The response of tumours with Gompertzian growth characteristics to fractionated radiotherapy

Mathematical modeling of the synergistic interplay of radiotherapy and immunotherapy in anti-cancer treatments

Introduction

While radiotherapy has long been recognized for its ability to directly ablate cancer cells through necrosis or apoptosis, radiotherapy-induced abscopal effect suggests that its impact extends beyond local tumor destruction thanks to immune response. Cellular proliferation and necrosis have been extensively studied using mathematical models that simulate tumor growth, such as Gompertz law, and the radiation effects, such as the linear-quadratic model. However, the effectiveness of radiotherapy-induced immune responses may vary among patients due to individual differences in radiation sensitivity and other factors.

Methods

We present a novel macroscopic approach designed to quantitatively analyze the intricate dynamics governing the interactions among the immune system, radiotherapy, and tumor progression. Building upon previous research demonstrating the synergistic effects of radiotherapy and immunotherapy in cancer treatment, we provide a comprehensive mathematical framework for understanding the underlying mechanisms driving these interactions.

Results

Our method leverages macroscopic observations and mathematical modeling to capture the overarching dynamics of this interplay, offering valuable insights for optimizing cancer treatment strategies. One shows that Gompertz law can describe therapy effects with two effective parameters. This result permits quantitative data analyses, which give useful indications for the disease progression and clinical decisions.

Discussion

Through validation against diverse data sets from the literature, we demonstrate the reliability and versatility of our approach in predicting the time evolution of the disease and assessing the potential efficacy of radiotherapy-immunotherapy combinations. This further supports the promising potential of the abscopal effect, suggesting that in select cases, depending on tumor size, it may confer full efficacy to radiotherapy.

Tumor Volume Regression during and after Radiochemotherapy: A Macroscopic Description

Tumor volume regression during and after chemo and radio therapy is a useful information for clinical decisions. Indeed, a quantitative, patient oriented, description of the response to treatment can guide towards the modification of the scheduled doses or the evaluation of the best time for surgery. We propose a macroscopic algorithm which permits to follow quantitatively the time evolution of the tumor volume during and after radiochemotherapy. The method, initially validated with different cell-lines implanted in mice, is then successfully applied to the available data for partially responding and complete recovery patients.

Non-Homogeneous Tumor Growth and Its Implications for Radiotherapy: A Phenomenological Approach

Tumor regrowth and heterogeneity are important clinical parameters during radiotherapy, and the probability of treatment benefit critically depends on the tumor progression pattern in the interval between the fractional irradiation treatments. We propose an analytic, easy-to-use method to take into account clonal subpopulations with different specific growth rates and radiation resistances. The different strain regrowth effects, as described by Gompertz law, require a dose-boost to reproduce the survival probability of the corresponding homogeneous system and for uniform irradiation. However, the estimate of the survival fraction for a tumor with a hypoxic subpopulation is more reliable when there is a slow specific regrowth rate and when the dependence on the oxygen enhancement ratio of radiotherapy is consistently taken into account. The approach is discussed for non-linear two-population dynamics for breast cancer and can be easily generalized to a larger number of components and different tumor phenotypes.

Comparison of CD38-Targeted α- Versus β-Radionuclide Therapy of Disseminated Multiple Myeloma in an Animal Model

Targeted therapies for multiple myeloma (MM) include the anti-CD38 antibody daratumumab, which, in addition to its inherent cytotoxicity, can be radiolabeled with tracers for imaging and with β- and α-emitter radionuclides for radioimmunotherapy. Methods: We have compared the potential therapeutic efficacy of β- versus α-emitter radioimmunotherapy using radiolabeled DOTA-daratumumab in a preclinical model of disseminated multiple myeloma. Multiple dose levels were investigated to find the dose with the highest efficacy and lowest toxicity. Results: In a dose–response study with the β-emitter 177Lu-DOTA-daratumumab, the lowest tested dose, 1.85 MBq, extended survival from 37 to 47 d but did not delay tumor growth. Doses of 3.7 and 7.4 MBq extended survival to 55 and 58 d, respectively, causing a small equivalent delay in tumor growth, followed by regrowth. The higher dose, 11.1 MBq, eradicated the tumor but had no effect on survival compared with untreated controls, because of whole-body toxicity. In contrast, the α-emitter 225Ac-DOTA-daratumumab had a dose-dependent effect, in which 0.925, 1.85, and 3.7 kBq increased survival, compared with untreated controls (35 d), to 47, 52, and 73 d, respectively, with a significant delay in tumor growth for all 3 doses. Higher doses of 11.1 and 22.2 kBq resulted in equivalent survival to 82 d but with significant whole-body toxicity. Parallel studies with untargeted 225Ac-DOTA-trastuzumab conferred no improvement over untreated controls and resulted in whole-body toxicity. Conclusion: We conclude, and mathematic modeling confirms, that maximal biologic doses were achieved by targeted α-therapy and demonstrated 225Ac to be superior to 177Lu in delaying tumor growth and decreasing whole-body toxicity.

