Towards Predicting the Response of a Solid Tumour to Chemotherapy and Radiotherapy Treatments: Clinical Insights from a Computational Model
[['Chemotherapy and radiotherapy play important roles in the primary treatment of many cancers and in improving the survival after cancer surgery .', 'Currently , numerous chemotherapeutic drugs and irradiation techniques are employed , which have evolved over several decades through empirical clinical usage .', 'New treatments , such as a novel drug or a change in the scheduling of radiotherapy take many years to assess by conducting a clinical trial and clinicians would benefit greatly from having an alternative scientific approach to decide on how to improve current treatment strategies , but also in arriving at good decisions more quickly .', 'Mathematical modelling of such complex , dynamic situations might provide one solution to this problem , and speed up delivery of efficacious treatments to patients while preventing the use of potentially sub-optimal treatment combinations .', 'The effectiveness of these treatment protocols is considerably affected by internal tumour heterogeneities caused by perturbations in the intracellular pathways as well as by dynamical changes in the tissue microenvironment , in particular the distribution of oxygen [1] .', 'Hence , it is important to consider such heterogeneity when studying various optimisation protocols , as this can help in improving the delivery of multi-modality treatments .', 'A common treatment modality for cancer is chemotherapy .', 'Its delivery is limited by toxicity to normal tissues , so is often delivered in cycles that allow recovery of normal cells , but also , unfortunately , of tumour cells , leading to treatment failure .', 'Chemotherapeutic drugs function by killing the tumour cells through interfering with the cell-cycle mechanism , which regulates complex intracellular processes such as proliferation , cell division and DNA replication [2] .', 'The cell-cycle mechanism is very dynamic in nature and is influenced by the surrounding microenvironment , which contributes to cell-cycle mediated drug resistance and poor treatment outcome [1] , [2] .', 'One way of overcoming this is by using an appropriate combination of chemotherapeutic drugs that target the cell at various cell-cycle phase points , thus interfering with tumour cell division .', 'Radiotherapy is curative for certain cancers when used as the sole treatment , but clinical trials conducted in the last thirty years suggest a synergistic effect of concomitant chemotherapy and radiotherapy .', "As with chemotherapy , the cell-cycle plays a vital role in mediating a cell's sensitivity towards radiation therapy , as the cell-cycle phase determines the cell's relative radiosensitivity [3] , [4] .", 'Moreover , various studies have shown that the cells that are in G2-M phase are more sensitive to the radiation than those that are in G1 phase [3] .', "Furthermore , irradiation can also alter a cell's cell-cycle dynamics through the activation of various intracellular pathways including the p53 and p21 pathways [4] .", "Activation of these pathways and related cell repair mechanisms can delay the rate of progression of a cell's cell-cycle , causing a group of tumour cells to accumulate either in the G1 or G2 phase , preventing them from undergoing mitosis and making them progress in a synchronous manner [3] , [4] .", 'The treatment-dependent perturbations of cell-cycle progression together with cell-cycle-dependent therapeutic sensitivity are some of the many rationales behind the use of kinetically-based administration protocols of chemotherapy and radiation therapy [3] , [5] , [6] .', 'Studies have shown that both radiation therapy and chemotherapeutic drugs can induce a cell-cycle synchrony and arrest cells at a particular cell-cycle phase which improves the effectiveness of the next dose of radiation/chemotherapy [3] , [5] .', 'For example , while the drug paclitaxel induces a cell-cycle arrest at the cell-cycle phase G2-M , Flavopiridol causes cells to accumulate in G1 and G2 phases , enhancing the radiation sensitivity [3] .', 'Alternatively , radiation-induced cell-cycle delay can help various cell-cycle phase-specific drugs to induce a higher cell kill .', 'Moreover , combination regimes can also provide benefit from spatial cooperation and tissue reoxygenation , which enhance therapeutic response .', "As the interdependency of such therapeutic protocols , the cell-cycle mechanism and the tumour microenvironment clearly affect a cell's response to therapy , it is important to carefully study optimal combination and sequencing of treatments in order to help clinicians design therapeutic protocols that improve survival rates , in which mathematical modelling can be very helpful .", 'Clinically driven mathematical models can be used as powerful tools to understand , study , and provide useful predictions related to the outcome of various treatment protocols used to treat human malignancies .', 'Although there are several models in the