MIDOS: a novel stochastic model towards a treatment planning system for microsphere dosimetry in liver tumors

Abstract

Purpose

Transarterial radioembolization (TARE) procedures treat liver tumors by injecting radioactive microspheres into the hepatic artery. Currently, there is a critical need to optimize TARE towards a personalized dosimetry approach. To this aim, we present a novel microsphere dosimetry (MIDOS) stochastic model to estimate the activity delivered to the tumor(s), normal liver, and lung.

Methods

MIDOS incorporates adult male/female liver computational phantoms with the hepatic arterial, hepatic portal venous, and hepatic venous vascular trees. Tumors can be placed in both models at user discretion. The perfusion of microspheres follows cluster patterns, and a Markov chain approach was applied to microsphere navigation, with the terminal location of microspheres determined to be in either normal hepatic parenchyma, hepatic tumor, or lung. A tumor uptake model was implemented to determine if microspheres get lodged in the tumor, and a probability was included in determining the shunt of microspheres to the lung. A sensitivity analysis of the model parameters was performed, and radiation segmentectomy/lobectomy procedures were simulated over a wide range of activity perfused. Then, the impact of using different microspheres, i.e., SIR-Sphere^®, TheraSphere^®, and QuiremSphere^®, on the tumor-to-normal ratio (TNR), lung shunt fraction (LSF), and mean absorbed dose was analyzed.

Results

Highly vascularized tumors translated into increased TNR. Treatment results (TNR and LSF) were significantly more variable for microspheres with high particle load. In our scenarios with 1.5 GBq perfusion, TNR was maximum for TheraSphere^® at calibration time in segmentectomy/lobar technique, for SIR-Sphere^® at 1–3 days post-calibration, and regarding QuiremSphere^® at 3 days post-calibration.

Conclusion

This novel approach is a decisive step towards developing a personalized dosimetry framework for TARE. MIDOS assists in making clinical decisions in TARE treatment planning by assessing various delivery parameters and simulating different tumor uptakes. MIDOS offers evaluation of treatment outcomes, such as TNR and LSF, and quantitative scenario-specific decisions.

Introduction

Transarterial radioembolization (TARE) is progressing from being a palliative option to potentially curative treatment for hepatocellular carcinoma (HCC) and other liver neoplastic diseases due to the ability to deliver a high dose to the tumor while sparing normal tissue [1-4]. Tumor-selective radioembolization is possible as hepatic primary tumors and hepatic metastases derive their blood supply mainly from the hepatic artery, whereas normal hepatic parenchyma primarily receives blood from the portal vein [5]. In this treatment, a microcatheter is used to inject radioactive microspheres into branches of the hepatic artery to reach and embolize tumor vessels, emit radiation to eradicate the tumor, and spare normal liver parenchyma. This technique can be used for radiation segmentectomy, to treat relatively small tumors located in 1–2 Coinaud liver segments, or lobar administration, which is employed when multiple tumors occupy several segments in the same lobe [6]. With radiation segmentectomy, the intent is to deliver an ablative amount of radiation to the entire segment, whereas, with lobar administration, the hepatic parenchyma needs to maintain its function. Microspheres can lodge into the tumor, capillaries within the healthy liver, or follow arteriovenous shunts ending up in the lung.

Currently, three different types of microspheres are available to use in the clinic. Yttrium-90, a pure beta-emitter with a half-life of 64.04 h, has become the most common radionuclide available in two forms approved by the EMA and FDA: resin-based (⁹⁰Y-Resin) microspheres (SIR-Sphere^®) and glass-based (⁹⁰Y-Glass) microspheres (TheraSphere^®) [7, 8]. Recently, the EMA approved the third type of treatment microsphere (QuiremSphere^®) by using holmium-166 in microspheres made of poly-L-lactic acid (¹⁶⁶Ho-PLLA) [8, 9]. For ⁹⁰Y-microsphere treatments, macro-aggregated albumin labeled with technetium-99m, [^99mTc]Tc-MAA, is injected as a surrogate in a prior procedure for treatment planning. [^99mTc]Tc-MAA is assumed to have a similar final biodistribution as the ⁹⁰Y-microsphere, although the correlation varies with sphere type [10, 11]. SPECT-CT imaging using [^99mTc]Tc-MAA can then be utilized to estimate the absorbed dose in tumors and normal tissues, including the lungs [12]. However, this method is subject to significant uncertainties due to differences in physical properties between [^99mTc]Tc-MAA and ⁹⁰Y-microsphere, operator variability in planning/treatment catheter position, and uncertainties inherent to angiography imaging [2, 3, 8, 10, 13-15]. In addition, ⁹⁰Y-Resin and ⁹⁰Y-Glass differ in terms of size and microsphere-specific activity, leading to a different number of microspheres perfused to achieve a prescribed dose [16]. How to optimize the ⁹⁰Y-microsphere density used for TARE is still an open question due to its multiparametric nature, including tumor-specific properties (such as size and vascularity) and the procedure approach (selective or lobar treatment) [17, 18].

