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British Journal of Radiology (2006) 79, 389-400
© 2006 British Institute of Radiology
doi: 10.1259/bjr/30604050
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Full paper

A four-dimensional computer simulation model of the in vivo response to radiotherapy of glioblastoma multiforme: studies on the effect of clonogenic cell density

G S Stamatakos, PhD1, V P Antipas, PhD1, N K Uzunoglu, PhD1 and R G Dale, PhD2

1 In Silico Oncology Group, Microwave and Fibre Optics Laboratory, School of Electrical and Computer Engineering, National Technical University of Athens, 9 Iroon Polytechniou St., GR 157 80 Greece, 2 Department of Radiation Physics and Radiobiology, Hammersmith Hospitals NHS Trust and Faculty of Medicine, Imperial College, Charing Cross Hospital, Fulham Palace Road, London W6 8RF, UK


   Abstract
Top
 Abstract
Introduction
 Tumour cell distribution
 Recent improvements
 Parameter value selection: the...
 Validation and results
 Discussion and conclusion
 References
 
Tumours behave as complex, self-organizing, opportunistic dynamic systems. In an attempt to better understand and describe the highly complicated tumour behaviour, a novel four-dimensional simulation model of in vivo tumour growth and response to radiotherapy has been developed. This paper presents the latest improvements to the model as well as a parametric validation of it. Improvements include an advanced algorithm leading to conformal tumour shrinkage, a quantitative consideration of the influence of oxygenation on radiosensitivity and a more realistic, imaging based description of the neovasculature distribution. The tumours selected for the validation of the model are a wild type and a mutated p53 gene glioblastomas multiforme. According to the model predictions, a whole tumour with larger cell cycle duration tends to repopulate more slowly. A lower oxygen enhancement ratio value leads to a more radiosensitive whole tumour. Higher clonogenic cell density (CCD) produces a higher number of proliferating tumour cells and, therefore, a more difficult tumour to treat. Simulation predictions agree at least semi-quantitatively with clinical experience, and particularly with the outcome of the Radiation Therapy Oncology Group (RTOG) Study 83-02. It is stressed that the model allows a quantitative study of the interrelationship between the competing influences in a complex, dynamic tumour environment. Therefore, the model can already be useful as an educational tool with which to study, understand and demonstrate the role of various parameters in tumour growth and response to irradiation. A long term quantitative clinical adaptation and validation of the model aiming at its integration into the treatment planning procedure is in progress.


   Introduction
Top
 Abstract
 Introduction
Tumour cell distribution
 Recent improvements
 Parameter value selection: the...
 Validation and results
 Discussion and conclusion
 References
 
The rapid growth and resilience of tumours make it difficult to believe that they behave as random, disorganized and diffuse cell masses and suggests instead that they are emerging, opportunistic systems [1, 2]. If this hypothesis holds true, the growing tumour and not only the single cell [3] must be investigated and treated as a self-organizing complex dynamic system. This cannot be done with currently available simple in vitro/in vivo models or common mathematical approaches. Therefore, there is a need for novel computational models to simulate the complexity of solid tumour growth and invasion, combining a range of disciplines including medical, biological, engineering and statistical physics research [1, 2]. Recent research efforts have focused on modelling of tumour response to various therapeutic modalities but, unfortunately, current models of dynamic processes need substantial improvement due to the complexity of the problem and the paucity of large series' of clinical data.

This paragraph provides a brief outline to several of the concepts and earlier research efforts. Duechting et al [4, 5] introduced a simulation model which concerns only the in vitro case or the early avascular stages of small in vivo tumours and is based on a consideration of the distinct phases of the cell cycle. Kocher et al [6, 7] developed a simulation model of the development of a tumour spheroid and its response to radiosurgery. However the detailed geometry of the clinical tumour as depicted by imaging data has not been considered in this model. Instead, an equivalent spherical tumour is considered in place of the generally arbitrarily shaped actual tumour. Additionally, detailed cell cycle phase biology (phases G1, S, G2, M) has not been taken into account, with grouping of the cells into only proliferating and dormant classes being considered instead. None of the above mentioned models have been applied to large clinical tumours of varied geometrical shapes, and none of them simulates conformal shrinkage for an arbitrarily shaped clinical tumour undergoing treatment. In the tumour growth models presented by Kansal et al [1, 2], a discretising grid is used in which each geometrical cell is able to contain a large number of biological cells, but the grid has not been used to discretise clinical tumours of arbitrary shape. Neither is the response of the tumour to irradiation addressed in this model. Swanson et al [8, 10] and Mandonnet et al [9] have developed clinically significant spatiotemporally models of tumour growth and invasion concerning glioblastoma multiforme (GBM). Nevertheless, although growth and invasion constitute fundamental phenomena related to GBM treatment optimization, the investigators have not focused on the radiobiological mechanisms underlying radiotherapy. Byrne et al and Alarcon et al [11–14] have developed mathematical models of avascular tumour growth and angiogenesis evolution pertinent mainly to the initial stages of tumour development. Although valuable insight can be gained using such models, extension to clinical voluminous tumours is not an a priori manageable task.

An effort to overcome these shortcomings has previously been made by our group through the development of a four-dimensional patient-specific in vivo simulation model [15–17]. All parameters used in the model have already been defined and can be determined (in principle) experimentally or clinically. Therefore, use of new mathematically dictated parameters of ambiguous physical meaning is avoided. Furthermore, the authors believe that the development of an experimental and clinical biology based model provides substantial insight into the interdependence of the mechanisms involved, even if some parameters cannot currently be accurately quantified for individual patients.

