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

Generalized, sublethal damage-based mathematical approach for improved modeling of clonogenic survival curve flattening upon hyperthermia, radiotherapy, and beyond

Objective. Mathematical modeling can offer valuable insights into the behavior of biological systems upon treatment. Different mathematical models (empirical, semi-empirical, and mechanistic) have been designed to predict the efficacy of either hyperthermia (HT), radiotherapy (RT), or their combination. However, mathematical approaches capable of modeling cell survival from shared general principles for both mono-treatments alone and their co-application are rare. Moreover, some cell cultures show dose-dependent saturation in response to HT or RT, manifesting in survival curve flattenings. An advanced survival model must, therefore, appropriately reflect such behavior.Approach. We propose a mathematical approach to model the effect of both treatments based on the general principle of sublethal damage (SLD) accumulation for the induction of cell death and irreversible proliferation arrest. Our approach extends Jung's model on heat-induced cellular inactivation by incorporating dose-dependent recovery rates that delineate changes in SLD restoration.Main results. The resulting unified model (Umodel) accurately describes HT and RT survival outcomes, applies to simultaneous thermoradiotherapy modeling, and is particularly suited to reproduce survival curve flattening phenomena. We demonstrate the Umodel's robust performance (R2 0.95) based on numerous clonogenic cell survival data sets from the literature and our experimental studies.Significance. The proposed Umodel allows using a single unified mathematical function based on generalized principles of accumulation of SLD with implemented radiosensitization, regardless of the type of energy deposited and the mechanism of action. It can reproduce various patterns of clonogenic survival curves, including any flattening, thus encompassing the variability of cell reactions to therapy, thereby potentially better reflecting overall tumor responses. Our approach opens a range of options for further model developments and strategic therapy outcome predictions of sequential treatments applied in different orders and varying recovery intervals between them.

Immune modulations of the tumor microenvironment in response to phototherapy

The tumor microenvironment (TME) promotes pro-tumor and anti-inflammatory metabolisms and suppresses the host immune system. It prevents immune cells from fighting against cancer effectively, resulting in limited efficacy of many current cancer treatment modalities. Different therapies aim to overcome the immunosuppressive TME by combining various approaches to synergize their effects for enhanced anti-tumor activity and augmented stimulation of the immune system. Immunotherapy has become a major therapeutic strategy because it unleashes the power of the immune system by activating, enhancing, and directing immune responses to prevent, control, and eliminate cancer. Phototherapy uses light irradiation to induce tumor cell death through photothermal, photochemical, and photo-immunological interactions. Phototherapy induces tumor immunogenic cell death, which is a precursor and enhancer for anti-tumor immunity. However, phototherapy alone has limited effects on long-term and systemic anti-tumor immune responses. Phototherapy can be combined with immunotherapy to improve the tumoricidal effect by killing target tumor cells, enhancing immune cell infiltration in tumors, and rewiring pathways in the TME from anti-inflammatory to pro-inflammatory. Phototherapy-enhanced immunotherapy triggers effective cooperation between innate and adaptive immunities, specifically targeting the tumor cells, whether they are localized or distant. Herein, the successes and limitations of phototherapy combined with other cancer treatment modalities will be discussed. Specifically, we will review the synergistic effects of phototherapy combined with different cancer therapies on tumor elimination and remodeling of the immunosuppressive TME. Overall, phototherapy, in combination with other therapeutic modalities, can establish anti-tumor pro-inflammatory phenotypes in activated tumor-infiltrating T cells and B cells and activate systemic anti-tumor immune responses.

Development of a coupled modeling for tumor growth, angiogenesis, oxygen delivery, and phenotypic heterogeneity

Analysis of the evolution and growth dynamics of tumors is crucial for understanding cancer and the development of individually optimized therapies. During tumor growth, a hypoxic microenvironment around cancer cells caused by excessive non-vascular tumor growth induces tumor angiogenesis that plays a key role in the ensuing tumor growth and its progression into higher stages. Various mathematical simulation models have been introduced to simulate these biologically and physically complex hallmarks of cancer. Here, we developed a hybrid two-dimensional computational model that integrates spatiotemporally different components of the tumor system to investigate both angiogenesis and tumor growth/proliferation. This spatiotemporal evolution is based on partial diffusion equations, the cellular automation method, transition and probabilistic rules, and biological assumptions. The new vascular network provided by angiogenesis affects tumor microenvironmental conditions and drives individual cells to adapt themselves to spatiotemporal conditions. Furthermore, some stochastic rules are involved besides microenvironmental conditions. Overall, the conditions promote some commonly observed cellular states, i.e., proliferative, migrative, quiescent, and cell death, depending on the condition of each cell. Altogether, our results offer a theoretical basis for the biological evidence that regions of the tumor tissue near blood vessels are densely populated by proliferative phenotypic variants, while poorly oxygenated regions are sparsely populated by hypoxic phenotypic variants.