literature that study chemotherapy and radiation therapy , very few of them analyse the effect of the cell-cycle in treatment response [7]–[14] .', 'Recently , Powathil et al . [15] developed a hybrid multiscale cellular automaton model incorporating the effects of oxygen heterogeneity and cell-cycle dynamics to study cell-cycle based chemotherapy delivery .', 'They have shown that an appropriate combination of cell-cycle specific chemotherapeutic drugs could effectively be used to control tumour progression .', 'Most of the mathematical models for radiation therapy are based on a linear quadratic ( LQ ) formulation [12] , [14] , [16]–[20] .', 'A brief summary of various approaches in modelling tumour dynamics and radiotherapy can be found in the review by Enderling et al . [21] .', 'Here , we use a discrete multiscale modelling approach to study the multiple effects of cell-cycle and radiotherapy .', 'Ribba et al . [11] proposed a multiscale model incorporating a discrete mathematical model for cell-cycle regulation and cell-cycle phase dependent radiation sensitivity .', 'Richard et al . [12] studied in vitro responses of cells with cell-cycle phase-specific sensitivity to targeted irradiation and analysed the so called “bystander effect” using a cellular automaton approach .', 'In the present paper , we study the multiple effects of radiation therapy when applied in combination with cell-cycle specific chemotherapy in the control of malignant cell growth by using a previously developed hybrid multiscale cellular automaton model for tumour cell growth [15] .', 'In particular , we are interested in studying the effects of cell-cycle regulation in radiation therapy and further , how radiation-induced cell-cycle heterogeneity can potentially be used to increase tumour control when radiotherapy is administered with chemotherapy ( “chemoradiotherapy” ) .', 'Moreover , as the radiation sensitivity is also affected by the surrounding tumour microenvironment , especially the oxygen distribution , we use a modified linear quadratic model to study the effects of radiation in a changing microenvironment .'], ['Cell-cycle phase-specific chemotherapeutic drugs are used in treating various human malignancies as they interfere with the rapidly proliferating mass of the cells by blocking their cell division cycle .', 'Some of these chemotherapeutic drugs are S phase-specific as they interfere with its replication ( e . g . topoisomerase or thymidylate synthase inhibitors ) , resulting in cell death or cell-cycle arrest at the intra-S checkpoint or at the G2/M checkpoint .', 'Some other drugs are M phase-specific as they damage the formation of the mitotic spindle or prevent it from disassociating ( e . g . taxanes , vinca alkaloids ) while some block phase transitions at G1/S or G2/M cell-cycle checkpoints ( e . g . CDKIs ) and other drugs are not necessarily phase-specific as they interact with the DNA irrespective of its cell-cycle phases .', 'Here , for simplicity we consider two types of phase-specific chemotherapeutic drugs that are either G1 specific or G2-S-M specific .', 'While the concept of phase-specific chemotherapy is useful , and although some drugs have specific effects on the machinery of mitosis ( e . g . ‘spindle’ poisons ) it is becoming clear that chemotherapy drugs may affect more than one aspect of the cell cycle , and so the concept of phase-specificity is somewhat of an over-simplification .', 'The effects of cell-cycle specific chemotherapeutic drugs on solid tumours with intracellular and oxygen heterogeneities are described using the same cellular automaton framework used by Powathil et al . [15] .', 'Using the mathematical model we have shown that the cytotoxic effectiveness of the cell-cycle phase-specific chemotherapeutic drugs is significantly dependent on the spatial distribution of the tumour cell mass , the timing of the drug delivery , the time between the doses of cytotoxic drugs , and also the cell-cycle and oxygen heterogeneity [15] .', 'We have assumed that the diffusion coefficients and supply rates depend on the location of tumour cells within the tumour , as observed experimentally and hence , the cell-kill due to the chemotherapeutic drugs that are introduced affect the diffusion and supply rates of the drug and nutrients in a favourable manner and thus help to redistribute the subsequent doses introduced [15] .', 'The study also highlighted the importance of considering intracellular and external heterogeneities while studying the potential effectiveness of chemotherapeutic drugs [15] .', 'The effectiveness of radiation therapy significantly depends on the intracellular and extracellular dynamics of the targeted tumour .', 'The key intracellular processes , such as cell-cycle dynamics and external factors including oxygen distribution play a vital role in determining the radiosensitivity of the cells that are irradiated [3] , [22] .', 'In addition , the radiation fractions ( treatments ) that are delivered further dynamically change this radiosensitivity over time by