This work presents the latest advances and improvements of a four-dimensional, patient individualized, in vivo simulation model of tumour response to radiotherapy and discusses attempts at parametric and clinical validation. To this end, the outcome of a pertinent clinical study is exploited. It is also noted that an eventual combination of the model with the approaches of the previously mentioned research groups might be worth a careful investigation. In such a case, a more scientifically complete and clinically meaningful picture of the various aspects of tumour modelling might emerge. It is suggested that the work presented might also be considered a step towards shaping an emerging analytical-computational discipline of "In Silico Radiation Oncology".


   Tumour cell distribution
Top
 Abstract
 Introduction
 Tumour cell distribution
Recent improvements
 Parameter value selection: the...
 Validation and results
 Discussion and conclusion
 References
 
In the following, a brief outline of the model's construction is given.

  1. A discretizing cubic mesh is superimposed upon a three-dimensional reconstruction of the tumour, including its necrotic region and the surrounding anatomical features, based on the imaging data.
  2. Within each geometrical cell of the mesh, a number of equivalence classes (compartments) of the contained biological cells are defined based on their distribution over the various phases within or out of the cell cycle. Sufficient registers are used in order to store the current state of each equivalence class (e.g. time spent in phase G1, etc.).
  3. The mesh is scanned every hour.
  4. The basic biological "laws" (metabolic activity, cell cycling, mechanical restrictions, cell survival probability following irradiation with dose D, etc.) are applied on each geometrical cell at each scanning.
  5. A spatial and functional restructuring of the tumour may take place during each scanning as e.g. new cells may be produced (leading to differential tumour growth) or as existing cells may disappear (leading to differential tumour shrinkage).
  6. The simulation predictions can be two- or three-dimensionally visualized at any simulated instant of interest.

The basic philosophy of our approach can also be found (http://www.in-silico-oncology.iccs.ntua.gr/) at [17].


   Recent improvements
Top
 Abstract
 Introduction
 Tumour cell distribution
 Recent improvements
Parameter value selection: the...
 Validation and results
 Discussion and conclusion
 References
 
Vessel network, neoangiogenesis and oxygen supply based on imaging data
The observation that angiogenesis occurs around tumours was made nearly 100 years ago [18–21]. The hypothesis that tumours produce a diffusible "angiogenic" substance was put forward in 1968 [22, 23]. Mammalian cells require oxygen and nutrients for their survival, and functional cells must therefore be located within a distance of 100–200 µm from the nearest capillary blood vessels, which is the diffusion limit for oxygen. Vessels in an embryo are assembled from endothelial precursors and this primitive network subsequently expands by sprouting (angiogenesis) or intussusception, in which interstitial tissue columns are inserted into the lumen of pre-existing vessels and partition the vessel lumen [24]. Tumour vessels develop by sprouting or intussusception from pre-existing vessels. Circulating endothelial precursors, shed from the vessel wall or mobilized from the bone marrow, can also contribute to tumour angiogenesis [25, 26]. Tumour cells can also grow around an existing vessel to form a perivascular cuff. Without blood vessels, tumours cannot grow beyond a critical size or metastasise to another organ [27].

In contrast to normal vessels, tumour vasculature has the following distinct characteristics:

  1. Tumour vessel ultrastructure is abnormal [27]
  2. Tumour vessel ultrastructure is highly disorganized [27]
  3. Tumour vessels are tortuous and dilated, with uneven diameter, excessive branching and shunts [27]
  4. Tumour blood flow is chaotic and variable [27, 28], and leads to hypoxic and acidic regions in tumours [27, 29]
  5. Vessel walls have numerous "openings", widened interendothelial junctions, and a discontinuous or absent basement membrane [27]
  6. Tumour vessels are "leaky" and have tremendous heterogeneity in leakiness over space and time [27, 30, 31]

Definition of the imaging based tumour layers dictates the number and the metabolic state of the individual biological cells included within each layer. During the simulation process, and in the case of tumour growth, the normal tissue capillaries are shifted away and tumour capillaries are generated in their place [27, 32]. Consequently, the new tumour cells are assumed to be sufficiently oxygenated and able to divide.

A "proliferating layer" is assumed to exist between the external surface of the gross tumour volume and a hypothetical surface (HYP) enclosing its necrotic kernel and lying 1.5 mm further out. The tumour volume contained between HYP and the surface of the necrotic region has been assumed to contain large numbers of dormant G0 cells; therefore this is called the "G0 cell layer". This layer contains a substantial number of dormant cells around the necrotic area of the tumour as it appears on the imaging data. We have assumed that the clonogenic cell density (CCD) in the "proliferating layer" is two times the CCD in the G0 cell layer. CCD in the necrotic or dead cell layer of the tumour has also been assumed to be one fifth of the CCD in the proliferating cell layer.

Consideration of oxic and hypoxic cells
The resistance of cells that are hypoxic at the time of therapy will influence the efficacy of treatment with radiation, chemotherapy and combined modality regimens. Tumour cell response to ionizing radiation is strongly dependent upon oxygen, any given dose killing substantially fewer hypoxic than oxic cells. The radiation dose that allows a particular level of survival tends to increase by the same factor at all levels of survival when oxygen is removed. This allows calculation of oxygen enhancement ratio (OER) for the same level of biological effect. For most cells, OER for X-rays is around 3.0 [6, 7, 33]. Some researchers [33–35] report that OER reduces for radiation doses to 3.0 Gy or less. In practical terms, within a tumour microenvironment the oxic cells are those which proliferate, whereas the hypoxic cells are dormant or G0 cells.

