the probability of choosing either vessel branch proportional to the squared radius of the vessel. This process was repeated until the BPs reached a terminal vessel from the HA and PV trees. Subsequently, BPs migrated to the closest terminal vessel on the HV tree and were then routed to the inferior vena cava following the vascular path of the HV tree. During the simulation, the spatial coordinates (x,y,z) of each BP was recorded every 0.1s.

Figure 3.

Schematic of the blood flow pathway from the heart to the liver

2.3. Patient specific blood DVHs

2.3.1. Patient cohort

Six patients (three male and three female) with HCC were selected covering a range of target volumes, target locations, and prescription doses (Table 1). All patients had complete livers and were treated with external beam radiotherapy. The study was approved by our institution’s internal review board.

Table 1.

Target specifications and beam delivery parameters of VMAT, IMRT, SOBP and PBS for each patient.

	Target volume (ml)	Organ volume (ml)	Target location	Prescription dose (Gy-RBE)	Number of fields				Beam-on time (s)
					VMAT	IMRT	SOBP	PBS	VMAT	IMRT	SOBP	PBS
AM P1	249	2572	Central	52.5	1	6	2	2	60	120	90	255
AM P2	603	2757	Central	42.0	1	6	4	3	60	120	180	375
AM P3	140	1481	Peripheral	67.5	1	6	2	2	60	120	90	255
AF P1	557	3980	Central	37.5	1	7	2	2	60	140	90	255
AF P2	141	2474	Peripheral	45.0	1	6	2	2	60	120	90	255
AF P3	194	1505	Central	45.0	1	6	2	2	60	120	90	255

Treatment plans from four treatment modalities were generated for each patient by an experienced treatment planner: volumetric-modulated arc therapy (VMAT), intensity-modulated radiation therapy (IMRT), passive scattered spread-out Bragg peak (SOBP) and active pencil beam scanning (PBS) proton therapy (Table 1). All patients were treated to the prescription dose in 15 fractions. The VMAT plan consisted of a single arc and was delivered in 60 s at maximum gantry rotation speed. IMRT plans were composed of 6–7 fields, where the beam-on time (BOT) for each field was estimated as 20 s, with 20 s break time between fields for gantry rotation. The majority of the proton plans had 2 fields. Each field in the SOBP plan was delivered in 45 s. The BOT of Field 1 and Field 2 in the PBS plan were 120 s and 135 s, respectively. The break time between proton fields were set as 5 min to account for the wait time before the beam would become available. For patients with total treatment time exceeding 10 min, the blood flow simulation was repeated to generate the BP paths for another 10 min. Since BP paths through the liver vascular trees are generated in real time probabilistically, the paths likely vary each time the simulation is run.

2.3.2. Image registration

To calculate patient specific DVHs, the dose distributions and reference livers were aligned using deformable image registration. The mesh format AM/AF reference livers were first voxelized using an algorithm developed in-house. Subsequently, patient CT images were registered to the voxelized reference livers using contour-based deformable registration in MIMVista (MIM Software, Cleveland, USA). The resulting deformation vector fields were used to deform the patients’ dose distributions from the treatment plans. The recorded spatial coordinates (x,y,z) of each simulated BP were also mapped onto the voxelized reference livers. As a result, the patient CT, treatment dose distributions and BP paths were all aligned with the reference liver.

2.3.3. Time-dependent blood DVH

Each treatment fraction is composed of multiple fields which require certain amount of time to deliver, with break times in between to prepare for the next field. Given the registered dose fields, BOTs and number of fractions, the dose rate was computed for each voxel along the BP path. The accumulated dose to a BP after one fraction was computed by integrating over all time steps k along the BP path (Eq.1).

$$d_i=\sum _1^k{\dot{d}}_k(\overrightarrow{x},t)\mathrm{\Delta }t$$ (1)

where ${\dot{d}}_k(\overrightarrow{x},t)$ represents the dose rate at $\overrightarrow{x}$ at time t, Δt represents the time step (set to 0.1s). The dose over multiple BPs were represented by a one-dimensional dose array D_j = {d₁, d₂, …,d_N}, where d_i was the dose to the i-th BP and N was the total number of BPs. d_i = 0 for BPs that did not cross any dose fields. Blood DVHs (bDVH) after a single fraction were then computed.