redistributing the tumour cells within the cell-cycle , by inducing repopulation of the tumour cell mass , by allowing reoxygenation of the tumour , and by causing the need for repair of the DNA damage induced by treatment [3] , [4] , [22] .', 'Clinically , a kinetically based administration of chemotherapy and radiation therapy is often used to achieve an improved therapeutic effect due to the processes of spatial cooperation , independent additive cell-kill and cellular , molecular and tissue level interaction between modalities [3] , [5] , [6] .', 'However , most of these interactions are dependent on the type of drugs given and the temporal separation between the drugs and radiation fractions , and hence an appropriate combination of these therapeutic modalities is an essential requirement to achieve maximum survival [33] , [34] .', 'Here , we show the analysis of the effects of four hypothetically-scheduled , clinically-used combinations ( adjuvant radiation , neo-adjuvant radiation , concurrent radiation and chemo-radiation-chemo ) of cell-cycle phase-specific chemotherapy and fractionated radiation therapy .', 'A representative result showing the changing dynamics of cells in various cell-cycle phases for the adjuvant therapy ( radiation is given after the chemotherapy ) is given in Figure 4 and the figures for the rest of the combinations can be found in the Supplementary Material , in Figure S2 , S3 and S4 .', 'Moreover , a comparison of the total number of cells for different combination protocols is given in Figure', '5 . Figure 4 shows the sequencing of two types of chemotherapeutic drugs followed by radiation therapy .', 'Two doses of cell-cycle phase-specific chemotherapy , specific to either G1 or G2 phases of the cell-cycle , are given at time\u200a=\u200a340 h and 370 h , followed by 5 fractions of radiation , given with a daily dose of d\u200a=\u200a2 . 5 Gy , starting at time\u200a=\u200a400 h .', 'The plots show that when the radiation therapy is given after the chemotherapy doses , the partial cell synchrony during the radiation is lost and a higher proportion of cells stay in G1 , except for the case where two G2 phase-specific drugs ( Figure 4b ) are combined .', 'In Figure S2 , we plot the effects of combination treatments when two doses of the chemotherapy drugs are given after the radiation therapy .', 'The radiation starts at time\u200a=\u200a340 h with a similar dose as the previous case and the doses of chemotherapeutic drugs are given at times\u200a=\u200a466 h and 496 h .', 'As we have seen in the previous section , the radiation given before two doses of chemotherapy introduces a partial cell-cycle synchrony of cell distribution that remains until the end of the therapy .', 'The cell-phase distribution for the case when the doses of chemotherapeutic drug are given before and after the radiation , is shown in Figure S3 .', 'The figure shows that the administration of a G2-specific drug , which kills fewer number of cells compared to the G1-phase-specific drug , helps to keep the cells in synchrony throughout the treatment time ( similar to Figure 4 ) .', 'In the last case , we studied the application of chemotherapy during the radiation schedule where the chemotherapeutic drugs are given at times 370 h and 400 h with the radiation , starting at time\u200a=\u200a340 h .', 'The plots in Figure S4 indicate that the increased cell-kill further reduces the cell-cycle synchrony with the number of G1 phase cells being dominant .', 'The total number of cells for all four cases of radiation and chemotherapy sequencing are compared in Figure', '5 . The plots show that in the absence of additional fractions of radiation and further doses of chemotherapy all the schedules perform in a similar fashion , although some give a better cell-kill .', 'However , the effects of the sequencing are critically dependent on the number of cells in various phases of the cell-cycle , as this determines how sensitive they are to the various therapeutic strategies .', 'In every combination except concomitant therapy ( chemotherapy given during radiation ) , we kept the total treatment time constant , if not identical , to compare the effects on tumour control .', 'We have also used the same set of parameter values and doses for each phase-specific chemotherapy [15] .', 'It can be seen from Figure 5 that when a G1-phase-specific drug is administrated , the cell-kill is usually higher than a G2-phase-specific drug .', 'This is mainly because , in most of the cases , the percentage of cells that are in G1 phase is higher than those in G2 .', 'The two factors that contribute to such a cell population distribution are hypoxia and space limitation , as hypoxic cells take a longer time to complete one full cell-cycle and the lack of space forces the cell to enter a resting phase .', 'The cell will re-enter the active phase of the cell-cycle when conditions become favourable .', 'In other words , resting tumour cells may , under favourable conditions created by the administration of the therapies , play