Extensive work has been done to measure hypoxia in human brain tumours (especially for gliomas) [36–41]. Different LQ {alpha} and beta parameter values for the oxic (G1, S, G2, M) and hypoxic (G0) cells are considered. The interrelation between the hypoxic and oxic LQ parameters is given by the following expressions [7]:


Formula 001



Formula 002

Nygren and Ahnstrom [42] suggest that OER can range from 2.0 to 3.0, and Palcic et al [34], Stuschke et al [43] and Speke and Hill [44] reported a value for OER of 2.3, 2.7, 2.75, respectively.

It should be noted that hypoxic cells in the clinical setting can become oxic again when either new microvasculature vessels have emerged in their vicinity, or when the space between them and the nearest blood vessels has been cleared of other cells already killed by irradiation.

An advanced algorithm leading to conformal tumour shrinkage
The tumour shrinkage process usually tends to behave as a conformal contraction [45]. The biological rationale for the cells to be "pulled" towards the centre-of-mass of the tumour is that the surrounding normal tissues exert a rather uniform pressure upon the tumour in such a way that the brain tends to recover its (physiological) normal shape (homeostasis). Nevertheless, deviations of this rule due to local inhomogeneities are to be expected.

In order to satisfactorily simulate this process in conjunction with the rest of the simulation strategy, the "centre-of-mass algorithm" (CMA) is introduced. Its primary mission is to "pull" individual tumour cells towards the centre-of-mass of the entire tumour. It is emphasised that the term "centre-of-mass" is not used in a strict context as it refers to a tumour of uniform mass density. The model does not consider the eventual appearance and behaviour of new tumour foci out of the space occupied by the primary tumour due to infiltration of adjacent tissues.

Any geometrical cell which is in two-dimensional side contact with at least one other geometrical cell of the main tumour mass is considered to belong to the main cohesive tumour mass. Let us suppose that the distance from the centre of an isolated geometrical cell C2 to the nearest geometrical cell occupied by the main tumour mass is larger than one geometrical cell. The content of the isolated geometrical cell tends to move towards the centre of the main cohesive tumour mass. Only shifts along the x+ or x- or y- or y+ or z- or z+ directions are allowed. During the application of CMA, the distance from the centre of C2 to the centre of the main tumour along each of the six possible directions is calculated. Subsequently, the least distance is selected. In cases where more than one distance has the same least value, the direction is selected randomly. After completion of the next discretization mesh scan, C2 will be closer to the tumour centre-of-mass and after a number of scans, C2 will be connected with the nearest cell of the main tumour mass.

A more realistic approach to tumour cell distribution
A random number generator is used to produce a uniform cell distribution over the individual time units constituting each phase within or out of the cell cycle (e.g. S, G0), this being a more realistic approach. This is because cells within the same cell cycle phase are not considered to be synchronized during the initialization process.


   Parameter value selection: the paradigm of in vivo glioblastoma multiforme
Top
 Abstract
 Introduction
 Tumour cell distribution
 Recent improvements
 Parameter value selection: the...
Validation and results
 Discussion and conclusion
 References
 
The selected paradigm is a recently irradiated glioblastoma multiforme tumour. An oncology specialist delineates the gadolinium enhanced T1 weighted MRI imaging-based apparent boundary of the tumour as well as the boundary of its necrotic region. As a first approach, the neovasculature field is assumed to coincide with the area of the tumour where pronounced metabolism is apparent on the imaging data. A cube defining the anatomical region of interest is superimposed on the three-dimensionally reconstructed tumour and surrounding anatomical features. The cube includes a volume discretizing mesh. The dimensions of each geometrical cell of the mesh, considered to be able to accommodate 106 biological cells [46] (NBC = 106), are 1 mmx1 mmx1 mm.

A number of researchers have focused on the measurement of the duration of cell cycle (Tc), especially for gliomas. Hoshino et al [47] have reported a mean Tc of 57 h. Crafts et al [48] suggested that Tc can range from 2 days to 3 days. Hoshino and Wilson [49] have mentioned a Tc of 75.6 h. Pertuiset et al [50] have found an average value of 1–2.5 days. The Tc values considered in this paper have been 24 h, 48 h and 72 h. Salmon et al [51] have suggested that proliferating tumour cells would spend their time in the various cell cycle phases as follows: fraction of time spent in G1: TG1 = 0.4 TC, S: TS = 0.39 TC, G2: TG2 = 0.19 TC and M: TM = 0.02 TC.

After irradiation, most often the reproductively dead cells will continue to cycle for (usually) 1–3 divisions before their ultimate (biological) death. In the model developed by our group, reproductively dead cells are assumed to undergo two mitoses before their biological death [17]. Reproductively dead cells and their offspring which are still cycling are considered proliferating until their ultimate biological death because they are detectable through imaging modalities. This point emphasises the conceptual differences that may arise between an engineering and a medical physics/clinical approach to the same biological phenomenon. Exploratory simulation runs have shown that if still-cycling reproductively (but not yet biologically) dead cells are added to the unaffected proliferating cells, an increase in the number of the latter by up to a factor of 10 (1 log) is to be expected during a typical radiotherapy course.