To address these uncertainties, efforts have been made to predict the distribution of microspheres and the dosimetric implications. Computational fluid dynamics (CFD) works have studied the hemodynamics of microsphere perfusion [19]. Nevertheless, as the microvasculature is not typically visible under angiographic imaging, CFD models cannot model the impact of the distribution of embolic microspheres at the microscopic scale, the critical objective in TARE [20, 21]. To overcome this, two main approaches can be used to link the ⁹⁰Y-microsphere density administered and the heterogeneity in the tissue: developing liver arterial tree models to simulate the effect of microspheres trapping in the liver parenchyma [22-24] and Monte Carlo simulations based on their position in the tumor [25]. These studies provided insights to understand better the parameters that impact TARE. However, the role of tumor vascularity was not included in these models, as well as the effect of specific tumor locations within the liver and the election of the catheter insertion position. Consequently, we currently lack comprehensive tools capable of replicating clinically relevant TARE scenarios to evaluate clinical decisions based on the type of microsphere perfused and the technique performed.

This study introduces MIDOS, a novel stochastic model for microsphere dosimetry. MIDOS integrates a very detailed and complete adult (male and female) liver vasculature, providing a realistic picture regarding flow directionality. In addition, MIDOS includes a tumor uptake model and dynamically considers perfusion through the vascular tree, allowing for the assessment of different treatment techniques in TARE. We evaluated our model to quantify the impact of diverse treatment choices, such as performing radiation segmentectomy/lobectomy together with the type of applied microspheres, and how these parameters might impact clinically used metrics.

Material and methods

Recently, Correa-Alfonso et al. [26] generated polygonmesh adult (male and female) liver vasculature models. The computational liver was divided into nine independent functional segments based on the Couinaud classification, including the primary vessels of the hepatic artery, portal, and hepatic veins. Hemodynamic and geometric parameters from these major vessels were used to create very detailed vascular trees using the constrained constructive optimization (CCO) algorithm. Poiseuille’s and Murray’s laws were used at each bifurcation to ensure blood conservation. About 6000 blood vessels were created in the vascular trees of both phantoms. More details on the vascular tree model can be found elsewhere [26].

In this work, we extended these vascular models to create about 18,000 blood vessels, ensuring a homogeneous blood supply across the entire liver (Fig. 1). The blood vessels were modeled as cylinders of various radii, with the smallest vessel having a radius of 0.05 mm. The hepatic artery and portal vein trees meet the hepatic venous to generate about 3000 arteriovenous junctions. Of note, these anatomical junctions are representations of the smaller-scale capillary beds in our computational model.

Fig. 1.

Adult male (left, 2.36 kg) and female (right, 1.81 kg) liver phantoms. The hepatic arterial, hepatic portal venous, and hepatic venous trees are represented in red, blue, and purple, respectively. The dark areas represent virtual tumors of 45 mm (male), 36 mm, and 44 mm (female), which were placed in both models to simulate radiation segmentectomy and radiation lobectomy, respectively. The catheter path is displayed in black through the arterial tree