Lattice or Oxygen-Guided Radiotherapy: What If They Converge? Possible Future Directions in the Era of Immunotherapy

Palliative radiotherapy has a great role in the treatment of large tumor masses. However, treating a bulky disease could be difficult, especially in critical anatomical areas. In daily clinical practice, short course hypofractionated radiotherapy is delivered in order to control the symptomatic disease. Radiation fields generally encompass the entire tumor mass, which is homogeneously irradiated. Recent technological advances enable delivering a higher radiation dose in small areas within a large mass. This goal, previously achieved thanks to the GRID approach, is now achievable using the newest concept of LATTICE radiotherapy (LT-RT). This kind of treatment allows exploiting various radiation effects, such as bystander and abscopal effects. These events may be enhanced by the concomitant use of immunotherapy, with the latter being ever more successfully delivered in cancer patients. Moreover, a critical issue in the treatment of large masses is the inhomogeneous intratumoral distribution of well-oxygenated and hypo-oxygenated areas. It is well known that hypoxic areas are more resistant to the killing effect of radiation, hence the need to target them with higher aggressive doses. This concept introduces the "oxygen-guided radiation therapy" (OGRT), which means looking for suitable hypoxic markers to implement in PET/CT and Magnetic Resonance Imaging. Future treatment strategies are likely to involve combinations of LT-RT, OGRT, and immunotherapy. In this paper, we review the radiobiological rationale behind a potential benefit of LT-RT and OGRT, and we summarize the results reported in the few clinical trials published so far regarding these issues. Lastly, we suggest what future perspectives may emerge by combining immunotherapy with LT-RT/OGRT.

Towards a Reduced In Silico Model Predicting Biochemical Recurrence After Radiotherapy in Prostate Cancer

OBJECTIVE

Purposes of this work were i) to develop an in silico model of tumor response to radiotherapy, ii) to perform an exhaustive sensitivity analysis in order to iii) propose a simplified version and iv) to predict biochemical recurrence with both the comprehensive and the reduced model.

METHODS

A multiscale computational model of tumor response to radiotherapy was developed. It integrated the following radiobiological mechanisms: oxygenation, including hypoxic death; division of tumor cells; VEGF diffusion driving angiogenesis; division of healthy cells and oxygen-dependent response to irradiation, considering, cycle arrest and mitotic catastrophe. A thorough sensitivity analysis using the Morris screening method was performed on 21 prostate computational tissues. Tumor control probability (TCP) curves of the comprehensive model and 15 reduced versions were compared. Logistic regression was performed to predict biochemical recurrence after radiotherapy on 76 localized prostate cancer patients using an output of the comprehensive and the reduced models.

RESULTS

No significant difference was found between the TCP curves of the comprehensive and a simplified version which only considered oxygenation, division of tumor cells and their response to irradiation. Biochemical recurrence predictions using the comprehensive and the reduced models improved those made from pre-treatment imaging parameters (AUC = 0.81 ± 0.02 and 0.82 ± 0.02 vs. 0.75 ± 0.03, respectively).

CONCLUSION

A reduced model of tumor response to radiotherapy able to predict biochemical recurrence in prostate cancer was obtained.

SIGNIFICANCE

This reduced model may be used in the future to optimize personalized fractionation schedules.

Radial basis function-generated finite difference scheme for simulating the brain cancer growth model under radiotherapy in various types of computational domains

BACKGROUND AND OBJECTIVES

We extend the original mathematical model, i.e., Swanson's reaction-diffusion equation to the surfaces with no boundary, and we find a new numerical method based on a meshless approach for solving numerically Swanson's reaction-diffusion model in the square and on the sphere.

METHODS

To solve numerically the Swanson's reaction-diffusion model and its extension version, a collocation meshless technique, namely radial basis function-generated finite difference (RBF-FD) scheme is employed for approximating the spatial variables in the square domain and on the sphere, respectively. Also, to approximate the time variable of the studied models, a first-order semi-implicit backward Euler scheme is used. The resulting fully discrete scheme is a linear system of algebraic equations per time step that is solved via the biconjugate gradient stabilized (BiCGSTAB) iterative algorithm with a zero-fill incomplete lower-upper (ILU) preconditioner.

RESULTS

The numerical simulations show the growth of untreated and treated brain tumors with radiotherapy using estimated and clinical data (given from magnetic resonance imaging (MRI) scans of patients). Moreover, the results reported here can be used for improving the treatment strategies of the invasive brain tumor.

CONCLUSIONS

Using the developed numerical scheme in this paper, we can simulate the behavior of the invasive form of brain tumor response to radiotherapy. Also, we can see the effects of radiation response on the brain tumor cell concentration of individual patients. The proposed meshless technique, which is applied for solving numerically the studied model, does not depend on any background mesh or triangulation for approximation in comparison with mesh-dependent methods. Moreover, we apply this technique to the sphere via any set of distributed points easily.

Approaches to combat hypoxia in cancer therapy and the potential for in silico models in their evaluation

AIM

The negative impact of tumour hypoxia on cancer treatment outcome has been long-known, yet there has been little success combating it. This paper investigates the potential role of in silico modelling to help test emerging hypoxia-targeting treatments in cancer therapy.

METHODS

A Medline search was undertaken on the current landscape of in silico models that simulate cancer therapy and evaluate their ability to test hypoxia-targeting treatments. Techniques and treatments to combat tumour hypoxia and their current challenges are also presented.

RESULTS

Hypoxia-targeting treatments include tumour reoxygenation, hypoxic cell radiosensitization with nitroimidazoles, hypoxia-activated prodrugs and molecular targeting. Their main challenges are toxicity and not achieving adequate delivery to hypoxic regions of the tumour. There is promising research toward combining two or more of these techniques. Different types of in silico therapy models have been developed ranging from temporal to spatial and from stochastic to deterministic models. Numerous models have compared the effectiveness of different radiotherapy fractionation schedules for controlling hypoxic tumours. Similarly, models could help identify and optimize new treatments for overcoming hypoxia that utilize novel hypoxia-targeting technology.

CONCLUSION

Current therapy models should attempt to incorporate more sophisticated modelling of tumour angiogenesis/vasculature and vessel perfusion in order to become more useful for testing hypoxia-targeting treatments, which typically rely upon the tumour vasculature for delivery of additional oxygen, (pro)drugs and nanoparticles.