A standard fractionation scheme (2 Gy once a day, 5 days per week, 60 Gy in total) has been simulated. The LQ model parameters of the tumour have been assumed to be {alpha}oxic = 0.17 Gy–1, betaoxic = 0.02 Gy–2, and {alpha}hypoxic = (0.17/OER) Gy–1, betahypoxic = (0.02/OER2) Gy–2 for a GBM with known mt p53 gene [52] and {alpha}oxic = 0.6 Gy–1, betaoxic = 0.06 Gy–2, {alpha}hypoxic = (0.6/OER) Gy–1, betahypoxic = (0.06/OER2) Gy–2 for a GBM with known wild type (wt) p53 gene [7, 53]. Both the {alpha}oxic and betaoxic parameters are assumed to remain constant throughout the cell cycle. For visualization purposes, cells are "painted as dead" during the time interval between a lethal cell hit and necrosis or apoptosis.

For the specific type of poorly differentiated tumour under consideration, and for simplification reasons, all non-clonogenic cells have been considered to be necrotic and sterile cells have not been taken into account. The contribution of the living non-clonogenic cells will be considered in a future version of the model.

A typical tumour volume of 20 cm3 contains 4–5% clonogenic tumour cells [54], i.e. the typical CCD is around 107 cm–3 [55]. Most calculations of the biological effect of radiation on tumours assume that the CCD is uniform. But in practice, tumours will almost certainly have a non-uniform CCD [56]. This factor has not been modelled except by Brahme and Agren [57] and Webb and Nahum [56]. At present, as there is paucity of experimental data in vivo, a range of reasonable values has been assumed. Webb and Nahum [56] have reported that the variation of CCD across tumours is a very important factor, especially for brain tumours. Hence the CCD has been assumed to be 1x104 cells mm–3, 2x104 cells mm–3, 3x104 cells mm–3, 4x104 cells mm–3.

Within each geometrical cell of the discretizing mesh, the initial distribution of the clonogenic cells throughout the cell cycle, the G0 and the necrotic phases depends on the layer of the tumour to which the geometrical cell under consideration belongs [17].

The cell loss factor [58] has been taken to be 0.3 as cell death products are removed from brain with substantial difficulty and it has been expressed as the sum of the cell loss factor due to necrosis (0.2) and the cell loss factor due to apoptosis (0.1).

It is noted that estimates of the percentage of proliferating, dormant and dead cells in the various tumour layers are included in Stamatakos et al [17] and are based on a rather semi-quantitative representation of the cycling status of tumour cells depending on the imaging-based layer in which the considered tumour cells lie. For example, proliferating cells included in the dark tumour areas that appear on gadolinium enhanced T1 weighted MRI slices are expected to be scant. Concerning cell density, the standard assumption of 106 cells mm–3 [46] has been made.


   Validation and results
Top
 Abstract
 Introduction
 Tumour cell distribution
 Recent improvements
 Parameter value selection: the...
 Validation and results
Discussion and conclusion
 References
 
In order to clinically evaluate the simulation model, several arm simulations of the Radiation Therapy Oncology Group (RTOG) study 83-02 [18] have been performed. The GBM imaging data considered throughout the whole paper have been used as the spatial basis for performing in silico experiments. The following typical parameter values have been adopted: OER = 3.0, clonogenic cell density = 1xCCD = 1x104 cells mm–3, cell cycle duration Tc = 30 h and the LQ parameters have been assumed to match the values of a GBM with mt p53 gene as previously mentioned.

Figure 1aGo shows the total number of proliferating and dormant tumour cells as a function of time for the hyperfractionated (1.2 Gy twice daily to the dose of 81.6 Gy, "HF-81.6") and accelerated hyperfractionated (1.6 Gy twice daily to the dose of 54.4 Gy, "AHF-54.4") radiotherapy schedules. All schemes considered in this paper start on the first day of the radiotherapy course. HF-81.6 is completed on day 46 after initiation of treatment whereas AHF-54.4 is completed on day 23. Figure 1bGo depicts the total number of proliferating and dormant tumour cells as a function of time for the hyperfractionated (1.2 Gy twice daily to the dose of 76.8 Gy, "HF-76.8") and accelerated hyperfractionated (1.6 Gy twice daily to the dose of 48 Gy, "AHF-48") radiotherapy schedules. Both irradiation schedules start on the first day of the first week of treatment. HF-76.8 is completed on day 44 after initiation of treatment whereas AHF-48 is completed on day 19. According to the graphs, before completion of the AHF course, cell kill due to AHF irradiation is more pronounced than cell kill induced by the HF scheme. This can be explained by the fact that a higher total dose has been administered to the tumour by the AHF scheme whereas, for the period under consideration, both schemes are characterized by the same time intervals between consecutive sessions. In cases where not all living cells have been killed by AHF irradiation, tumour repopulation is considerable so that, by the time the HF scheme is completed, living tumour cells and their progeny which have escaped AHF irradiation outnumber tumour cells which have escaped HF irradiation. Improved tumour control following HF irradiation in comparison with the AHF scheme is in agreement with the conclusions of the clinical trial RTOG 83-02.


