Ellis Formula Research Articles

Feasibility of pulse oxygen saturation for the condition evaluation of acute respiratory distress syndrome patients in the Lijiang region

To explore the feasibility of using pulse oxygen saturation (SpO2) to evaluate the condition of patients with acute respiratory distress syndrome (ARDS) in the Lijiang region. Patients with ARDS who visited the department of emergency of People's Hospital of Lijiang from August to December 2020 were selected as study subjects. Patients were divided by severity into mild ARDS group [200 mmHg (1 mmHg = 0.133 kPa) ≤ oxygenation index (PaO2/FiO2, P/F) ≤ 300 mmHg] and moderate to severe ARDS group (P/F ≤ 200 mmHg). The general condition, clinical diagnosis, arterial blood gas analysis results of the patients were recorded, and the differences of the above indexes between the two groups of ARDS were compared. Spearman correlation analysis was used to analyze the correlation between SpO2 and arterial oxygen saturation (SaO2). SpO2 was carried into the Ellis equation and the Rice equation to calculate the derived P/F and analyze the correlation between the derived P/F and the P/F measured in arterial blood gas analysis; receiver operator characteristic curve (ROC curves) were plotted, the sensitivity and specificity of SpO2/fraction of inspiration oxygen (SpO2/FiO2, S/F) instead of P/F to assess oxygenation in patients with ARDS was calculated. To evaluate the feasibility of SpO2 for the condition evaluation of patients with ARDS in the Lijiang region. Compared with the mild ARDS group, the arterial partial pressure of oxygen (PaO2), SaO2 and hemoglobin (Hb) were significantly decreased in the moderate to severe ARDS group [PaO2 (mmHg): 50.5 (39.3, 56.5) vs. 60.0 (55.0, 67.5), SaO2: 0.86 (0.73, 0.91) vs. 0. 93 (0.90, 0.96), Hb (g/L): 142±27 vs. 156±24, respectively, all P < 0.05]. Correlation analysis revealed a significant positive correlation between SpO2 and SaO2 in ARDS patients residing at high altitude (R = 0.650, P = 0.000). The P/F derived by the Rice formula was significantly and positively correlated with the P/F derived from arterial blood gas analysis (R = 0.802, P = 0.000). The deduced P/F in mild and moderate to severe ARDS groups were all significantly correlated with the measured P/F (R values were 0.562, 0.647, both P = 0.000). The P/F derived using the Ellis formula showed a significant positive correlation with the P/F derived from arterial blood gas analysis (R = 0.822, P = 0.000). The deduced P/F of mild ARDS group and moderate to severe ARDS group were all positively correlated with the measured P/F (R values were 0.556, 0.589, P values were 0.000, 0.010). There was a significant positive correlation between S/F and P/F in ARDS patients (R = 0.828, P = 0.000), and the regression equation was S/F = 1.33 P/F+52.41. ROC curve analysis showed that S/F had some predictive value for patients with mild and moderate to severe ARDS, and area under ROC curve (AUC) and 95% confidence interval (95%CI) were 0.903 (0.829-0.977), 0.936 (0.870-1.000), both P = 0.000. When the cut-off value was 452 mmHg, S/F had a sensitivity of 100% and a specificity of 80.9% for predicting mild ARDS. When the cut-off value was 319 mmHg, S/F predicted moderate to severe ARDS with 95.1% sensitivity and 86.2% specificity. At high altitude, SpO2 and SaO2 have been correlated in patients with ARDS, and P/F derived using SpO2 and measured P/F were significantly correlated in patients with ARDS, especially in those with moderate to severe ARDS. SpO2 may be useful in the assessment of severity of illness in patients with ARDS at high altitude.

The sensitivity of human tissues to changes in dose fractionation: Deductions from the RCR survey among UK radiotherapists

The dosage prescriptions reported in the Royal College of Radiologists' fractionation survey among radiotherapists have been further analysed using model equations in order to deduce estimates of fractionation-sensitivity parameters for each of the six cases under consideration. For example, using the linear-quadratic model, including a (significant) time factor, with radical treatments of a T 2 breast carcinoma or a T 1N 0 squamous carcinoma of the vocal cord, the (α/β) ratios were 26±20 Gy and 37±46 Gy, respectively. The values of the time factor, expressed as the maximum extra dose required per day to counteract the decrease in effect with increasing overall time (γ/α), were 0.60 and 0.45 Gy/day respectively. Using the Ellis formula, which provided a significantly better fit to the dosage prescription ( P=0.003), the exponents of N were calculated to be 0.24±0.05 and 0.27±0.07, respectively. The corresponding values of the T exponent were 0.16±0.06 and 0.014±0.075. About 20% of radiotherapists prescribed doses greater than ±10% from the mean fitted values for the breast treatment, and about 6% of them in the case of the vocal cord.

