https://orcid.org/0009-0002-3092-8750
Figures
Abstract
The aim of this study is to demonstrate the enhanced efficiency of combined therapeutic strategies for the treatment of growing tumors, based on computational experiments of a continuous-level modeling framework. In particular, the tumor growth is simulated within a host tissue and treated as a multiphase fluid, with each cellular species considered as a distinct fluid phase. Our model integrates the impact of chemical species on tumor dynamics, and we model –through reaction-diffusion equations– the spatio-temporal evolution of oxygen, vascular endothelial growth factor (VEGF) and chemotherapeutic agents. Simulations of a growing tumor exposed to external radiation showcase the rapid impact of radiotherapy on tumor suppression, however this effect diminishes over time. To enhance the therapeutic efficiency of radiotherapy, we investigate the combination of external radiation with the anti-VEGF drug bevacizumab and the cytotoxic drug docetaxel. Our simulations demonstrate that this synergistic approach integrates the immediate effectiveness of radiation therapy with the enduring tumor-suppressive capabilities of chemotherapy.
Citation: Lampropoulos I, Koutsi M, Kavousanakis ME (2025) Modeling of chemo-radiotherapy targeting growing vascular tumors: A continuum-level approach. PLoS ONE 20(1): e0301657. https://doi.org/10.1371/journal.pone.0301657
Editor: Saiedeh Razi-Soofiyani, Tabriz University of Medical Sciences, IRAN, ISLAMIC REPUBLIC OF
Received: March 19, 2024; Accepted: December 29, 2024; Published: January 15, 2025
Copyright: © 2025 Lampropoulos et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript and its Supporting information files.
Funding: Basic Research Program, NTUA, PEVE (No. 65232000).
Competing interests: The authors have declared that no competing interests exist.
Introduction
Cancer treatment often combines multiple therapies to enhance efficacy and mitigate resistance. Since the 1965 pioneering clinical trial by Frei et al. [1] on chemotherapeutic combinations for leukemia, efforts have expanded to optimize multi-modal strategies, integrating chemotherapy and radiotherapy [2, 3].
Radiotherapy, delivered as high-energy external beams, remains a cornerstone for many treatment regimens, and is estimated to be administered to over half of all cancer patients during their course of treatment [4]. It primarily induces mitotic death disrupting cellular division in tumors [5–8]. Chemotherapy complements radiotherapy by disrupting cancer cell cycle, with taxanes (e.g., docetaxel) stabilizing microtubules to block division and induce apoptosis [9, 10]. Anti-angiogenesis agents like bevacizumab disrupt the tumor’s vascular support targeting the vascular endothelial growth factor (VEGF), which is essential for the survival of proliferating endothelial cells [11–13]. However, these drugs pose challenges as they can potentially increase tumor invasiveness or hinder drug delivery to the affected site [14, 15].
To improve the delivery of radiotherapeutic and chemotherapeutic agents, many clinical trials and investigations have been made regarding combining treatment modalities. Current studies examine the effects of new combination treatments, especially the treatment modality that combines radiation, the agents of taxane, and bevacizumab [16]. Hainsworth et al. [17] performed a phase II trial of combined modality treatment with chemotherapy, radiation therapy, bevacizumab, and erlotinib (a cytotoxic drug used to treat non-small cell lung cancer among others, brand name Tarceva) in patients with locally advanced squamous carcinoma of the head and neck. Another pilot trial was performed by Wozniak et al. [18] regarding the incorporation of bevacizumab with concurrent radiotherapy in the treatment of locally advanced stage III non-small-cell lung cancer (NSCLC). Apart from clinical trials, radiotherapy is combined with taxane chemotherapy in clinical practice, with the LULACATRT protocol for non-small cell lung cancer [19] and the GIENACTRT protocol for esophageal and gastroesophageal carcinomas [20]. On the other hand, docetaxel and bevacizumab is also a common combination in several cancer treatments [21–24].
Radiation therapy and chemotherapy have been extensively studied through computational modeling. The linear-quadratic (LQ) model proposed by Fowler [25] remains a foundational framework for predicting tumor cell survival post-irradiation influencing various simulation approaches [26–31]. The impact of cytotoxic chemotherapy, anti-angiogenesis therapy and combinations with radiotherapy has been investigated in several computational studies [32–37], using diverse techniques like population balance models [38, 39] and Monte Carlo extensions [40]. Finally, partial differential equations (PDE)-based models have further detailed spatial and temporal tumor dynamics [26, 41, 42].
In this study, we build upon existing literature [43–46] to develop a unified computational model that integrates radiotherapy, cytotoxic chemotherapy and anti-angiogenic therapy. Our extended PDE-based framework simulates the spatio-temporal dynamics of tumor cells and their microenvironment under various standalone and combination treatments for vascular tumors, providing detailed insights into therapy interactions. A notable feature of this model is its ability to analyze the radiosensitization phenomenon, where agents such as taxanes and oxygen enhance radiotherapy efficacy [47, 48]. Radiosensitization is incorporated by computing the spatio-temporal evolution of cellular and chemical species within an affected two-dimensional model tissue, and adjusting the radiotherapy-induced killing rate accordingly.
The paper is structured as follows: in Section Models, we describe the mathematical formulation of our model, focusing on the implementation/integration of radiotherapy in the proposed computational framework. Section Results presents simulations elucidating the impact of standalone radiation treatments and the combined approach of radio-chemotherapy on tumor growth and expansion. Section Conclusion encapsulates a summary of our results, a discussion on the merits and drawbacks inherent in this modeling approach, and recommendations for future research.
Models
In this section, we consolidate the basic mathematical relationships that underpin our two-dimensional multiphase model. We conceptualize the host tissue as a fluid composed of distinct, non-miscible, and interacting cellular-liquid phases. These phases include: I. Healthy cells, denoted with θh. II. Cancer cells, denoted as θc, forming the building blocks of the malignant tumor. III. Young (immature) blood vessels, represented with θyv, that are products of pathological angiogenesis. IV. Mature blood vessels, denoted as θmv, shaping the tissue’s capillary network. V. Interstitial fluid, represented with θint, containing water and solutes.
In addition to cellular phases, the presented model considers four distinct chemical species integral to the described phenomena. These chemicals include: I. Oxygen, which represents nutrients and is denoted with, c. II. The vascular endothelial growth factor (VEGF), denoted with g, a principal factor in angiogenesis and peritumoral vascular network formation. III. The anti-angiogenesis drug bevacizumab, represented with a, which inhibits VEGF. IV. The cytotoxic drug docetaxel, denoted with w, which belongs to the taxane medications family, and exhibits cytotoxic properties.
The model’s core elements encompass mass balance equations for both fluid/cellular and chemical species, as well momentum balance equations employed to compute the macroscopic flow of each fluid phase. Momentum balance equations are complemented by auxiliary algebraic expressions, as detailed bellow.