Simulation of head and neck cancer oxygenation and doubling time in a 4D cellular model with angiogenesis

Tumor oxygenation has been correlated with treatment outcome for radiotherapy. In this work, the dependence of tumor oxygenation on tumor vascularity and blood oxygenation was determined quantitatively in a 4D stochastic computational model of head and neck squamous cell carcinoma (HNSCC) tumor growth and angiogenesis. Additionally, the impacts of the tumor oxygenation and the cancer stem cell (CSC) symmetric division probability on the tumor volume doubling time and the proportion of CSCs in the tumor were also quantified. Clinically relevant vascularities and blood oxygenations for HNSCC yielded tumor oxygenations in agreement with clinical data for HNSCC. The doubling time varied by a factor of 3 from well oxygenated tumors to the most severely hypoxic tumors of HNSCC. To obtain the doubling times and CSC proportions clinically observed in HNSCC, the model predicts a CSC symmetric division probability of approximately 2% before treatment. To obtain the doubling times clinically observed during treatment when accelerated repopulation is occurring, the model predicts a CSC symmetric division probability of approximately 50%, which also results in CSC proportions of 30-35% during this time.

Modeling the Cellular Response of Lung Cancer to Radiation Therapy for a Broad Range of Fractionation Schedules

Purpose: To demonstrate that a mathematical model can be used to quantitatively understand tumor cellular dynamics during a course of radiotherapy and to predict the likelihood of local control as a function of dose and treatment fractions.Experimental Design: We model outcomes for early-stage, localized non-small cell lung cancer (NSCLC), by fitting a mechanistic, cellular dynamics-based tumor control probability that assumes a constant local supply of oxygen and glucose. In addition to standard radiobiological effects such as repair of sub-lethal damage and the impact of hypoxia, we also accounted for proliferation as well as radiosensitivity variability within the cell cycle. We applied the model to 36 published and two unpublished early-stage patient cohorts, totaling 2,701 patients.Results: Precise likelihood best-fit values were derived for the radiobiological parameters: α [0.305 Gy^-1; 95% confidence interval (CI), 0.120-0.365], the α/β ratio (2.80 Gy; 95% CI, 0.40-4.40), and the oxygen enhancement ratio (OER) value for intermediately hypoxic cells receiving glucose but not oxygen (1.70; 95% CI, 1.55-2.25). All fractionation groups are well fitted by a single dose-response curve with a high χ² P value, indicating consistency with the fitted model. The analysis was further validated with an additional 23 patient cohorts (n = 1,628). The model indicates that hypofractionation regimens overcome hypoxia (and cell-cycle radiosensitivity variations) by the sheer impact of high doses per fraction, whereas lower dose-per-fraction regimens allow for reoxygenation and corresponding sensitization, but lose effectiveness for prolonged treatments due to proliferation.Conclusions: This proposed mechanistic tumor-response model can accurately predict overtreatment or undertreatment for various treatment regimens. Clin Cancer Res; 23(18); 5469-79. ©2017 AACR.

A review of the development of tumor vasculature and its effects on the tumor microenvironment

Background

The imbalance of angiogenic regulators in tumors drives tumor angiogenesis and causes the vasculature to develop much differently in tumors than in normal tissue. There are several cancer therapy techniques currently being used and developed that target the tumor vasculature for the treatment of solid tumors. This article reviews the aspects of the tumor vasculature that are relevant to most cancer therapies but particularly to vascular targeting techniques.

Materials and methods

We conducted a review of identified experiments in which tumors were transplanted into animals to study the development of the tumor vasculature with tumor growth. Quantitative vasculature morphology data for spontaneous human head and neck cancers are reviewed. Parameters assessed include the highest microvascular density (h-MVD) and the relative vascular volume (RVV). The effects of the vasculature on the tumor microenvironment are discussed, including the distributions of hypoxia and proliferation.

Results

Data for the h-MVD and RVV in head and neck cancers are highly varied, partly due to methodological differences. However, it is clear that the cancers are typically more vascularized than the corresponding normal tissue. The commonly observed chronic hypoxia and acute hypoxia in these tumors are due to high intratumor heterogeneity in MVD and lower than normal blood oxygenation levels through the abnormally developed tumor vasculature. Hypoxic regions are associated with decreased cell proliferation.

Conclusion

The morphology of the vasculature strongly influences the tumor microenvironment, with important implications for tumor response to medical intervention such as radiotherapy. Quantitative vasculature morphology data herein may be used to inform computational models that simulate the spatial tumor vasculature. Such models may play an important role in exploring and optimizing vascular targeting cancer therapies.

Development of an in silico stochastic 4D model of tumor growth with angiogenesis

PURPOSE

A stochastic computer model of tumour growth with spatial and temporal components that includes tumour angiogenesis was developed. In the current work it was used to simulate head and neck tumour growth. The model also provides the foundation for a 4D cellular radiotherapy simulation tool.

METHODS

The model, developed in Matlab, contains cell positions randomised in 3D space without overlap. Blood vessels are represented by strings of blood vessel units which branch outwards to achieve the desired tumour relative vascular volume. Hypoxic cells have an increased cell cycle time and become quiescent at oxygen tensions less than 1 mmHg. Necrotic cells are resorbed. A hierarchy of stem cells, transit cells and differentiated cells is considered along with differentiated cell loss. Model parameters include the relative vascular volume (2-10%), blood oxygenation (20-100 mmHg), distance from vessels to the onset of necrosis (80-300 μm) and probability for stem cells to undergo symmetric division (2%). Simulations were performed to observe the effects of hypoxia on tumour growth rate for head and neck cancers. Simulations were run on a supercomputer with eligible parts running in parallel on 12 cores.