Figure 1
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  		Figure 1. (a) Total number of proliferating and dormant tumour cells as a function of time for the hyperfractionated (1.2 Gy twice daily, 5 days per week to the dose of 81.6 Gy, "HF-81.6") and accelerated hyperfractionated (1.6 Gy twice daily, 5 days per week to the dose of 54.4 Gy, "AHF-54.4") radiotherapy schedules. HF-81.6 is completed on day 46 after initiation of treatment whereas AHF-54.4 is completed on day 23. In all fractionation schedules considered in this paper, no radiation is administered on Saturdays or Sundays. (b) Total number of proliferating and dormant tumour cells as a function of time for the hyperfractionated (1.2 Gy twice daily, 5 days per week to the dose of 76.8 Gy, "HF-76.8") and accelerated hyperfractionated (1.6 Gy twice daily, 5 days per week to the dose of 48 Gy, "AHF-48") radiotherapy schedules. Both irradiation schedules start on the first day of the first week of treatment. HF-76.8 is completed on day 44 after initiation of treatment whereas AHF-48 is completed on day 19.

 
Figure 2aGo shows the total number of proliferating and dormant tumour cells as a function of time for the hyperfractionated (1.2 Gy twice daily to the dose of 72 Gy, "HF-72") and accelerated hyperfractionated (1.6 Gy twice daily to the dose of 48 Gy, "AHF-48") radiotherapy schedules. HF-72 is completed on day 40 after initiation of treatment whereas AHF-48 is completed on day 19. Figure 2bGo shows the total number of proliferating and dormant tumour cells as a function of time for the hyperfractionated (1.2 Gy twice daily to the dose of 64.8 Gy, "HF-64.8") and accelerated hyperfractionated (1.6 Gy twice daily to the dose of 48 Gy, "AHF-48") radiotherapy schedules. HF-64.8 is completed on day 37 after initiation of treatment whereas AHF-48 is completed on day 19. Both irradiation schedules start on the first day of the first week of treatment. According to the graphs, before completion of the AHF course, cell kill due to AHF irradiation is more pronounced than cell kill induced by the HF scheme. This can be explained by the fact that higher total dose has been administered to the tumour by the AHF scheme with the same fractionation as the one followed by the HF scheme for the period under consideration. In cases where not all living cells have been killed by AHF irradiation, tumour repopulation is considerable so that, by the time the HF scheme is completed, living tumour cells and their progeny which have escaped AHF irradiation outnumber tumour cells which have escaped HF irradiation.


Figure 2
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  		Figure 2. (a) Total number of proliferating and dormant tumour cells as a function of time for the hyperfractionated (1.2 Gy twice daily, 5 days per week to the dose of 72 Gy, "HF-72") and accelerated hyperfractionated (1.6 Gy twice daily, 5 days per week to the dose of 48 Gy, "AHF-48") radiotherapy schedules. HF-72 is completed on day 40 after initiation of treatment whereas AHF-48 is completed on day 19. (b) Total number of proliferating and dormant tumour cells as a function of time for the hyperfractionated (1.2 Gy twice daily, 5 days per week to the dose of 64.8 Gy, "HF-64.8") and accelerated hyperfractionated (1.6 Gy twice daily, 5 days per week to the dose of 48 Gy, "AHF-48") radiotherapy schedules. HF-64.8 is completed on day 37 after initiation of treatment whereas AHF-48 is completed on day 19. Both irradiation schedules start on the first day of the first week of treatment.

 
Figure 3Go depicts the total number of tumour cells (proliferating, dormant and dead cells) as a function of time for the hyperfractionated (1.2 Gy twice daily to the dose of 76.8 Gy, "HF-76.8") and accelerated hyperfractionated (1.6 Gy twice daily to the dose of 48 Gy, "AHF-48") radiotherapy schedules. Irradiation starts on the first day of the first week. HF-76.8 is completed on day 44 after initiation of treatment whereas AHF-48 is completed on day 19. Both irradiation schedules start on the first day of the first week of treatment. At the end of week 10, repopulation following AHF-48 treatment is more pronounced than repopulation following HF-76.8, in accordance with Figure 1bGo.


Figure 3
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  		Figure 3. Total number of tumour cells(proliferating, dormant and dead cells) as a function of time for the hyperfractionated (1.2 Gy twice daily, 5 days per week to the dose of 76.8 Gy, "HF-76.8") and accelerated hyperfractionated (1.6 Gy twice daily, 5 days per week to the dose of 48 Gy, "AHF-48") radiotherapy schedules. Irradiation starts on the first day of the first week. HF-76.8 is completed on day 44 after initiation of treatment whereas AHF-48 is completed on day 19. Both irradiation schedules start on the first day of the first week of treatment.

 
All of the above mentioned observations are in agreement with the statistically significant results of the RTOG 83-02 study [18] which refer to the survival for the following radiotherapy fractionations: high dose hyperfractionation (HF, total doses: 76.8 Gy and 81.6 Gy), low dose hyperfractionation (HF, total doses: 64.8 Gy and 72.0 Gy) and accelerated hyperfractionation (AHF, total doses: 48.0 Gy and 54.4 Gy). It should be noted that carmustine (BCNU), which was also administered to all patients included in that study, was assumed not to significantly modify the effectiveness relations among the different radiotherapeutic schemes considered. The reason for this is that the same chemotherapy administration schedule was followed for all patients regardless of the specific radiotherapeutic scheme administered to each individual patient. All patients received BCNU 80 mg m–2, intravenously, on days 1, 2 and 3 of the first week of radiotherapy and subsequently on 3 consecutive days every 8 weeks for a period of 1 year, to a maximum dose of 1440 mg m–2. Only limited modifications of the schedule based on age and not on the radiotherapy scheme administered were made during the trial. Agreement with the results of the RTOG 83-02 study implies that the simulation model has successfully captured and integrated the critical biological aspects that determine the clinical outcome. Furthermore, it strengthens the proposition that advanced cancer integrative (radio)biology might explain and predict the therapy outcome, despite the complexity of the clinical setting.