Cellular phases
The volume fraction of each cellular phase, i, considered in the model is computed through the solution of their mass balance equations. In addition, each phase possesses its velocity field, denoted with , and pressure, pi. The velocity fields are computed by solving momentum balance equations for incompressible fluids in a creeping flow, while pressures are determined through the continuity equation (mass balance for all cellular species) complemented by algebraic equations of state. Inertia is negligible in biological growth processes due to the low Reynolds numbers involved [43, 49].
Mass balance equations for cellular phases.
It is reasonable to assume a uniform density for the tissue being modeled at the spatial scales we are considering. Following this assumption, the mass balance equation can be formulated for each cellular phase (healthy and cancer cells, mature vessels, and interstitial fluid) as follows:
(1)
where θi denotes the volume fraction of cellular phase, i. The term qi denotes the source term for the cellular species, and encompasses all processes related to mass transfer between them.
While the primary transport mechanism for healthy cells, cancer cells, mature vessels, and interstitial fluid is convection (the second term in Eq (1)), for young vessels we introduce chemotaxis as an additional mechanism. In particular, the mass balance equation for young vessels reads:
(2)
where χg denotes the chemoattraction potency parameter. The vascular network in healthy tissues is fully developed and primarily consists of endothelial cells [50]. The endothelial cells at the vascular stalk’s tip exhibit sensitivity to chemotaxis, driven by the presence of vascular endothelial growth factor (VEGF), leading to directed migration towards regions with higher VEGF, g, concentrations [51].
A fundamental premise of our model is based on the assumption that the tissue is a closed system, ensuring zero net mass transfer between cellular species (a common assumption of a wide selection of relevant models [31, 46, 52–55]):
(3)
Lastly, the no-void assumption is incorporated:
(4)
indicating that the tissue domain exclusively comprises the cellular/fluid phase components outlined in the model.
Cellular phases sources terms.
Before we continue with the momentum balance equations, the source/sink terms that correspond to each fluid phase are presented. The source term for healthy cells, qh is:
(5)
Here, the two terms correspond to mitosis and cellular death. The mitosis rate is proportional to the rate constant km,h while kd,h represents the rate constant for death of healthy cells. The mitosis process is influenced by the interstitial fluid θint, which provides necessary materials to the cells. The nutrient concentration is indicated by c, and cp is the parameter for the saturated kinetic term, equal to the oxygen concentration at which the mitosis rate is half-maximal. Additionally, the death rate of healthy cells depends on specific threshold nutrient concentration values, and
. By ensuring
, we confirm that a lower nutrient concentration positively influences the cellular death rate.
Similarly, the source term for cancer cells, qc includes mitosis and cellular death:
(6)
Here, km,c and kd,c are the rate constants for cancer cell mitosis and death, respectively. The described rates are determined by the parameters cp, , and
, which are consistent for both healthy and cancerous cells. These three parameters are assumed to be equal for both healthy and cancer cells to reduce the number of model parameters. Additionally, the rapid proliferation of cancer cells and their high resilience to hypoxia are considered since we stipulate km,c > km,h and kd,c < kd,h.
For young vessels, the source term includes angiogenesis, maturation and cellular occlusion:
(7)
It is evident that angiogenesis is regulated by the rate constant kang. Interstitial material is used for the production of new vessels. A saturated kinetic term is included, with the parameter ϵ denoting the interstitial fluid’s volume fraction at which the angiogenesis rate becomes half maximal. Due to the exerted pressures, the vasculature experiences occlusion proportional to the rate . Assuming that the vascular pressure is uniform and equal to a reference pressure, pref = 0, i.e., pyv = pmv = pref = 0, vessel occlusion starts when the sum of partial pressures, θhph + θcpc, approaches the set threshold,
. This effect is modeled utilizing the function
:
(8)
where h is the steepness parameter.
Lastly, the maturation rate is proportional to and characterised by several parameters (kg1, kg2, m, ν) that ensure the approach’s realistic character [45].
For mature vessels, the maturation term is mirrored and the appropriate occlusion term completes qmv:
(9)
Unlike young vessels, a smaller rate constant and a higher threshold
are used to reflect the mature vasculature’s enhanced mechanical properties.
Momentum balance equations for cellular phases.
It is crucial to emphasize that the primary mode of cellular species transportation is convection. To delve deeper into this process, it becomes essential to recognize that the cellular movement is more accurately characterized by creeping flow, a low Reynolds number flow. This approach allows the formulation of momentum balance equations capturing the intricate dynamics of cellular species transportation:
(10)
Here, the stress tensor, denoted by σi, represents that of a high viscosity, incompressible fluid:
(11)
where μi denotes dynamic viscosity.
(12)
with dij being the drag coefficient between phases i and j.
denotes inter-phase drag and the pressure’s impact on the interface between cellular species [43] Finally, in our computations, we assume the viscosity of the different cellular phases to be uniform, considering their similar chemical composition.
In addition, it is important to highlight that computing of pressure fields, pi, for each cellular phase, involves employing carefully selected algebraic equations, commonly referred to as closures. One such equation, integrated into our model, originates from the assumption of a closed system. This assumption allows us to sum up the mass balance equations of all cellular species, Eqs (1) and (2), and incorporate Eq (3) into the analysis. This approach ensures proper consideration of various components, providing a thorough understanding of pressure fields and their impact on the overall dynamics of the system:
(13) χg denotes the parameter regulating the strength of chemoattraction between the young vessels and VEGF.
Completing the set of algebraic equations for pressure fields involves a specific condition where vascular phase pressures, pyv and pmv, are equated to a reference pressure, pref, since our model does not consider the details of the flow through individual blood vessels, and neglects pressure drops. This reference pressure, which is exerted onto the tissue, is set at zero (pref = 0). The algebraic constraints for pressure fields of healthy and cancer cells follow the work of Hubbard and Byrne [43]:
(14)
This assumes that healthy and cancer cells act like isotropic fluids. When their cellular density exceeds their natural density, their membranes deform and experience stress. This is formulated with Σ(θ) becoming non-zero when the cellular density θ surpasses the numerical value θ*, which is maintained within a healthy tissue:
(15)
The larger the deviation θ − θ* is, the greater the stress, with Λ being a tension constant measuring the tendency of cells to restore their natural density.
Chemical species
In our modeling approach, we have opted to represent nutrient chemical species by focusing on oxygen. While the significance of the nutrient, c, and VEGF, g, remains paramount as inherent components of the studied ecosystem, the inclusion of anti-angiogenesis drug (bevacizumab), and cytotoxic drug (docetaxel) is pivotal for the advancement of our model extension. This extension enables us to investigate the intricate interplay of radiation with anti-VEGF chemotherapy, radiation with cytotoxic chemotherapy, as well as the combined effects of all therapies. The chemical species under consideration are assumed to be soluble in all fluid phases, and their relatively small size renders any contribution to the fluid volume negligibly small. Additionally, the dynamics of these chemical species are significantly faster compared to the cellular phases. More specifically, the processes involving the auxiliary chemical species are completed within minutes to hours, whereas the processes involving cellular phases last from days to weeks. For this reason, it is reasonable to adopt the quasi-steady state approximation for all chemical species.
