Linear Quadratic Model in the Clinical Practice via the Web-Application

Abstract

The modern development and improvement of mathematical models that describe the radiobiology of processes in the body occurring under the influence of radiation every year lead to more complicated calculations related to the estimation of its impact both on the effectiveness of radiotherapy and on the possibility of making changes to the radiation treatment regimen. This significantly increases the time spent by medical physics and radiation oncologists and also requires special training of qualified specialists capable of performing such calculations. The aim of the study is to optimize calculations related to the estimation of radiation doses when the radiation treatment schedule changes, by modeling such changes with specially designed software on the basis of the theory of a linear-quadratic radiobiological model. The Web application is accessed via the Internet link https://hypo-calc.github.io/. As an example of using the Web application, the possible cases in clinical practice are considered.

1. Introduction

The history of the development of radiobiological models began immediately after the discovery of X-rays and is rapidly continuing at present time, overcoming an increasing number of restrictions. The practical application of biological radio models is a typical clinical practice when treating oncological diseases.

Owing to the development of radiobiological models, it has become possible to mathematically describe the biological phenomena that occur in the body under the influence of ionizing radiation. They allow for predicting the event that causes ionizing radiation in a particular cell. The practical application of radiobiological models makes it possible to calculate radiation doses and the number of fractions, compare the biological effect of irradiation under different dose fractionation regimes, and present physical quantities in the form of clinical indicators. Radiation therapy (RT) is characterized by physical and mathematical values, which are expressed by specific numerical values (dose per fraction, total radiation dose, number of RT sessions, time interval between RT sessions, etc.). But when developing and optimizing radiation treatment plans, doctors and physicists use clinical indicators (biologically effective dose, linear-quadratic equivalent dose for 2 Gy fractions, tumor control probability, normal tissue complication probability, etc.) [1, 2].

Such radiobiological models as NSD, KRE, and TDF are considered out-of-date and can be useful only for the prevention of radiation complications, but they are not effective for the destruction of malignant tumors. Also, they cannot be used to calculate the biological effect on parenchymal tissues (lungs, nervous tissue, intestines, liver, kidneys, etc.).

To date, the LQ model is the most commonly used model in clinical practice [3], but it also has limitations being a simplified model of cell damage, and it should be used with caution considering the assessment of the possible risks of complications from the dose and irradiated volume, based on the QUANTEC findings in the conditions of modern medical imaging, optimization of dosimetric planning of exposure, and new approaches to conducting RT sessions. Today, there are modifications of the LQ model [4, 5, 6], which allow calculating tolerant doses, as well as the probability of occurrence of radiation complications in tissues as a function of the volume of exposure, and single and total dose.

To achieve the main goal of radiation therapy (tumor eradication, alleviation of the patient’s condition), it is necessary to deliver a dose of radiation, which is sufficient to destroy the tumor, to the volume of radiation exposure [7]. This occurs at the cost of acceptable toxicity of normal tissue, which is associated with radiation complications. The rapid development and improvement of RT planning technologies significantly affect the reduction of the negative consequences of the influence of radiation on healthy tissues and organs at risk without worsening the results of the treatment of cancers. But even with the use of the best planning technologies on modern radiotherapy equipment in accordance with high standards of treatment, for many sites, there is a high frequency of relapses and mortality from the underlying disease. A key role in this belongs to an increase in the duration of the general course of RT [8, 9, 10, 11, 12].

The problem of estimation of the negative impact of interruptions in radiation treatment and the ways of their compensation is regularly raised at the training courses by the International Atomic Energy Agency (IAEA) in cooperation with the Government of Russian Federation through the State Research Centre—Burnasyan Federal Medical Biophysical Centre of Federal Medical Biological Agency and the Association of Medical Physicists of Russia (AMPR). At the same time, at the present stage, it is proposed to rely on the linear-quadratic radiobiological model (LQM) theory, which has a long and complex history [13, 14, 15].

The practical application of the LQM in many institutions is an integral part of the clinical practice of cancer therapy. However, calculations related to the estimation of radiation doses when the radiation treatment schedule changes during the course of RT lead to a significant increase in the working time of medical physicists and radiation oncologists and also require special training of qualified specialists capable of conducting them.