A mathematical model for predicting the changes of non-small cell lung cancer based on tumor mass during radiotherapy

This study aims to build a feasible mathematical model to analyze the mass evolution of NSCLC during standard fractionated radiotherapy. Seventy-three cases of NSCLC who were received radiotherapy with prescription dose of 2 Gy  ×  30 fx were selected retrospectively and divided into adenocarcinoma (ADC) group and squamous cell carcinoma (SCC) group according to the pathological type. A total of six sets of CT/CBCT images were collected. The tumor masses were measured according to each set of images. We build a mathematical model (Linear Quadratic_Repopulation&Reoxygenation& Dissolution model, LQ_RRD model), which was used to fit the first five sets of measured mass into a smooth curve. By adjusting the model parameters (λ, ν and µ), the optimal fitting results can be obtained. In order to verify the accuracy of model prediction, we measured the mass of the review images (MV, measured values), and found out the estimate point of the corresponding time (EV, estimated value) on the fitting curve. The difference and correlation between MV and EV were compared. It was found that the model could substantially simulate the tumor mass changes during radiotherapy, and it had a good fit to the clinical data (%RMSE-Median  =  5.52, %RMSE-Range  =  [3.19, 10.73]). Comparing the differences of model parameters between ADC and SCC group, there was no significant difference in λ (t  =  1.622, p   =  0.109), but the difference was significant in ν and µ (z  =  -7.270, p   =  0.000 and t  =  -10.205, p   =  0.000). Moreover, linear correlation analysis showed that there was a linear correlation between MV and EV no matter mass or volume (r  =  0.960, p   =  0.000 versus r  =  0.926, p   =  0.000). Nevertheless, the deviation between MV and EV of volume was larger than that of mass (z  =  -1.897, p   =  0.058 versus z  =  -3.387, p   =  0.001), and the deviation was more pronounced in larger tumors. We suggest that this mathematical model is more suitable to predict the tumor mass than volume for NSCLC during radiotherapy.

Mathematical Modeling of Preclinical Alpha-Emitter Radiopharmaceutical Therapy

Preclinical studies, in vivo, and in vitro studies, in combination with mathematical modeling can help optimize and guide the design of clinical trials. The design and optimization of alpha-particle emitter radiopharmaceutical therapy (αRPT) is especially important as αRPT has the potential for high efficacy but also high toxicity. We have developed a mathematical model that may be used to identify trial design parameters that will have the greatest impact on outcome. The model combines Gompertzian tumor growth with antibody-mediated pharmacokinetics and radiation-induced cell killing. It was validated using preclinical experimental data of antibody-mediated ²¹³Bi and ²²⁵Ac delivery in a metastatic transgenic breast cancer model. In modeling simulations, tumor cell doubling time, administered antibody, antibody specific-activity, and antigen-site density most impacted median survival. The model was also used to investigate treatment fractionation. Depending upon the time-interval between injections, increasing the number of injections increased survival time. For example, two administrations of 200 nCi, ²²⁵Ac-labeled antibody, separated by 30 days, resulted in a simulated 31% increase in median survival over a single 400 nCi administration. If the time interval was 7 days or less, however, there was no improvement in survival; a one-day interval between injections led to a 10% reduction in median survival. Further model development and validation including the incorporation of normal tissue toxicity is necessary to properly balance efficacy with toxicity. The current model is, however, useful in helping understand preclinical results and in guiding preclinical and clinical trial design towards approaches that have the greatest likelihood of success. SIGNIFICANCE: Modeling is used to optimize αRPT.