Mass balance equations for chemical species.
The diffusion is recognized as the primary mechanism governing the transportation of chemical species [56–58]; considering their rapid dynamics, we solve the associated mass balance equations in steady-state conditions. Consequently, the general equation governing mass transfer for chemical species can be formulated as:
(16)
where, Di denotes the diffusion coefficient of chemical species, i, and si represents the corresponding source term associated with said species. The mass balance equations for all chemical species incorporated in our model are complemented by zero-flux (Neumann) boundary conditions.
For the nutrient/oxygen, the source term, sc, is:
(17) krep denotes the nutrient’s replenishment rate constant from the vasculature. cv is the nutrient’s concentration in the vasculature. The nutrient is consumed for cellular sustenance, at a rate proportional to kc,i(i = h, c), and mitosis, at a rate proportional to kcm,i(i = h, c).
There is also, a second chemical species whose source term has to be defined, an that is of course VEGF:
(18)
Here, VEGF’s secretion is performed by the cells, where ω denotes the secretion multiplier for cancer cell secretion. ksec is this process’ rate constant. VEGF production is designed to accelerate in hypoxia and ca is the oxygen concentration for which it reaches its maximum rate [45]. VEGF gets bound by endothelial cell receptors, thus; the parameter kcang denotes the appropriate rate constant for this phenomenon. Lastly, the substance’s decay is proportional to the rate constant kd.g.
Therapy-free model and parameters
For a complete description of the model and a comprehensive listing of parameters pertinent to the therapy-free model, we refer the reader to the S1 File of the manuscript, chapters II and III. We also refer the readers to previous works [44, 45] for a more in-depth analysis of both the therapy-free and the combination chemotherapy model. In the present study, we concentrate on novel enhancements introduced, specifically addressing adaptations necessary for incorporating radiation within our model, particularly concerning the cancerous cells phase. In light of this particular rationale, it is worth noting that the current research abstains from presenting any form of source terminology related to equilibrium equations for diverse cellular and chemical entities, as this is not the central focus of the present investigation.
Radiotherapy, anti-VEGF and cytotoxic chemotherapy
The integration of radiation is a significant expansion to our established model, as elaborated by Lampropoulos et al. [44, 45]. We incorporate both standalone radiation therapy and combined radio-chemotherapy, involving two drugs: the anti-VEGF drug bevacizumab and the cytotoxic drug, docetaxel. Radiation administration involves the precise application of an external beam, targeting the affected region of the malignant growth in the patient’s organism [59].
External radiation therapy often divides the total dose into smaller fractions, typically administered over several weeks to minimize collateral damage to normal tissues. Each therapeutic session typically lasts approximately 30 to 45 minutes, with treatments administered five days a week and weekend breaks, spanning five to eight weeks [60]. Conventional fractionation treatment in radiotherapy employs fraction sizes of 1.8 to 2 Gy resulting in a total weekly dose of 9 to 10 Gy [61]. When used as adjuvant therapy, external beam radiation doses typically range from 45 to 60 Gy for various cancer types such as breast, head, and neck cancers [60].
On the other hand, both pharmaceutical agents are introduced into the human body through intravenous administration [62] and eventually reach the affected tissue through the vascular network. We model the extra-vascular transportation of drugs within the tissue through diffusion.
Radiation.
In the present study, we propose that radiation exclusively impacts the cellular phase of cancer cells. In particular, Eq (19) models the cancer cell death rate due to radiotherapy [46], and formulates an exponential decay over time, with a half-life of for successive radiation fractions administered at preset intervals. By the term fraction, we refer to an individual therapeutic session for radiotherapy administration. Therefore, the radiation kinetic term is formulated as follows:
(19)
Here, the parameter krad quantifies the strength of radiotherapy dose, and trad,i signifies the initiation time of each radiotherapy session, i, ranging from i = 1 to Nrad. H denotes the Heaviside function activating each radiotherapy session for times, t > trad,i. The parameter rt indicates the half-life associated with the death of tumor cells due to radiotherapy. Finally, the time of the ith radiation administration, trad,i, is calculated using:
(20)
with
denoting the time of the first radiation and,
the period between two radiotherapy sessions.
Incorporating radiation in our model involves modifying the cancer cell source term, qc, to accommodate the radiation-induced cellular death kinetics described in Eq (19):
(21)
Radiation therapy parameter determination.
In this section, we provide details for the determination of parameter values used for the implementation of our radiotherapy model. We start our analysis by re-visiting the LQ model that serves as a pivotal mathematical formulation quantifying the survival fraction (SF) of cell colonies under specific radiation doses. LQ provides a practical and straightforward means of establishing the relationship between SF and radiation dose, taking the form of an exponential function encompassing both linear and quadratic components [39, 63]. The pioneering work of Fowler [25] involved calculating the SF for individual cells at a specific dose level, denoted as d, utilizing the LQ cell survival model:
(22)
Here n represents the number of fractions, while d denotes the energy absorbed for each individual dose fraction (measured in Gy). Consequently, the product of n × d signifies the overall dose measured in Grays. Furthermore, α and β, represent the linear and quadratic coefficients, respectively. Notably, the values of α and β are intrinsically linked to the radio-sensitivity of cells. For numerous tumors, the ratio can be estimated from empirical observations, providing valuable insights into treatment [64].
In our simulations, numerical values for α and β are assigned based on the intrinsic radio-sensitivity of the cell type being modeled [65]. In particular, the α/β ratio typically ranges from 3 to 10 Gy for various tumors [66] thus, we utilize the ratio:
(23)
This choice is supported by the LQ model’s ability to accurately replicate in vitro cell survival and depict the impact of clinical fractionation, offering researchers and medical professionals a consistent and reliable means of interpreting the effects of fractionation on tissues and tumors [67].
In a study by Higashi et al. [68], an irradiation dose of 30 Grays (d = 30 Grays) for a single fraction (n = 1) resulted in a significant and exponential decrease in viable cells, emphasizing the profound impact of radiotherapy on cellular viability. This reduction can be quantified by the SF of the cells (SF = 0.16 = 16%), which serves as a metric for evaluating radiotherapy efficacy. Using Eqs (22) and (23) we can thus derive the values of parameters α = 0.015 Gy and β = 0.0015 Gy.
Assuming cancer cells are evenly spread throughout the body, we explore the potential impact of radiotherapy on the cell death rate. The term for cancer cell death caused by radiotherapy Eq (19) can be formulated as:
(24)
Integrating Eq (24):
(25)
Based on the findings of Higashi et al. [68], tumor cells experienced a decline of 84% over the course of a span of 12 days. This implies, that we can adjust the values of parameters, krad and rt, which gives when t = 12. The fitting process produces the following set of parameter values: rt = 0.5 and krad = 0.91, and in Fig 1 we illustrate the evolution of SF according to Eq (25). In our model, the parameter rt, denoting the half-life of cell death induced by radiotherapy, remains constant as it is contingent upon the intricate interplay between tumor cells. The krad value quantifies the radiotherapy intensity, and can be adjusted based on the radiation dosage administered in each treatment session, allowing for modifications in case of a different radiotherapy schedule.
