Theoretical "iso-survival" formulae for fractionated radiation therapy

[Flash radiotheray at very high dose-rate: A brief account of the current situation]

In the last decade, major advances in high precision treatment delivery and multimodal imaging allowed radiotherapy to be more efficient and better tolerated. However, the technology of the accelerators used to generate X-ray beams is outdated and does not allow to explore the tolerance to novel approaches in terms of dose-rate. We have been the first to propose a completely novel modality of irradiation, named Flash radiotherapy, in which the dose per pulse and the instant dose-rate during the pulses is 10³ to 10⁴ higher than those used in conventional facilities. Flash has been shown to spare mouse lung from radio-induced fibrosis, whilst leaving unchanged the antitumor potential. Other teams have shown that the advantage of Flash in terms of reduced complications extends to normal brain and intestinal crypts. The goal of this paper is to review the progress of studies dealing with very high dose-rate "Flash" irradiation, describe the theoretical models proposed to explain the underlying mechanisms, and discuss the prospects for clinical applications of this emerging technique.

[Mathematical models of dose fractionation based on LQ function. (population-tissue models)]

A mathematical model was developed to calculate the probability of tumor tissue sterilization. It is assumed that tumor tissue contains normal and radio-resistant tumor cells and the survival of both types of tumor cells can be described by LQ functions. A package of programmes was created to solve the extreme problems in the determination of the parameters of LQ functions and the relative count of radio-resistant cells in the volume of tumor tissue. A programme complex was devised to solve practical tasks in radiological care. The series of tasks, which illustrate various aspects of determination of the parameters of the mathematical model by using clinical data and calculating the probability of tumor tissue radiation sterilization.

Consideration of tissue response in the application of the two-mutation model to radiation carcinogenesis

The Moolgavkar-Venzon-Knudson (MVK) two-mutation model of carcinogenesis is an analytical model that predicts the variation of cancer yield-rate with time, and with dose of a carcinogen. The model is biologically based, and assumes that a specific mutation in a stem-cell will increase its rate of proliferation compared with that of unmutated cells, so that a clone of pre-malignant cells develops; a second specific mutation in any one of these will make it malignant, and a cancer will start to grow. The model has been used in recent years to analyse a number of sets of epidemiological data on carcinogenesis. The purpose of this paper is to point to a problem in the use of this model for radiation-induced carcinogenesis, namely that ionizing radiation causes reproductive death of stem cells, which leads to regenerative division and hence a change in the number of stem-cells at risk. The possible effects of such changes on the predictions of the model are discussed. At low dose-rates of continuous or chronic irradiation and at low doses of acute irradiation, it is expected that pre-malignant cells will be killed along with the unmutated cells, and that the regenerative division of the surviving pre-malignant cells will restore the numbers of both stem cells and pre-malignant cells to what they would have been in the absence of cell killing; hence, no net effect of the tissue regeneration is expected. At high dose-rates, the initial delay in regenerative division and subsequent faster proliferation are expected to lead to an initial reduction in tumour yield-rate with time (compared with that predicted by the MVK model) followed by a faster increase. For acute irradiation, in the particular case of beta-particle irradiation of the skin, at high doses where there are practically no surviving cells in the irradiated area, repopulation by unirradiated cells from the margin is predicted to lead to a decrease in tumour yield-rate with dose. The predictions have been compared with published data on the induction of osteosarcoma in mouse by repeated injection of 89Sr, the induction of skin tumours in rat by acute and chronic irradiation with electrons, and the induction of skin tumours in mouse by acute irradiation with beta-particles. At low doses and dose-rates the basic MVK model fitted the data well. At higher doses and dose-rates the expected effects of tissue regeneration were observed qualitatively, although there were some discrepancies in detail; these are discussed.