A stochastic approach for TARE

The objective of the model is to predict the amount of activity delivered to the tumor(s), normal hepatic parenchyma, and lung compartments, given the following initial clinical conditions defined by the user: planned activity to perfuse, type of microsphere, catheter’s injection point, time passed since calibration to adjust for physical decay, and lung shunt fraction estimated with [^99mTc]Tc-MAA in planning. To this end, our approach incorporates processes for microsphere perfusion, tumor shunts, and tumor uptake. The model computes the total number of microspheres to be injected using the microsphere-specific activity and the extended shelf life, adjusting for physical decay [27]. After the simulation, the model calculates the tumor-to-normal ratio (TNR) and lung shunt fraction (LSF) to quantify the activity in the tumor, normal tissue, and lung, defined as

$$\text{TNR}=\frac{A_{\text{tumors}}∕m_{\text{tumors}}}{A_{\text{normal}\_\text{liver}}∕m_{\text{normal}\_\text{liver}}}$$ (1)

where $A_{\text{tumors}}$ and $m_{\text{tumors}}$ are the activity and the mass of all the tumors and $A_{\text{normal}\_\text{liver}}$ and $m_{\text{normal}\_\text{liver}}$ are the activity and mass of the normal liver, respectively, and

$$\text{LSF}=\frac{A_{\text{lung}}}{A_{\text{liver}}}\times 100$$ (2)

where $A_{\text{lung}}$ is the activity in both lungs, while $A_{\text{liver}}$ is the activity in the entire liver. Additionally, the model offers the possibility to integrate the Medical Internal Radiation Dose (MIRD) formalism [28] to convert activity into absorbed dose values in each compartment (tumor(s), normal liver parenchyma or lung), as follows:

$$\mathrm{D}[\text{Gy}]=\text{CF}[\mathrm{J}∕\text{GBq}]\times \frac{\mathrm{A}[\text{GBq}]}{\mathrm{m}[\text{kg}]}$$ (3)

where D [Gy] is the absorbed dose in the selected compartment; CF [J/GBq] is the absorbed dose conversion factor; A [GBq] is the activity contained in the compartment; and m [kg] is the mass of the compartment.

Modeling microsphere perfusion

Microspheres tend to form clusters, as observed ex vivo [16, 29-32]. To account for this fact, we considered the perfusion of clusters of microspheres, simulated cluster by cluster. For each new cluster, we first sampled the number of microspheres per cluster from a lognormal probability density function (PDF), using the reported data by Pasciak et al. [16]. The size of each microsphere was randomly sampled from a Gaussian distribution with mean and variance in accordance with vendor specifications or the literature [3]. Starting from a selected injection point, a Markov chain was subsequently used to transport clusters perfused through the hepatic artery: at each bifurcation, a probability proportional to the flow in each branch is assigned to the cluster to decide what vessel is next selected. The cluster then flows as a train of microspheres through the arterial tree, ending up in the normal tissue (getting stuck in the arteriovenous junctions), shunting to the lung, or lodging in the tumor, according to adjustable probabilities.

Modeling tumor shunts

Tumor local environment may induce abnormalities in the normal liver vasculature, creating direct paths or shunts to the hepatic vein. Then, microspheres can flow to the heart and be pumped to the lungs through the pulmonary artery [33], potentially causing radiation-induced lung disease, such as pneumonitis [34]. To consider this phenomenon in our model, a probability for shunting out from the tumor (P_shunt) was included for each tumor-incoming cluster. This probability can be adjusted to match the LSF measured in the planning stage by SPECT-CT imaging based on [^99mTc]Tc-MAA perfusion.

Modeling tumor uptake

We integrated a tumor uptake model to simulate various tumor types. To this end, the tumor uptake is modeled as a capacitor, defined by a tumor-specific uptake constant, $\rho$. This parameter represents the resistance of the tumor to be filled with cluster, which is related to the grade of tumor vascularization. Whether a cluster of microspheres does not flow to the lung, the probability of being trapped is defined as follows:

$${\mathrm{P}}_{\text{cluster}\_\text{trapped}}=\text{exp}\left(-{\rho \mathrm{N}}_{\text{tumor}}\right)$$ (4)

where $N_{tumor}$ is the number of clusters lodged in the tumor at a given point. Following this approach, the model implies that the tumor can store clusters until it reaches a saturation level, i.e., a maximum number of embolizing clusters. The uptake constant, $\rho$, determines how fast the tumor is reaching saturation. For higher values of $\rho$, $P_{cluster\_trapped}$ tends to zero faster, meaning a high resistance of the tumor to store clusters. In the same line, as $N_{tumor}$ approaches the tumor saturation, $P_{cluster\_trapped}$ becomes very low, and the following clusters perfused will flow to the normal tissue until reaching an arteriovenous junction and getting lodged.