In the following, a parametric analysis is carried out in order to study the effect of the clonogenic cell density as well as that of the OER and cell cycle duration. The model code has been executed for a simulated period of up to 6 weeks, an interval which normally covers the treatment period of the radiotherapy course and may extend to some days after its completion. The cell cycle duration has been assumed to be 48 h, OER equal to 3.0, CCD equal to 1x104 cells mm–3 and the LQ parameters have been assumed to match the values of a GBM with known mt p53 gene, unless otherwise stated.

The simulation results of Figure 4Go demonstrate the ability of the algorithm to effectively simulate the tumour response to a standard irradiation scheme under different values of CCD (1x104 cells mm–3, 2x104 cells mm–3, 3x104 cells mm–3, 4x104 cells mm–3). Higher values of CCD not only produce a higher number of proliferating tumour cells, but also affect the entire tumour composition. Such behaviour is in accordance with the previously described interdependence among the various metabolic layers of the tumour (dormant, dead, proliferating). Figure 5Go provides a two-dimensional visualization of the simulated response of a clinical glioblastoma multiforme tumour to the standard fractionation scheme for different CCD values. At the end of day 3, the tumour with 1xCCD appears to be more radiosensitive compared with tumours with 2xCCD, 3xCCD and 4xCCD as its population of proliferating cells is lower compared with the others (Figure 5a–dGo). At the end of day 5, all tumours are strongly affected by radiation treatment, whereas the highest number of proliferating cells is contained in tumour with 4xCCD (Figure 5hGo). At the beginning of the first day of the second week (day 8), newly-produced proliferating cells are present in sufficient numbers to be apparent in Figure 5k,lGo. The population of the proliferating cells is larger in Figure 5lGo compared with Figure 5kGo while the majority of the slice of tumour Figure 5lGo is painted as "proliferating". For this specific case, the LQ model parameters of the tumour are in accordance with a GBM with known mt p53 gene.


Figure 4
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  		Figure 4. Simulation predictions of(a) the number of proliferating and (b) the total number of proliferating, dormant and dead mt p53 tumour cells in the case of the standard fractionation scheme (2 Gy once a day, 5 days per week, 60 Gy in total) for different clonogenic cells densities (CCD = 104 cells mm–3). Irradiation schedule starts on the first day of the first week of treatment. It should be stressed that reproductively dead cells and their offspring that are still cycling are considered proliferating until their ultimate biological death. The periodicity noticed on all graphs reflects the weekly irradiation periodicity. It is noted that no irradiation takes place during the weekend.

 

Figure 5
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  		Figure 5. Two-dimensional visualization of the simulated response of a radiosensitive clinical glioblastoma multiforme tumour to the standard fractionation scheme, for a range of clonogenic cell density (CCD). The figure shows a centrally located horizontal slice of a tumour with clonogenic cell density equal to 1xCCD( = 104 cells mm–3) (a) 3 fictitious days after the beginning of the radiotherapy course (e) 5 fictitious days after the beginning of the radiotherapy course, (i) 8 fictitious days after the beginning of the radiotherapy course. A centrally located horizontal slice of a tumour with clonogenic cell density equal to 2xCCD (b) 3 fictitious days after the beginning of the radiotherapy course (f) 5 fictitious days after the beginning of the radiotherapy course (j) 8 fictitious days after the beginning of the radiotherapy course. A centrally located horizontal slice of a tumour with clonogenic cell density equal to 3xCCD, (c) 3 fictitious days after the beginning of the radiotherapy course, (g) 5 fictitious days after the beginning of the radiotherapy course, (k) 8 fictitious days after the beginning of the radiotherapy course. A centrally located horizontal slice of a tumour with clonogenic cell density equal to 4xCCD, (d) 3 fictitious days after the beginning of the radiotherapy course, (h) 5 fictitious days after the beginning of the radiotherapy course, (l) 8 fictitious days after the beginning of the radiotherapy course. For this specific case, the LQ model parameters of the tumour are in accordance with a GBM cell line with known mt p53 gene. Irradiation schedule starts on the first day of the first week of treatment. Colour Code: dark grey: proliferating cell layer, light grey: dormant cell layer (G0), white: dead cell layer. The colouring criterion "99.8%" used to visualize the predictions has been defined as follows. "For a geometrical cell of the discretising mesh, if the percentage of dead cells is lower than 99.8% then {if percentage of proliferating cells > percentage of G0 cells, then paint the geometrical cell dark grey (proliferating cell layer)}, else paint the geometrical cell light grey (G0 cell layer) else paint the geometrical cell white (dead cell layer)".

 
The simulation results in Figure 6Go demonstrate the ability of the algorithm to effectively simulate the tumour response to a standard irradiation scheme for different OER values (1.0, 2.0, 3.0). Higher OER values describe a more radioresistant tumour.


Figure 6
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  		Figure 6. Simulation predictions of the number of(a) proliferating and (b) dormant mt p53 tumour cells in the case of standard fractionation for different OER (1.0, 2.0, 3.0) values. It is stressed that reproductively dead cells and their offspring that are still cycling are considered proliferating until their ultimate biological death. Irradiation schedule starts on the first day of the first week of treatment.