A new theoretical formula for fractionated radiotherapy based on a saturable cellular repair mechanism

Recently we have published a new model of survival cellular response to radiation based on the existence of a saturable cellular repair mechanism. In the present paper we extend the predictions of this model of calculating the change in total dose necessary to achieve an equal response in a tissue when the dose per fraction in a radiotherapy fractionation schedule is varied. The model provides a new explanation of the difference between late and early radiation reactions. Results obtained from the model for different tissues and standard fractionation schedules are approximately equal to those obtained by the L.Q. (linear-quadratic) model. The model is compatible with in vitro survival curves that are straight at high doses.

Evaluation of time-dose and fractionation for 252Cf neutrons in preoperative bulky/barrel-cervix carcinoma radiotherapy

Time-dose fractionation factors (TDF) were calculated for 252Cf (Cf) neutron therapy versus 137Cs for intracavitary use in the preoperative treatment of bulky/barrel-shaped Stage IB cervix cancers. The endpoint assessed was gross and microscopic tumor eradication from the hysterectomy specimen. We reviewed the data obtained in clinical trials between 1976-1987 at the University of Kentucky Medical Center. Preoperative photon therapy was approximately 45 Gy of whole pelvis irradiation in 5 weeks for both 137Cs and Cf treated patients. 137Cs implant was done after pelvic irradiation x1 to a mean dose of 2104 +/- 36 cGy at point A at a dose rate of 50.5 cGy/h. There were 37.5% positive specimens. Using Cf intracavitary implants, dose varied from 109 to 459 neutron cGy in 1-2 sessions. Specimens were more frequently cleared of tumor (up to 100% at appropriate dose) and showed a dose-response relationship, both by nominal dose and by TDF adjusted analysis of dose, dose-rate, number of sessions, and overall time. Limited understanding of relative biological effectiveness, schedule, effect of implants, and dose rate all made it difficult to use TDF to study neutron effects. Relative biological effectiveness (RBE) was estimated and showed that for Cf, RBE was a complex function of treatment variables. In the pilot clinical studies, a value of 6.0 had been assumed. The present findings of RBE for tumor destruction are larger than those assumed. Cf was effective for cervix tumor therapy and produced control without significant side effects due to the brachytherapy method used. The TDF model was of limited value in the present analysis and more information is still needed for RBE, dose-rate, and fractionation effects for Cf neutrons to develop a more sophisticated and relevant model.

The "second" side of the neck in supraglottic cancer

The purpose of this study was to assess the impact of decisions made at operation for primary neck tumor on survival and cause of death-specifically, whether bilateral dissection, unilateral dissection, or delayed dissection influences eventual outcome. Of 244 patients with primary cancer of the supraglottis who were treated surgically, 22 (9%) had no neck treatment, 188 (77%) had unilateral neck dissection, and 34 (14%) had bilateral simultaneous neck dissection. Dissection on one or both sides of the neck was required later in 6 (27%) of the 22 patients with no neck treatment and in 32 (17%) of the 188 patients who had unilateral dissection. There were no differences among the treatment groups in the incidence of death from any cause or in the incidence of death from cancer, although neck stage did differ from group to group. The influence of delayed metastasis on survival was analyzed to explain these findings. Multivariate analysis demonstrated that death occurred at a rate 1.81 times higher in a person with a positive neck stage than in a person of the same age who had a negative stage. Also, death occurred at a rate 1.5 times higher for a person 10 years older than for a person with the same stage of disease. The influence of stage on the probability of need for a second dissection indicated that a patient with a positive neck stage had a 6.3 times greater likelihood of requiring a subsequent neck dissection than a person with a negative neck stage.

Linear-quadratic model isoeffect relations for proliferating tumor cells for treatment with multiple fractions per day

A modified linear-quadratic model isoeffect relation that includes the effect of proliferation is proposed. As for a planned course of therapy, the treatment time T and the number of fractions N are not independent variables; the new isoeffect relation involves only the fraction size d and the total dose D, but differs from the unmodified linear-quadratic model isoeffect relation and predicts higher isoeffect doses for small dose fractions. Using the new isoeffect relation it is explicitly shown, for a simple model, that decreasing the fraction size will improve the therapeutic ratio only if multiple fractions per day are given.