The parameters rt and krad were calibrated to produce a survival fraction, SF = 0.16 after 12 days, in order to fit the observations of Higashi et al. [68].
More specifically, in the case of 30 fractions of 2 Gy, the LQ model predicts a SF ≈ 34%. Through trial and error, we compute that krad = 0.23 leads to a tumor that at the end of the therapeutic course, has a surviving fraction . The deviation between LQ and our model is approximately 4.6%, and can be considered as acceptable, given that in our model, the surviving cells undergo several mitosis cycles during this time interval (the LQ model does not account for mitosis cycles). Hence, the value krad = 0.23 was selected for our simulations.
Table 1 summarises the parameters relevant to the implementation of radiotherapy.
Docetaxel-induced radiosensitization.
Certain chemotherapeutic and other agents possess the capability to augment the effects of radiation in tumors, thereby amplifying the DNA damage caused by radiation [69]. Taxanes, such as docetaxel or paclitaxel, are among these agents. Their mechanism of action entails arresting the cell cycle during the G2/M phase, a phase when cells are highly radiosensitive [70–73]. Recent clinical trials have corroborated this finding by demonstrating promising results regarding the inclusion of docetaxel as a radiosensitizer in patients ineligible for cisplatin-based chemoradiation. Specifically, this therapeutic regimen has shown improvements in both overall survival and progression-free survival among the studied patients [74].
We formulate radiosensitization in our model, by modifying the kinetic term describing the cellular death caused by external radiotherapy, as presented in Eq (19):
(26) ξdoc serves as the radiation’s potency amplifier due to the presence of docetaxel; hs is an appropriately selected smoothing parameter determined by doectaxel’s concentration. According to Eq (26), the radiation’s potency is amplified with increasing values of docetaxel’s concentrations, w. In addition, the radiotherapy-induced killing rate is amplified only for highly proliferative cancer cells the cycle of which is disrupted by the cytotoxic drug. Targeting of highly proliferative cancer cells is modeled utilizing the smoothened Heaviside function,
.
Oxygen radiosensitization.
Another important parameter influencing the effectiveness of radiation is the level of oxygenation in the targeted cells. Oxygen acts as a potent radiosensitizer, significantly boosting the ability of radiation to eradicate cancer cells [48]. In fact, hypoxic cells can exhibit resistance to radiation therapy, contrary to well-oxygenated cells [75, 76].
To incorporate the impact of oxygen-induced radiosensitization into our model, we modify the existing kinetic term, . In contrast to Eq (26), where the initial term remained unaltered and was multiplied by an additional factor, here, the initial krad is substituted to generate a term with an equivalent effect during standalone/monotherapy radiation treatment:
(27)
The updated kinetic term for cellular death induced by external radiation integrates its heightened efficacy against well-oxygenated cells. The term ensures that the death rate constant,
, is amplified with increasing oxygen concentration relative to its initial value.
To ensure an equivalent outcome between the kinetic terms and
when standalone radiotherapy is administered, we derived a new rate constant
, so as to ensure that krad and
produce a comparable killing effect. For a more comprehensive description of the derivation steps for the value of
, we refer the reader to Chapter IV of SI.
Anti-VEGF drug (Bevacizumab).
The mass balance equation for the anti-vascular endothelial growth factor (anti-VEGF) drug bevacizumab is expressed by Eq (16). Within this equation, the source term, sa, plays a pivotal role in formulating and understanding the drug’s behavior and effectiveness:
(28)
where krep,a represents the rate constant for the replenishment of the drug by the vasculature, indicating the efficiency of drug supply to the targeted area; ac represents the drug concentration in the bloodstream; kd,a denotes the drug’s decay rate constant in the tissue; kn,a denotes to the rate constant for drug consumption utilized for neutralizing VEGF. a50 denotes the drug concentration corresponding to the half maximal rate of VEGF neutralization, commonly referred to as the median effective dose. Finally, kcap,yv represents the rate constant at which the drug is consumed, inducing endothelial cells apoptosis. The model does not account for the drug’s transport mechanisms within the bloodstream.
In our proposed model, bevacizumab is administered intravenously during Nbev scheduled sessions. If we denote with the time of administration i (i = 1, …, Nbev), we model the dynamics of the drug concentration within the bloodstream as follows:
(29)
Here, ainj denotes the drug concentration in the bloodstream at the time of injection, and kel,a represents the drug elimination rate constant in the bloodstream (determines how quickly the drug is cleared from the body). Eq (29) can be interpreted as a mathematical model generating a series of exponentially decaying pulses, representing the periodic administration of bevacizumab. In our modeling approach, the time interval between two consecutive sessions is denoted with Tbev, thus .
Including bevacizumab affects the source term of both VEGF (g) and young vessels, (yv):
(30)
with kap,yv being the rate constant of young vessel apoptosis induced by bevacizumab.
(31)
where kn,g is the VEGF neutralization rate constant.
Cytotoxic drug (Docetaxel).
To integrate the cytotoxic drug docetaxel into the model, we need first to formulate the mass balance equation for docetaxel adopting the general mass balance equations for chemical species (Eq (16)). In particular, the source term of docetaxel is formulated as follows:
(32)
Here, w represents the concentration of docetaxel, and krep,w denotes docetaxel’s replenishment constant by the vasculature, representing the rate at which the drug is supplied to the tissue. wc is the concentration of the drug in the bloodstream. The drug decay rate constant in the tissue, denoted as kd,w, indicates the rate at which the drug is metabolized or eliminated from the tissue. The drug consumption rate constants by cancer and healthy cells are denoted as, kcn,c and kcn,h, respectively. Additionally, wm signifies the concentration at which the drug consumption rate by cells reaches its half-maximal point. li represents the proliferation rate of healthy cells (i = h) and cancer cell (i = c), and lcr is a critical value that serves as the criterion for rapidly proliferating cells. Finally, hd is a steepness parameter for the smoothed Heaviside function used to characterise rapidly proliferating cells. The model does not take into account the transport mechanisms of the drug in the blood.
Similarly to bevacizumab, the total number of docetaxel administrations is denoted by Ndoc. The elimination rate constant of the drug in the bloodstream is denoted by kel,w, and the time of the ith docetaxel administration is denoted by, . Then, the evolution of docetaxel concentration in the bloodstream, wc is modelled using:
(33)
where winj is the initial concentration of docetaxel in the bloodstream. If the time interval between two successive administrations is denoted as, Tdoc, then:
.