RESULTS

Using biologically plausible model parameters for head and neck cancers, the tumour volume doubling time varied from 45 ± 5 days (n = 3) for well oxygenated tumours to 87 ± 5 days (n = 3) for severely hypoxic tumours.

CONCLUSIONS

The main achievements of the current model were randomised cell positions and the connected vasculature structure between the cells. These developments will also be beneficial when irradiating the simulated tumours using Monte Carlo track structure methods.

Tumor radio-sensitivity assessment by means of volume data and magnetic resonance indices measured on prostate tumor bearing rats

PURPOSE

Radiation therapy is one of the most common treatments in the fight against prostate cancer, since it is used to control the tumor (early stages), to slow its progression, and even to control pain (metastasis). Although many factors (e.g., tumor oxygenation) are known to influence treatment efficacy, radiotherapy doses and fractionation schedules are often prescribed according to the principle "one-fits-all," with little personalization. Therefore, the authors aim at predicting the outcome of radiation therapy a priori starting from morphologic and functional information to move a step forward in the treatment customization.

METHODS

The authors propose a two-step protocol to predict the effects of radiation therapy on individual basis. First, one macroscopic mathematical model of tumor evolution was trained on tumor volume progression, measured by caliper, of eighteen Dunning R3327-AT1 bearing rats. Nine rats inhaled 100% O2 during irradiation (oxy), while the others were allowed to breathe air. Second, a supervised learning of the weight and biases of two feedforward neural networks was performed to predict the radio-sensitivity (target) from the initial volume and oxygenation-related information (inputs) for each rat group (air and oxygen breathing). To this purpose, four MRI-based indices related to blood and tissue oxygenation were computed, namely, the variation of signal intensity ΔSI in interleaved blood oxygen level dependent and tissue oxygen level dependent (IBT) sequences as well as changes in longitudinal ΔR1 and transverse ΔR2(*) relaxation rates.

RESULTS

An inverse correlation of the radio-sensitivity parameter, assessed by the model, was found with respect the ΔR2(*) (-0.65) for the oxy group. A further subdivision according to positive and negative values of ΔR2(*) showed a larger average radio-sensitivity for the oxy rats with ΔR2(*)<0 and a significant difference in the two distributions (p < 0.05). Finally, a leave-one-out procedure yielded a radio-sensitivity error lower than 20% in both neural networks.

CONCLUSIONS

While preliminary, these specific results suggest that subjects affected by the same pathology can benefit differently from the same irradiation modalities and support the usefulness of IBT in discriminating between different responses.

A mathematical model of tumor growth and its response to single irradiation

BACKGROUND

Mathematical modeling of biological processes is widely used to enhance quantitative understanding of bio-medical phenomena. This quantitative knowledge can be applied in both clinical and experimental settings. Recently, many investigators began studying mathematical models of tumor response to radiation therapy. We developed a simple mathematical model to simulate the growth of tumor volume and its response to a single fraction of high dose irradiation. The modelling study may provide clinicians important insights on radiation therapy strategies through identification of biological factors significantly influencing the treatment effectiveness.

METHODS

We made several key assumptions of the model. Tumor volume is composed of proliferating (or dividing) cancer cells and non-dividing (or dead) cells. Tumor growth rate (or tumor volume doubling time) is proportional to the ratio of the volumes of tumor vasculature and the tumor. The vascular volume grows slower than the tumor by introducing the vascular growth retardation factor, θ. Upon irradiation, the proliferating cells gradually die over a fixed time period after irradiation. Dead cells are cleared away with cell clearance time. The model was applied to simulate pre-treatment growth and post-treatment radiation response of rat rhabdomyosarcoma tumors and metastatic brain tumors of five patients who were treated with Gamma Knife stereotactic radiosurgery (GKSRS).

RESULTS

By selecting appropriate model parameters, we showed the temporal variation of the tumors for both the rat experiment and the clinical GKSRS cases could be easily replicated by the simple model. Additionally, the application of our model to the GKSRS cases showed that the α-value, which is an indicator of radiation sensitivity in the LQ model, and the value of θ could be predictors of the post-treatment volume change.

CONCLUSIONS

The proposed model was successful in representing both the animal experimental data and the clinically observed tumor volume changes. We showed that the model can be used to find the potential biological parameters, which may be able to predict the treatment outcome. However, there is a large statistical uncertainty of the result due to the small sample size. Therefore, a future clinical study with a larger number of patients is needed to confirm the finding.

A voxel-based multiscale model to simulate the radiation response of hypoxic tumors

PURPOSE

In radiotherapy, it is important to predict the response of tumors to irradiation prior to the treatment. This is especially important for hypoxic tumors, which are known to be highly radioresistant. Mathematical modeling based on the dose distribution, biological parameters, and medical images may help to improve this prediction and to optimize the treatment plan.

METHODS

A voxel-based multiscale tumor response model for simulating the radiation response of hypoxic tumors was developed. It considers viable and dead tumor cells, capillary and normal cells, as well as the most relevant biological processes such as (i) proliferation of tumor cells, (ii) hypoxia-induced angiogenesis, (iii) spatial exchange of cells leading to tumor growth, (iv) oxygen-dependent cell survival after irradiation, (v) resorption of dead cells, and (vi) spatial exchange of cells leading to tumor shrinkage. Oxygenation is described on a microscopic scale using a previously published tumor oxygenation model, which calculates the oxygen distribution for each voxel using the vascular fraction as the most important input parameter. To demonstrate the capabilities of the model, the dependence of the oxygen distribution on tumor growth and radiation-induced shrinkage is investigated. In addition, the impact of three different reoxygenation processes is compared and tumor control probability (TCP) curves for a squamous cells carcinoma of the head and neck (HNSSC) are simulated under normoxic and hypoxic conditions.