Comparison of hyper- and hypofractionated radiation schemes with IMRT technique in small cell lung cancer: Clinical outcomes and the introduction of extended LQ and TCP models

PURPOSE

To evaluate the outcomes of 45 Gy/15 fractions/once-daily and 45 Gy/30 fractions/twice-daily radiation schemes utilizing intensity-modulated radiation therapy (IMRT) in extensive stage small cell lung cancer (SCLC), and to build up a new radiobiological model for tumor control probability (TCP) considering multiple biological effects.

METHODS

Fifty-eight consecutive patients diagnosed with extensive stage SCLC, treated with chemotherapy and chest irradiation, were retrospectively reviewed. Thirty-seven received hyperfractionated IMRT (Hyper-IMRT, 45 Gy/30 fractions/twice-daily) and 21 received hypofractionated IMRT (Hypo-IMRT, 45 Gy/15 fractions/once-daily). Local progression-free survival (LPFS) and overall survival (OS) were calculated and compared. An extended linear-quadratic (LQ) model, LQRG, incorporating cell repair, redistribution, reoxygenation, regrowth and Gompertzian tumor growth was created based on the clinical data. The TCP model was reformulated to predict LPFS. The classical LQ and TCP models were compared with the new models. Akaike information criterion (AIC) was used to assess the quality of the models.

RESULTS

The 2-year LPFS (34.1% vs 27.9%, p = 0.44) and OS (76.9% vs 76.9%, p = 0.26) were similar between Hyper- and Hypo-IMRT patients. According to the LQRG model, the α/β calculated was 9.2 (95% confidence interval: 8.7-9.9) Gy after optimization. The average absolute and relative fitting errors for LPFS were 9.1% and 18.7% for Hyper-IMRT, and 8.8% and 16.2% for Hypo-IMRT of the new TCP model, compared with 29.1% and 62.3% for Hyper-IMRT, and 30.7% and 65.3% for Hypo-IMRT of the classical model.

CONCLUSIONS

Hypo- and Hyper-IMRT resulted in comparable local control in the chest irradiation of extensive stage SCLC. The LQRG model has better performance in predicting the TCP (or LPFS) of the two schemes.

A radiobiological model of radiotherapy response and its correlation with prognostic imaging variables

Radiobiological models of tumour control probability (TCP) can be personalized using imaging data. We propose an extension to a voxel-level radiobiological TCP model in order to describe patient-specific differences and intra-tumour heterogeneity. In the proposed model, tumour shrinkage is described by means of a novel kinetic Monte Carlo method for inter-voxel cell migration and tumour deformation. The model captures the spatiotemporal evolution of the tumour at the voxel level, and is designed to take imaging data as input. To test the performance of the model, three image-derived variables found to be predictive of outcome in the literature have been identified and calculated using the model's own parameters. Simulating multiple tumours with different initial conditions makes it possible to perform an in silico study of the correlation of these variables with the dose for 50% tumour control ([Formula: see text]) calculated by the model. We find that the three simulated variables correlate with the calculated [Formula: see text]. In addition, we find that different variables have different levels of sensitivity to the spatial distribution of hypoxia within the tumour, as well as to the dynamics of the migration mechanism. Finally, based on our results, we observe that an adequate combination of the variables may potentially result in higher predictive power.

Kinetic Models for Predicting Cervical Cancer Response to Radiation Therapy on Individual Basis Using Tumor Regression Measured In Vivo With Volumetric Imaging

This article describes a macroscopic mathematical modeling approach to capture the interplay between solid tumor evolution and cell damage during radiotherapy. Volume regression profiles of 15 patients with uterine cervical cancer were reconstructed from serial cone-beam computed tomography data sets, acquired for image-guided radiotherapy, and used for model parameter learning by means of a genetic-based optimization. Patients, diagnosed with either squamous cell carcinoma or adenocarcinoma, underwent different treatment modalities (image-guided radiotherapy and image-guided chemo-radiotherapy). The mean volume at the beginning of radiotherapy and the end of radiotherapy was on average 23.7 cm(3) (range: 12.7-44.4 cm(3)) and 8.6 cm(3) (range: 3.6-17.1 cm(3)), respectively. Two different tumor dynamics were taken into account in the model: the viable (active) and the necrotic cancer cells. However, according to the results of a preliminary volume regression analysis, we assumed a short dead cell resolving time and the model was simplified to the active tumor volume. Model learning was performed both on the complete patient cohort (cohort-based model learning) and on each single patient (patient-specific model learning). The fitting results (mean error: ∼ 16% and ∼ 6% for the cohort-based model and patient-specific model, respectively) highlighted the model ability to quantitatively reproduce tumor regression. Volume prediction errors of about 18% on average were obtained using cohort-based model computed on all but 1 patient at a time (leave-one-out technique). Finally, a sensitivity analysis was performed and the data uncertainty effects evaluated by simulating an average volume perturbation of about 1.5 cm(3) obtaining an error increase within 0.2%. In conclusion, we showed that simple time-continuous models can represent tumor regression curves both on a patient cohort and patient-specific basis; this discloses the opportunity in the future to exploit such models to predict how changes in the treatment schedule (number of fractions, doses, intervals among fractions) might affect the tumor regression on an individual basis.