Furthermore, the inclusion of docetaxel requires appropriate modifications in the source terms for both healthy and cancer cells, denoted as qh and qc, respectively:
(34)
Here, kap,i represents the rate constants for the docetaxel induced apoptosis in healthy (kap,h) and cancer cells (kap,c) alike. We also note that docetaxel specifically targets rapidly proliferating cells [77]. To distinguish between rapidly proliferating cells and those with a lower rate of proliferation, a critical proliferation rate value, lcr, is employed. The smoothened Heaviside function , activates docetaxel’s killing efficacy (last term of Eq (34)) for cells with proliferation rate, li, exceeding the threshold lcr. The proliferation rate is determined by:
(35)
Here, km,i, i = h, c, is the mitosis rate constant for healthy and cancer cells respectively, and cp denotes the nutrient concentration at which mitosis reaches half its maximum rate. To model the higher mitosis rate of cancer cells, we set km,c > km,h. More details are provided in our recent publication [44]. The parameters values associated with chemotherapeutic treatments are provided in the S1 File, chapter II.
Computational framework
Before presenting the results derived from the application of the aforementioned treatment modalities, it is important to provide a concise overview of the computational mesh and solvers used to address the underlying computational problem. In this study, all computations were performed using Comsol Multiphysics, which is grounded in the principles of the Finite Elements Method (FEM). For the calculation of the volume fractions θi, the pressures pi, i = h, c, yv, mv, int, and the dimensionless concentrations j, j = c, g, a, w, linear basis functions were selected. Quadratic basis functions were selected for the discretization of the velocity fields . The Stabilized Convection-Diffusion Equation module from Comsol’s Mathematics library was used for the volume fraction calculations, with consistent stabilization options enabled. For the remaining equations, we utilized the Coefficient form PDE from the Mathematics library. The numerical scheme for time integration was 2nd order BDF, and the direct solver for the solution of the linearized problem was the PARDISO solver. At each time step, the system was solved with a segregated solver consisting of three steps. The first step involved solving the mass balance equations. These solutions were then used to update the velocity and pressure fields in the second step. Finally, the chemical species’ concentrations were calculated in the third step. Anderson acceleration was enabled for the segregated steps.
The affected tissue is modeled by a circular domain with a dimensionless radius, Rtissue, set at 30, unless stated otherwise. One unit of length roughly corresponds to L = 400 μm. This particular length was selected so as to simulate spheroids that are large enough to commence angiogenesis. In addition, one dimensionless unit of time coincides with the time required for a healthy cell’s mitotic division (24 h) [78]. Every calculation performed concerns non-dimensionalised units. The computational domain is discretized using an unstructured mesh generated through the Delaunay Triangulation method, resulting in approximately 70,000 elements and 2,000,000 degrees of freedom. When performing a full simulation using an AMD Ryzen 9 3900X 12-Core Processor for a dimensionless time up to t = 650, the required computational time is approximately 70 hours.
Results
We initiate our investigation by determining the initial conditions imposed at the start of the simulations. These conditions involve inserting a cancerous seed into an otherwise healthy tissue. With the initial conditions set, the next step is to examine how the inclusion of radiotherapy influences the dynamics of a growing tumor as predicted from the proposed mathematical model. Next, we integrate radiotherapy with chemotherapy schedules, specifically in conjunction with both a cytotoxic and anti-VEGF agent. This combination not only broadens the scope of our study but also enables us to investigate potential synergistic effects arising from the simultaneous implementation of these two therapeutic modalities. This approach seeks to capture the intricate interplay treatment modalities and explore the benefits of a multi-faceted therapeutic strategy within the context of our investigation. To provide a comprehensive overview of the various therapeutic protocols employed in our study, we detail the specifics of these treatment regimens in Table 2.
Initial conditions
The computational experiments begin with the insertion of a cancerous seed into the healthy and stable tissue. The initial conditions for all variables are summarized in Table 3. These values represent the solution of a steady-state problem, reflecting the healthy tissue maintained in a stable condition (homeostasis) [43–45].
Radiotherapy simulations
In this section, we investigate the impact of radiotherapy on cancerous tumors. As previously outlined in the introduction, our computational model incorporates an irradiation dose of 2 Gy per fraction. The term “fraction” refers to each session during which irradiation is administered to the malignant region. The radiotherapy treatment spans a therapeutic duration of six weeks, ensuring a comprehensive approach to addressing the malignancy [60].
In Fig 2(a), we depict the evolution of the fraction of the computational domain, S, that is covered by cancer cells throughout the simulation following radiotherapy. We refer to this metric as Surface Coverage (SC), and is calculated as follows:
(36) S denotes the total surface area of the computational domain, and θc is the volume fraction of cancer cells.
To assess the efficacy of radiation therapy, we perform a comparison between simulations of untreated tumors and tumors treated with radiotherapy over time. This comparison is performed for both the surface coverage, SC (indicative of the total number of surviving cancer cells) and the tumor’s radius as depicted in Fig 2(b).
The cancer cell surface coverage in the untreated tumor remains higher at all times compared to the irradiated tumor scenario, underscoring the effectiveness of radiotherapy in reducing the number of cancer cells. Notably, one can observe that at the end of all radiotherapy sessions, the SC value drops to 0.75% (t = 120 dimensionless units), which compares to SC = 7% for the treatment-free simulation. Unfortunately, the tumor experiences a relapse shortly after the completion of the treatment regimen, and indicatively, we report that at dimensionless time, t = 350, the surface coverage of cancer cells for the treatment-free simulation is 28.8%, whereas for the radiotherapy scenario reduces to 22.1%.
The oscillations during radiotherapy sessions that are depicted in Fig 2(a) are attributed to the rapid cancer cell decrease during irradiation, which is followed by short relapses (surviving cancer cells start to multiply again when irradiation ceases).
Upon careful examination of Fig 2(a), one can observe a brief interval, immediately following the completion of all radiotherapy sessions, during which tumor proliferation appears to accelerate before returning to rates akin to those computed for the untreated tumor’s case. This stimulation of cancer growth is attributed to radiation-induced apoptosis within the model. In particular, the material produced by radiation-induced apoptosis enriches the interstitial fluid, thereby providing nourishment to cancer cells. Additionally, the elimination of cancer cells through radiation creates more space for cell expansion and increases oxygen levels within that space.
A similar trend is observed in experiments. Indeed, studies have shown that radiotherapy and cytotoxic chemotherapy-induced apoptosis can stimulate tumor growth [80, 81], partially due to the secretion of pro-apoptotic proteins [82, 83]. Furthermore, stochastic models by Zupanc et al. [84] suggest that the surplus space resulting from radiation treatment can trigger a proliferation response in cancer cells, as they seek to avoid inhibition of cell differentiation caused by encapsulation from neighboring cells.
In Fig 2(b) we show the tumor radius evolution, Rt, which is computed as the average maximum distance at which the cancer cell volume fraction reaches the critical value, θc = 0.01. Examining Fig 2(b), it is clear that radiotherapy effectively stabilizes the tumor radius throughout the treatment sessions. However, once the therapy is completed the surviving cancer cells resume their growth at a rate similar to that observed prior the initiation of therapy.
Additionally, in Fig 3, we present the surface distributions of the interstitial fluid at t = 80, 120 and 200. At t = 80, we observe the interstitial fluid distribution prior to the initiation of radiotherapy. Our model suggests that the increased proliferative capacity of cancer cells leads to a scarcity of interstitial fluid within the spheroid, as cancer-induced inflammation expands and cancer cells consume the fluid’s resources. The second snapshot, at t = 120, marks the end of radiation therapy administration. Here, radiation-induced cancer cell death allows the interstitial fluid to increase into the tumor’s interior. Finally, the third image (right panel) corresponding to t = 200, shows the abundance of interstitial fluid within the tumor’s interior following the formation of the necrotic core.