Repair capacity and kinetics of human skin during fractionated radiotherapy: erythema, desquamation, and telangiectasia after 3 and 5 year's follow-up

Prospective clinical fractionation studies on acute and late reactions in skin have been going on since 1972 at the Radiotherapy Department in Gothenburg. The clinical assay consisted of breast cancer patients irradiated postoperatively to the internal mammary nodes from unilateral or bilateral fields exposed to various dose schedules. 750 fields in 450 patients have been analysed. Schedules with 1, 2 or 5 fractions per week and 2 or 3 fractions per day were evaluated with erythema, desquamation and telangiectasia as endpoints. For some schedules a dose-response relationship was established in a limited dose range, but often there was only one dose group per schedule. These data are suited to analysis by the method of direct analysis of quantal response. This was used in the present analysis, along with the linear quadratic (LQ) model and its generalization, the incomplete repair (IR) model. The repair capacity was similar for erythema and desquamation, with alpha/beta ratios between 7.5 and 11.2 Gy. Unexpectedly, there was more significant time factor during radiotherapy courses up to 6 weeks for erythema and desquamation, but the repair capacity was changed after 4 weeks for both endpoints, and alpha/beta increased to between 18.3 and 34.5 Gy. The repair capacity for late telangiectasia differed significantly from that for erythema and desquamation, with alpha/beta values between 2.8 and 4.3 Gy. There was a significant time factor for telangiectasia with characteristic doubling time of about 16 days, when an exponential function for time was used. Concerning the repair kinetics in skin, there were insufficient data to obtain precise estimates, but there was a suggestion of two components of repair. This was inferred from higher-than-predicted recovery with 15-min intervals, when the data were fitted with the monoexponential model. The monoexponential fit gave t1/2 between 1.1 and 1.3 h for acute effects and 3.5 h for late effects. Recovery after 15-min fractionation intervals, if it resulted from a fast repair component, would be consistent with a half-time of 0.3-0.4 h. The time factor and the relative long half-time for repair for late effects have important implications for multiple-fraction-per-day treatment, and imply that interfraction intervals of 4 h or less, as commonly used, will be insufficient. Instead, intervals of 6 h or longer are recommended. Using accelerated fractionation with a significant reduction in overall treatment time a dose reduction is still necessary to take into account the time factor for late effects.(ABSTRACT TRUNCATED AT 400 WORDS)

A comparison of mathematical models for regeneration in acutely responding tissues

A mathematical model is presented to describe the regenerative response of mouse gut epithelal cells to radiation. The model, derived from radiobiological principles, predicts the cellular surviving fraction following any irradiation regimen. There are three basic elements to the model (a) a single dose survival curve, either the linear-quadratic or the two-component model, (b) a part to incorporate the regenerative response, either a Gompertzian or a logistic growth and (c) a part to accomodate the delayed onset of regeneration, including either a mitotic delay, a fixed time delay, both, or neither. The models are similar in spirit, but different in detail to the model proposed by Cohen. The various models are evaluated on three large datasets, where the response is cell survival in the jejunum or the colon measured using the crypt colony assay. The models were fit and the parameter estimates and standard errors were obtained from the raw observations using non-linear least squares. It is concluded that Gompertzian growth gives a better fit to the data than logistic growth; the delayed onset of regeneration in these tissues can be best accounted for by a mitotic delay or a mitotic delay plus a fixed time delay, and there is little to choose between the linear-quadratic and the two-component model. There was a strong relationship between the tissue cell cycle time and the regenerative response, the mitotic delay being longer and the rate of regeneration slower for the colon than for the jejunum.