To compute bDVH for multiple fractions, the same radiation fields were delivered at different starting times with different sets of BPs. For example, if the first field is delivered at t = 40s during the first fraction, the same field would be delivered at a different time (e.g., t = 45s) for the second fraction. As BPs flow through the liver vasculature, the BP pool in the liver changes over time. Delivering at a different starting time yields a different set of BPs. The bDVH after F fractions was calculated as $D_F={\displaystyle {\sum }_1^F{\tilde{D}}_j}$, where ${\tilde{D}}_j$ represented the one-dimensional dose array of D_j at different starting times.

2.3.4. Dosimetric analysis

The impact of treatment modalities, BOT, break time between fields, and fractionation on bDVH metrics was studied. The bDVH metrics computed included the mean dose to blood 𝜇_dose,b, the percentage of blood receiving non-zero dose (V_>0Gy) and at least 0.125Gy (V_>0.125Gy), as well as the 2% of BPs which received the highest dose (D_2%). To assess the effect of only treatment modality without the confounding factor of varying time structure of delivery, all four treatment modalities were evaluated with identical BOT and break time at 60 s and 0 s, respectively. The impact of BOT was assessed by varying the delivery time for each modality to achieve different BOTs ranging from 30 s to 240 s in 30 s intervals, while keeping the break time at 0s. Using BOT of 60 s, the effect of break-time (0, 25, 50, 100, 200, or 300 s) was evaluated for different modalities. In addition, bDVHs at 1, 3, 5, 10, 15 fractions were generated, where the changes in DVH metrics over the number of fractions were investigated. Finally, the bDVH and DVH metrics were generated using clinical treatment parameters stated in Table 1 to investigate the compound effect of the treatment modalities, BOT and the break time. The whole-body blood flow simulation was performed in Python v3.9. All other simulations and analyses in this study were performed in MATLAB 2020a (MathWorks, Natick, US).

3 |. RESULTS

3.1. Blood particle simulation and equilibrium state

The BPs flowing through the liver vasculatures were explicitly tracked. At t = 0.1s, BPs started to enter the liver through the HA and PV trees. BPs propagated along their paths and partially filled the liver at t = 10s (Figure 4). An equilibrium state was reached at t = 40s, where the number of BPs exiting through the HV tree and inferior vena cava was approximately equal to the number of BPs entering the liver. At the equilibrium state, ~10,000 BPs were present in the liver at any time, constituting ~10% of the total blood volume in the whole-body simulation, consistent with reported values in ICRP Publication 89. The average BP path length for AM and AF livers were 38 cm and 33 cm respectively, leading to a blood speed of 2 cm/s for the AM liver and 2.3 cm/s for the AF liver. All dose fields were delivered after reaching the equilibrium state to prevent the underestimation of dose to BPs.

Figure 4.

Left: 4D blood flow model simulation in the liver at 0.1s, 10s and > 40s, where it reaches equilibrium state. Right: The number of BPs tracked through the liver plotted as a function of time.

3.2. Effect of treatment modalities on dose received by circulating blood

Proton treatments achieved lower integral dose to circulating blood compared to photon treatments. While keeping the BOT and break time constant for all modalities, the bDVHs after 1 and 15 fractions (Figure 5A, B) demonstrated reduced integral dose to blood for PBS and SOBP compared to VMAT and IMRT. The highest integral dose was observed with VMAT. The values of 𝜇_dose,b, V_>0.125Gy, D_2% for the photon treatments were greater than those for the proton treatments for 5 out of 6 patients (Figure 5 C, E, F), with a mean percent difference ± standard deviation (SD) of 45% ± 44%, 53% ± 48%, 19% ± 14%, respectively across all 6 patients. V_>0Gy was least affected by the difference in treatment modalities, with a mean percent difference ± SD of 6% ± 7%.

Figure 5.