a vital role in cell synchronisation and knowledge of this should inform the design of better combinations of cell-cycle phase-specific chemotherapy and fractionated radiation therapy .', 'Moreover , it can be seen from Figure 5 that although the various combination regimes mostly show varying results immediately after the treatment , in the absence of further treatments , these lead to a similar end point with same number of cells as time increases as observed in most clinical situations .', 'However , these differences in the proportion of cells in various cell-cycle phases immediately after each treatment protocol might be vital in designing further treatment plans for that individual patient .', 'Additionally , another factor that plays an important role in therapeutic intervention is the spatial distribution of cells in the tumour mass and its blood supply as these determine the nature of the tumour microenvironment .', "The tumour microenvironment , in particular oxygen distribution , can significantly affect a cell's radiation sensitivity and thereby introduce cell-cycle heterogeneity throughout the tumour .", 'On the other hand , the vascular distribution within and surrounding the tumour mass determines the effectiveness of the spatial distribution and supply of the chemotherapeutic drugs which determine the cell-kill .', 'A representative figure showing the spatial distribution of the cells in various cell-cycle phases at different treatment time points is given in Figure S5 .', 'In this section , we illustrate the potential of our multiscale computational model to compare different treatment regimens and test the predictions against what happens in a real tumour model in a prospective manner .', 'In doing so we provide a ranking of each regimen in terms of overall treatment efficacy .', 'This highlights the potential of our model to produce an optimal treatment regimen for a given patient .', 'We have compared two different treatment protocols currently used in oesophageal cancer , namely Herskovic , modified Herskovic ( both currently used in clinical practice ) and also a third experimental protocol which is currently not used in clinical practice .', 'Oesophageal cancer is in the “top 10” most common malignancies worldwide , and is the fifth highest in terms of mortality [35] .', 'The treatment of oesophageal cancer used to be primarily surgical or palliative .', 'In the 1980s , clinical trials started that were designed to look at radiation treatment of these tumours and determine whether better results could be obtained by combining radiation therapy with chemotherapy .', 'A seminal trial that commenced in that decade ( “Herskovic” ) reported two groups of patients who received either radiation therapy alone or radiation therapy with chemotherapy [36]–[38] .', 'Although the trial was not conducted with the same rigour of a modern-day trial ( numbers were relatively small and not all the patients were randomized , leading to the possibility of confounding the results ) , the authors reported that none of the radiotherapy alone group but a significant minority of the combined modality group was alive several years after treatment .', 'The method of giving “chemo-radiotherapy” for non-operable cases was then adopted and a dose of radiation and chemotherapy using Cisplatin and 5-FU was used as most patients could tolerate this combination without severe or life-threatening complications .', 'A later modification , not supported by any trial evidence , was to change the extra chemotherapy given in this regimen from adjuvant ( after the chemo-radiotherapy , to neo-adjuvant ( before the radiotherapy ) as this was better tolerated [38] .', 'This “modified-Herskovic” regimen was then used for the best part of a decade or more , before further clinical studies were done to evaluate new chemotherapy agents and different radiotherapy planning techniques [38] .', 'One of these , the SCOPE-1 study , a two arm , open , randomised multicentre Phase II/III trial , is designed to investigate the effect of the drug cetuximab on chemoradiotherapy [39] .', 'This study has recently stopped recruiting and involves hundreds of patients .', 'For the Herskovic treatment protocol , chemotherapy treatment is given on weeks 1 , 5 , 8 and 11 of the treatment period with Cisplatin 80 mg/m2 on day 1 ( D1 ) as a single dose , and 5FU , from day 1 to 4 ( D1-4 ) as a continuous infusion .', 'Radiation therapy is given from weeks 1 to 5 , in 25 fractions of 2 Gy .', 'Similarly , for the modified Herskovic regimen , chemotherapy is given on weeks 1 , 3 , 6 and 10 and radiation therapy is given from week 6 to 10 in 25 fractions of 2 Gy dose .', 'Finally , for the experimental protocol , which has been designed for this comparison study , chemotherapy is given on weeks 1 , 3 , 6 and 9 and radiation is given from weeks 12 to 15 in 20 fractions of 2 . 