Simulations of clinical scenarios

To reproduce clinical scenarios, tumors can be arbitrarily placed within either or the two liver phantoms at the user’s discretion with a specific size. Next, a simulation begins with a user-definable activity corresponding to the type of microsphere selected and a specific catheter position in the hepatic arterial tree. Mass metrics for the tumor and normal liver were obtained using a density factor of 1.03 g/mL for both volumes, as suggested elsewhere [13]. Table 1 shows four different types of microspheres included in the simulations and their physical characteristics. Table 2 summarizes all the parameters defined in our model.

Table 1.

Types of microspheres used for our models to simulate clinical scenarios

	Planning microsphere	Treatment microspheres
Isotope	99mTc	90Y (TheraSphere®)	90Y (SIR-Sphere®)	166Ho (QuiremSphere®)
Material	Albumin	Glass	Resin	poly (L-lactic acid)
Diameter	10–90 mm	20–30 mm	20–60 mm	16–60 mm
Activity per microsphere (microsphere-specific activity) at calibration time	*300 Bq	2500 Bq	50 Bq	240–375 Bq
# beads in 1 GBq at calibration time	-	400,000	20,000,000	3,252,033
Half-life	6 h	64.04 h	64.04 h	26.8 h

Data for each type were adopted from the literature [2, 3, 7]. *The microsphere-specific activity has been calculated to deliver 350,000–700,000 [^99mTc]Tc-MAA microspheres in ~140–150 MBq [3]

Table 2.

Overview of the parameters defined in the model

Parameter	Definition
P shunt	Lung shunt fraction measured in treatment planning, sampled to each cluster that meets the tumor periphery
$\rho$	Tumor resistance to uptake clusters, related to the tumor vascularity
$P_{cluster\_trapped}$	The probability assigned to a cluster to become lodged in the tumor, following Eq. (1)

In the scenarios described below, data for tumors size, activity delivered for each type of microsphere perfused, and LSF measured in treatment planning (P_shunt) were taken from clinical population studies reported in the literature [13-15, 35-37].

Sensitivity analysis

Different clinical scenarios were evaluated to analyze the sensitivity of the parameters in the model. Table 3 shows the specifications of the clinical scenarios simulated and the variables analyzed. We simulated a segmentectomy technique for a 45-mm tumor diameter located in the sixth hepatic segment (Fig. 1) through the perfusion of 148 MBq of [^99mTc]Tc-MAA planning tracer and 1.5 GBq of ⁹⁰Y-Resin, ⁹⁰Y-Glass, and ¹⁶⁶Ho-PLAA treatment microspheres at calibration time. The catheter was placed in the corresponding sixth arterial branch. Then, TNR and LSF were calculated for different combinations of the model parameters. Simulations were repeated 15 times to obtain the statistical variability of each scenario.

Table 3.

Different clinical scenarios simulated to analyze the model sensitivity

Treatment modality	Phantom	Tumor size & location	Activity perfused (GBq)	# Beads injected	Pshunt	Tumor uptake constant	Output	Sims.
Radiation segmentectomy	Adult male	45 mm 6th Seg.	[99mTc]Tc-MAA, 0.148 90Y-Resin, 1.5 90Y-Glass, 1.5 166Ho-PLLA, 1.5	[99mTc]Tc-MAA, 493,333 90Y-Resin, 30,000,000 90Y-Glass, 600,000 166Ho-PLLA, 4,878,049	8%	$\rho$, [2 · 10−4, 2 · 10−3]	TNR vs $\rho$	15
					4 - 12%	$\rho$, 3.3 · 10−4	LSF vs Pshunt

The activity was perfused at calibration time, according to the microsphere-specific activity indicated in Table 1. Data for tumor size, activity delivered for each type of microsphere perfused, and LSF measured in treatment planning (P_shunt) were taken clinical population studies [13-15, 35-37]. The parameter $\rho$ is the tumor uptake constant; output refers to the clinical variables to be analyzed after simulations: TNR and LSF