On the probability of cure for heavy-ion radiotherapy

The probability of a cure in radiation therapy (RT)-viewed as the probability of eventual extinction of all cancer cells-is unobservable, and the only way to compute it is through modeling the dynamics of cancer cell population during and post-treatment. The conundrum at the heart of biophysical models aimed at such prospective calculations is the absence of information on the initial size of the subpopulation of clonogenic cancer cells (also called stem-like cancer cells), that largely determines the outcome of RT, both in an individual and population settings. Other relevant parameters (e.g. potential doubling time, cell loss factor and survival probability as a function of dose) are, at least in principle, amenable to empirical determination. In this article we demonstrate that, for heavy-ion RT, microdosimetric considerations (justifiably ignored in conventional RT) combined with an expression for the clone extinction probability obtained from a mechanistic model of radiation cell survival lead to useful upper bounds on the size of the pre-treatment population of clonogenic cancer cells as well as upper and lower bounds on the cure probability. The main practical impact of these limiting values is the ability to make predictions about the probability of a cure for a given population of patients treated to newer, still unexplored treatment modalities from the empirically determined probability of a cure for the same or similar population resulting from conventional low linear energy transfer (typically photon/electron) RT. We also propose that the current trend to deliver a lower total dose in a smaller number of fractions with larger-than-conventional doses per fraction has physical limits that must be understood before embarking on a particular treatment schedule.

Modelling the interplay between hypoxia and proliferation in radiotherapy tumour response

A tumour control probability computational model for fractionated radiotherapy was developed, with the goal of incorporating the fundamental interplay between hypoxia and proliferation, including reoxygenation over a course of radiotherapy. The fundamental idea is that the local delivery of oxygen and glucose limits the amount of proliferation and metabolically-supported cell survival a tumour sub-volume can support. The model has three compartments: a proliferating compartment of cells receiving oxygen and glucose; an intermediate, metabolically-active compartment receiving glucose; and a highly hypoxic compartment of starving cells. Following the post-mitotic cell death of proliferating cells, intermediate cells move into the proliferative compartment and hypoxic cells move into the intermediate compartment. A key advantage of the proposed model is that the initial compartmental cell distribution is uniquely determined from the assumed local growth fraction (GF) and volume doubling time (TD) values. Varying initial cell state distributions, based on the local (voxel) GF and TD, were simulated. Tumour response was simulated for head and neck squamous cell carcinoma using relevant parameter values based on published sources. The tumour dose required to achieve a 50% local control rate (TCD50) was found for various GFs and TD's, and the effect of fraction size on TCD50 was also evaluated. Due to the advantage of reoxygenation over a course of radiotherapy, conventional fraction sizes (2-2.4 Gy fx(-1)) were predicted to result in smaller TCD50's than larger fraction sizes (4-5 Gy fx(-1)) for a 10 cc tumour with GFs of around 0.15. The time to eliminate hypoxic cells (the reoxygenation time) was estimated for a given GF and decreased as GF increased. The extra dose required to overcome accelerated stem cell accumulation in longer treatment schedules was estimated to be 0.68 Gy/day (in EQD26.6), similar to published values derived from clinical data. The model predicts, for a 2 Gy/weekday fractionation, that increased initial proliferation (high GF) should, surprisingly, lead to moderately higher local control values. Tumour hypoxia is predicted to increase the required dose for local control by approximately 30%. Predicted tumour regression patterns are consistent with clinical observations. This simple yet flexible model shows how the local competition for chemical resources might impact local control rates under varying fractionation conditions.