5 Gy dose .', 'Note that , in all these three treatment protocols the total amount of chemotherapeutic drug and the radiation dosage is kept the same ( although their biological effect are different ) .', 'We simulated treatment with each of the three regimens using our multiscale model over a period of 17 weeks .', 'The simulations were carried out using precisely the same set of parameter values for all three treatment regimens and the results showing the total cell kill over time are given in Figure', '6 . First of all we note , as can be seen by comparing the green and blue curves , that our computational simulation results show that the modified Herskovic regimen gives a better final outcome than that of the Herskovic treatment plan as suspected clinically [38] .', 'This gives a degree of confidence in our multiscale model , since this is effectively a “blind test” between the two different regimens .', 'However , interestingly , as seen by comparing the red curve with the green and blue curves , our multiscale model predicts that the new experimental regimen is more effective than either the Herskovic or Modified Herskovic regimen .', 'Although more clinical and experimental studies would be required to confirm these predications , these results highlight the predictive power of our model and its ability to distinguish between a number of different regimens and rank them in terms of overall efficacy or indeed even to predict an optimal treatment strategy .', 'One of the crucial steps towards the successful delivery of anti-cancer treatment is the optimal scheduling and sequencing of different therapeutic modalities , in particular radiotherapy and chemotherapy .', 'The roles of cell-cycle phases as well as tissue hypoxia are believed to be critical in determining the radiation sensitivity of cells and the action of several chemotherapeutic drugs [3] .', 'Changes in the treatment of cancer are currently driven largely by the products emerging from the pharmaceutical industry and although some thought and time is devoted to understanding how best to schedule , combine , and deliver the anti-cancer treatments that are currently available in order to increase their effectiveness , it would be helpful to be able to make rapid rationale treatment choices when designing new treatments based on currently available knowledge rather than take several years to test two treatment regimens in the clinical setting , as currently happens with clinical phase III studies .', 'We believe in silico experiments may help in this regard .', 'In this paper , we have presented a hybrid multiscale cellular automaton model to study the effects of radiotherapy , alone and in combination with cell-cycle specific chemotherapeutic drugs , in controlling the growth of a solid tumour .', 'We have also incorporated the heterogeneities in cell-cycle dynamics and oxygen distribution into our hybrid cellular automaton model as they play an important role in therapeutic effectiveness .', 'The effect of radiation therapy is studied using a modified linear quadratic model that incorporates some of the important factors responsible for radiation sensitivity such as cell-cycle phase-specific radiation sensitivity , improved survival due to DNA repair , and hypoxia .', 'The simulation results from the model showed very good agreement with previous biological experimental results in predicting the cell-cycle dynamics after the irradiation of the tumour cell mass with a single dose of the radiation .', 'The results predicted an increase in the number of G2 phase cells and a possible scenario of partial synchronisation of the cell-cycle , while the control cell population remained in a more or less G1 phase cells dominated proportion .', 'When the cells are irradiated with fractionated radiation , the results showed that the cell-kill enhances the reoxygenation of the tumour mass but also allows the re-entry of resting cells into the active cell-cycle .', 'The study of various factors affecting radiation sensitivity indicated that cell-cycle phase-specific sensitivity and survival due to DNA repair mechanisms could play a vital role in improving radiation cell-kill .', 'Using the present model we have also analysed various possible combinations of cell-cycle phase-specific chemotherapeutic drugs and fractionated radiotherapy .', 'The results show that the sequencing and the type of the chemotherapeutic drugs can significantly affect the cell-cycle and oxygen heterogeneities of the tumour mass which will further affect the effectiveness of the entire therapeutic strategy when they are given in several doses and/or fractions and are consistent with various experimental results [33] , [34] .', 'Overall , the results from the model show its potential usefulness in studying and understanding a kinetic administration of cell-cycle phase-specific chemotherapeutic drugs in combination with radiation therapy .', 'The results from the current and previous studies [15] also confirmed the importance of temporally changing spatial dynamics to improve therapeutic strategies .', 'In future studies , we would like adapt the current model to address the interactions between tumour cells and normal cells and to study how