Example blood DVHs from VMAT, IMRT, SOBP and PBS after 1 (dotted line) and 15 fractions (solid lines) demonstrate greater integral dose to blood from photon treatments compared to proton treatments for both AM (A) and AF (B) patients. The average 𝜇_dose,b (C), V_>0Gy (D), V_>0.125Gy (E), D_2% (F) of VMAT and IMRT (grey bar) are compared with the average 𝜇_dose,b, V_>0Gy, V_>0.125Gy, D_2% of SOBP and PBS (black bar) for all six patients.

3.3. Effect of Beam-on time on dose received by circulating blood

The total beam delivery time is determined by values of the BOT and the break time between fields. For a given break time, V_>0Gy increased with BOT quadratically for all modalities, by approximately 55% over 200s for both AM and AF patients (Figure 6). V_>0.125Gy also increased with BOT, with a steeper slope at low BOTs (<120s for AF, <180s for AM), and subsequently reached a plateau or decreased with BOT at high BOTs. The opposite trend was observed for D_2%, which decreased by approximately 63% to 68% and 53% to 60% with increasing BOTs for AM and AF patients, respectively. Negligible effect of BOTs on 𝜇_dose,b was observed for all treatment modalities (data not shown).

Figure 6.

The dependence of V_>0Gy, V_>0.125Gy, D_2% on beam on time (BOT) plotted across different treatment modalities including VMAT, IMRT, SOBP and PBS after 1 fraction for representative adult male (AM) and adult female (AF) patients.

3.4. Effect of Break time on dose received by circulating blood

The total break time between fields had a substantial impact on V_>0Gy, V_>0.125Gy, D_2% for IMRT at break times <100s, with little impact for the other modalities. When the total break time between IMRT fields varied from 0s to 100s, V_>0Gy and V_>0.125Gy increased by approximately 25% and 8% respectively, whereas D_2% decreased by ~10% (Figure 7). For the proton treatments, an initial increase was observed in V_>0Gy, V_>0.125Gy, D_2% from zero break time to non-zero break times, with little variations among non-zero break times. Similar patterns were seen in both AM and AF patients. IMRT was most influenced because the break time between each field is impacted by both the total break time and the number of fields. While the total break times were kept consistent among different modalities, IMRT had much more fields, causing much shorter break time between each field.

Figure 7.

The dependence of V_>0Gy, V_>0.125Gy, D_2% on total break time between fields were plotted across different treatment modalities including IMRT, SOBP and PBS after 1 fraction for representative adult male (AM) and adult female (AF) patients.

3.5. Effect of fractionation on dose received by circulating blood

The V_>0Gy, V_>0.125Gy, and D_2% increased rapidly during fractionated delivery. V_>0Gy increased from 80%, 48%, 73%, 37% after the first fraction to 100% after 15 fractions for PBS, SOBP, IMRT and VMAT, respectively (Figure 8, 1^st row). V_>0Gy rapidly reached 100% after 5 fractions for PBS, SOBP and IMRT. Similarly, V_>0.125Gy increased from 52% (PBS), 34% (SOBP), 46% (IMRT), 31% (VMAT) after 1 fraction to 100% after 10 fractions. In addition, D_2% increased linearly with the number of fractions for all treatment modalities (Figure 8, 2^nd row).

Figure 8.

Blood DVHs metrics V_>0Gy, V_>0.125Gy (1^st row), and D_2% (2^nd row) over 1, 3, 5, 10, 15 fractions were plotted for representative adult male (AM) patient with different treatment modalities including VMAT, IMRT, SOBP and PBS. V_>0Gy, V_>0.125Gy, D_2% increased with increasing number of fractions.

