Mathematical models of tumour and normal tissue response

The historical application of mathematics in the natural sciences and in radiotherapy is compared. The various forms of mathematical models and their limitations are discussed. The Linear Quadratic (LQ) model can be modified to include (i) radiobiological parameter changes that occur during fractionated radiotherapy, (ii) situations such as focal forms of radiotherapy, (iii) normal tissue responses, and (iv) to allow for the process of optimization. The inclusion of a variable cell loss factor in the LQ model repopulation term produces a more flexible clonogenic doubling time, which can simulate the phenomenon of 'accelerated repopulation'. Differential calculus can be applied to the LQ model after elimination of the fraction number integers. The optimum dose per fraction (maximum cell kill relative to a given normal tissue fractionation sensitivity) is then estimated from the clonogen doubling times and the radiosensitivity parameters (or alpha/beta ratios). Economic treatment optimization is described. Tumour volume studies during or following teletherapy are used to optimize brachytherapy. The radiation responses of both individual tumours and tumour populations (by random sampling 'Monte-Carlo' techniques from statistical ranges of radiobiological and physical parameters) can be estimated. Computerized preclinical trials can be used to guide choice of dose fractionation scheduling in clinical trials. The potential impact of gene and other biological therapies on the results of radical radiotherapy are testable. New and experimentally testable hypotheses are generated from limited clinical data by exploratory modelling exercises.     

Radiobiological modeling and clinical trials

PURPOSE: Standard clinical trial designs can lead to restrictive conclusions: the "best recommended treatments" based on trial results, although generally applicable to patient populations, do not necessarily apply to individual patients. In theory, radiobiological modeling, coupled with reliable predictive assays, can be used to rationalize the selection of patients for particular schedules in trials. MATERIALS AND METHODS: Linear-quadratic modeling of radiotherapy can be used to simulate a clinical trial. This is achieved by random sampling techniques where the key radiobiological parameters (alpha, beta, T(pot) and clonogen number) are selected from known or expected ranges. Clinical trial design in radiotherapy may be improved by formal radiobiological assessment designed to estimate the likely changes in tumor cure probability (TCP) and the likely normal tissue biologically effective dose (BED). Modeling may also be used to rationalize the allocation of patients to a test or standard schedule or for individual optimization of a treatment schedule. Such approaches depend on there being reliable predictive assays of the radiobiological parameters in individual patients. The influence of variations in predictive assay accuracy on the improved outcomes are assessed. RESULTS: Clinical trials, which have been preceded by modeling simulation, offer potentially substantial improvements in the results of cancer treatment by radiotherapy. These exceed the usual gains found in standard clinical trials. CONCLUSION: Future preclinical trial design should include modeling assessments that indicate how best to structure the trial.     

Effets biologiques des hautes doses par fraction

The radiobiological concepts described for conventional doses per fraction (1.8 to 2 Gy) seem difficult to translate to high doses per fraction radiobiology. In fact, specific mechanisms are involved during high dose per fraction irradiation, involving vascular microenvironment damage and anti tumor immune response. The “5R's” of “classical” radiobiology (factors influencing the response of healthy or cancer cells to irradiation) seem to play a less important role in case of high doses per fraction. In addition, applicability of the linear quadratic model in this context is debated. It is therefore difficult to obtain reliable equivalent doses, hence the importance of including our patients in clinical trials, especially in case of concomitant systemic treatments. In addition to stereotactic radiotherapy, flash irradiations defined by a dose rate approximately 2000 times faster than “conventional” irradiation can also deliver high doses per fraction, with a much better tolerance for normal tissue without loss of anti tumor efficacy. Finally, availability of robust prospective data is a prerequisite to answer the question of short and long-term risk/benefit ratio of these different irradiation techniques.     

Superfractionation as a potential hypoxic cell radiosensitizer: prediction of an optimum dose per fraction

Purpose: A dose “window of opportunity” has been identified in an earlier modeling study [1] if the inducible repair variant of the LQ model is adopted instead of the pure LQ model, and if all survival curve parameters are equally modified by the presence or absence of oxygen. In this paper we have extended the calculations to consider survival curve parameters from 15 sets of data obtained for cells tested at low doses using clonogenic assays.

Methods and Materials: A simple computer model has been used to simulate the response of each cell line to various doses per fraction in multifraction schedules, with oxic and hypoxic cells receiving the same fractional dose. We have then used pairs of simulated survival curves to estimate the effective hypoxic protection (OER′) as a function of the dose per fraction.

