Fig 1.
(A) A whole-body physiologically based pharmacokinetic (PBPK) model, consisting of 19 compartments, including normal organs and a tumor (prostate cancer), was developed. This model calculates absorbed doses in all orans including tumor and organs-at risk, and determines drug concentration within vascular space of tumor. In this framework, oxygenated blood enters the arterial compartment, from which it is distributed to all tissue compartments based on organ-specific blood flow rates. Blood exiting the tissues flows into the venous compartment and is routed back to the lungs for reoxygenation, thereby completing the circulatory loop. The systemic circulation is simplified into two main compartments—arterial and venous blood—with oxygenated blood from the lungs entering the arterial side directly, bypassing explicit modeling of the heart’s pumping function. Image from Pixabay (https://www.pixabay.com), used under the Pixabay License. (B) A schematic of capillary growth driven by the movement of individual tip endothelial cells (tECs). The migration of each cell, located at the nodes of a computational lattice (finite difference discretization), is governed by five states. These states are determined by coefficients representing the probabilities of the cell remaining stationary (P0) or moving in one of four directions: right (P1), up (P2), left (P3) or down (P4). The processes of anastomosis and branching occur as part of angiogenesis. Tumor induced angiogenesis creates a nonhomogeneous microvessels distribution within the tumor and it defines as tumor vascular space. Image created using Servier Medical Art (https://smart.servier.com/), licensed under CC BY 4.0. (C) The computational domain designed to simulate the spatiotemporal distribution of radiopharmaceuticals, encompassing parent vessels, tumor microvasculature, tumor interstitial space, and surrounding normal host tissue. Using the drug concentration in the tumor’s vascular space, derived from PBPK model, and the synthesized microvasculature, the CDR equations simulate spatiotemporal distribution of radiopharmaceuticals in order to calculate delivered dose to tumor more accurately. More information about mathematical modeling of angiogenesis process and PBPK model provided in S1 Text.
Table 1.
Parameters considered in CDR equation for simulating spatiotemporal distribution in tumor microenvironment.
Table 2.
Measured and estimated parameters of 4 patients for PBPK Model.
Fig 2.
Multi‑Compartment PBPK model and spatial simulation domain for radiopharmaceutical transport in tumor tissue.
(A) Solution domain and microvascular network distribution and Schematic of the Multi-Compartment Model for tumor in radiopharmaceutical therapy. This diagram represents a multi-compartmental physiologically based pharmacokinetic model detailing the distribution and interaction radiopharmaceuticals within four distinct physiological spaces: vascular, interstitial, cell membrane, and intracellular compartments. The transient mass transfer simulation incorporates convective contributions and applies the multi-compartment model to track solute dynamics across the biophysical domains. Key Steps: 1: Physical decay of radionuclide; 2 & 3 Exchange of radiopharmaceuticals between the vascular and interstitial compartments; 4 & 5: Binding and subsequent unbinding of the free drug to receptors on cell membrane; 6: Internalization of peptides that bound to receptors into intracellular space; 7: clearance of radiopharmaceuticals. Image created using Servier Medical Art (https://smart.servier.com/), licensed under CC BY 4.0. (B) Illustration of Solution Domain with Microvascular Network Distribution and Boundary Conditions. The simulation focuses on a tumor with an approximate diameter of 16 mm, embedded in a domain that also includes adjacent normal tissue. The microvascular network comprises vessels with diameters around 20 μm, branching from larger parent vessels of about 400 μm in diameter [118]. Vascular architecture is generated based on the sprouting angiogenesis model originally formulated by Anderson and Chaplain [57,58], with vessel density increasing in response to local concentrations of tumor angiogenic factors. This leads to a complex capillary network with enhanced branching and loop formation in the tumor region. The domain extends 2 mm beyond the tumor margin to capture the full therapeutic range of 177Lu, with emphasis placed on modeling both intratumoral and peritumoral transport phenomena.
Fig 3.
The modeling of angiogenesis process over a span of 30 days is depicted and compared qualitatively with actual microvasculature surrounding and within tumors.