their combined spatial dynamics affect their therapeutic responses .', 'Furthermore , our general computational model can be easily adapted to reflect the behaviour in real clinical scenarios by appropriate validation and comparison with experimental and clinical data .', 'One such comparison on the treatment protocols used in the treatment of oesophageal cancer indicated a therapeutic benefit of the modified Herskovic treatment protocol over the previously used Herskovic protocol .', 'Moreover , the simulations indicated that a suggested new experimental treatment protocol might be an even better strategy than currently used treatment options , but clearly more detailed studies are necessary to validate this prediction .', 'As discussed earlier , the evolution of clinical treatment is slow and takes place over many decades , for the reasons that clinicians must be cautious when introducing new treatments in case of poor efficacy or excess and unexpected toxicity or intracellular and extracellular heterogeneities , and that once a preferred treatment route has been started on , it is generally modified in an incremental fashion .', 'It is very unusual for clinicians to start a completely new way of treating tumours without evidence .', 'The computaional simulation results of our multiscale mathematical model indicates a way for doctors to test the efficacy of new treatment strategies , to allow them to plan more adventurous treatments in silico , prior to beginning actual testing and long and costly clinical trials .', 'This departure may help relieve some of the stagnation in treatment strategy for tumours that have a poor prognosis , and allow medicine to move forward to more innovative treatments that can be evaluated for potential efficacy , prior to clinical testing .'], ['The computational model is simulated on a spatial grid of size grid points and each automaton element whether it is empty or occupied , has a physical size of , where , simulating a cancer tissue of area .', 'If the element is occupied by a cancer cell , the evolution of this cancer cell is based on the decisions made by the cell-cycle mechanism within the cell .', 'To model the cell-cycle dynamics within each cell , we use a very basic model originally developed by Tyson and Novak [40] , [41] that includes only the interactions which are considered to be essential for cell-cycle regulation and control , as given below .', '( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) ( 6 ) where are the rate constants and the values are chosen in proportional to those in Tyson and Novak [40] , [41] as given in Powathil et al . [15] .', 'In addition to the cancer cells and the empty spaces , the spatial domain also consists of a random distribution of blood vessel cross sections with density , where is the number of vessel cross sections ( Figure S6 ) .', 'This can be justified if we assume that the blood vessels are perpendicular to the cross section of interest and there are no branching points through the plane of interest [42] , [43] .', 'Moreover , the tumour vessel network is irregular , chaotic and abnormal as compared to that of normal vascular network [44] , [45] .'], ['Chemotherapy is a commonly used treatment for cancer .', 'Chemotherapeutic drugs act on rapidly proliferating cells , such as cancer cells , by interfering with the cell-cycle and other cell-cycle specific targets .', 'Hence , it might be more effective to use a combination of chemotherapeutic drugs that targets the cells in different phases of the cell-cycle .', 'The distribution of chemotherapeutic drug type , can be modelled by a similar equation as that of oxygen distribution ( Eq . 7 ) , given by: ( 8 ) where is the diffusion coefficient of the drug type , is the rate by which the drug is take in by a cell ( assumed to be zero as it is very negligible when compared to oxygen uptake ) , is the drug supply rate by the pre-existing vascular network and is the drug decay rate [32] .', 'Here , as similar to the oxygen distribution , the diffusion and the supply rate of the drugs are spatially varied depending on the location in the computational domain .', 'The details can be found in [15] .', 'Radiation therapy is often used in combination with chemotherapy with the intention of increased therapeutic gain for patients with cancer .', 'The survival probability of the cells after they are irradiated are traditionally calculated using linear quadratic ( LQ ) model [46] , given by ( 9 ) where is the radiation dose and and are sensitivity parameters , taken to be and [32] .', "It has been observed that the radiation sensitivity varies with the cell's oxygenation status [47] , [48] and the effect of changing tissue oxygen levels on the radiation sensitivity can be incorporated into the LQ model ( Eq . 