Radiation segmentectomy and lobar administration

The impact of treatment microspheres (⁹⁰Y-Glass, ⁹⁰Y-Resin, and ¹⁶⁶Ho-PLLA) was studied under segmentectomy and lobar administration. Table 4 provides an overview of the parameters used for simulations. The catheter was placed in the sixth hepatic arterial branch and the right branch of the hepatic artery, respectively (Fig. 1). Several independent perfusions of activity in a range of 0.5–3.5 GBq for ⁹⁰Y-Glass and ⁹⁰Y-Resin and 0.5–12.5 GBq for ¹⁶⁶Ho-PLLA were carried out per each type of microsphere and correlated to TNR. For each value of activity administered, the corresponding PDF for cluster formation was utilized. Activity was translated into absorbed dose by calculating Eq. (5) considering the specific conversion factors [8]: CF (⁹⁰Y) = 50 J/GBq and CF (¹⁶⁶Ho) = 15.87 J/GBq.

Table 4.

Radiation segmentectomy/lobectomy technique simulations for different treatment microspheres

Treatment modality	Phantom	Area perfused	Bead type	Pshunt	Tumor(s) size	Tumor uptake constant	A (GBq)	ESL	Output	A (GBq)	ESL	Output	Sims.
Radiation segmentectomy	Adult male	6th Seg.	90Y-Resin 90Y-Glass	8%	45 mm	$\rho$, 3.3 · 10−4	90Y-Glass/90Y-Resin, 0.5–3.5	0	TNR/mean dose vs activity injected	1.5	0-9	TNR vs ESL	15
Radiation lobectomy	Adult female	Right lobe	166Ho-PLLA	9%	36 mm	$\rho$, 4 · 10−4	166Ho-PLLA, 0.5–12.5
					44 mm	$\rho$, 2.8 · 10−4

The activity was perfused at calibration time, according to the microsphere-specific activity in Table 1. A_injected, activity perfused (GBq); ESL extended shelf life (days post-calibration). Data for tumor size, activity delivered for each type of microsphere perfused, and LSF measured in treatment planning (P_shunt) were taken from clinical population studies [13-15, 35-37]. The parameter $\rho$ is the tumor uptake constant; output refers to the clinical variable to be analyzed after simulation: TNR and mean dose

Extended shelf life

The effect of the physical decay in the specific activity of treatment microspheres (⁹⁰Y-Glass, ⁹⁰Y-Resin, and ¹⁶⁶Ho-PLLA) was evaluated by simulating different days that passed since calibration. All the scenarios described in Table 4 were performed under perfusion of 1.5 GBq for 0–9 days post-calibration, and the impact in TNR per each treatment bead was evaluated.

Results

Model parameter sensitivity

Figure 2 presents the sensitivity of the model parameters with respect to the clinical metrics TNR and LSF. The TNR decreases for all types of microspheres with the tumor uptake constant ($\rho$), (Fig. 2, left). It can be observed that for the same $\rho$, the TNR varies with the type of microsphere. In particular, ⁹⁰Y-Resin shows the highest range of variability. The right panel presents a linear correlation between P_shunt and LSF for all types of microspheres. For a fixed value of P_shunt, LSF fluctuated more across simulations for ⁹⁰Y-Resin than for[^99mTc]Tc-MAA, ⁹⁰Y-Glass, and ¹⁶⁶Ho-PLLA.

Fig. 2.

Model sensitivity analysis. The left panel displays the impact of the tumor uptake constant ($\rho$) on the tumor-to-normal ratio. The right panel shows the correlation between the lung shunt fraction quantified in treatment planning (P_shunt) and the lung shunt fraction after treatment (LSF). Perfusions of 148 MBq [^99mTc]Tc-MAA, 1.5 GBq of ⁹⁰Y-Glass, ⁹⁰Y-Resin, and ¹⁶⁶Ho-PLLA were performed. Boxplots indicate the range of results after repeating the same simulations N = 15 times. In every column, four boxplots corresponding to the type of microsphere are displayed. From left to right, [^99mTc] Tc-MAA, ⁹⁰Y-Glass, ⁹⁰Y-Resin, and ¹⁶⁶Ho-PLLA