The model muddle: in search of tumor growth laws

In this article, we will trace the historical development of tumor growth laws, which in a quantitative fashion describe the increase in tumor mass/volume over time. These models are usually formulated in terms of differential equations that relate the growth rate of the tumor to its current state and range from the simple one-parameter exponential growth model to more advanced models that contain a large number of parameters. Understanding the assumptions and consequences of such models is important, as they often underpin more complex models of tumor growth. The conclusion of this brief survey is that although much improvement has occurred over the last century, more effort and new models are required if we are to understand the intricacies of tumor growth.

In silico modelling of treatment-induced tumour cell kill: developments and advances

Mathematical and stochastic computer (in silico) models of tumour growth and treatment response of the past and current eras are presented, outlining the aims of the models, model methodology, the key parameters used to describe the tumour system, and treatment modality applied, as well as reported outcomes from simulations. Fractionated radiotherapy, chemotherapy, and combined therapies are reviewed, providing a comprehensive overview of the modelling literature for current modellers and radiobiologists to ignite the interest of other computational scientists and health professionals of the ever evolving and clinically relevant field of tumour modelling.

THREE TYPES OF SIMPLE DDE'S DESCRIBING TUMOR GROWTH

In this paper, we compare three types of dynamical systems used to describe tumor growth. These systems are defined as solutions to three delay differential equations: the logistic, the Gompertz and the Greenspan types. We present analysis of these systems and compare with experimental data for Ehrlich Ascites tumor in mice.

Radiotherapy and risks of tumor regrowth or inducing second cancer

Considerable research is aimed at determining the mechanism by which tumor cures, or regrows or second cancer develops, to be predictable and controllable. The wide range of doses, from low to very high, estimated statistically is responsible for such risks. A mathematical model is presented that describes both: the growth due to lower or over irradiated doses or the post therapy relapse of human cancer, and the shrinkage due to either of over irradiated doses, or appropriate irradiated doses. Simulations of the presented model showed that the initial tumor energy, administered dose energy, and their subsequent summation of tumor regrowth energy are always balanced with summation of Whole Body Cell Energy Burden during all treatment phases. Tumor regrows if its energy is higher than that of the dose, or if the increase of dose energy from that of the tumor is less than the one required to complete its shrinkage path. Patient-specific approaches that account for variations in tumor energies should enable more accurate dose estimates and, consequently, better protection against either lower or over irradiation that could lead to tumor regrowth and increase risks of second cancer.

A mathematical model for brain tumor response to radiation therapy

Gliomas are highly invasive primary brain tumors, accounting for nearly 50% of all brain tumors (Alvord and Shaw in The pathology of the aging human nervous system. Lea & Febiger, Philadelphia, pp 210-281, 1991). Their aggressive growth leads to short life expectancies, as well as a fairly algorithmic approach to treatment: diagnostic magnetic resonance image (MRI) followed by biopsy or surgical resection with accompanying second MRI, external beam radiation therapy concurrent with and followed by chemotherapy, with MRIs conducted at various times during treatment as prescribed by the physician. Swanson et al. (Harpold et al. in J Neuropathol Exp Neurol 66:1-9, 2007) have shown that the defining and essential characteristics of gliomas in terms of net rates of proliferation (rho) and invasion (D) can be determined from serial MRIs of individual patients. We present an extension to Swanson's reaction-diffusion model to include the effects of radiation therapy using the classic linear-quadratic radiobiological model (Hall in Radiobiology for the radiologist. Lippincott, Philadelphia, pp 478-480, 1994) for radiation efficacy, along with an investigation of response to various therapy schedules and dose distributions on a virtual tumor (Swanson et al. in AACR annual meeting, Los Angeles, 2007).

Critical parameters determining standard radiotherapy treatment outcome for glioblastoma multiforme: a computer simulation

The aim of this paper is to investigate the most critical parameters determining radiotherapy treatment outcome in terms of tumor cell kill for glioblastoma multiforme tumors by using an already developed simulation model of in vivo tumor response to radiotherapy.