The sensitivity of human tissues to changes in dose fractionation: Deductions from the RCR survey among UK radiotherapists

The dosage prescriptions reported in the Royal College of Radiologists' fractionation survey among radiotherapists have been further analysed using model equations in order to deduce estimates of fractionation- sensitivity parameters for each of the six cases under consideration. For example, using the linear-quadratic model, including a (significant) time factor, with radical treatments of a T2 breast carcinoma or a T1N0 squamous carcinoma of the vocal cord, the (alpha/beta) ratios were 26 +/- 20 Gy and 37 +/- 46 Gy, respectively. The values of the time factor, expressed as the maximum extra dose required per day to counteract the decrease in effect with increasing overall time (gamma/alpha), were 0.60 and 0.45 Gy/day respectively. Using the Ellis formula, which provided a significantly better fit to the dosage prescription (P = 0.003), the exponents of N were calculated to be 0.24 +/- 0.05 and 0.27 +/- 0.07, respectively. The corresponding values of the T exponent were 0.16 +/- 0.06 and 0.014 +/- 0.075. About 20% of radiotherapists prescribed doses greater than +/- 10% from the mean fitted values for the breast treatment, and about 6% of them in the case of the vocal cord

Isoeffect models and fractionated radiation therapy.

Since the advent of the Ellis formula, a number of isoeffect formulae have been proposed for estimating the total dose required in different fractionation schedules to produce equivalent biological effects.‘*3*” Most often these formulae are applied retrospectively to clinical data thus making it impossible to control factors such as the endpoint for assessing damage, non-biased scoring of the damage by (preferably) more than one independent observer, the criteria for patient eligibility, and adequate follow-up records for assessment. Thus, these formulae often are viewed skeptically by clinicians. However, there is an urgent need to accurately and concisely define isoeffect relationships for human normal tissues in vivo because of the implied steepness of dose-response curves as derived from animal data. Also, the initiation of new fractionation schedules, for example, hyperfractionation, are all based on experimental data and, although a comparison of mouse and human data indicates that the shape of the dose-response curves for the two species is similar, the absolute dose level for response varies, generally in the direction of a lower isoeffect dose in humans. The lung is one dose-limiting normal tissue for which there are some dose-response data available for humans. The lung is dose-limiting not only in the treatment of malignant diseases of the thorax but in the use of total body irradiation as a preparative regimen for bone marrow transplantation. This latter treatment and the use of upper half body irradiation have resulted in published dose-response curves for the incidence of radiation pneumonitis in humans, at least after large single doses of radiation. However, limited data are available after fractionated doses and these are, again, from retrospective studies.

Lack of evidence for increased tolerance of rat spinal cord with decreasing fraction doses below 2 Gy

The radiation tolerance of the spinal cord, both in man and in rats, has been shown to depend strongly on the size of the dose per fraction. With fraction doses down to about 2 Gy, the spinal cord tolerance can be predicted by a modified Ellis formula: D ∼ N 0.43. More recently alternative isoeffect formulas were based on the linear-quadratic (LQ) model of cell survival where the effect of dose fractionation is characterized by the ratio α β which varies from tissue to tissue. For the spinal cord, as well as for other late responding tissues, the ratio α β is small, in contrast to most acutely responding tissues. Both the Ellis-type formula, and to a lesser extent the LQ-model, predict a continuously increasing tolerance dose with decreasing fraction size. From the LQ model, the concept of “flexure dose” has been derived, which proposes the limit of effective fractionation to be about 0.1 α β . At this dose per fraction no significant further gain in tolerance would be detected. From previous experiments on the rat cervical spinal cord with doses per fraction down to about 2 Gy, the ratio α β was determined to be 1.7 Gy, and the LQ-model would predict a rise in tolerance with a reduction in fraction size to far below 2 Gy. Based on these predictions clinical studies have been initiated assuming a significantly increased tolerance by reduction of fraction size to about 1 Gy. However, in the present experiments no evidence was found for such an increase in tolerance with fraction sizes below 2 Gy.