RESULTS

The model describes the spatiotemporal behavior of the tumor on three different scales: (i) on the macroscopic scale, it describes tumor growth and shrinkage during radiation treatment, (ii) on a mesoscopic scale, it provides the cell density and vascular fraction for each voxel, and (iii) on the microscopic scale, the oxygen distribution may be obtained in terms of oxygen histograms. With increasing tumor size, the simulated tumors develop a hypoxic core. Within the model, tumor shrinkage was found to be significantly more important for reoxygenation than angiogenesis or decreased oxygen consumption due to an increased fraction of dead cells. In the studied HNSSC-case, the TCD50 values (dose at 50% TCP) decreased from 71.0 Gy under hypoxic to 53.6 Gy under the oxic condition.

CONCLUSIONS

The results obtained with the developed multiscale model are in accordance with expectations based on radiobiological principles and clinical experience. As the model is voxel-based, radiological imaging methods may help to provide the required 3D-characterization of the tumor prior to irradiation. For clinical application, the model has to be further validated with experimental and clinical data. If this is achieved, the model may be used to optimize fractionation schedules and dose distributions for the treatment of hypoxic tumors.

Modeling the Interplay Between Tumor Volume Regression and Oxygenation in Uterine Cervical Cancer During Radiotherapy Treatment

This paper describes a patient-specific mathematical model to predict the evolution of uterine cervical tumors at a macroscopic scale, during fractionated external radiotherapy. The model provides estimates of tumor regrowth and dead-cell reabsorption, incorporating the interplay between tumor regression rate and radiosensitivity, as a function of the tumor oxygenation level. Model parameters were estimated by minimizing the difference between predicted and measured tumor volumes, these latter being obtained from a set of 154 serial cone-beam computed tomography scans acquired on 16 patients along the course of the therapy. The model stratified patients according to two different estimated dynamics of dead-cell removal and to the predicted initial value of the tumor oxygenation. The comparison with a simpler model demonstrated an improvement in fitting properties of this approach (fitting error average value <5%, p < 0.01), especially in case of tumor late responses, which can hardly be handled by models entailing a constant radiosensitivity, failing to model changes from initial severe hypoxia to aerobic conditions during the treatment course. The model predictive capabilities suggest the need of clustering patients accounting for cancer cell line, tumor staging, as well as microenvironment conditions (e.g., oxygenation level).

Modeling of liver metastatic disease with applied drug therapy

Colorectal carcinoma is acknowledged as the second leading cause of total cancer-related death in the European Region. The majority of deaths related to colorectal carcinoma are connected with liver metastatic disease. Approximately, in 25% of all patients, liver metastatic disease is diagnosed at the same time as the primary diagnosis, while up to a quarter of others would develop liver metastases in the course of the illness. In this study, we developed reaction-diffusion model and analyzed the effect of drug therapy on liver metastatic disease for a specific patient. Tumor volumes in specific time points were obtained using CT scan images. The nonlinear function for cell proliferation rate as well as data about clinically applied drug therapy was included in the model. Fitting procedure was used for parameter estimation. Good agreement of numerical and experimental results shows the feasibility and efficacy of the proposed system.

A model to simulate the oxygen distribution in hypoxic tumors for different vascular architectures

PURPOSE

As hypoxic cells are more resistant to photon radiation, it is desirable to obtain information about the oxygen distribution in tumors prior to the radiation treatment. Noninvasive techniques are currently not able to provide reliable oxygenation maps with sufficient spatial resolution; therefore mathematical models may help to simulate microvascular architectures and the resulting oxygen distributions in the surrounding tissue. Here, the authors present a new computer model, which uses the vascular fraction of tumor voxels, in principle measurable noninvasively in vivo, as input parameter for simulating realistic PO2 histograms in tumors, assuming certain 3D vascular architectures.

METHODS

Oxygen distributions were calculated by solving a reaction-diffusion equation in a reference volume using the particle strength exchange method. Different types of vessel architectures as well as different degrees of vascular heterogeneities are considered. Two types of acute hypoxia (ischemic and hypoxemic) occurring additionally to diffusion-limited (chronic) hypoxia were implemented as well.

RESULTS

No statistically significant differences were observed when comparing 2D- and 3D-vessel architectures (p>0.79 in all cases) and highly heterogeneously distributed linear vessels show good agreement, when comparing with published experimental intervessel distance distributions and PO2 histograms. It could be shown that, if information about additional acute hypoxia is available, its contribution to the hypoxic fraction (HF) can be simulated as well. Increases of 128% and 168% in the HF were obtained when representative cases of ischemic and hypoxemic acute hypoxia, respectively, were considered in the simulations.

CONCLUSIONS

The presented model is able to simulate realistic microscopic oxygen distributions in tumors assuming reasonable vessel architectures and using the vascular fraction as macroscopic input parameter. The model may be used to generate PO2 histograms, which are needed as input in models predicting the radiation response of hypoxic tumors.

A mathematical model of tumor volume changes during radiotherapy

PURPOSE

To develop a clinically viable mathematical model that quantitatively predicts tumor volume change during radiotherapy in order to provide treatment response assessment for prognosis, treatment plan optimization, and adaptation.