The models demonstrate that newly formed blood vessels near and within the tumor exhibit significant branching and looping, consistent with in vivo observations. Comparing angiogenesis models that correspond with in vivo studies highlights that newly formed blood vessels near and within the tumor exhibit significant branching and looping [67]. The timeline includes snapshots at specific intervals: (A) Day 5, (B) Day 25, and (C) Day 30, showing the progression of vascular development in the model. Additionally, (D) represents tumor vasculature captured through optical frequency domain imaging [119], serving as a reference for comparison.
Fig 4.
Interstitial fluid pressure (IFP) and intravascular fluid pressure (IVP) validation; (A) Simulated distribution of IFP and (B) IVP within the tumor microenvironment.
These simulations illustrate the elevated internal pressures characteristic of solid tumors. Notably, when an inlet pressure of 25 mmHg is applied, the corresponding microvascular pressure reaches 19.8 mmHg, while the interstitial fluid pressure rises to 14.8 mmHg, indicating a significant internal buildup of pressure within the tumor tissue. (C) An analysis correlating intravascular and interstitial pressure across various tumors in realistic models suggests that both pressures are elevated with a minimal disparity between them [122]. This trend is attributed to structural abnormalities in the tumor vasculature, dysfunctional lymphatic drainage, and the dense packing of tumor cells. Elevated IFP impedes convective transvascular transport and can hinder the delivery and retention of therapeutic agents by promoting outward fluid convection, potentially facilitating drug efflux from the tumor. While this study does not account for lymphatic drainage, monitoring IFP and IFV remains vital for understanding drug delivery dynamics and improving therapeutic strategies in solid tumors.
Fig 5.
Validation of PBPK model and parameter estimation.
By employing post-treatment data obtained from γ-camera images for four patients [42], the designed PBPK model is fitted to the acquired data, facilitating the customization of RPT for personalized treatment strategies. Details of the fitting procedures—including optimization techniques and convergence criteria—as well as the statistical outcomes of the parameter estimation are provided in Figs A-D and Tables C-F in S2 Text.
Fig 6.
The spatiotemporal distribution of radiopharmaceuticals in tumor interstitial space at (A) 500 min, (B) 1000 min, (C) 3000 min, (D) 6000 min, and (E) 30000 min post-treatment for patient 1.
(F) Different absorbed doses (due to contribution of peptides in interstitial space) calculated by PBPK vs. CDR models.
Fig 7.
The spatiotemporal distribution of radiopharmaceuticals that bound to receptors at (A) 500 min, (B) 1000 min, (C) 3000 min, (D) 6000 min, and (E) 30000 min post-treatment for patient 1.
(F) Different absorbed doses (due to the contribution of peptides bound to receptors) calculated by PBPK vs. CDR models.
Fig 8.
The spatiotemporal distribution of radiopharmaceuticals that internalized into tumor cells at (A) 500 min, (B) 1000 min, (C) 3000 min, (D) 6000 min, and (E) 30000 min post-treatment for patient 1.
(F) Different absorbed doses (due to the contribution of peptides internalized into cells) calculated by PBPK vs. CDR models.
Fig 9.
The spatiotemporal distribution of radiopharmaceuticals is shown for (A) the tumor vascular space and (B) the amount of radiopharmaceuticals in muscle that affect tumor cells, for patient 1, at different time points during treatment.
Fig 10.
Comparison of radiopharmaceutical concentration in tumor vascular space, with and without amino acid administration, across 4 mCRPC patients.
The figures show a decrease in the concentration of radiopharmaceuticals in the tumor vasculature following the administration of amino acids.
Table 3.
Impact of amino acid administration on RPT Efficacy. Kidney and salivary gland organs are individually designated as organs-at-risk, determining the maximum safe injected activity. Subsequently, the absorbed dose to tumors is computed. Left column indicates the maximum possible activity that can be injected and the tumor absorbed dose when the kidney is the organ at risk, with and without amino acid administration. Right column indicates the maximum possible activity that can be injected and the tumor absorbed dose when the salivary gland is the organ at risk, with and without amino acid administration.