Local stem cell depletion model for radiation myelitis

We propose a model for normal tissue damage based on the assumption that adult mammalian stem cells have limited mobility and, consequently, for each organ, there is a maximum volume (the "critical volume," Vc), that can be repopulated and repaired by a single surviving stem cell. This concept is applied to a simple, 1-dimensional model of the spinal cord, where the critical volume is a "slice" of "thickness," t, assumed to be small compared to lengths of spinal cord usually irradiated clinically. The probability of myelitis is explicitly obtained as a function of the dose, dose per fraction, length of cord irradiated, slice thickness, number of stem cells per slice and parameters alpha and beta of the stem cell survival curve. The complication probability is expressed as a triple negative exponential function of dose analogous to the double negative exponential function for tumor control, resulting in a steep dose-response curve with short tails in both the high dose and low dose regions. We show that the model predictions are compatible with the experimental data for radiation myelitis in the rat. We discuss how this concept can be applied to other organs such as skin and to organs composed of structurally and functionally distinct subunits, such as the kidney.

Does parasternal irradiation reduce thoracic vertebral metastases in breast cancer?

A review of 197 patients with bone metastases from breast carcinoma was undertaken to assess the effect of adjuvant parasternal irradiation on the distribution of those metastases. A total of 128 patients (Group I) received radiation. The control group consisted of 69 patients (Group II) who did not receive adjuvant radiation. The dose absorbed by the mid-thoracic vertebrae (T3 to T8) varied from 10 Gy in 3 weeks to 20 Gy in 4 weeks. The distribution of bone metastases was analyzed at the initial development in the 197 patients. Repeat assessments (X ray/scans) were available for analysis in 132 patients. Patients in Group I had less metastases from T3 to T8: 37/128 (29%), vs 30/69 (43.5%) for patients in Group II: (chi 2 = 3.62; p less than 0.10). For the 132 patients with serial assessments, the difference at last evaluation was more significant: 45/86 (52%) in Group I, vs 36/46 (78%) in Group II: (chi 2 = 7.44; p less than 0.01). The data shows that patients receiving low-dose exit beam irradiation from the parasternal field have fewer mid-thoracic bone metastases. Potential implications are discussed.

A unified approach to dose-effect relationships in radiotherapy. I: Modified TDF and linear quadratic equations

A linear quadratic factor analogue (LQF) to the variable-exponent TDF model is introduced. In both of these models, account is taken of the volume of tissue irradiated. Scaling factors are used such that an LQF or a TDF of 100 represents tolerance for each volume or partial volume of each tissue or organ irradiated. These models are sufficient for tissues which are irradiated fairly homogeneously. Examples illustrate the use of these models.

Some implications of the Linear Quadratic model for tumor control probability

To define an optimal radiation therapy strategy, the dependence of the probability of cure and of significant complications, on the parameters controlled by the radiotherapist must be determined. The recent success of the Linear Quadratic (LQ) model in constructing isoeffect relations for normal tissue damage and in describing in vitro cell survival curves, indicate that this model could be used to determine this dependence. The problem of tumor control is addressed. Using LQ model parameters obtained from human tumor cell lines, the sigmoid dose-response curves for controlling tumors of fixed size but of several histologies, with a fractionated course of radiotherapy is obtained. Except for squamous cell carcinoma, the calculated average tumor control doses (TCD37 or TCD50) are unrealistically low, but the model can be made more realistic by including inhomogeneities in the spatial dose distribution and heterogeneous tumor cell populations. The slope of the dose response curves are determined and the significance of the "relative slope" parameter rho as a measure of the number of cells in a tumor's most radioresistant clone is noted. The relation of the model's predictions to qualitative features of the experimental animal data for both tumor control and for normal tissue damage is discussed. Experiments to test the validity of this type of model are suggested.