 
Figure 7Go depicts the corresponding simulation results of a clinical glioblastoma tumour to the standard fractionation scheme for different Tc values. For the short period simulated (10 days) the tumour with Tc = 48 h appears to be more difficult to treat than the one with Tc = 60 h. It can nevertheless be noted that during most of the working days the tumour with Tc = 48 h shows a better response to irradiation whereas the tumour with Tc = 60 h shows a slower response to radiation due to its long Tc.


Figure 7
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  		Figure 7. Simulation predictions of the number of(a) proliferating and (b) dormant mt p53 tumour cells in the case of standard fractionation for different Tc (cell cycle time) values. It is stressed that reproductively dead cells and their offspring that are still cycling are considered proliferating until their ultimate biological death. Irradiation schedule starts on the first day of the first week of treatment.

 
Figure 8Go is a three-dimensional reconstruction of the simulated response of a GBM with known mt p53 gene for Tc = 24 h (Figure 8aGo) and Tc = 72 h (Figure 8bGo) at the end of day 8. Figure 8aGo contains few geometrical cells which have been painted as "proliferating", in contrast to Figure 8bGo in which all geometrical cells have been painted as "dead". During the interval in which the tumour is not irradiated (days 6 and 7), the proliferating tumour cells with Tc = 24 h multiply prominently faster than in the case of the tumour with Tc = 72 h.


Figure 8
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  		Figure 8. Three-dimensional visualization of the simulated response of a radiosensitive clinical glioblastoma multiforme tumour to the standard fractionation scheme, for (a) Tc = 24 h and (b) Tc = 72 h at the end of day 8. It is pointed out that total tumour cells include all morphologically existing cells, living (proliferating and quiescent) and dead (but not yet lysed or fragmented) alike. Irradiation schedule starts on the first day of the first week of treatment. Colour Code: red: proliferating cell layer, green: G0 layer, blue: dead cell layer. The colouring criterion "99.8%" was used to visualize the predictions (Figure 5Go).

 
Concerning the model sensitivity issue, GoGoFigures 4, 6 and 7Go can also be used in order to quantitatively visualize the effect of perturbing critical model parameters on the simulation predictions. For example, according to Figure 4aGo, on the 30th day after initiation of the radiotherapy course the number of proliferating tumour cells for the case of CCD equal to 4xCCD (CCD = 104 cells mm–3) is as high as 270% of the number of proliferating cells for the case of 1xCCD on the same day. According to Figure 6aGo, on the 30th day after initiation of the radiotherapy course the number of proliferating tumour cells for the case of OER = 3.0 is as high as 127% of the number of proliferating cells for the case of OER = 1.0 on the same day. According to Figure 7aGo, on the 30th day after initiation of the radiotherapy course the number of proliferating tumour cells for the case of Tc = 48 h is as high as 339% of the number of proliferating cells for the case of Tc = 60 h on the same day. Concerning estimation of confidence intervals it is noted that, due to the limited information contained in the RTOG 83-02 clinical study data and the small number of complete sets of clinical data available by our group, no strict confidence limits can be estimated as yet. It is pointed out that cancer is inherently a partly stochastic phenomenon and, therefore, an accurate estimation of the confidence limits of its predicted in vivo behaviour is not always feasible. It is also noted that, although the simulation model predicts the obvious, it provides a means for quantitatively studying the response of real clinical malignant tumours to radiotherapeutic schemes.


   Discussion and conclusion
Top
 Abstract
 Introduction
 Tumour cell distribution
 Recent improvements
 Parameter value selection: the...
 Validation and results
 Discussion and conclusion
References
 
The results presented are in agreement with both qualitative clinical experience and the outcome of the RTOG 83-02 clinical study. Predictably, a whole tumour with shorter Tc tends to repopulate faster and therefore is more difficult to treat.

Greater CCD produces a higher number of proliferating tumour cells, and therefore a tumour which is more difficult to treat, and eventually produces a lower tumour control probability (TCP) [59]. Webb and Nahum [56] report that the TCP is a complicated function of the variation in both dose and CCD and TCP will depend on the assumption made about CCD. It is pointed out that reproductively dead cells can cycle only for a very limited number of divisions (one to three) before they and their progeny die biologically. Therefore, re-growth of reproductively dead cells or reproductively "killed" clonogenic cells is unimportant for clinical outcome [60].

Although simulation prediction curves look quite similar (GoGoFigures 4, 6 and 7Go), a closer observation reveals several differences among them. Random numbers have actually been used, but the fact that all biological parameters for all curves in Figure 4Go are the same except for the CCD has led to a pretty similar (analogous) behaviour of the composite biological system. Furthermore, a careful and successful application of the Monte Carlo technique has led to a remarkably stable numerical behaviour of the simulation model.

It has been experimentally demonstrated that most solid tumours contain resistant hypoxic cells, with estimates of the hypoxic fractions ranging from below 1% to well over 50% of the total viable cell population [33]. Additionally, evidence that hypoxia exists in human tumours to a degree that can influence radiation response comes from those clinical trials in which some form of hypoxia modification has been attempted and found to improve tumour response [61]. Knisely and Rockwell [62] have reported that the resistance of gliomas to treatment with radiation and antineoplastic drugs may result in part from the effects of the extensive and severe hypoxia that is present in such tumours. They have emphasised that the brain tumours contain extensive regions in which the tumour cells are subjected to an unphysiological degree of hypoxia, this being involved in the evolution of cells in low-grade malignancies to the resistant, aggressive phenotype characteristic of glioblastomas. Furthermore, the results of the parametric OER study agree with data presented by Horsman and Overgaard [33]. It is stressed that although the model confirms the obvious, it allows a quantitative study of the inter-relationship between the competing influences in a complex, dynamic tumour environment. A possible future refinement of the model would include a detailed description of the modulation of cell cycling by oxygen tension by taking into account the latest pertinent experimental observations. It is also noted that, according to our model, intrinsic sensitivity, at least as modified, by OER, is an important determinant of the poor outcome of glioblastoma multiforme irradiation, which appears to be in disagreement with Taghian et al [63].