Observations on pulmonary metastases in patients after single doses and multiple fractions of fast neutrons and cobalt-60 gamma rays

Pulmonary metastases have been irradiated with single and fractionated doses of fast neutrons and cobalt- 60 gamma rays. The response to radiation was measured on volume changes of the lesions and thus RBE values could be derived. A correlation was found between grading of the tumour and volume doubling time and also between RBE and volume doubling time. This suggests an advantage for high LET radiation of slowly growing, well differentiated tumours. Furthermore the RBE for multiple fractions tends to be higher than for single doses. Calculation of the N exponent of the Ellis formula indicated that hardly any shoulder exists when neutrons are applied.

The relationship between fractionation and total dose for X ray induced brain damage

An isoeffect curve for X ray induced brain damage in the rat has been obtained by comparing the effects of fractionation of dose into 8,15 or 30 equal fractions given in the same overall treatment time of 6 weeks. The slope of the isoetfect curve is 0.37. This data on the brain is compared with other published data for radiation damage to the rat spinal cord and a composite isoeffect curve for damage to the central nervous system (CNS) is shown. From this we suggest a modified formula for the CNS that is in agreement with that suggested by van der Kogel 7 for the spinal cord, and is more compatible with present clinical experience than the normal Ellis formula.

Therapeutic irradiation and brain injury

This is a review and reanalysis of the literature on adverse effects of therapeutic irradiation on the brain. Reactions have been grouped and considered according to time of appearance. The emphasis of the analysis is on delayed reactions, especially those that occur from a few months to several years after irradiation. Over 100 such cases are reported in the literature. Eighty cases were identified in which the patient was given a single course of radiation therapy and in which reasonable estimates of the time-dose-fractionation regimen could be made. Eleven of these patients had been irradiated for cancer of the skin, 17 for other extracranial tumors, 20 for either a pituitary tumor or a craniopharyngioma and 32 for primary brain tumors. All dose specifications were converted into equivalent megavoltage rads. The data were analyzed in terms of total dose, overall treatment time and number of treatment fractions. The data were also analyzed in terms of time-dose-fractionation (TDF) and nominal standard dose (NSD). NSD calculations were done according to the usual Ellis formula and also according to a modification in which the N exponent was −0.44 and the T exponent −0.06. When total dose was plotted against number of fractions, a line with slope 0.44 fitted well with the lower dose limits at which brain necrosis has been reported. Also discussed were acute radiation reactions, early delayed radiation reactions, somnolence and leukoencephalopathy post-irradiation/chemotherapy and combined effects of radiation and chemotherapy.

Time dependent repair of radiation damage in the rat spinal cord after X-rays and neutrons

The increases in total dose required to cause paralysis after irradiation of the cervical cord or lumbar cord by two equal doses of X-rays or neutrons separated by times from 1 to 120 days have been measured. No increase was needed until the doses were separated by times in excess of 15 days. The timing was the same for irradiation of the lumbar and cervical cord in the same rat strain and was different from that observed in a different rat strain. The time course of repair was the same after X rays or neutrons. The data show that for irradiation induced myelopathy the exponent of T in the Ellis formula is zero until 15 days and is then, between 15 and 120 days, 0.08 for both X-rays and neutrons. Comparison with results of Van der Kogel indicate that the time for which the exponent of T is zero varies with rat strain. It is suggested that the time dependent repair in the spinal cord, which is the same after X-rays or neutrons, is due to repopulation and not slow repair. The recovery from sublethal damage after neutrons is about half that observed after X-rays.

Isoeffect curve for radiation myelopathy.

In a previous publication on the effect of X-ray dose fractionation on the total dose required to produce radiation myelopathy in rats, we showed that the slope of the Strandqvist-type isoeffect curve between one and 30 fractions to the cervical or lumbar cord was 0.4 (White and Hornsey, 1978). This was in good agreement with the observation of Van der Kogel (1977) on a different strain of rat, who obtained a value of 0.44 between one and 20 fractions. The slopes of these isoeffect curves are all steeper than that for skin damage which is 0.34, decreasing to 0.27 as fractionation is increased (Fowler, 1971). The slope of the curve for radiation myelopathy between four and 30 fractions for fractionation given in the same overall treatment time, which gives the exponent of N in the Ellis formula (total dose αNαTβ, Ellis 1969), was 0.35 (White and Hornsey, 1978). There was no indication of reducing slope with increasing number of fractions at least up to 30 fractions as is seen for both skin damage (Fowler, ...