Isoeffect curves for radiation-induced cardiomyopathy in the dog

Canine hearts were irradiated with a range of total doses given in 2, 3, or 4 Gy per fraction. Echocardiography was done before irradiation and at 3 and 6 months after irradiation. Histologic analyses were done of tissues taken at necropsy 6 months after irradiation. The percentage vascular component in the ventricles decreased with increasing total doses. The connective tissue component increased at lower doses and then decreased at higher doses. There was more fibroblastic proliferation and collagen production in the lower dose range and there may have been more cell killing by the higher doses. There was some evidence of myocardial hypertrophy at higher doses, which would have caused an apparent decrease in the connective tissue component. In either case, surviving fibroblasts would be expected to continue proliferating and producing collagen. At longer time intervals after irradiation the connective tissue component would likely continue to increase as observed clinically. That increase would be proportional to dose, but might not be closely related to initial killing of fibroblasts. Many factors such as changes in blood supply, continued loss of myocardium, and other stresses on the heart could influence the degree of fibrosis at later times. A relationship of response to cell killing appeared to exist based on alpha/beta ratios that were less than 3 Gy whether determined at the mid-range or for the greatest response of vasculoconnective tissues. Alpha/beta ratios ranged from 2.7 to 5 Gy for increases in diastolic wall thickness of the left ventricle at 3 and 6 months after irradiation. The low alpha/beta ratios reflect relatively steep isoeffect curves and have important implications for use of coarser fractionation schedules for treatment volumes that include the heart. The risk of cardiac damage could be significantly increased.

Lack of correlation between basal cell survival and gross response in irradiated swine skin

The relationship between basal cell survival and gross response in irradiated swine skin was tested by comparing dose survival curves derived from time-dose isoeffect data with curves obtained directly from basal cell counts in histological sections. Assuming equal effect per exposure and constant cell survival at isoeffect, best-fitting single-hit multi-target and linear-quadratic response curves were determined for time-dose schedules resulting in non-healing of 50% of irradiated fields. Basal cell survivals for single doses of 970, 1649, 2231, and 2619 rad were estimated 1) by counting regenerating islands and 2) by monitoring total basal cell counts through time. The dose survival curve derived from the isoeffect data was steeper than the curve obtained from direct basal cell counts. Furthermore, the direct basal cell survival curve extrapolates to less than 100% at zero dose, indicating the presence of a resistant basal cell subpopulation. The data show that the isoeffect in this case is not strongly coupled to basal cell survival. Rather, the probability of healing of an irradiated field is more sensitive to the dose per fraction than is basal cell survival, implying a contribution to non-healing from damage to stromal elements such as the capillary endothelium.

Implications of tissue target-cell survival-curve shape for values of split-dose recovery doses: late versus early effects

Recent data from this laboratory on split-dose recovery for early and late effects in pig skin are consistent with the linear-quadratic model for cell survival, and with relative cell survival-curve shapes for early- and late-effect target cells where the early-effect cells have an initially steeper and straighter survival-curve than the late-effect cells.

An evaluation of protracted low dose rate irradiation from an acute dose survival curve

A method of transformation from an acute survival curve to a protracted low dose rate survival curve based on the concept of the dose equivalent sublethal damage advanced by Oliver is introduced. The transformation was done by solving a differential equation which was formulated between the slope of both survival curves and which was able to be applied to the situation where the dose rate might change with time, e.g., in using short-life radioisotope or the complexity of the recovery of sub-lethal damage. The effects of the short-life radioisotopes (radon seeds and radiogold grains) in radiotherapy were estimated by using this method and applying it to Lajtha-Oliver's model; our data are compared with the data obtained by Orton.

Dose distribution in 42 MV roentgen irradiation of cervical carcinoma

The dose distribution obtained with two different techniques for external irradiation of cervical carcinoma is described. The dose to the central area is somewhat higher with the multiple beam technique compared with the newly introduced shielding block technique. The difference between the calculated CRE values for the two techniques is small. When the shielding block is not placed over the area corresponding to the position of the applicators from the previous intracavitary treatment the considerable difference in absorbed dose between the two techniques does not correspond to the difference in the calculated CRE values. A comparison of the relative distributions laterally, in total dose and CRE, shows that for central volumes the relative CRE is much higher than the relative dose, when the normalization is made at the pelvic wall.