Semi-mechanistic modelling of the tumour growth inhibitory effects of LY2157299, a new type I receptor TGF-beta kinase antagonist, in mice

Human xenografts Calu6 (non-small cell lung cancer) and MX1 (breast cancer) were implanted subcutaneously in nude mice and LY2157299, a new type I receptor TGF-beta kinase antagonist, was administered orally. Plasma levels of LY2157299, percentage of phosphorylated Smad2,3 (pSmad) in tumour, and tumour size were used to establish a semi-mechanistic pharmacokinetic/pharmacodynamic model. An indirect response model was used to relate plasma concentrations with pSmad. The model predicts complete inhibition of pSmad and rapid turnover rates [t(1/2) (min)=18.6 (Calu6) and 32.0 (MX1)]. Tumour growth inhibition was linked to pSmad using two signal transduction compartments characterised by a mean signal propagation time with estimated values of 6.17 and 28.7 days for Calu6 and MX1, respectively. The model provides a tool to generate experimental hypothesis to gain insights into the mechanisms of signal transduction associated to the TGF-beta membrane receptor type I.

Growth laws in cancer: implications for radiotherapy

Comparing the conventional Gompertz tumor growth law (GL) with the "Universal" law (UL), which has recently been proposed and applied to cancer, we have investigated the implications of the growth laws for various radiotherapy regimens. According to the GL, the surviving tumor cell fraction could be reduced ad libitum, independent of the initial tumor mass, simply by increasing the number of treatments. In contrast, if tumor growth dynamics follows the Universal scaling law, there is a lower limit of the surviving fraction that cannot be reduced further regardless of the total number of treatments. This finding can explain the so-called tumor size effect and re-emphasizes the importance of early diagnosis because it implies that radiotherapy may be successful provided that the tumor mass at treatment onset is rather small. Taken together with our previous work, the implications of these findings include revisiting standard radiotherapy regimens and treatment protocols overall.

Mathematical modeling of brain tumors: effects of radiotherapy and chemotherapy

Gliomas, the most common primary brain tumors, are diffusive and highly invasive. The standard treatment for brain tumors consists of a combination of surgery, radiation therapy and chemotherapy. Over the past few years, mathematical models have been applied to study untreated and treated brain tumors. In an effort to improve treatment strategies, we consider a simple spatio-temporal mathematical model, based on proliferation and diffusion, that incorporates the effects of radiotherapeutic and chemotherapeutic treatments. We study the effects of different schedules of radiation therapy, including fractionated and hyperfractionated external beam radiotherapy, using a generalized linear quadratic (LQ) model. The results are compared with published clinical data. We also discuss the results for combination therapy (radiotherapy plus temozolomide, a new chemotherapy agent), as proposed in recent clinical trials. We use the model to predict optimal sequencing of the postoperative (combination of radiotherapy and adjuvant, neo-adjuvant or concurrent chemotherapy) treatments for brain tumors.

Investigation of various growth mechanisms of solid tumour growth within the linear-quadratic model for radiotherapy

The standard linear-quadratic survival model for radiotherapy is used to investigate different schedules of radiation treatment planning to study how these may be affected by different tumour repopulation kinetics between treatments. The laws for tumour cell repopulation include the logistic and Gompertz models and this extends the work of Wheldon et al (1977 Br. J. Radiol. 50 681), which was concerned with the case of exponential re-growth between treatments. Here we also consider the restricted exponential model. This has been successfully used by Panetta and Adam (1995 Math. Comput. Modelling 22 67) in the case of chemotherapy treatment planning. Treatment schedules investigated include standard fractionation of daily treatments, weekday treatments, accelerated fractionation, optimized uniform schedules and variation of the dosage and alpha/beta ratio, where alpha and beta are radiobiological parameters for the tumour tissue concerned. Parameters for these treatment strategies are extracted from the literature on advanced head and neck cancer, prostate cancer, as well as radiosensitive parameters. Standardized treatment protocols are also considered. Calculations based on the present analysis indicate that even with growth laws scaled to mimic initial growth, such that growth mechanisms are comparable, variation in survival fraction to orders of magnitude emerged. Calculations show that the logistic and exponential models yield similar results in tumour eradication. By comparison the Gompertz model calculations indicate that tumours described by this law result in a significantly poorer prognosis for tumour eradication than either the exponential or logistic models. The present study also shows that the faster the tumour growth rate and the higher the repair capacity of the cell line, the greater the variation in outcome of the survival fraction. Gaps in treatment, planned or unplanned, also accentuate the differences of the survival fraction given alternative growth dynamics.