METHOD AND MATERIALS

The correction factors containing hypoxia, DNA single strand breaks, potentially lethal damage, and other factors were used to develop an improved cell survival model based on the popular linear-quadratic model of cell survival in radiotherapy. The four-level cell population model proposed by Chvetsov et al. was further simplified by removing the initial hypoxic fraction and reoxygenation parameter, which are hard to obtain in routine clinics, such that an easy-to-use model can be developed for clinical applications. The new model was validated with data of nine lung and cervical cancer patients.

RESULTS

Out of the nine cases, the new model can predict tumor volume change in six cases with a correlation index R (2) greater than 0.9 and the rest of three with R (2) greater than 0.85.

CONCLUSION

Based on a four-level cell population model, a more practical and simplified cell survival curve was proposed to model the tumor volume changes during radiotherapy. Validation study with patient data demonstrated feasibility and clinical usefulness of the new model in predicting tumor volume change in radiotherapy.

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.

Model-based drug development applied to oncology

Model-based drug development (MBDD) is an approach that is used to organize the vast and complex data streams that feed the drug development pipelines of small molecule and biotechnology sponsors. Such data streams are ultimately reviewed by the global regulatory community as evidence of a drug's potential to treat and/or harm patients. Some of this information is captured in the scientific literature and prescribing compendiums forming the basis of how new and existing agents will ultimately be administered and further evaluated in the broader patient community. As this data stream evolves, the details of data qualification, the assumptions and/or critical decisions based on these data are lost under conventional drug development paradigms. MBDD relies on the construction of quantitative relationships to connect data from discrete experiments conducted along the drug development pathway. These relationships are then used to ask questions relevant at critical development stages, hopefully, with the understanding that the various scenarios explored represent a path to optimal decision making. Oncology, as a therapeutic area, presents a unique set of challenges and perhaps a different development paradigm as opposed to other disease targets. The poor attrition of development compounds in the recent past attests to these difficulties and provides an incentive for a different approach. In addition, given the reliance on multimodal therapy, oncological disease targets are often treated with both new and older agents spanning several drug classes. As MBDD becomes more integrated into the pharmaceutical research community, a more rational explanation for decisions regarding the development of new oncology agents as well as the proposed treatment regimens that incorporate both new and existing agents can be expected. Hopefully, the end result is a more focussed clinical development programme, which ultimately provides a means to optimize individual patient care.

Altered fractionation outcomes for hypoxic head and neck cancer using the HYP-RT Monte Carlo model

OBJECTIVE

Altered fractionation radiotherapy is simulated on a set of virtual tumours to assess the total doses required for tumour control compared with clinical head and neck data and the doses required to control hypoxic vs well-oxygenated tumours with different radiobiological properties.

METHODS

The HYP-RT model is utilised to explore the impact of tumour oxygenation and the onset times of accelerated repopulation (AR) and reoxygenation (ROx) during radiotherapy. A biological effective dose analysis is used to rank the schedules based on their relative normal tissue toxicities.

RESULTS

Altering the onset times of AR and ROx has a large impact on the doses required to achieve tumour control. Immediate onset of ROx and 2-week onset time of AR produce results closely predicting average human outcomes in terms of the total prescription doses in clinical trials. Modifying oxygen enhancement ratio curves based on dose/fraction significantly reduces the dose (5-10 Gy) required for tumour control for hyperfractionated schedules. HYP-RT predicts 10×1.1 Gy per week to be most beneficial, whereas the conventional schedule is predicted as beneficial for early toxicity but has average-poor late toxicity.

CONCLUSION

HYP-RT predicts that altered radiotherapy schedules increase the therapeutic ratio and may be used to make predictions about the prescription doses required to achieve tumour control for tumours with different oxygenation levels and treatment responses.

ADVANCES IN KNOWLEDGE

Oxic and hypoxic tumours have large differences in total radiation dose requirements, affected by AR and ROx onset times by up to 15-25 Gy for the same fractionation schedule.

A hybrid cellular automaton model of solid tumor growth and bioreductive drug transport

Bioreductive drugs are a class of hypoxia selective drugs that are designed to eradicate the hypoxic fraction of solid tumors. Their activity depends upon a number of biological and pharmacological factors and we used a mathematical modeling approach to explore the dynamics of tumor growth, infusion, and penetration of the bioreductive drug Tirapazamine (TPZ). An in-silico model is implemented to calculate the tumor mass considering oxygen and glucose as key microenvironmental parameters. The next stage of the model integrated extra cellular matrix (ECM), cell-cell adhesion, and cell movement parameters as growth constraints. The tumor microenvironments strongly influenced tumor morphology and growth rates. Once the growth model was established, a hybrid model was developed to study drug dynamics inside the hypoxic regions of tumors. The model used 10, 50 and 100 \mu {\rm M} as TPZ initial concentrations and determined TPZ pharmacokinetic (PK) (transport) and pharmacodynamics (cytotoxicity) properties inside hypoxic regions of solid tumor. The model results showed that diminished drug transport is a reason for TPZ failure and recommend the optimization of the drug transport properties in the emerging TPZ generations. The modeling approach used in this study is novel and can be a step to explore the behavioral dynamics of TPZ.

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.