Combination of radiotherapy with cytotoxic and anti-VEGF chemotherapy
In this paragraph, we focus on the application of a synergistic approach involving the concurrent administration of anti-VEGF and cytotoxic chemotherapy drugs, combined with the influential effects of radiotherapy on malignant tumors. The mathematical model detailing the integration of anti-VEGF and cytotoxic agents has been previously presented in [45], where we demonstrated the enhanced efficacy of combined treatment leading to significant tumor shrinkage. Once again in this study, combination therapy was modeled based on typical clinical practices. The therapy scheduling and the doses administered were calculated and non-dimensionalised according to relevant protocols [21, 22]. As mentioned in the previous paragraph, Table 2 summarises the schedule and dosage of each treatment. A detailed description of the dosing and its non-dimensionalisation can be found in our previous work [45]. Here, we take a step forward to investigate the effect of radiochemotherapy treatments. For a more comprehensive understanding of the combination therapy model and a detailed table containing all non-dimensionalised parameter values utilized, we refer the reader to S1 File, chapters II and III.
Fig 4 presents crucial time stamps relevant to the implementation of combination therapy in our model. As illustrated in the figure, we established a shared starting point for the initiation of each therapy at dimensionless time, t = 80. At this time, all three treatments are simultaneously administered and subsequently, each unfolds independently. We define the therapy duration to extend until the day of the final therapeutic intervention, specifically the last injection of bevacizumab or docetaxel.
Similar to the discussion in the previous section, we present SC and Rt evolution with time in Fig 5. To provide additional insight, beyond standalone radiotherapy, and the cumulative impact of all examined therapies, we explore simulations involving tumors treated with docetaxel, bevacizumab, as well as combined treatments using both agents [45]. As shown in both Fig 5(a) and 5(b), the combined effect of the three therapeutic agents (combined radiochemotherapy) yields the most favorable outcomes among the considered scenarios. More specifically, referring to Fig 5(a), it is evident that radiotherapy plays a pivotal role in the therapeutic regiment by hastening treatment results and significantly reducing the cancer cell count. While its long-term impact may not be as pronounced, it imparts a unique benefit by leaving a more suppressed tumor for other agents to manage. Consequently, this results in a small tumor throughout the entire simulation course (green curve with open triangles) compared to the one associated with the combined chemotherapy (purple curve with stars).
For all simulations, therapies start at dimensionless time, t = 80.
These observations are reinforced by examining Fig 5(b). As we previously established, radiotherapy has the ability to maintain the tumor’s radius stable for its duration, enabling the co-administration of the other two therapeutic agents to achieve substantial tumor shrinkage. This established baseline has enduring effects, as the tumor requires a considerable amount of time to return to its original size, thanks to the lasting benefits provided by the combination of bevacizumab and docetaxel.
In Fig 6, we depict snapshots of cancer cell surface distributions for: (i) a therapy-free simulation, (ii) an irradiated tumor, (iii) a docetaxel/bevacizumab combined chemotherapy regimen and (iii) a combined radiochemotherapy simulation. In the first column, the surface distributions depict a growing tumor without any therapeutic interventions.
The first horizontal line depicts cancer cell volume fraction distributions at dimensionless time, t = 100, and the second horizontal line for dimensionless time, t = 300. By dimensionless time t = 500, the therapy-free and radiotherapy simulations show tumors that approach the computational domain boundaries. In contrast, the distributions for combined chemotherapy and radiochemotherapy at dimensionless time, t = 500, reveal tumors with the lowest growth dynamics. For the calculations presented in this figure, the domain’s radius is set equal to Rtissue = 50.
One can observe three distinct tumor zones: i) the necrotic innermost region of the tumor, where cancer cells have died due to a lack of nutrients; ii) the quiescent zone surrounding the necrotic core, containing living cancer cells that are not actively dividing due to limited access to nutrients; and iii) the proliferating outermost layer of the tumor, where cancer cells have better access to nutrients, supporting their rapid division. In the cases, where the cancer cell volume fraction in the innermost region is practically zero (necrotic core), the only phase present is solely the interstitial fluid. Moving to the second column, the surface distributions depict a tumor subjected to radiation. Although the three tumor zones remain distinguishable, it is apparent that both the tumor’s radius and the total number of cancer cells have been reduced due to radiotherapy. This tumor exhibits morphological similarity to the untreated tumor, albeit with a delay attributable to the application of radiotherapy.
The third column showcases a tumor treated with the bevacizumab/docetaxel combination therapy. Notably, the tumor is suppressed for a significantly prolonged period compared to the tumors in the first two columns. While the short-term effects of radiotherapy are undeniably more potent, the chemical agents appear to hold an advantage in the long term. Finally, the last column presents a tumor treated with all three therapeutic agents (radiochemotherapy). As previously emphasized, triple therapy seamlessly combines the benefits of both radiation therapy -with its immediate results- and the chemical agents -with their sustained efficacy over time. Undoubtedly, this combination promptly yields the highest reduction in cell count and the most effective containment of the tumor radius. At this point, it should also be noted that in all cases, the growing tumor front is not perfectly circular. This irregularity can be attributed to the unstructured computational mesh, as well as the discretised initial conditions that lack radial symmetry. It is important to note that a structured mesh allows for the computational of perfectly radially symmetric structures (see S2 Fig in S1 File). However, our goal is to represent “more realistic” tumor growth patterns that naturally exhibit irregularities. Furthermore, to compute radially symmetric structures one can bypass the use of structured 2D meshes and use more efficient 1D simulations. However, this approach limits the model’s ability to simulate tumor growth in e.g., in complex, radially asymmetric domains or in cases where tumor morphology arises from multiple initial cancerous seeds.
Let us now shift our focus on evaluating the impact of each therapeutic regimen on the healthy cells of the tissue. We introduce a new metric, namely the recession of healthy tissue denoted as ϕ(t). This metric quantifies the difference between the initial surface area covered by healthy cells and the corresponding surface are at each time point, t:
(37)
Fig 7 illustrates ϕ(t) for each treatment, including the untreated, therapy-free tumor. The dynamics of healthy tissue recession are influenced by two primary factors. Firstly, the inflammatory properties of cancer play a pivotal role, leading to nutrient deprivation and the displacement of healthy cells. Secondly, the impact of bevacizumab, which binds to VEGF, and impedes the proliferation of endothelial cells, resulting in vascular recession and subsequent tissue damage [13].
We also illustrate the therapy-free simulation with blue line and open circles. Anti-VEGF therapy (crossed orange line) induces the highest healthy cell recession dynamics, suppressing the nutrient supply. The cytotoxic therapy (yellow line with open diamonds) limits only the short-term damage, however not preventing the long term healthy tissue recession. Combination chemotherapy (purple line with open stars), and radiochemotherapy (green line with open triangles) mitigate the negative impact of anti-VEGF therapy. For the calculations presented in this figure, the domain’s radius is set equal to Rtissue = 50.