Results: The resistance of hypoxic cells is reduced by using smaller doses per fraction than 2 Gy in all these fractionated clinical simulations, whether using a simple LQ model, or the more complex LQ/IR model. If there is no inducible repair, the optimum dose is infinitely low. If there is inducible repair, there is an optimum dose per fraction at which hypoxic protection is minimized. This is usually around 0.5 Gy. It depends on the dose needed to induce repair being higher in hypoxia than in oxygen. The OER′ may even go below unity, i.e. hypoxic cells may be more sensitive than oxic cells.

Conclusions: If oxic and hypoxic cells are repeatedly exposed to doses of the same magnitude, as occurs in clinical radiotherapy, the observed hypoxic protection varies with the fractional dose. The OER′ is predicted to diminish at lower doses in all cell lines. The loss of hypoxic resistance with superfractionation is predicted to be proportional to the capacity of the cells to induce repair, i.e. their intrinsic radioresistance at a dose of 2 Gy.     

Biological factors influencing optimum fractionation in radiation therapy.

Optimum fractionation in radiotherapy occurs when tumor control is improved without enhancement of complications. The main influence on choice of overall time, total dose and fraction size is biological: the proliferation status of tumors. For rapidly proliferating tumors, shorter schedules than 6 to 8 weeks are necessary. Optimum overall time is similar to Tk, the time after beginning cytotoxic treatment when rapid proliferation in tumors starts: 21 to 35 days in head and neck tumors. These, and non-small cell lung tumors, have a clonogenic cell doubling time during radiotherapy of about 3 days. New developments in designing optimum schedules for such tumors are presented: carefully regulated hypofractionation (CRH). For slowly proliferating tumors, especially prostate adenocarcinoma, intracellular repair is large, so larger doses per fraction will be necessary. New evidence is presented showing that their alpha/beta ratio may indeed be lower than 3 Gy. For an entirely different reason from that above, hypofractionation should be tested.     

Some factors of importance in the radiation treatment of malignant melanoma

The results of radiotherapy in 204 lesions of malignant melanoma in 114 patients were analysed with regard to radiobiological parameters such as total dose, dose per fraction, treatment time, tumour volume and also by various fractionation models. Ninety-seven of 204 lesions showed a complete response (CR) which was persistent in 80. Neither total dose, treatment time nor various modifications of the nominal standard dose (NSD) concept showed any well-defined correlation with response. There was, however, a significant relationship between dose per fraction and response so that high doses per fraction yielded a significantly better response (24% CR for doses less than 4 Gy vs. 57% CR for doses greater than or equal to 4 Gy, p less than 0.001). The lack of influence of treatment time influence allowed an analysis of the data according to the linear-quadratic model yielding an alpha/beta ratio of 2.5 Gy. Using this ratio, an iso-effect for different fractionation schedules could be estimated by the extrapolated total dose (ETD). This was further improved when corrected for the other important parameter which was tumour volume. Thus, an iso-effect formular for malignant melanomas could be calculated as ETDvol = D X ((d + 2.5)/2.5) X M-0.33 where d is total dose and dose per fraction in Gy, respectively and M is mean diameter (cm). The 50% response for ETDvol was found to be 86 Gy. This formular seems to be currently the best way to determine an optimal radiation schedule for effective radiation treatment of malignant melanoma. Treatment of 45 patients with only local or regional disease showed 26 patients who achieved local tumour control with a 56% 3-year survival compared to no survivors among 19 patients in whom the disease could not be controlled locally. This indicated that proper attention should be given to the local treatment of recurrent melanoma since this has important implications for survival. Successful treatment of malignant melanoma may be possible when the special radiobiological features of the disease are taken into account.     

Fractionated regimens for stereotactic radiotherapy of recurrent tumors in the brain

Radiosurgery (single-fraction stereotactic radiotherapy) was initially developed to treat non-malignant arteriovenous malformations, but there is growing interest in its use for the treatment of recurrent brain tumors. We suggest that there are sound reasons to expect improved results for tumor radiotherapy, in terms of late effects, if a fractionated regimen is used. At present, no published guidelines are available for choosing appropriate doses for fractionated regimens. We present two sets of guidelines, based on experimentally derived radiobiological parameters: first, we estimate gamma-ray doses which, if delivered in various numbers of fractions, should produce equivalent early effects to 70 Gy of 125I X rays delivered at low dose rate; this latter regimen is currently used in RTOG interstitial brachytherapy trials. Second, we estimate doses for multi-fractioned stereotactic radiotherapy which may be advantageous alternatives to particular doses of single-fractioned radiosurgical therapy. As the appropriate hardware is available, the use of fractionated stereotactic radiotherapy deserves serious consideration for the treatment of recurrent tumors in the brain.     