The CMA tumour shrinkage algorithm applied in this paper is more realistic than the one previously described by Stamatakos et al [17], as CMA is able to substantially conserve the tumour shape (conformal shrinkage) in accordance with Perez and Brady [45]. Additionally, the imaging based neovasculature distribution has been in agreement with Horsman and Overgaard [33]. Finally, the uniform cell distribution algorithm produces a realistic initial cell phase distribution. The predictions of all indicative simulations performed agree at least qualitatively with the clinical experience and with the data presented by Duechting et al [1, 5], Kocher et al [6, 7] and Horsman and Overgaard [33].

It is stressed that the model presented addresses the imageable gross tumour. In order to take into account the brain micrometastases (diffuse invasion), an approach similar to the one developed by Swanson et al [8, 10] and Mandonnet et al [9] or Kansal et al [1, 2] must be considered. It is also noted that the end points of most clinical trials have not been the volumetric and/or metabolic response of tumours, but rather the overall response to treatment such as survival and tumour relapse interval. Consequently, currently available clinical data can be exploited for the validation of the models only in a rough (approximate) way. This implies that a better survival following treatment scheme X compared with survival following treatment scheme Y can be roughly used as an indicator of better tumour control achievable with scheme X. Obviously this can be the case if radiation toxicity is within acceptable limits for both treatment schemes. Concerning fluctuations of radiosensitivity throughout the cell cycle, differing radiosensitivity of cells in the various phases of the cell cycle has been considered and successfully simulated by our group in the case of small in vitro or in vivo pre-angiogenetic/avascular tumour spheroids [17]. Nevertheless, although easily includable, this variation has not been addressed in the present version of the in vivo gross tumour model in order to keep computer memory and execution time requirements as low as possible.

Modelling of the irradiation effects on the surrounding normal tissues in vivo, modelling of the tumour response to chemotherapy in vivo and further enhancement of our models with more genetic data are currently under way. Systematic comparison with clinical data is expected to lead to more clinically adapted parameter values. The clinical validation procedure is in progress and involves comparison of the model predictions with pertinent clinical data before, during and after the radiotherapy course. The easily adjustable, modular simulation model "follows" the clinical practice and activates a self-optimization procedure.

It should also be noted that very rare deviations from the poor standard prognosis of glioblastoma multiforme do exist. Nevertheless, the aim of the simulation model is to predict the most likely time course of the treatment response and, consequently, it is unlikely that an extreme scenario would be predicted. Execution of the computer code on supercomputer systems where a more dense discretizing mesh could be considered might refine the model's prediction accuracy.

The validation process in conjunction with generic parameter estimation techniques (neural networks technique, taboo searching, etc.) can be used to achieve better estimates of the input parameters. Agreement with clinical observations strengthens the applicability of the model to real situations. An integrated and patient- individualized decision support and spatiotemporal treatment planning system is expected to emerge after completion of the necessary clinical adaptation and validation processes. Such a system could also serve as an educational platform for professionals and patients by means of virtual reality demonstrations of the likely natural development and treatment responsiveness of specific cancers so that all groups might positively contribute to the discussion about treatment procedure.

Use of genotyping data might enhance the potential of the model, as more accurate estimates of the patient's individualized linear quadratic radiobiological parameters might be achieved through the use of molecular interaction networks.

A more quantitative validation can be achieved using the patient data to be collected and applying multiple parameter adaptation methods such as genetic algorithms or neural networks. The imaging (e.g. MRI, PET, etc.), histopathological and, eventually, genetic data are introduced into the simulation model and a candidate radiotherapeutic or chemotherapeutic scheme is defined.

The output of the simulation run, which is the prediction of the tumour and the most affected normal tissue response to the treatment scheme, is then evaluated by the supervising doctor. If a further scheme is to be tested in silico, the simulation run is repeated with the same imaging and radiobiological/pharmacodynamic data as previously.

In the end, the modelling platform might serve as a generic "decision-support system". In this way, the medical doctor might make his or her final decision on the selection of the most promising therapeutic scheme by taking into account both the predicted outcomes of all simulated regimens as well as his or her own medical knowledge and expertise. This computational platform does not therefore intend to replace the medical doctor's input, but to add the possibility of investigating the impact of specific treatment-induced perturbations.


   Acknowledgments
 
The project has been supported by the Hellenic Ministry of Education and Religious Affairs under the program "Irakleitos: Fellowships for Research of the National Technical University of Athens". The project has been co-funded by the European Social Fund (75%) and National Resources (25%).

Received for publication June 7, 2004. Revision received August 8, 2005. Accepted for publication August 30, 2005.


   References
Top
 Abstract
 Introduction
 Tumour cell distribution
 Recent improvements
 Parameter value selection: the...
 Validation and results
 Discussion and conclusion
 References
 

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