3.6. Blood DVHs with clinical parameters

The patient specific blood DVH metrics 𝜇_dose,b, V_>0Gy, V_>0.125Gy, D_2% computed with clinically realistic delivery times (for our center) were influenced by the combined effect of treatment modalities and BOT to various degrees. As BOT had negligible effect on 𝜇_dose,b, variations in 𝜇_dose,b reflected the sole effect of treatment modality, with lower 𝜇_dose,b from proton treatment than from photon treatment (Figure 9A). V_>0Gy increased from VMAT to PBS (Figure 9B), as the BOT increased from 60s for VMAT to 255s for PBS (Table 2). The difference in integral dose between proton and photon treatments had little effect on V_>0Gy. On the other hand, although V_>0.125Gy followed an overall increasing trend as BOT increased, a dip in value was observed for the proton treatments in three patients (Figure 9 C), influenced by the reduced integral dose from proton treatments. Similarly, D_2% generally decreased with increasing BOT (Figure 9 D). However, while the BOT of IMRT was greater than BOT of SOBP, higher D_2% was found for IMRT. In addition, patient gender related trend is observed, where V_>0Gy is consistently greater for female patients than that of male patients. The specific values of 𝜇_dose,b, V_>0Gy, V_>0.125Gy, D_2% for all patients are listed in the Appendix.

Figure 9.

Blood DVH metrics 𝜇_dose,b, V_>0Gy, V_>0.125Gy, D_2% of VMAT, SOBP, IMRT and PBS using actual clinical BOTs and break times (Table 2) after 1 fraction for six patients indicated a combined effect of BOT and integral dose. The treatment modalities were ordered based on ascending BOTs. The dotted and solid line represent the DVH metrics from male and female patients, respectively.

4 |. DISCUSSION

This study introduced a 4D liver blood flow model capable of estimating dose to circulating blood during fractionated radiotherapy. The model vastly improves upon existing methods (Basler et al., 2018; Jin et al., 2020) by including realistic vasculature trees and re-entry of blood into the liver as well as accounting for the time structure of radiation dose delivery, which turned out to be essential. The model allowed us to study the impact of photon (VMAT and IMRT) and proton (passive SOBP and active PBS) treatment modalities, delivery time and fractionation on the dose to circulating blood. Our non-invasive approach of calculating dose to blood can provide insights on how to reduce radiation-induced lymphopenia and guide radiotherapy treatment planning.

The DVH metrics 𝜇_dose, V_>0Gy, V_>0.125Gy and D_2% were analyzed in this study to evaluate the impact of different treatment modalities and parameters on the dose to circulating blood. It is important to note that although in-vitro measurements on the radiation response of various lymphocytes demonstrated observed apoptosis after 0.125Gy, there was no clear threshold dose for apoptosis (Heylmann et al., 2021). Therefore, defining an optimal dose threshold to evaluate remains a challenging topic. V_>0Gy was used in this study to quantitatively assess the amount of circulating blood irradiated, i.e., dose-sparing effect; whereas V_>0.125Gy was chosen to evaluate potential lymphocytes apoptosis based on the experimental observation. In addition, D_2% was reported to evaluate the dose required to kill 2% of lymphocytes. The threshold of 2% was selected based on the clinical data of lymphocyte counts over the course of fractionated radiotherapy in liver patients, where approximately 2% of lymphocytes was depleted per fraction (De et al., 2021).

Compared to photon treatments, the integral dose and mean dose to circulating blood 𝜇_dose were noticeably reduced for proton therapy for all six patients in this study. This difference was caused by the dosimetric advantage of proton therapy with superior dose conformality and the lack of exit dose (Yoo et al., 2018; Chuong et al., 2020). This observation is consistent with previous reports of integral dose to circulating blood in the brain from proton and IMRT treatments, where 50% increase in integral dose was found after IMRT (Hammi et al., 2020). In addition, blood DVH metrics V_>0Gy, V_>0.125Gy, and D_2% were lower for proton treatments (Figure 5), consistent with the lymphocyte sparing effect of proton therapy reported in the literature (Fang et al., 2017; Ko et al., 2018). Interestingly, our study suggested a much greater impact on V_>0.125Gy and D_2% from treatment modalities than on V_>0Gy (Figure 5).