Modeling and computer simulations of tumor growth and tumor response to radiotherapy

A model of tumor growth and tumor response to radiation is introduced in which each tumor cell is taken into account individually. Each cell is assigned a set of radiobiological parameters, and the status of each cell is checked in discrete intervals. Tumor proliferation is governed by the cell cycle times of tumor cells, the growth fraction, the apoptotic capacity of the tumor, and the degree of tumor angiogenesis. The response of tumor cells to radiation is determined by the radiosensitivities and the oxygenation status. Computer simulation is performed on a 3D rigid cubic lattice, starting out from a single tumor cell. Random processes are simulated by Monte Carlo methods. Short cell cycle time, high growth fraction, and tumor angiogenesis all increase tumor proliferation rates. Accelerated time-dose patterns result in lower total doses needed for tumor control, but the extent of dose reduction depends on the kinetics and the radiosensitivities of tumor cells. Tumor angiogenesis alters fully oxygenated and hypoxic fractions within the tumor and subsequently affects the radiation response. It is demonstrated for selected radiobiological parameters that the simulation tools are suitable to quantitatively assess the total doses needed for tumor control. Using the simulation tools, it is feasible to simulate time-dependent effects during fractionated radiotherapy and to compare different time-dose patterns in terms of their tumor control.

Effect on tumour control of time interval between surgery and postoperative radiotherapy: an empirical approach using Monte Carlo simulation

In this work, a procedure, based on Monte Carlo techniques, to analyse the effect on the tumour control probability of the time interval between surgery and postoperative radiotherapy is presented. The approach includes the tumour growth as well as the survival of tumour cells undergoing fractionated radiotherapy. Both processes are described in terms of the binomial distribution. We have considered two different growth models, exponential and Gompertz, the parameters of which have been fixed to reproduce the clinical outcome corresponding to a retrospective study for patients with head and neck squamous cell carcinomas. In the cases analysed, we have not found significant differences between the results obtained for both growth models. The mean doubling times found for residual clonogens after surgery are less than 40 days. The rate of decrease in local control is around 0.09% per day of delay between surgery and radiotherapy and the corresponding time factor is about 0.11 Gy per day.

Dose dependence of the growth rate of multicellular tumour spheroids after irradiation

The present study investigated differences in the growth rate of multicellular tumour spheroids of the MCF-7 line of human breast cancer before and after their irradiation. Growth of the spheroids was analysed according to a model based on a Gompertz function. In this model, normalization to a common initial volume is achieved in a way that enables meaningful comparisons to be made between the results obtained for each spheroid. For irradiated spheroids the model includes an additional term to take account of sterilized cells. We found that the growth rate observed before irradiation is not fully recovered by irradiated spheroids and that growth recovery reduces with higher irradiation doses. Surviving fractions obtained at doses below 3 Gy are comparable with those found in clonogenic assays on spheroids of the same cellular line. At larger doses, discrepancies between the different studies are considerable.

The use of the linear quadratic model in radiotherapy: a review

To be able to predict the impact of any radiotherapy treatment the physics of radiation interactions and the expected biological effect for any radiotherapy treatment situation (dose, fractionation, modality) must be both understood and modelled. This review considers the current use and accuracy of the linear quadratic model which can be used to consider the variation in tissue response with fraction size. Cell kill following radiation damage results from damage to the DNA which can take a variety of forms. In many cases the linear quadratic model is used to estimate the relative impact for different situations especially clinical studies relating to fraction size. This is mainly undertaken using parameters derived from the linear quadratic model such as biological effective dose and standard effective dose. The model has also been adapted to consider the effect of overall treatment time, repair during treatment (as occurs for brachytherapy treatments) and other situations. There are some concerns over its use, mainly in the small dose ranges (both total low doses and low doses per fraction) where studies have shown its inaccuracy. In other situations however it does appear to provide a reasonable estimate of relative clinical effect. As with all models, however results should never be considered out of clinical context.

Dynamic optimization of a linear-quadratic model with incomplete repair and volume-dependent sensitivity and repopulation

PURPOSE

The linear-quadratic model typically assumes that tumor sensitivity and repopulation are constant over the time course of radiotherapy. However, evidence suggests that the growth fraction increases and the cell-loss factor decreases as the tumor shrinks. We investigate whether this evolution in tumor geometry, as well as the irregular time intervals between fractions in conventional hyperfractionation schemes, can be exploited by fractionation schedules that employ time-varying fraction sizes.