A new approach to dose escalation in non-small-cell lung cancer

PURPOSE: To describe the radiobiological rationale for dose-per-fraction escalation in non-small-cell lung cancer (NSCLC) and to devise a novel Phase I scheme to implement this strategy using advanced radiotherapy delivery technologies. METHODS AND MATERIALS: The data from previous dose escalation trials in NSCLC are reanalyzed to establish a dose-response relationship in this disease. We also use data relating prolongation in treatment time to survival to compute the potential doubling time for lung tumors. On the basis of these results, and using a Bayesian model to determine the probability of pneumonitis as a function of mean normalized lung dose, a dose-per-fraction escalation strategy is developed. RESULTS: Standard approaches to dose escalation using 2 Gy per fraction, five fractions per week, require doses in excess of 85 Gy to achieve 50% long-term control rate. This is partly because NSCLCs repopulate rapidly, with a 1.6% per day loss in survival from prolongation in overall treatment time beyond 6 weeks, and a cell doubling time of only 2.5 to 3.3 days. A dose-per-fraction escalation strategy, with a constant number of fractions, 25, and overall time, 5 weeks, is projected to produce tumor control rates predicted to be 10%-15% better than 2 Gy per fraction dose escalation, with equivalent late effects. This Phase I clinical study is divided into three parts. Step 1 examines the feasibility of the maximum breath-holding technique and junctioning of tomotherapy slices. Step 2 treats 10 patients with 30 fractions of 2 Gy over 6 weeks and then reduces duration to 5 weeks using fewer but larger fractions in 10 patients. Step 3 will consist of a dose-per-fraction escalation study on roughly 50 patients, maintaining 25 fractions in 5 weeks. Bayesian methodology (a modification of the Continual Reassessment Method) will be used in Step 3 to allow consistent and efficient escalation within five volume bins. CONCLUSION: A dose-per-fraction escalation approach in NSCLC should yield superior outcomes, compared to standard dose escalation approaches using a fixed dose per fraction, for a given level of pneumonitis and late toxicity. Highly conformal radiotherapy techniques, such as intensity modulated radiotherapy (IMRT) and helical tomotherapy with its adaptive capabilities, will be necessary to achieve significant dose-per-fraction escalation without unacceptable lung and esophageal morbidity.     

A simple method of obtaining equivalent doses for use in HDR brachytherapy

PURPOSE: To develop a simple program that can be easily used by clinicians to calculate the tumor and late tissue equivalent doses (as if given in 2 Gy/day fractions) for different high-dose-rate (HDR) brachytherapy regimens. The program should take into account the normal tissue sparing effect of brachytherapy. METHODS AND MATERIALS: Using Microsoft Excel, a program was developed incorporating the linear-quadratic (LQ) formula to calculate the biologically equivalent dose (BED). To express the BED in terms more familiar to all clinicians, it was reconverted to equivalent doses as if given as fractionated irradiation at 2 Gy/fraction. Since doses given to normal tissues in HDR brachytherapy treatments are different from those given to tumor, a normal tissue dose modifying factor (DMF) was applied in this spreadsheet (depending on the anticipated dose to normal tissue) to obtain more realistic equivalent normal tissue effects. RESULTS: The spreadsheet program created requires the clinician to enter only the external beam total dose and dose/fraction, HDR dose, and the number of HDR fractions. It automatically calculates the equivalent doses for tumor and normal tissue effects, respectively. Generally, the DMF applied is < 1 since the doses to normal tissues are less than the doses to the tumor. However, in certain circumstances, a DMF of > 1 may need to be applied if the dose to critical normal tissues is higher than the dose to tumor. Additionally, the alpha/beta ratios for tumor and normal tissues can be changed from their default values of 10 Gy and 3 Gy, respectively. This program has been used to determine HDR doses needed for treatment of cancers of the cervix, prostate, and other organs. It can also been used to predict the late normal tissue effects, alerting the clinician to the possibility of undue morbidity of a new HDR regimen. CONCLUSION: A simple Excel spreadsheet program has been developed to assist clinicians to easily calculate equivalent doses to be used in HDR brachytherapy regimens. The novelty of this program is that the equivalent doses are expressed as if given at 2 Gy per fraction rather than as BED values and a more realistic equivalent normal tissue effect is obtained by applying a DMF. Its ease of use should promote the use of LQ radiobiological modeling to determine doses to be used for HDR brachytherapy. The program is to be used judiciously as a guide only and should be correlated with clinical outcome.     