The therapeutic regimes presented above, effectively manage to mitigate inflammation (since they mitigate recession of the healthy tissue), thereby safeguarding the surrounding tissue. Radiation therapy aligns with the results depicted in Fig 2, affording short-term protection to the tissue. In contrast, cytotoxic chemotherapy (docetaxel) significantly limits short-term tissue damage while maintaining a less damaged state for an extended time period. However, at later stages of the simulation the tumor relapse is accompanied by significant healthy tissue damage. Notably, one can observe that the anti-VEGF therapy (bevacizumab) intensifies the healthy tissue damage even when compared to the therapy-free simulation, since its primary functionality is to impede the nutrient supply. These findings are in line with clinical findings. The meta-analysis by Zhao et al. [85] underlines the significant risks of hypertension and proteinuria associated with the use of bevacizumab. Specifically, hypertension risks were attributed to a few factors. Bevacizumab, functioning as a VEGF inhibitor, induces cell apoptosis and decreases endothelial renewal; consequently, this leads to reduction of capillary density and elevation of the peripheral vascular resistance, thereby impeding blood circulation. In the present model, the reduced vascular density and endothelial renewal are illustrated. Furthermore, Fig 7 illustrates the negative impact of bevacizumab on healthy cells, depicting a significant healthy tissue recession.
The combination of bevacizumab with the more favorable docetaxel regimen effectively mitigates the former’s side effects. Even though, the healthy tissue recession grows continuously during the entire course of the simulation, at the final stages of the simulation (t ≈ 600) the combination of chemotherapies yields results comparable to cytotoxic monotherapy (ϕ ≈ 90%). Finally, one can observe that the application of chemoradiotherapy reduces the healthy tissue recession compared to the combined chemotherapy regimen.
After presenting the surface plots of the cancerous tissue, we now provide a snapshot (t = 200) of the cancer cell volume fraction alongside the other cellular species. In Fig 8, we observe the surface distributions of healthy and cancerous cells, as well as mature and young vessels at t = 200. Notably, the healthy tissue recedes in response to the expanding cancerous tissue, and the inclusion of the anti-VEGF drug bevacizumab significantly impacts the vasculature. Young vessels are particularly affected, with their volume fractions reduced by approximately 70%.
Returning to the quantification of combination therapy’s efficacy, Fig 9(a) presents the surface coverage, SC, for tumors being treated with a combination of cytotoxic drug, anti-VEGF drug and external radiotherapy. However, in this case, radiotherapy is administered either for half the fractions or, with radiotherapy starting at t = 186, after all chemotherapeutic sessions have concluded. Reducing radiation by half significantly mitigates the anti-tumor effect, especially in the later stages of tumor development. Although the tumor’s response remains roughly homologous with the default regime until t ≈ 400, it experiences a notable relapse thereafter. The combined administration of chemotherapies increases the residence time of docetaxel [45], and delays the onset of tumor relapse. As soon as the concentration of the cytotoxic drug drops below a certain threshold, the surviving cancer cells begin to proliferate unhindered.
(a) Evolution of cancer cells surface area, SC, over time for the following treatments: tumor treated with docetaxel, bevacizumab and radiotherapy all commencing at t = 80 (blue curve with circles- Triple basic); the same therapeutic schedule but with half the radiotherapy fractions (red curve with squares—Triple half); and the initial regimen but with radiotherapy beginning after the chemotherapy sessions conclude (t = 186) (yellow curve with triangles- Triple late). (b) Surface distribution of cancer cells at t = 120 and t = 220 for tumor treated with concurrent docetaxel, bevacizumab and radiotherapy. (c) Surface distribution of cancer cells at t = 225 and t = 325 for tumor treated with delayed radiotherapy. To better display the cancer cell distribution at t = 225, the bottom panel corresponds to the distribution of cancer cells using different coloring scale (with a maximum value of 1.5 × 10−4).
On the other hand, delaying radiation treatment, results in almost total tumor eradication. It is noteworthy that between t = 200 and t = 230, tumor concentrations universally drop below θ = 0.01, rendering Rt (tumor radius) undetectable. However, despite the tumor remaining contained for an extended period, it experiences an aggressive relapse. Admittedly, administering all therapies in parallel leads to a more synergistic action, with docetaxel and bevacizumab targeting a more vulnerable tumor and better managing long-term containment.
Fig 9(b) and 9(c) depict the surface distributions of cancer cells for concurrent radiotherapy (commencing at t = 80) and delayed radiotherapy (commencing at t = 186), respectively. The left column shows cancer cell distributions right after the completion of each radiotherapy administration (t = 120 and t = 225, respectively). The right column presents cancer cell distributions after an interval of 100 dimensionless time points.
Of particular note in this figure is the significant disparity in therapeutic outcomes between the same treatment regimen administered with and without delay. Notably, early/concurrent administration results in the tumor radius remaining relatively stable after a Δt = 100, with a slight increase in cellular density. Conversely, delayed radiotherapy exhausts the tumor, leaving only a small seed capable of spawning a new tumor. By inspecting the tumor at t = 225 (left panel of Fig 9(c)), one can observe that the combined effect of chemo-radiotherapy is particularly effective on the tumor’s periphery. This region, abundant in cellular debris and oxygen, becomes highly conducive to restoration, as evident after Δt = 100. The tumor’s proliferative zone shows signs of restoration, indicating an impending relapse.
Lastly, we are enhancing our understanding of each therapy’s contribution by pairing them up two at a time and comparing their effectiveness. This comparison is illustrated in Fig 10, where the combinations of docetaxel-radiation, bevacizumab-radiation and bevavicumab-docetaxel are administered in the simulated tumors. It becomes evident that inclusion of docetaxel achieves a long term, tumor suppressive environment. Both combinations that exclude docetaxel exhibit poor long-term efficacy. Furthermore, bevacizumab-radiation leads to immediate tumor relapse after completing the radiation fractions. Interestingly, both radiation and bevacizumab function effectively as adjuncts to cytotoxic chemotherapy. Radiation weakens the tumor while docetaxel concentration builds up, making its target more vulnerable. Bevacizumab, on the other hand, prolongs docetaxel’s residence time in the tissue [45].
Radiosensitization
Docetaxel-induced radiosensitization.
Fig 11 highlights the enhanced efficacy of triple-combined therapy, considering the phenomenon of radiosensitization. In particular, we examine the impact of radiosensitization in two cases: (a) when all treatments are administered simultaneously, and (b) when radiation follows the completion of chemotherapy. For each of the two cases, we simulated tumor growth both with and without considering cancer cell radiosensitization. In both scenarios, radiosensitization enhances the efficacy of the administered therapies, as evidenced by consistently lower SC values. Particularly in the case of delayed radiation regimen, radiosensitization extends the duration during which the tumor approaches eradication and mitigates the aggressiveness of tumor relapse.