METHODS

We construct a mathematical model of a spherical tumor with a hypoxic core and a viable rim, which is most appropriate for a prevascular tumor, and is only a caricature of a vascularized tumor. This model is embedded into the traditional linear-quadratic model by assuming instantaneous reoxygenation. Dynamic programming is used to numerically compute the fractionation regimen that maximizes the tumor-control probability (TCP) subject to constraints on the biologically effective dose of the early and late tissues.

RESULTS

In several numerical examples that employ five or 10 fractions per week on a 1-cm or 5-cm diameter tumor, optimally varying the fraction sizes increases the TCP significantly. The optimal regimen incorporates large Friday (afternoon, if 10 fractions per week) fractions that are escalated throughout the course of treatment, and larger afternoon fractions than morning fractions.

CONCLUSION

Numerical results suggest that a significant increase in tumor cure can be achieved by allowing the fraction sizes to vary throughout the course of treatment. Several strategies deserve further investigation: using larger fractions before overnight and weekend breaks, and escalating the dose (particularly on Friday afternoons) throughout the course of treatment.

Rapid growth of microscopic rectal cancer as a determinant of response to preoperative radiation therapy

PURPOSE

To quantify the dose-time fractionation factors in preoperative radiation therapy for microscopic pelvic deposits of rectal cancer. This provides a biologic basis for understanding and improving the results of adjuvant therapies for this disease.

METHODS

The reduction in incidence of pelvic relapses as a function of radiation dose and overall treatment time was determined from the literature. The displacement of dose-response curves to higher doses reflects the growth during radiation treatment of subclinical pelvic deposits which are beyond the future surgical margins.

RESULTS

Dose-response curves are steep if the effect of overall duration of radiation therapy is accounted for. The time-related displacement of these steep dose-response curves is consistent with a median doubling time for malignant clonogenic cells of about 4 or 5 days, much faster than the growth rate of the average primary tumor at diagnosis. This rapid growth is evident within the first few days of irradiation, implying that the natural growth rate of these microscopic deposits if fast, and/or that an acceleration of growth follows initiation of radiation injury with a very short lag time.

CONCLUSION

Subclinical pelvic deposits of rectal cancer grow rapidly during preoperative radiation therapy with an adverse influence on the rate of pelvic tumor control from protracting the duration of adjuvant treatment. Low doses only offer clinically relevant reduction in risk of pelvic relapses if the overall radiation treatment time is short. For a given overall treatment duration there is a relatively steep dose-response curve, predicting that significant improvements in tumor control are possible.

Mathematical model of 5-[125I]iodo-2'-deoxyuridine treatment: continuous infusion regimens for hepatic metastases

PURPOSE

Due to the cytotoxicity of DNA-bound iodine-125, 5-[125I]Iodo-2'-deoxyuridine ([125I]IUdR), an analog of thymidine, has long been recognized as possessing therapeutic potential. In this work, the feasibility and potential effectiveness of hepatic artery infusion of [125I]IUdR is examined.

METHODS

A mathematical model has been developed that simulates tumor growth and response to [125I]IUdR treatment. The model is used to examine the efficacy and potential toxicity of prolonged infusion therapy. Treatment of kinetically homogeneous tumors with potential doubling times of either 4, 5, or 6 days is simulated. Assuming uniformly distributed activity, absorbed dose estimates to the red marrow, liver and whole-body are calculated to assess the potential toxicity of treatment.

RESULTS

Nine to 10 logs of tumor-cell kill over a 7- to 20-day period are predicted by the various simulations examined. The most slowly proliferating tumor was also the most difficult to eradicate. During the infusion time, tumor-cell loss consisted of two components: A plateau phase, beginning at the start of infusion and ending once the infusion time exceeded the potential doubling time of the tumor; and a rapid cell-reduction phase that was close to log-linear. Beyond the plateau phase, treatment efficacy was highly sensitive to tumor activity concentration.

CONCLUSIONS

Model predictions suggest that [125I]IUdR will be highly dependent upon the potential doubling time of the tumor. Significant tumor cell kill will require infusion durations that exceed the longest potential doubling time in the tumor-cell population.