Introducing LQM into practice for estimation of radiation doses taking into account the loss of the biological effect when modifying radiation treatment regimens, specialists face the above-mentioned difficulties. Therefore, to solve the identified issues, we have proposed the Web application that allows us to optimize the processes associated with the estimation of radiation doses when modifying the radiation treatment schedule for patients.

The aim of our study was to optimize calculations related to the estimation of radiation doses when the radiation treatment schedule changes by simulating such changes in special software created on the basis of the theory of a linear-quadratic radiobiological model.

2. Description and features of the web application

The development of the application was carried out by specialists in the field of radiobiology, medical physics, and practicing radiation oncologists on the basis of the International Sakharov Environmental Institute of Belarusian State University and N.N. Alexandrov National Cancer Centre of Belarus. The source code of the program was written by an IT developer, a specialist in applied mathematics and actuarial sciences, using JavaScript (52.2%) and HTML (47.8%) programming languages. The technical requirement for the user is to have a browser that supports JavaScript. The program is accessed via the Internet link https://hypo-calc.github.io/.

Web application features:

• calculation of isoeffective doses;

• calculation of the number of fractions;

• calculation of amendments for the modified treatment regimen;

• calculation of EQD₂ taking into account the interruptions in RT course;

• calculation of EQD₂ taking into account the reduction of days of treatment;

• accounting for incomplete reparation with multi-fraction irradiation per day;

• possibility of correcting errors in the release of the dose, etc.

The application is divided logically into three areas. These are the data entry area, the area of calculated values, and the treatment schedule. In the data entry area, the user sets the parameters he needs. The following cells are required to be filled in: Dose per Fraction, Number of Fractions, Fractions proceed, α/β ratio. Fields Start of treatment, Recovery halftime T_1/2, and Use D_prolif are filled in when it is necessary to take into account the duration of the course, interruptions, and incomplete reparation with multifraction irradiation per day. The Use D_prolif field becomes active when the number of days of the RT course exceeds 21 days. The appearance of the application is shown in Figure 1.

Figure 1.

Appearance of the web-application.

In the area of calculated values, the values of Overall treatment days, Total dose, Biological Effective Dose (BED), and Equivalent dose EQD₂ depend on the entered values of the Dose per fraction and the Fractions. The values in the cells Factual gap days and Factual treatment days depend on the changes made to the Treatment Schedule. The Equivalent dose owing to proliferation is calculated on condition that the cell Use D_prolif is filled.

The Treatment schedule is filled in automatically if the input fields are filled in correctly. Clicking on filled cells makes them empty; clicking on empty cells adds fractions. With a forced change in the number of fractions, a dose per fraction is recalculated inside the calendar cells. If the number of already treated fractions is set, they are displayed in the Treatment schedule in gray. The dose in these cells is not recalculated when the number of remaining fractions changes. When pointing to a cell with the mouse cursor, it becomes available to add several factions per day by clicking on the “+” inside the cell. By clicking on 𝛥t, the time interval between fractions can be set.

A digital copy of the application, as well as the necessary documents and materials about it, are registered and transferred for storage to the National Center for Intellectual Property of the Republic of Belarus (certificate of voluntary registration and deposit of the copyright object No. 1487-KP, act No. d20220013 dated 03/25/2022; the authors are Orgish A.N., Batyan A.N., Dziameshka P.D., Hancharova K.V., Haida A.V.).

3. Modeling clinical cases in the web application

As an example of how the application works, the following case, which is possible in clinical practice, is considered below (tasks are taken from the lectures from the training courses by the International Atomic Energy Agency (IAEA) in cooperation with the Government of Russian Federation through the State Research Centre—Burnasyan Federal Medical Biophysical Centre of Federal Medical Biological Agency “Regional Training Course on Radiobiology for Radiation Oncologists and Medical Physicists” 2018 and the Association of Medical Physicists of Russia (AMPR) “Virtual Regional Training Course on Transition from 3D Conformal Radiation Therapy to Intensity Modulated Radiation Therapy” 2021).