The HYP-RT hypoxic tumour radiotherapy algorithm and accelerated repopulation dose per fraction study

The HYP-RT model simulates hypoxic tumour growth for head and neck cancer as well as radiotherapy and the effects of accelerated repopulation and reoxygenation. This report outlines algorithm design, parameterisation and the impact of accelerated repopulation on the increase in dose/fraction needed to control the extra cell propagation during accelerated repopulation. Cell kill probabilities are based on Linear Quadratic theory, with oxygenation levels and proliferative capacity influencing cell death. Hypoxia is modelled through oxygen level allocation based on pO₂ histograms. Accelerated repopulation is modelled by increasing the stem cell symmetrical division probability, while the process of reoxygenation utilises randomised pO₂ increments to the cell population after each treatment fraction. Propagation of 10⁸ tumour cells requires 5–30 minutes. Controlling the extra cell growth induced by accelerated repopulation requires a dose/fraction increase of 0.5–1.0 Gy, in agreement with published reports. The average reoxygenation pO₂ increment of 3 mmHg per fraction results in full tumour reoxygenation after shrinkage to approximately 1 mm. HYP-RT is a computationally efficient model simulating tumour growth and radiotherapy, incorporating accelerated repopulation and reoxygenation. It may be used to explore cell kill outcomes during radiotherapy while varying key radiobiological and tumour specific parameters, such as the degree of hypoxia.

Monte Carlo role in radiobiological modelling of radiotherapy outcomes

Radiobiological models are essential components of modern radiotherapy. They are increasingly applied to optimize and evaluate the quality of different treatment planning modalities. They are frequently used in designing new radiotherapy clinical trials by estimating the expected therapeutic ratio of new protocols. In radiobiology, the therapeutic ratio is estimated from the expected gain in tumour control probability (TCP) to the risk of normal tissue complication probability (NTCP). However, estimates of TCP/NTCP are currently based on the deterministic and simplistic linear-quadratic formalism with limited prediction power when applied prospectively. Given the complex and stochastic nature of the physical, chemical and biological interactions associated with spatial and temporal radiation induced effects in living tissues, it is conjectured that methods based on Monte Carlo (MC) analysis may provide better estimates of TCP/NTCP for radiotherapy treatment planning and trial design. Indeed, over the past few decades, methods based on MC have demonstrated superior performance for accurate simulation of radiation transport, tumour growth and particle track structures; however, successful application of modelling radiobiological response and outcomes in radiotherapy is still hampered with several challenges. In this review, we provide an overview of some of the main techniques used in radiobiological modelling for radiotherapy, with focus on the MC role as a promising computational vehicle. We highlight the current challenges, issues and future potentials of the MC approach towards a comprehensive systems-based framework in radiobiological modelling for radiotherapy.

Computer simulation of tumour control probabilities after irradiation for varying intrinsic radio-sensitivity using a single cell based model

BACKGROUND

Currently, optimisation of the dose distribution and clinical acceptance are almost entirely based on the physical dose distribution and tumour control probability modelling is far from being routinely used as objective in treatment planning. For future individualised radiotherapeutic strategies, a reliable patient specific simulation model, taking into account customised tumour features, is needed to predict and improve treatment outcome.

MATERIALS AND METHODS

To approach these demands, a single cell and Monte-Carlo based model was developed, which enables three-dimensional tumour growth and radiation response simulation. Tumour cells were characterised by cell-associated features such as age, intrinsic radio-sensitivity, proliferation ability, and oxygenation status, while capillary cells were considered as sources of a radial-dependent oxygen profile. Response to radiation was simulated by the linear-quadratic model, taking into account the lower radio-sensitivity of poorly oxygenated tumour cells.

RESULTS

The present study shows the influence of the model components and demonstrates the impact of the intra- and inter-tumoural radio-sensitivity heterogeneity on the treatment response.

CONCLUSION

The simulation model adequately delineates the importance of the above described selected parameters on tumour control probability, providing an insight into the interplay of different physical and biological parameters, and its relevance for an individual tumour response.

Tumor-volume simulation during radiotherapy for head-and-neck cancer using a four-level cell population model

PURPOSE

To develop a fast computational radiobiologic model for quantitative analysis of tumor volume during fractionated radiotherapy. The tumor-volume model can be useful for optimizing image-guidance protocols and four-dimensional treatment simulations in proton therapy that is highly sensitive to physiologic changes.

METHODS

The analysis is performed using two approximations: (1) tumor volume is a linear function of total cell number and (2) tumor-cell population is separated into four subpopulations: oxygenated viable cells, oxygenated lethally damaged cells, hypoxic viable cells, and hypoxic lethally damaged cells. An exponential decay model is used for disintegration and removal of oxygenated lethally damaged cells from the tumor.

RESULTS

We tested our model on daily volumetric imaging data available for 14 head-and-neck cancer patients treated with an integrated computed tomography/linear accelerator system. A simulation based on the averaged values of radiobiologic parameters was able to describe eight cases during the entire treatment and four cases partially (50% of treatment time) with a maximum 20% error. The largest discrepancies between the model and clinical data were obtained for small tumors, which may be explained by larger errors in the manual tumor volume delineation procedure.

CONCLUSIONS

Our results indicate that the change in gross tumor volume for head-and-neck cancer can be adequately described by a relatively simple radiobiologic model. In future research, we propose to study the variation of model parameters by fitting to clinical data for a cohort of patients with head-and-neck cancer and other tumors. The potential impact of other processes, like concurrent chemotherapy, on tumor volume should be evaluated.

An imaging-based tumour growth and treatment response model: investigating the effect of tumour oxygenation on radiation therapy response

A multiscale tumour simulation model employing cell-line-specific biological parameters and functional information derived from pre-therapy PET/CT imaging data was developed to investigate effects of different oxygenation levels on the response to radiation therapy. For each tumour voxel, stochastic simulations were performed to model cellular growth and therapeutic response. Model parameters were fitted to published preclinical experiments of head and neck squamous cell carcinoma (HNSCC). Using the obtained parameters, the model was applied to a human HNSCC case to investigate effects of different uniform and non-uniform oxygenation levels and results were compared for treatment efficacy. Simulations of the preclinical studies showed excellent agreement with published data and underlined the model's ability to quantitatively reproduce tumour behaviour within experimental uncertainties. When using a simplified transformation to derive non-uniform oxygenation levels from molecular imaging data, simulations of the clinical case showed heterogeneous tumour response and variability in radioresistance with decreasing oxygen levels. Once clinically validated, this model could be used to transform patient-specific data into voxel-based biological objectives for treatment planning and to investigate biologically optimized dose prescriptions.