The same regimen but with radiotherapy starting after the completion of chemotherapy sessions (delayed) is depicted with red curves when neglecting radiosensitization, and with red dotted curves incorporating radiosensitization.
Oxygen radiosensitization.
To highlight the impact of oxygen on radiation therapy, we present the (average) radial distribution of cancer cells at t = 120 (immediately after the last fraction has been administered) in Fig 12 for tumors treated with radiation therapy, with and without oxygen sensitization. While the overall radiation effect is similar in both scenarios, maintaining comparable cancer cell surface coverage (SC) and tumor radius (Rt), the spatial distribution of cancer cells differs notably. For clarity, we show the average cancer cell volume fraction for each azimuthal angle across the tumor’s radius and rescale the volume fractions for each tumor with Fig 12 illustrates the significant impact of radiosensitization on tumor morphology. In comparison to the default scenario without radiosensitization, well-oxygenated regions near the tumor’s periphery show a more pronounced reduction in cancer cells, while the interior shows up to a 50% lesser degree of effect.
The red line illustrates the corresponding distribution incorporating oxygen induced radiosensitization. Both distributions are computed at dimensionless time, t = 120, following the last fraction of administered radiotherapy.
We now examine the incorporation of oxygen radiosensitization in triple-combined therapy for treating tumors. In Fig 13(a) and 13(b), we present the cancer cell surface coverage, SC, for simulations that disregard radiosensitization (blue curves), simulations that incorporate docetaxel-induced sensitization (red curves), and simulations incorporating oxygen-induced sensitization (yellow curves). Fig 13(a) presents simulations when radiation treatment occurs concurrently with combined chemotherapy, all starting at t = 80. Fig 13(b) illustrates the comparison for radiation therapy that begins after the administration of chemotherapy. As tumors progress and depletes tissue oxygen, radiation efficacy diminishes due to increased radio-resistance in hypoxic conditions. This is supported by Fig 13(c) and 13(d). Fig 13(c) presents the surface distribution of the dimensionless concentration of oxygen at t = 80, corresponding to the beginning of concurrent radiochemotherapy. Fig 13(d) shows the surface distribution of the dimensionless concentration of oxygen at t = 186, which marks the beginning of delayed radiotherapy. Delayed radiotherapy encounters a more oxygen-depleted system, resulting in less radiosensitivity.
The first horizontal line ((a) & (b)) illustrates the evolution of cancer cells surface area, SC, over time for tumors treated with triple-therapeutic schemes. The blue lines with circles show the evolution of SC when radiosensitization is not incorporated in the model, the squared red lines represent SC for the case of docetaxel-induced radiosensitization and the yellow lines with triangles illustrate SC for the scenario of oxygen-induced radiosensitization. The second horizontal line ((c) & (d)) displays the distribution of oxygen concentration. Panel (c) corresponds to the time of administration of concurrent triple radiochemotherapy (t = 80), while panel (d) corresponds to t = 186, the time of delayed radiation therapy administration.
The new parameters associated with radiosensitization are presented in Table 4.
Model limitations
At this point, we should acknowledge certain limitations of the proposed model. Firstly, a significant constraint lies in the large number of parameters incorporated in the model, and this complexity is inherent to capturing a wide array of phenomena. Calibrating the model with extensive experimental data poses a substantial challenge in its current form. However, addressing this limitation could pave the way for future improvements beyond the scope of this study. Specifically, conducting a sensitivity analysis that identifies the parameters with the most significant impact on the model output is a promising research direction. Additionally, while the computational demands of our current work are not prohibitive, they are not trivial either and could pose a challenge for translational research. The current model is solved within a two dimensional domain, which may impose certain limitations. As indicated in the literature, interactions occurring in a 3D environment significantly contribute to tumor growth [86, 87]. Extending the model to three spatial dimensions would considerably increase the system’s degrees of freedom, thereby escalating computational costs. This underscores the importance of considering parallel processing for future research endeavors. Finally, it should be noted that the model formulating the formation and occlusion of vessels can be enhanced by employing discrete modeling approaches, where blood vessels are represented as a series of discrete points and incorporating stochastic growth of blood vessels [88, 89]. Such hybrid models have the potential to offer a more detailed framework for understanding the interactions involved in anti-angiogenic treatments on vasculature.
Conclusion
Our study centers on modeling host tissue as a highly viscous and multiphase fluid. Utilizing a continuous model framework, we employ mass and momentum balance equations solving them with the Finite Elements Method (FEM) through the commercial software Comsol Multiphysics ® to ensure both accuracy and efficiency.
A distinctive feature of our model is its versatility, seamlessly integrating various therapeutic agents, including cytotoxic chemotherapy (docetaxel), and anti-VEGF (bevacizumab), with external radiation. This is the first study integrating all three therapies (cytotoxic, anti-angiogenic and radiation) in one model and monitoring their interplay and short vs long-term intricacies of this schema. When studying single-radiotherapy, our model manages to capture the accelerated tumor growth that is experimentally reported after the completion of radiotherapy [80, 81, 84]. In addition, the model showcases the synergistic effects of combined radiochemotherapy, elucidating the intricate interplay between the immediate action of radiation and the long-term effectiveness of chemotherapy. That synergistic effect between therapeutic modalities is further explored after taking into consideration the phenomena of oxygen and taxane induced radiosensitization. We showcase the enhanced cancer cell killing efficacy of radiation therapy when combined with docetaxel [70–73] and on well oxygenated targets [75, 76]. This observation qualitatively aligns with experimental data, supporting the effectiveness of the radiation-taxane combination and the inclusion of bevacizumab, as reported in studies such as [90, 91]. It should be noted that the treatment-schedule adopted in our model follows the protocol for non-small lung cancer [19].
Although our research has advanced our understanding of combination therapy and tumor response, there remains significant potential for exploration. Platinum-based drugs, frequently used with bevacizumab and taxanes, warrant further investigation [21, 22, 92, 93], alongside integrating non-classic chemotherapy forms like immunotherapy (e.g., virotherapy, T-cell therapy) [94]. While this study focuses on specific cancer categories such as non-small cell lung and gynecological cancers, additional clinical data are needed to refine the model for other cancer types and improve variable accuracy. Moreover, the presented radiochemotherapy scheme does not optimize treatment parameters or schedule complexities. Previous findings highlighted the importance of therapy scheduling [45], suggesting that Gaussian (or Bayesian) optimization could systematically enhance treatment timing and success [95]. This method, leveraging probabilistic models and efficient evaluation, could optimize initiation intervals in triple-therapy schemes or explore fractionated radiotherapy protocols with varied dose distributions.
Supporting information
S1 File. Supplementary material.
This file includes: Section I: Boundary conditions; Section II: Parameters for therapy-free model, bevacizumab and docetaxel implementation; Section III: Non-dimensionalized model; Section IV: Oxygen Sensitization parameter determination, Section V: Simulations in structured and unstructured meshes.
https://doi.org/10.1371/journal.pone.0301657.s001
(PDF)
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