Heterogeneity of radiation response of a parent human epidermoid carcinoma cell line and four clones

We examined the radiobiological parameters of a parent tumor cell line and four tumor clones of a human skin squamous cell carcinoma. The parent line and all clones have a tumor morphology, aneuploid karyotype, and the ability to passage infinitely in vitro. The parent cell line and three of four clones formed tumors in nude mice. The parent line, SCC-12, has a D0 of 1.59 Gy and an n of 7.5. In the four clones, D0 ranges from 1.31 Gy to 2.66 Gy and n ranges from 2.1 to 22.8. Potentially lethal damage repair (PLDR) following 11 Gy ranges from 1.7 to 13.1, at 24 hours, however PLDR following equitoxic doses of radiation is similar. The mean inactivation dose value (D) in the parent tumor cell line is 3.48 and ranges from 3.31 to 4.17 in the tumor clones. Radiobiological interpretation of heterogeneity may complicate predictive assays for clinical radiotherapy.     

Comparison between Fractionated High Dose Rate Irradiation and Continuous Low Dose Rate Irradiation in Spheroids

Recent interest in clinical brachytherapy focuses on the possible radiobiological equivalence between fractionated high dose rate (HDR) and continuous low dose rate (LDR) irradiations. This study is designed to compare the radiobiological effects between the two in vitro using multicellular spheroids of human tumor. Both HDR and LDR irradiations were delivered by ¹³⁷Cs source, the dose rates of which were as 1.18 Gy/min and 5.5 mGy/min, respectively. Fractionated HDR irradiation of various fraction sizes was applied twice a day. We found that: (1) The fractionated HDR irradiation (8 Gy/2 fr/day) was more effective radiobiologically than continuous LDR irradiation (8 Gy/day) and the ratio of radiobiological effects of these irradiations was estimated as 0.82, based on the 50% spheroid cure dose (SCD₅₀); (2) the radiobiological effectiveness was independent of the fraction size of HDR irradiation administrated, and the repair of sublethal damage (SLD) was absent, suggesting that the sparing effect of fractionated HDR irradiations was absent in spheroids. Our findings could provide important information for the clinical usage of the fractionated HDR radiotherapy to replace continuous LDR radiotherapy.     

The role of radiotherapy in recurrent and metastatic malignant melanoma: a clinical radiobiological study

A review of the literature and our data has been completed to analyze the clinical radiobiology of malignant melanoma. Six hundred eighteen radiotherapy-treated malignant melanoma lesions were analyzed with regard to radiobiological parameters such as total dose, dose per fraction, treatment time, tumor volume, and various fractionation models. Forty-eight per cent of the treated tumors achieved complete response, which was persistent in 87% after 5 years. Neither total dose, treatment time, nor various modifications of the NSD concept showed any well-defined correlation with response. There was, however, a significant relationship between dose per fraction and response, and a high dose per fraction yielded a significantly better response (59% CR for doses greater than 4 Gy versus 33% CR for doses per fraction less than or equal to 4 Gy). The lack of treatment time influence allowed analysis of the data according to the linear-quadratic model, resulting in an alpha/beta ratio of 2.5 Gy. Using this ratio, an iso-effect for different fractionation schedules could be estimated by the extrapolated total-dose (ETD). The ratio was further improved when corrected for the tumor volume. Thus, an iso-effect formula for malignant melanoma could be calculated as: ETDvol (Gy) = D X [d + 2.5)/2.5) X M-.33, where D and d are total dose and dose per fraction in Gy, respectively, and M is the mean tumor diameter in cm. Based on a logit analysis, a complete response level of 50% appeared at an ETDvol value of 83 Gy. The formula is currently the best way to determine an optimal radiation schedule for an effective radiation treatment of malignant melanoma. The tumor response was further improved in 134 additional cases receiving adjuvant hyperthermia. Here, a thermal enhancement ratio (TER) of 2.0 was observed. In a group of 131 patients with only local or regional disease, a 5 year survival rate of 49% was observed in 77 patients with persistent local tumor control, but only 3% survived among the 54 patients in whom local therapy failed. It is therefore, highly important to the probability of survival in recurrent melanoma that proper local treatment be performed.     

Hypofractionated radiotherapy in prostate cancer

In regards to prostate cancer, the classic radiotherapy dose ranges from 70–80 Gy, administered in daily 2-Gy fractions. However, when taking into account the particular radiobiological model of prostate cancer cells, one realizes that there is a potential theoretical advantage to delivering a greater biological effective dose per treatment in a lower number of fractions. Both recent and older publications have attempted to explore this treatment option. This critical review comprehensively examines the current state of knowledge concerning hypofractionated radiotherapy in prostate cancer.     