3.1 Calculation of isoefficient doses

It is necessary to find the value of dose per fraction for a regimen isoeffective to the classical (2 Gy per fraction in 30 fractions), implemented in 18 fractions every other day, taking into account early reactions (α/β = 10 Gy) and late complications (α/β = 3 Gy). The total treatment time does not change.

For the solution, it is necessary to fill in the active cells with data in the form as it is required in the task (Figure 2). Further, after filling in the Treatment schedule, simulate irradiation every other day. To do this, unnecessary fractions must be removed with a mouse click. The result is the value inside the calendar cells. This is 2.85 Gy for late complications of alpha beta 3 Gy. For early reactions, change the value of alpha beta to 10 and get the result of 3.06 Gy.

Figure 2.

Calculation of isoefficient doses.

3.2 Calculation of the number of fractions

It is necessary to find the number of fractions during irradiation of the mammary gland at 2.67 Gy, so that this regimen is isoeffective to the classical one at 2 Gy per fraction up to 50 Gy daily, without taking into account proliferation. For the mammary gland alpha beta is 4.6 Gy; alpha beta of early skin reactions is 8.8 Gy; for late complications is 1.7 Gy.

The data of a dose per fraction, the passed fractions, and the alpha beta coefficient are entered. Next, by the selection method, the number of fractions, at which the value of the equivalent dose will be closest to 50 Grays, is substituted. In the first case, these are 17 fractions (Figure 3).

Figure 3.

Calculation of the number of fractions. Alpha beta 4.6 Gy.

Then, the alpha beta for early skin reactions is changed, and the value of 18 fractions by the selection method is received (Figure 4).

Figure 4.

Calculation of the number of fractions. Alpha beta 8.8 Gy.

For an alpha beta coefficient of 1.7 Gy, the number of fractions is 16 (Figure 5).

Figure 5.

Calculation of the number of fractions. Alpha beta 1.7 Gy.

3.3 Calculation of corrections for the modified treatment regimen

The patient has prescribed five sessions of preoperative radiation therapy dose per fraction of 5 Gy. On Monday and Tuesday, everything went as had been planned. On Wednesday, there was a break in the treatment. What dose should be given for the last two fractions to complete RT as planned on Friday? α/β = 10 Gr.

The data from the condition of the problem are entered. Simulate a situation in which the third fraction is skipped is simulated, and the result of 6.73 Gy per fraction is obtained (Figure 6).

Figure 6.

Calculation of corrections for a modified treatment regimen.

3.4 Calculation of EQD₂ taking into account the interruption in RT course

Irradiation of tumors of the head and neck. The maximum dose to the spinal cord is 45 Gy. Dose on the main target is 70 Gy. Due to reactions after fraction 25, the patient was placed on a two-week break. EQD₂ is to be calculated.

The data from the condition of the problem are entered, the situation of a two-week break is simulated, and the answer of 59.5 Gy is obtained (Figure 7).

Figure 7.

Сalculation of EQD₂ taking into account the interruption in RT course.

3.5 Calculation of EQD₂ taking into account the reduction of days of treatment

Irradiation is carried out according to the scheme of 6 fractions per week for 5 weeks. Dose per fraction is 2 Gr. EQD₂ needs to be calculated.

The available data are entered; factions are mandatorily transferred to Saturdays. And the answer of 64.5 Gy is obtained (Figure 8).

Figure 8.

Calculation of EQD₂ taking into account the reduction of days of RT course.

3.6 Accounting for incomplete reparation with multi-fraction irradiation per day

Irradiation of the head and neck tumor was planned with the parameters of a dose per fraction of 2 Gy 35 fractions, 5 fractions per week. The spinal cord accounts for 50 Gy (1.43 Gy per fraction). In order to reduce late complications before the treatment, it was decided to switch to 2 fractions per day with a six-hour break. It is necessary to calculate the dose per fraction and the equivalent dose to the spinal cord.

The solution to this problem consists of two stages. In the first stage, we find what dose per fraction is necessary to irradiate the tumor with an increase in the number of fractions by 2 times. To do this, we enter the data from the condition of the problem are entered. The situation, in which the number of days of treatment is doubled, is simulated. The desired value of 1.08 Gy is obtained (Figure 9).