High-dose rate intracavitary radiation therapy for advanced head and neck tumors

Fifty-seven treatments were performed on 27 head and neck pateints with recurrent or residual tumors on a high dose rate, remote controlled afterloading unit: There were 16 cases of maxillary sinus tumors, 6 epipharynx, 3 alveolar ridge, 1 hard palate and 1 floor of mouth. All patients have been followed up more than 2 years except one. Five patients are alive without local recurrence for more than 2 years. In 13 patients local tumors disappeared once and normal mucosa covered the tumor sites. Two patients died from local bleeding. In six patients this method failed to destroy tumors. Our prupose was palliative local control: therefore, in two-thirds of cases treated we were successful with this easy method of nonfractionated acute intracavitary radiation. This result is favorable, considering that all cases treated here were failures following full dose external radiation, although the treatment had to be repeated more than twice in 15 cases. Relief of symptoms is excellent when this therapy is used.

A prospective clinical trial of post-operative radiotherapy delivered in three fractions per week versus two fractions per week in breast carcinoma

The results are presented of a five-year prospective trial of 411 patients with breast carcinoma treated by mastectomy and post-operative radiotherapy. Twice or thrice weekly dose-fractionation techniques were used, i.e. 12 fractions in 28 days and six fractions in 18 days. The results show that, in terms of survival and control of local disease, the two techniques are very similar. The early and late radiation effects on the normal tissues up to five years are similar and acceptable. Dose levels thought to be biologically equivalent on the basis of experience at St Thomas's Hosiptal were selected. It is stressed that there is a narrower margin of normal tissue tolerance when six fractions over 18 days are used, that it is essential to treat all fields on each treatment occasion and that the total maximum tissue dose of 3600 R is not exceeded. Attention is drawn to the fact that the dose for six fractions, derived from the NSD formula which would be predicted to be equivalent to the 12-fraction dose is, in fact, over 10% higher than the dose used here and likely to produce unacceptable late radiation changes.

Errors and uncertainties in external radiation therapy. A system analysis with a cell kinetic model

The relative importance of accurate knowledge and control of various biologic and physical factors in external radiation therapy was estimated with a simple cell kinetic treatment response model for a simulated tumour therapy situation. A set of parameter values was chosen for a reference case so that the calculated probability of zero surviving tumour cells was about 0.5. The parameters were varied, one at a time, over a range that was considered representative with respect to uncertainty in dose calculation, dose delivery and the description of the pertinent patient data. Also parameters in the cell kinetic model were varied. The calculations demonstrated a strong influence on the number of surviving cells from even moderate changes in the parameters which affect the tumour dose. Predictions with regard to curability are heavily dependent on the reliability of the biologic parameters in the model.

The influence of dose fractionation on acute and late reactions in patients with postoperative rasiotherapy for carcinoma of the breast

The clinical courses of 418 patients with breast carcinoma who received postradical-mastectomy radiation therapy to the internal mammary and supraclavicular are were revied in order to determine the normal tissue tolerance with various time-dose fractionation radiation regimens. Four different dose fractionation schedules wereemplyed during the period 1958-1968 at Memorial Hospital: 4900 rads/10f/32d; 4410 rads/9f/30d; 5000 rads/13f/18d; and 4600 rads/20f/28d. There was no difference in the acute reaction or local recurrence rate of the tumor in the irradiated area among the four fractionation schemes, but the late effects were quite different. Patients who received 4900 rads in 10 fractions in 32 days had a 33% incidence of subcutaneous fibrosis.A similiar dose in 13 fractions in 21/2-3 weeks resulted in about 17% subcutaneous fibrosis. No significance of these clinical findings is discussed in relation to the various formulae currently being used to express time-dose equivalence.