Efficient Monte Carlo modelling of individual tumour cell propagation for hypoxic head and neck cancer

A Monte Carlo tumour model has been developed to simulate tumour cell propagation for head and neck squamous cell carcinoma. The model aims to eventually provide a radiobiological tool for radiation oncology clinicians to plan patient treatment schedules based on properties of the individual tumour. The inclusion of an oxygen distribution amongst the tumour cells enables the model to incorporate hypoxia and other associated parameters, which affect tumour growth. The object oriented program FORTRAN 95 has been used to create the model algorithm, with Monte Carlo methods being employed to randomly assign many of the cell parameters from probability distributions. Hypoxia has been implemented through random assignment of partial oxygen pressure values to individual cells during tumour growth, based on in vivo Eppendorf probe experimental data. The accumulation of up to 10 million virtual tumour cells in 15 min of computer running time has been achieved. The stem cell percentage and the degree of hypoxia are the parameters which most influence the final tumour growth rate. For a tumour with a doubling time of 40 days, the final stem cell percentage is approximately 1% of the total cell population. The effect of hypoxia on the tumour growth rate is significant. Using a hypoxia induced cell quiescence limit which affects 50% of cells with and oxygen levels less than 1 mm Hg, the tumour doubling time increases to over 200 days and the time of tumour growth for a clinically detectable tumour (10(9) cells) increases from 3 to 8 years. A biologically plausible Monte Carlo model of hypoxic head and neck squamous cell carcinoma tumour growth has been developed for real time assessment of the effects of multiple biological parameters which impact upon the response of the individual patient to fractionated radiotherapy.

Time-dependent cell disintegration kinetics in lung tumors after irradiation

We study the time-dependent disintegration kinetics of tumor cells that did not survive radiotherapy treatment. To evaluate the cell disintegration rate after irradiation, we studied the volume changes of solitary lung tumors after stereotactic radiotherapy. The analysis is performed using two approximations: (1) tumor volume is a linear function of the total cell number in the tumor and (2) the cell disintegration rate is governed by the exponential decay with constant risk, which is defined by the initial cell number and a half-life T(1/2). The half-life T(1/2) is determined using the least-squares fit to the clinical data on lung tumor size variation with time after stereotactic radiotherapy. We show that the tumor volume variation after stereotactic radiotherapy of solitary lung tumors can be approximated by an exponential function. A small constant component in the volume variation does not change with time; however, this component may be the residual irregular density due to radiation fibrosis and was, therefore, subtracted from the total volume variation in our computations. Using computerized fitting of the exponent function to the clinical data for selected patients, we have determined that the average half-life T(1/2) of cell disintegration is 28.2 days for squamous cell carcinoma and 72.4 days for adenocarcinoma. This model is needed for simulating the tumor volume variation during radiotherapy, which may be important for time-dependent treatment planning of proton therapy that is sensitive to density variations.

Single-cell-based computer simulation of the oxygen-dependent tumour response to irradiation

Optimization of treatment plans in radiotherapy requires the knowledge of tumour control probability (TCP) and normal tissue complication probability (NTCP). Mathematical models may help to obtain quantitative estimates of TCP and NTCP. A single-cell-based computer simulation model is presented, which simulates tumour growth and radiation response on the basis of the response of the constituting cells. The model contains oxic, hypoxic and necrotic tumour cells as well as capillary cells which are considered as sources of a radial oxygen profile. Survival of tumour cells is calculated by the linear quadratic model including the modified response due to the local oxygen concentration. The model additionally includes cell proliferation, hypoxia-induced angiogenesis, apoptosis and resorption of inactivated tumour cells. By selecting different degrees of angiogenesis, the model allows the simulation of oxic as well as hypoxic tumours having distinctly different oxygen distributions. The simulation model showed that poorly oxygenated tumours exhibit an increased radiation tolerance. Inter-tumoural variation of radiosensitivity flattens the dose response curve. This effect is enhanced by proliferation between fractions. Intra-tumoural radiosensitivity variation does not play a significant role. The model may contribute to the mechanistic understanding of the influence of biological tumour parameters on TCP. It can in principle be validated in radiation experiments with experimental tumours.

A multiscale model for avascular tumor growth

The desire to understand tumor complexity has given rise to mathematical models to describe the tumor microenvironment. We present a new mathematical model for avascular tumor growth and development that spans three distinct scales. At the cellular level, a lattice Monte Carlo model describes cellular dynamics (proliferation, adhesion, and viability). At the subcellular level, a Boolean network regulates the expression of proteins that control the cell cycle. At the extracellular level, reaction-diffusion equations describe the chemical dynamics (nutrient, waste, growth promoter, and inhibitor concentrations). Data from experiments with multicellular spheroids were used to determine the parameters of the simulations. Starting with a single tumor cell, this model produces an avascular tumor that quantitatively mimics experimental measurements in multicellular spheroids. Based on the simulations, we predict: 1), the microenvironmental conditions required for tumor cell survival; and 2), growth promoters and inhibitors have diffusion coefficients in the range between 10(-6) and 10(-7) cm2/h, corresponding to molecules of size 80-90 kDa. Using the same parameters, the model also accurately predicts spheroid growth curves under different external nutrient supply conditions.