In addition to treatment modalities, the beam-on time influenced V_>0Gy, V_>0.125Gy and D_2% for all treatment modalities. V_>0Gy, V_>0.125Gy both increased considerably with BOTs, while the opposite trend was observed for D_2% (Figure 6). This demonstrates a treatment time dependent tradeoff between low-dose to a large fraction of blood and high-dose to a small fraction of blood. Basler et al. also reported an elevated percentage of blood receiving low-dose with longer BOTs, where the percentage of blood receiving > 0.5 Gy (V_>0.5Gy) increased by 15% as BOT varied from 27 s to 253 s (2018). The opposite trend was observed in our study, whereas V_>0.5Gy decreased with increasing BOTs (data not shown). This discrepancy comes from the fact that 0.5 Gy was located near the high-dose tail of the blood DVH in our study, but was at the low-dose bath in the study of Basler et al. The cut-off value between low-dose bath and high-dose depends on the prescribed dose per fraction and blood speed used in the simulation. The treatment plans used in Basler et al. had substantially higher dose per fraction and assumed slower blood speed (1 cm/s) as compared to values applied in our study (2 cm/s for AM and 2.3 cm/s for AF), leading to a greater dose to circulating blood. These results could be of practical importance in determining the optimal treatment plans. For example, if the goal is to minimize the percentage of blood receiving >0.5 Gy, BOT should be increased in some protocols but decreased in others.

The patient specific DVH metrics 𝜇_dose,b, V_>0Gy, V_>0.125Gy, D_2% computed with clinically realistic delivery times were influenced by the combined effect of treatment modalities and BOT to various degrees. As BOT had little effect on 𝜇_dose,b, the difference in 𝜇_dose,b (Figure 9A) was mostly contributed by the reduced integral liver dose from proton treatment (Table A1 in Appendix). V_>0Gy was predominately affected by the BOT (Figure 9B), which shows that minimizing the fraction of circulating blood irradiated by shortening the treatment delivery time outweighs the integral dose advantage of proton therapy. Thus, it cannot be generally assumed that proton treatments are advantageous in terms of dose to the blood simply based on integral dose considerations. Patient gender related trend was also observed for V_>0Gy, where V_>0Gy appeared higher for female patients. This could be caused by the higher blood circulation speed in the female liver compared to the male liver, leading to more blood cells receiving dose. Interestingly, the trend of V_>0.125Gy shared similarities to both 𝜇_dose,b and V_>0Gy (Figure 9C). For patients with obvious reduction in 𝜇_dose,b from proton treatments (AM, P2, AF P1, AF, P3), difference in integral dose seem to have more impact on V_>0.125Gy. When 𝜇_dose,b is comparable among treatment modalities (AM, P1, AM P3, AF, P2), BOT becomes the major contributor, causing the upward trend of V_>0.125Gy. Similarly, contributions from both treatment modalities and BOT are found for D_2% (Figure 9D).

This study has several limitations. The physiological factor variations including age, blood speed and anatomical variation of hepatic vasculature among patients (Sahani et al., 2004; Catalano et al., 2008; Debbaut et al., 2014) were not considered. In addition, the model did not include patient-specific altered tumor vasculature, which can be highly heterogeneous. HCC tumors are well vascularized with an arterialized blood supply (Zhu et al., 2009; Sahani et al., 2013). The blood flow in certain regions might be faster as compared to normal tissue (Abdullah et al., 2008). More complex models with adjustable blood speed in certain areas corresponding to the tumor location could be developed to imitate the presence of tumor. Furthermore, contour-based deformable registration used in this study requires complete patient liver contours, which might be unavailable for patients with missing liver segments due to prior resection. In this case, the liver must be re-contoured to include the resected component. Validation of the proposed 4D liver model presents another challenge, as dose to circulating blood cannot be directly measured in the human body and there are no anatomical physical phantoms that include blood circulation available for experimental validation. Indirect validation of the proposed model is being conducted by correlating estimated dose to circulating blood to the observed lymphocyte depletion in liver patients (Sung et al., 2020). For further validation, quantitative in-vitro studies could be conducted to study the radiation response of lymphocytes with different treatment modalities and parameters (Heylmann et al., 2021). In addition, cytogenetic bio-dosimetry methods such as counting the frequency of dicentric chromosomes in peripheral blood lymphocytes could be considered, where the frequency of dicentric chromosomes increases with radiation dose (Kelly et al., 1965; Ludovici et al., 2021). With development of detailed vasculature also in animal models (Fung et al., 2011; Peeters et al., 2017), validation with in vitro and in vivo experiments would be of great interest to evaluate the accuracy of dose estimation and the clinical translation feasibly.