9 ) by using the concepts of an “oxygen enhancement ratio” or “oxygen modification factor” [32] , defined as ( 10 ) where is the oxygen concentration at position , OER is the ratio of the radiation doses needed for the same cell kill under anoxic and oxic conditions , is the maximum ratio and mm Hg is the pO2 at half the increase from 1 to [32] , [49] .", "Hence , the modified LQ model for the survival probability , incorporating the effects of oxygen distributions can be written as: ( 11 ) The relative radiosensitivity of an individual cell is also partially determined by the cell's cell-cycle phase and studies show that the cells are more sensitive when in the S-G2-M phase as compared with the G1 phase [3] .", 'We have incorporated this varying sensitivity due to the changes in cell-cycle phase by an additional term in the equation for survival probability ( Eq .', "11 ) [20] , which gives: ( 12 ) The parameter varies from 0 to 1 , depending on the individual cell's position at the time of the irradiation .", 'Here , we assumed that the cells in S-G2-M phase has maximum sensitivity with while the cells in G1 phase and the resting phase has relative sensitivities of and , respectively .', 'In the current model , although we are not considering the individual cell repair , the studies suggest that 98% of damage caused by the radiation is likely to be repaired within few hours of radiation , if they are treated with low dose radiation ( ) [50] , [51] .', 'If the radiation dosage is higher , this repair mechanism may not be sufficient to repair all the DNA damage .', 'Considering these aspects , Endering et al . [16] , introduced some correctional terms into the LQ model to accommodate these effects due to the cellular repair during low dose radiation treatment .', 'Using these modifications ( allowing for less repair ) into the above LQ model ( Eq .', '11 ) with the effects of hypoxia , we obtain the survival probability of the cell as: ( 13 ) This survival probability is then used to calculate the survival chances of each cell when they are irradiated with the radiation rays .', 'To study this survival chance of an individual cell , a random number is drawn for each cell at every time when they are irradiated and compared against the calculated survival probability .', 'The irradiated cell survives if the random number is smaller than the survival probability and die otherwise .', 'Here , we also consider the effects of radiation on cell-cycle as irradiation results in a divisional delay , and , in particular , G2 phase delay/arrest in many cell lines [3] , [4] .', 'Experimental results show that 
['Introduction', 'Results/Discussion', 'Methods', 'Microenvironment and tumour growth']
['In this paper we use a hybrid multiscale mathematical model that incorporates both individual cell behaviour through the cell-cycle and the effects of the changing microenvironment through oxygen dynamics to study the multiple effects of radiation therapy .', 'The oxygenation status of the cells is considered as one of the important prognostic markers for determining radiation therapy , as hypoxic cells are less radiosensitive .', 'Another factor that critically affects radiation sensitivity is cell-cycle regulation .', 'The effects of radiation therapy are included in the model using a modified linear quadratic model for the radiation damage , incorporating the effects of hypoxia and cell-cycle in determining the cell-cycle phase-specific radiosensitivity .', "Furthermore , after irradiation , an individual cell's cell-cycle dynamics are intrinsically modified through the activation of pathways responsible for repair mechanisms , often resulting in a delay/arrest in the cell-cycle .", 'The model is then used to study various combinations of multiple doses of cell-cycle dependent chemotherapies and radiation therapy , as radiation may work better by the partial synchronisation of cells in the most radiosensitive phase of the cell-cycle .', 'Moreover , using this multi-scale model , we investigate the optimum sequencing and scheduling of these multi-modality treatments , and the impact of internal and external heterogeneity on the spatio-temporal patterning of the distribution of tumour cells and their response to different treatment schedules .']
['Anti-cancer treatments such as radiotherapy and chemotherapy have evolved through clinical trial-and-error over decades , and although they cure some cases and are partially effective in many , the majority of such cancers ultimately recur .', 'Doctors turn to new , expensive drugs as they emerge , but perhaps fail to study and learn how to use the therapies they already have most effectively .', 'This is partly because clinical trials are expensive to conduct , both in terms of time and money .', 'The cancer cell is complicated , but many mechanisms that control its response to treatment are now understood .', 'We show here how a mathematical model accurately reproduces the results of previous biological experiments of cancer treatment , opening up the possibility of using it to predict which combinations of drugs and radiotherapy would be best for patients .']
['oncology', 'medicine', 'mathematics', 'radiation therapy', 'applied mathematics', 'cancer treatment', 'chemotherapy and drug treatment', 'biology', 'computational biology']