Treatment evaluation for radiation segmentectomy and lobar administration

We evaluated the impact of different treatment microspheres (⁹⁰Y-Glass, ⁹⁰Y-Resin, and ¹⁶⁶Ho-PLLA) under two common techniques used in TARE: segmentectomy and lobar administration. TNR for ⁹⁰Y-Glass decreased as more activity was administered (Figs. 3 and 4, left panel). Regarding ⁹⁰Y-Resin, TNR increased in segmentectomy and lobar technique up to 2 and 2.5 GBq perfusions, respectively. Activity in tumors was lower for ¹⁶⁶Ho-PLLA beads in this activity range. The small tumor for lobar administration showed a higher dose than the large tumor. It is important to note that the calculated values for the mean dose in each compartment are not necessarily clinically applicable since the characteristics of the tumor employed in this study were arbitrarily selected.

Fig. 3.

Impact of different treatment microspheres in radiation segmentectomy. Per each value of activity perfused, fifteen simulations were performed using the corresponding PDF for cluster formation. Solid lines represent the mean value, and the area filled shows the range between all simulations

Fig. 4.

Impact of different treatment microspheres in lobar administration. Per each value of activity perfused, fifteen simulations were performed using the corresponding PDF for cluster formation. Solid lines represent the mean value and the area filled shows the range between all simulations

Determining the optimal microsphere type and day

We simulated the impact of the extended shelf life on TNR in segmentectomy and lobar techniques for ⁹⁰Y-Glass, ⁹⁰Y-Resin, and ¹⁶⁶Ho-PLLA (Fig. 5), according to the characteristics in Table 1. For 1.5 GBq perfusions, TNR for ⁹⁰Y-Glass was the maximum at calibration time, whereas ⁹⁰Y-Resin showed the maximum TNR 1 day and 3 days post-calibration for segmentectomy and lobectomy, respectively. Regarding ¹⁶⁶Ho-PLLA beads, the maximum TNR was observed at 3 days post-calibration in both clinical procedures.

Fig. 5.

Impact of different treatment days on the microsphere-specific activity of treatment beads. Boxplots indicate the range of results after repeating the same simulations N = 15 times. In every column, three boxplots corresponding to the type of microsphere are displayed. From left to right: ⁹⁰Y-Glass, ⁹⁰Y-Resin, ¹⁶⁶Ho-PLLA

Discussion

This work represents the foundation of a novel model to be integrated into a clinical treatment planning system for radioembolization. This version of MIDOS allows the assessment of TNR for different beads and clinical procedures (Figs. 3 and 4), enabling the possibility of finding the optimal treatment window based on the activity delivered to the tumor(s) and normal liver. Moreover, the model adjusts the microsphere-specific activity of each type of bead, accounting for the time passed since calibration (Fig. 5). Therefore, the impact of the particle load on TNR can be evaluated with our model for different clinical procedures, leading to a potential clinical application of the MIDOS for treatment optimization.

However, two main additional steps are required before clinical translation. First, the adult human liver phantom models must be replaced by patient-specific data. An easy-to-implement first approximation might consist of scaling tumors from patients to the phantom scale and extrapolating results to the patient geometry. A more direct patient-specific but time-consuming strategy may include perfusing microspheres in the patient space using patient-specific vasculature and tumor. For this purpose, the CCO algorithm [26] must be applied to generate the vasculature from the patient-specific major vessels, which could be extracted from angiography or cone beam computed tomography (CBCT) images. Second, the current version of the MIDOS model predicts the amount of activity delivered to the tumor(s), normal tissue, and lung taken as compartments, and the corresponding dose absorbed values are calculated by applying the MIRD formalism. Nonetheless, this dosimetry approach disregards the spatial distribution of microspheres and consequently the heterogeneity associated [2, 4, 8, 28], which is critical to understanding the biological effect [16, 25]. To overcome this limitation, future steps should prioritize converting these activity tallies predicted into non-uniform dose distributions. To this end, clusters of microspheres must be spatially distributed in the normal liver and tumor compartment. For the normal liver, clusters might be statistically distributed around the arteriovenous junctions generated in the vascular tree. Regarding the tumor, a liver-specific tumor vasculature model should be developed to distribute the clusters according to the specific vasculature of the tumor. Then, a dose calculation algorithm like Monte Carlo can be integrated into the MIDOS model [12]. This will open the door to obtaining heterogeneous absorbed dose distributions as an output from the type of microsphere and activity perfused. In general, additional treatment information, such as lung shunting measured with angiography, catheter position, activity delivered, and type of microsphere perfused, would enable optimizing patient-specific scenarios.