Postoperative split course radiotherapy of squamous cell carcinoma of the oral tongue

Forty nine patients with squamous cell carcinoma of the oral tongue were irradiated postoperatively using a split course technique for external irradiation alone or in combination with intraoral cone X-ray therapy. After a 3 year follow-up the results of local control were compared with those of other authors, who irradiated initially with combined external and interstitial therapy in one series. For two series the nominal standard dose (NSD) was calculated with the modified Ellis formula given by Gabriel-Jurgens et al. 10 and Gremmel et al. 11 In our patient collective the recurrence rate was higher in the early T-stages, because the dosage applied was lower than in the majority of the cited publications. The combined modality of surgery and following external irradiation could not improve the outstanding results of interstitial therapy in T1 and T2 tumors. When the split course technique is used, the total dose and treatment time must be increased to obtain the same local control rate as in one series irradiation. Especially in advanced cases the split dose regimen seems to be an alternative treatment, because severe complications were not observed in our patients. For split course therapy we recommend the application of 1700 ret for Stage T1 oral tongue carcinoma 1800 ret for Stage T2 and 1900 ret for T3 tumors. For treatment of recurrences, surgery appears to be superior to irradiation.

Treatment of grades I and II brain astrocytomas. the role of radiotherapy

This retrospective analysis of 68 patients with Grade I or II brain astrocytomas included the years 1958-74. Two-thirds or 45 of the 68 cases received adjuvant irradiation. Fifty-six of the 68 tumors were histologic Grade II, the remaining twelve were Grade I. Comparison of all prognostically important parameters between the 45 irradiated and 23 non-irradiated patients revealed minimal but statistically insignificant bias in favor of the “grossly” excised, Grade II, and non-irradiated cases. Cumulative life-table survivals revealed excellent 5-year results for the 22 “grossly” excised cases (90%), with progressive decline in survival rates for the 45 irradiated patients (54%), the 23 non-irradiated cases (32%), and for the 45 incompletely excised cases with (41%) or without (13%) adjuvant radiotherapy. Ten-year cumulative survivals declined to 20–30% range regardless of treatment and prognostic parameters. An important exception is the 100% survival of the seven childhood cerebellar astrocytomas. Conclusions, based on analysis of treatment data, are as follows: Incompletely excised Grade I astrocytomas require adjuvant irradiation commensurate to Grade II lesions. Irradiation does not improve upon the 5-year survival rate when tumor resection is grossly complete. Conversely, the improvement in survival of incompletely excised cases by adjuvant radiotherapy is highly significant. Local tumor control rates show progressive improvement with increasing radiation doses in the range of 800–1500 ret, using the Ellis formula.

The response of human skin to irradiation with X-rays or fast neutrons

Before attempting a full scale therapeutic trial of fast neutrons it was necessary to irradiate normal human skin in order to confirm relative biologic effectiveness (RBE) values measured on animals. The lateral side of each thigh was used, with three irradiation fields on each. Single doses, 6 fractions or 12 fractions were given twice weekly with either X-rays or fast neutrons. The reacttons were scored for about 3 months after irradiation and the average values were used to derive dose effect curves. From these curves the RBE was obtained, as a function of dose per fraction, and was found to be in excellent agreement with values of relative biologic effectiveness (RBE) obtained from pigs, rats and mice over a wide range of doses per fraction. The Ellis formula was found to predict the results reasonably well for X-rays and a modified version was an excellent fit for neutrons.

Cellular survival and isoeffect formulae: a study in applied radiobiology.

The Ellis formula for the sterilization of tumors is compared with isoeffect curves for sterilizing doses as a function of the fractionation number N. Using cell parameters from human kidney cells irradiated in cell cultures we get a good agreement over a wide range of N values. The exponents obtained from the calculations are 0,26-0,30 and 0,36 for 200 keV X-rays, Co-60 rays and high energy electrons respectively. A reduction of tumor cells to 10(-8) seems sufficient for tumor sterilization. For normal tissue Ellis has proposed another formula (for X-rays) where a time factor T0,11 concerns homeostatic repair and proliferation of normal alls. The Ellis formula can be simplified when we assume that all treatment schemes be given with the same weekly dose-rate, i.e. have the same "hardness". Cell survivals according to this formula have been calculated and tolerance doses for other radiations giving the same effect computed. Only for electrons do the tolerance doses exceed the sterilizing doses. The Ellis formula does not represent isoeffects and can only be regarded as an approximation. Tolerance doses can be calculated giving the same cell survivals for all N-values, i.e. isoeffects, and a comparison with sterilizing doses shows a good agreement with clinical experience. However, the influence of the overall-dose rate still is an unsolved problem which can only be solved experimentally.