5 |. CONCLUSION

We have developed a 4D liver blood flow model capable of estimating the dose to circulating blood during radiation therapy. The model includes realistic vasculature trees and re-entry of blood to the liver, as well as the time structure of radiation dose delivery. We applied the model to assess different treatment modalities including volumetric-modulated arc therapy (VMAT), intensity-modulated radiation therapy (IMRT), passive scattered spread-out Bragg peak (SOBP) and active pencil beam scanning (PBS) proton therapy. We found that both the treatment modalities and the BOT impacted the shape and magnitude of the blood DVHs. Proton treatments reduced the integral dose, the mean dose, and highest 2% dose (D_2%) to circulating blood, as well as the fraction of circulating blood receiving >0.125 Gy, which has been considered as a threshold dose for apoptosis. V_>0.125Gy increased with BOT, whereas D_2% decreased, suggesting a BOT dependent tradeoff between low-dose bath to a large fraction of circulating blood and high-dose tail to a small fraction of circulating blood. The circulating blood receiving any dose (V_>0Gy), however, is hardly affected by different treatment modalities, but predominantly influenced by the beam-on time. These results suggest that proton treatments with shorter beam-on times could reduce dose to circulating blood. Our framework potentially provides insights into the clinical impact of lymphopenia as a guide to improved external beam radiotherapy treatment planning.

APPENDIX

Table A1.

The specific values of μ_dose,b, V_>0Gy, V_>0.125Gy, D_2% for all patients, planned with VMAT, SOBP, IMRT and PBS.

		VMAT	SOBP	IMRT	PBS
AM Patient 1	μdose,liver [Gy]	22.06	21.39	22.16	19.38
	μdose,b [Gy]	0.16	0.15	0.16	0.14
	V > 0Gy [%]	37.0	50.4	73.2	80.2
	V > 0.125Gy [%]	32.3	37.5	45.5	49.1
	D2% [Gy]	0.8	0.7	0.7	0.4
AM Patient 2	μdose,liver [Gy]	28.90	22.86	27.67	22.75
	μdose,b [Gy]	0.16	0.10	0.14	0.10
	V > 0Gy [%]	36.9	51.0	73.2	80.7
	V > 0.125Gy [%]	32.3	24.3	41.7	33.9
	D2% [Gy]	0.9	0.5	0.6	0.4
AM Patient 3	μdose,liver [Gy]	24.06	21.77	24.88	20.32
	μdose.b [Gy]	0.18	0.18	0.19	0.18
	V > 0Gy [%]	37.0	48.4	73.3	80.1
	V > 0.125Gy [%]	31.4	33.6	46.2	52.3
	D2% [Gy]	1.1	1.0	0.9	0.6
AM Patient 1	μdose,liver [Gy]	17.61	12.60	17.20	11.84
	μdose,b [Gy]	0.07	0.04	0.08	0.03
	V > 0Gy [%]	43.7	55.2	82.8	86.8
	V > 0.125Gy [%]	24.6	10.1	24.0	2.2
	D2% [Gy]	0.4	0.2	0.3	0.1
AM Patient 2	μdose,liver [Gy]	16.78	15.05	16.84	12.42
	μdose,b [Gy]	0.16	0.17	0.16	0.15
	V > 0Gy [%]	43.8	59.8	78.0	87.9
	V > 0.125Gy [%]	40.5	49.0	48.7	55.0
	D2% [Gy]	0.7	0.7	0.6	0.4
AM Patient 3	μdose,liver [Gy]	21.45	15.87	20.04	16.36
	μdose,b [Gy]	0.10	0.05	0.09	0.05
	V > 0Gy [%]	43.7	50.0	78.0	83.7
	V > 0.125Gy [%]	34.9	12.7	29.1	12.4
	D2% [Gy]	0.5	0.3	0.4	0.2