MIDOS is based on a stochastic approach to reproduce TARE treatments. The nature of TARE is very complex since it depends on many physical and biological variables. Deterministic models based on CFD techniques have shown interesting insights addressing hemodynamics perfusion variables present in RMT on a very detailed scale [38-42]. However, they also required simplifications since boundary conditions and the effect of other parameters are either unknown or unmanageable, in addition to being time-consuming and impractical for clinical implementation [20]. In view of these limitations, a model for the tumor microvasculature, the critical point in TARE [1-3], is not addressable through CFD. Therefore, the rationale for incorporating a stochastic approach to our model comes from the randomness inherent in TARE. Moreover, this approach allows a wide variability of clinical scenarios to be adapted for simulations regardless of boundary/initial conditions, potentially providing us with practical information to optimize treatment planning.

Our model incorporates a macroscale tumor uptake model based on the uptake constant $\rho$. Our sensitivity analysis showed that the TNR decreases for tumors with higher $\rho$ (Fig. 2, left). Tumors with low values of $\rho$ imply a low resistance of the tumor to store clusters, meaning a high uptake of the activity perfused. This represents a highly vascularized tumor with a high TNR. Different levels of vasculature will result in different types of tumor uptakes with a specific $\rho$, i.e., different tumors. Therefore, primary liver tumors, such as hepatocellular carcinoma or intrahepatic cholangiocarcinoma, together with secondary liver tumors, can be assessed with our model [17]. Patient-specific SPECT-CT or PET/CT imaging post-treatment would be useful to estimate $\rho$ from the TNR calculated, allowing the quantification of tumor inter-patient and intra-patient variability. In addition to this, further studies are required to link $\rho$ with a microscale tumor model considering tumor-specific vasculature properties, such as the vessel density, grade of heterogeneity, and the size of the tumor vessels, for instance. All these properties contribute to a specific grade of vascularity, leading to a specific $\rho$. This will allow correlating the TNR with detailed tumor-specific vasculature properties, providing a better understanding of the tumor vasculature and its role in TARE.

An interesting aspect of stochastic simulations is the possibility of estimating statistical variabilities when the same treatment conditions are reproduced independently. In this sense, TNR and LSF were found to be more variable for ⁹⁰Y-Resin microspheres (Fig. 2). This can be explained by the microsphere-specific activity of each type of bead. The lower it is, the greater the number of microspheres per mL injected to achieve the target dose. This leads to a more variable cluster pattern, with a larger variation in the number of microspheres per cluster across different simulations, which is translated into a more variable outcome for a fixed amount of activity perfused [16]. To account for this phenomenon, we incorporated the cluster patterns observed by Pasciak et al. [16]. However, they only analyzed ⁹⁰Y-Glass microspheres in normal tissue, while we assume that all types of microspheres (Table 1) satisfy these cluster patterns. In this sense, further human histological studies analyzing cluster patterns for each type of microsphere perfused might improve the results obtained in our model since they have different mechanical properties and potentially could lead to a different cluster pattern.

Conclusion

MIDOS allows to reproduce a variety of clinical scenarios, providing practical and quantitative information to optimize treatment planning. To this end, MIDOS incorporates the fundamental phenomena governing TARE: perfusion of microspheres through the vascular tree, accounting for their specific physical properties (size, activity, and extended shelf life), tumor shunts, and tumor uptake. Moreover, MIDOS evaluates clinical scenarios based on standard metrics, such as TNR and LSF.

Data Availability

The model employed and data utilized to show results in this paper are available upon reasonable request from the corresponding author, subject to privacy/ethical considerations.