Early and late radiation changes following a second course of irradiation.

The effect of two 10-fraction courses of irradiation separated by intervals varying from 1 to 10 months on both the early skin reaction and late deformity of the leg was studied in mice. It was found that the "memory" of the first course was much greater for the late end point (deformity) than for early skin reactions at all intervals between the two courses. The extent of the late deformity reaction following a second course was predicted fairly well from the Ellis formula for decay of partial tolerance, but it also appeared to be more complex than is evident from the formula. Evaluation of early vs. late reactions showed that the extent of the deformity is not governed entirely by the level of the early skin reaction; rather, it depends critically on the previous radiation history of the tissue.

The response of mouse skin to fractionated X rays

The feet of mice were irradiated with single doses or with 4, 8, 16 or 32 fractions of X rays, all in the same overall time of 43 days. Skin reactions and late deformities were scored and the results are discussed in terms of cell survival curve models. It is concluded that at low doses per fraction there is a significant component of irreparable damage, but it is not clear whether this is due to a high LET component of the radiation or to a variable sensitivity of the cells of the skin. The applicability of the Ellis formula is discussed and is found to be a reasonable approximation over a limited range of dose per fraction.

Measurement of Blood Flow in the Mouse Tail after Irradiation

The measurement of the local clearance rate after subcutaneous injection of ${}^{133}{\rm Xe}$ and $[{}^{51}{\rm Cr}]{\rm EDTA}$ provided a way to obtain information on blood flow and capillary diffusion capacity (CDC) in the mouse tail. These studies were performed under normal resting conditions and after the induction of maximal blood flow. A decrease in blood flow and CDC was observed about a year after tail irradiation with a single dose of 3000 rad X rays. This decrease was most significant under maximal blood flow conditions. The dose-effect curves for single doses when measured at 14-17 mo after irradiation showed a decrease in blood flow up to 2500 rad; whereas, after doses higher than 2500 rad, an increase in blood flow was observed. The same phenomenon was observed with the CDC; however, it was less pronounced. Studies of blood flow and CDC after fractionated irradiations demonstrated no significantly different curves, if the doses were expressed in ret. This indicates that the Ellis formula is...

The Ellis formula.

With regard to Ellis's now famous formula: it is fair to say that the indices are a bit off-putting to the average radiotherapy practitioner, while the notion of a “single dose equivalent”, be it true or false, is at least mildly attractive. The calculation is tedious. You can buy a special slide rule if you like.

An examination of some dose-time relationships in therapeutic radiology.

Abstract We have compared three representations of dose-time (or fraction number) relations in radiation therapy on published data for cancer of the breast, Hodgkin's disease, squamous cell carcinoma of the aerodigestive tract, and pericardial effusion. The comparison was on the basis of how well each representation separated the two groups of outcomes, success or failure, in each clinical series. As a measure of separation we have used a dimensionless coefficient and calculated its value as a function of the choice of recovery coefficient in the Strandqvist relations and for the Ellis formula. In general, the Strandqvist relations with an optimal value of the recovery coefficient—defined as that value which produces the maximum separation of the two groups of outcomes—give a better separation than the Ellis formula. Our method of analysis provides a unique procedure for estimating the best value for the recovery coefficient. More importantly, it shows whether the separation is sensitive to the choice of ...

Nominal standard dose and the ret.

Abstract 1. The criticisms adduced by Liversage of the N.S.D. concept and the Ellis formula are based on misunderstanding and insufficient consideration of the basic reasoning and evidence. 2. In clinical practice, by providing a single figure representing biological effectiveness, the concept and the formula are useful for teaching and comparison of treatment schedules besides being safe for prescribing. 3. Virtual coincidence of isoeffect values for normal tissue (N.S.D.) (Table I) and for Hodgkin's disease (tumour S.D.) (Table V) support the concept and the formula. For most other tumours the tumour effect is incidental to normal tissue tolerance. 4. Optimum schedules for various types of tumour might be possible to derive when more is known of the mitotic cycle times of the affected normal and malignant cells underradiation treatment. 5. Continual testing and sifting of data are desirable to verify the concept and to adjust the exponents of the formula.