Figure 9.

Accounting for incomplete reparation with multi-fraction irradiation per day. The first stage.

At the second stage, it is necessary to pre-calculate from the proportion, which in this case is equal to the dose per fraction for the spinal cord. It is 0.77 Gy per fraction. Next, we simulate a situation in which irradiation is carried out 2 times a day. The value of a dose per fraction is changed until the values in the cells of the calendar are equal to 0.77 Gy. And the answer that the equivalent dose to the spinal cord in this case is 42.7 Gy is obtained (Figure 10).

Figure 10.

Accounting for incomplete reparation with multi-fraction irradiation per day. The second stage.

3.7 Correction of errors in dose dispensing

Irradiation of a lung tumor. Dose per fraction of 2 Gy for 33 fractions is planned. After the twentieth fraction, it was found that due to an error (prescription or normalization), 1.8 Gy was supplied instead of 2 Gy. How to correct the treatment?

To find the value to which it is necessary to correct the radiation dose, it is necessary to carry out several stages of working with the application. In the first stage, the values of the already treated 20 fractions of 1.8 Gy per fraction are entered and the equivalent dose is defined (Figure 11).

Figure 11.

Correction of errors in dose dispensing. The first stage.

Further, using an intermediate calculation, it is necessary to find the difference in equivalent doses between the value of the equivalent dose planned for the end of the course and the value for the first 20 fractions of 1.8 Gy: ∆EQD = 66–35.4 = 30.6 Gy. After that, the values for the remaining 13 fractions are entered into the program and the value of a dose per fraction is selected, which will correspond to the obtained value of the equivalent dose of 30.6 Gy.

The equivalent dose is to be tracked. The value of the dose per fraction is changed so that the equivalent value is 30.6 Gy (Figure 12). In our case, this is 2.3 Gy. This means that it is necessary to adjust the dose per fraction to a value of 2.3 Gy.

Figure 12.

Correction of errors in dose dispensing. The second stage.

4. Conclusions

Calculations related for the evaluation of the effectiveness of radiotherapy and the possibility of making changes to the radiation treatment regimen require special training of qualified specialists who are able to carry out such calculations. We have developed a computer program that allows us to optimize the work associated with the estimation of radiation doses. The application was developed with no funding. In this version, not all functions related to the evaluation of radiation doses are implemented.

The tasks in this chapter are taken from the lectures from the training courses by the IAEA in cooperation with the Government of the Russian Federation through the State Research Centre—Burnasyan Federal Medical Biophysical Centre of Federal Medical Biological Agency “Regional Training Course on Radiobiology for Radiation Oncologists and Medical Physicists” 2018 and the AMPR “Virtual Regional Training Course on Transition from 3D Conformal Radiation Therapy to Intensity Modulated Radiation Therapy” 2021. The calculation of isoeffective doses, calculation of the number of fractions, calculation of amendments for the modified treatment regimen, calculation of EQD₂ taking into account the interruptions in RT course, calculation of EQD₂ taking into account the reduction of days of treatment, accounting for incomplete reparation with multi-fraction irradiation per day, and the possibility of correcting errors in the release of the dose are considered in detail.

Description of the multicomponent and multidirectional response of the body to the action of ionizing radiation in the form of simple mathematical expressions, aimed at the treatment of malignant neoplasms within an acceptable range of complications, is a difficult task. Each modification of radiobiological models allows going deeper into biology, expanding the boundaries of applicability, and overcoming an increasing number of shortcomings. The treatment of tumor diseases is currently different for adults and children, but often there is no difference in the treatment of men and women. In addition, great importance is given to genetics, but the issues of epigenetics remain aside.

The emergence of new and advanced parameters that should be taken into account when modeling the outcomes of radiation treatment leads to an increase in the volume of calculations and additional time spent on them, so now there is an increasing need to create programs with complex logic and algorithms to optimize the assessment of the radiation dose in the tumor and surrounding normal tissues. The future belongs to personalized medicine and artificial intelligence.

Acknowledgments

The authors express their gratitude to everyone, who took part in the development of this Web application.

Conflict of interest

The authors declare no conflict of interest.