The evolving role of external beam radiotherapy in localized prostate cancer

Primary organ-confined prostate cancer is curable with external-beam radiotherapy. However, prostate cancer expresses a unique radiobiological phenotype, and its ablation requires doses at the high-end range of clinical radiotherapy. At this dose level, normal tissue radiosensitivity restricts the application of curative treatment, and mandates the use of the most advanced high-precision treatment delivery techniques to spare critical organs at risk. The efficacy and tolerance of dose-escalated conventional fractionated radiotherapy and of the biological equivalent doses of moderate and extreme hypofractionation are reviewed. Current studies indicate that novel risk-adapted techniques to spare normal organs at risk are still required to deploy high-biological equivalent dose extreme hypofractionation, while affording preservation of quality of life and cost-effectiveness.     

Hypofractionated radiotherapy in prostate cancer

In regards to prostate cancer, the classic radiotherapy dose ranges from 70-80 Gy, administered in daily 2-Gy fractions. However, when taking into account the particular radiobiological model of prostate cancer cells, one realizes that there is a potential theoretical advantage to delivering a greater biological effective dose per treatment in a lower number of fractions. Both recent and older publications have attempted to explore this treatment option. This critical review comprehensively examines the current state of knowledge concerning hypofractionated radiotherapy in prostate cancer.     

Development of radiation therapy optimization

The principal radiobiological problems in the treatment of advanced tumors and the solution of many of them by radiobiologically optimized intensity-modulated radiation therapy are presented. Considerable improvements of the treatment outcome using radiobiologically optimized intensity-modulated treatments are achieved by: (a) increasing the tumor dose and dose per fraction; (b) keeping constant or even reducing slightly the dose and dose per fraction to organs at risk; (c) reducing the overall treatment time and the number of treatment fractions. The merits of the new radiation modalities and advanced intensity-modulated treatment techniques are compared in terms of equipment costs per patient cured. It is predicted that the new development of radiobiologically optimized intensity-modulated radiation therapy will rapidly become an important clinical tool, increasing the efficiency of the collaboration between radiation physicists, radiation biologists and radiation oncologists. Not only does it allow the optimal treatment of every patient, but it also promotes an efficient feedback of treatment outcome and complication data to improve the accuracy of known dose response relations to further augment future treatment results. Equipment costs may go up during a transition period until efficient interfaces between new diagnostic equipment, treatment-planning systems and intensity-modulated treatment units are fully developed. From then onwards the cost of high quality biologically optimized intensity-modulated treatments will decrease and so will the treatment time and personnel requirements, at the same time as the treatment quality is greatly improved particularly for more advanced tumors.     

Use of normalized total dose to represent the biological effect of fractionated radiotherapy

There are currently a number of radiobiological models to account for the effects of dose fractionation and time. Normalized total dose (NTD) is not another new model but is a previously reported, clinically useful form in which to represent the biological effect, determined by any specific radiobiological dose-fractionation model, of a course of radiation using a single set of standardized, easily understood terminology. The generalized form of NTD reviewed in this paper describes the effect of a course of radiotherapy administered with nonstandard fractionation as the total dose of radiation in Gy that could be administered with a given reference fractionation such as 2 Gy per fraction, 5 fractions per week that would produce an equivalent biological effect (probability of complications or tumor control) as predicted by a given dose-fractionation formula. The use of normalized total dose with several different exponential and linear-quadratic dose-fraction formulas is presented.     

The role of radiotherapy in the treatment of liver metastases

Radiotherapy has historically played a minor role in the treatment of patients with unresectable liver metastases from colorectal cancer and other malignancies. This can be attributed chiefly to the low tolerance of the whole liver to radiation. High-precision radiotherapy planning techniques have allowed much higher doses of radiation to be delivered safely to focal liver metastases, while sparing most of the normal liver. When combined with hepatic arterial fluorodeoxyuridine, high-dose focal liver radiotherapy is associated with excellent response rates, local control, and survival in patients with unresectable liver metastases from colorectal cancer. Radiotherapy, with and without concurrent systemic chemotherapy, has also been used with encouraging outcomes for patients with liver metastases from colorectal cancer and other cancers. There appears to be a radiation dose response for liver metastases; tumors treated with doses of 70 Gy or greater are likelier to have durable local control. Advancements in tumor imaging, in radiotherapy techniques that will allow the safe delivery of higher doses of radiation, and in novel tumor radiation sensitizers and normal tissue radioprotectors should substantially improve the outcome of patients with unresectable liver metastases treated with radiotherapy.