Computational simulator that calculates tumor control probability in a tumor heterogeneously irradiated for fractionated radiation oncology treatments

Abstract

Although in the ionizing radiation field many concepts and processes are currently recognized as radiobiological, there are also probabilistic ones, and a probabilistic treatment makes a better understanding about them. The purpose of this study is to develop a new radiobiological simulator that calculates the tumor control probability (TCP) for a tumor heterogeneously irradiated from a fractioned treatment. The three possible types of cells and the results of interactions of ionizing radiation with each cell of a determined volume are analyzed. For an irradiated region with a dose per fraction d, the simulator determines the radiation biological effects using the cell kill (K) and cell sub-lethal damage, volume, cell density, cell repair of damaged cells during the interfractions, and number of fractions. K is determined from its probabilistic complement, the cell survival (S), described with the linear-quadratic (LQ) S(d) model as K = 1 −LQ S(d). TCP is calculated from computational simulations as in the ratio of simulations with K = 100% and their total. This application opens new avenues for theoretical and experimental investigations concerning simulations of radiation treatments, and methodologies for therapy optimizations. Our simulator represents a novel methodology as TCP is calculated without analytical formulas, but based on its own probabilistic definition.

Keywords

Tumor control probability (TCP)Poisson distribution computational simulation Monte Carlo method linear-quadratic model biologically effective dose (BED)

1. Introduction

Although cancer can be effectively controlled through early diagnosis, followed by effective treatment, the main efforts should be focused on better and effective detection Appendix 1.

Computer-aided diagnosis (CAD) systems have become essential and crucial in the breast cancer classification problem, to facilitate the diagnosis process and increase survival chances. Deep learning techniques, such as the convolutional neural network (CNN), as developed in Davoudi and Thulasiraman,1 due to their classification capabilities on learned feature methods and ability of working with complex images, have been widely adopted in CAD systems.

The property of ionizing radiation (IR) for killing cells is used in radiation oncology therapies for treating cancer patients. The radiation treatments are performed with fractioned regimens for avoiding complications of irradiated normal tissues near to the treated zone.

It is well known that in radiation therapy, the main challenge is to deliver the dose to the tumor, while sparing the healthy tissues around it. One important decision to make is the beam configuration, which has motivated the development of optimizations like those of Yarmand and Craft,2 where they have proposed novel heuristic approaches for beam angle optimization in radiation therapy. The delivery of dose to the tumor should be as major as possible in all the tumor regions, and as discussed in our study, the zone around it should have the minimum dose.

The effects of IR are stochastic for the values of doses used in radiation oncology treatments; for this reason, many current radiobiological concepts and processes are also probabilistic metrics (PMs), like S, termed as the cell survival probability. The stochastic processes in the IR field are better undertaken when they are treated with probabilistic foundations. Although S is modeled with a decreasing exponential function, its probabilistic complement, K, or better termed as the cell kill probability, has been modeled with a probabilistic model in Frometa-Castillo.3 As a probabilistic metric, in a tissue volume irradiated with a dose d, K represents the probability of each healthy cell being transformed into a killed cell.

Given some properties of radiation, these may or may not interact with all the cells of a determined living tissue volume, and may or may not produce effects as a result of their interactions. Generally, the first fraction of a fractioned treatment produces a partial number of killed and sub-lethally damaged cells (SLDCs) from initially undamaged ones of an irradiated living tissue region. During the second and successive fractions, the radiation can interact with these three classes of cells, where the interactions can produce the same effects as that of the first fraction from the undamaged and SLDCs. The probability of interaction with a type of cell in a determined irradiated volume depends on its amount and the number of fractions, and its final result depends on the dose per fraction and its radiosensitivity for cell kill and sub-lethal damage.

If a tumor tissue is heterogeneously irradiated with a distribution of dose d_i, the region with dmin has a minimum probability of killing healthy cells. For this reason, the TCP should be determined with the dmin.

Nowadays, as shown in previous works4,5 computational simulations (CSs) have increased their roles in the detection and therapy of treatments to cancer patients. The way of obtaining the simulated experiments is actually an interesting method in a field like IR, where the availability of real data or feasibility of performing real experiments is little or none.

Our application simulates possible situations of an irradiated living tissue in a fractioned regimen, and is based on radiation biological effect (RBEf) calculations. Cell repair is considered as a temporal process during the interfractions. Our simulated calculations have not been previously applied by the current Monte Carlo (MC) methods. MC has been applied for calculating the dose, which is used in the analytical TCP expressions, or in the propagation of uncertainties. The MC studies have been focused on determining the dose distributions, and our application is for calculation of the RBEfs.

In a fractionated treatment, the first fraction in a living tissue produces a partial number of killed and SLDCs. The mean numbers of these cells depend on the cellular radiosensitivities. In the second and successive fractions, the radiation can interact with these three kinds of cells. When one analyzes all the cells of a determined volume of a living tissue, the probability of meeting one of these cells depends on its mean number. As the final result of interactions of cells with radiation with, the undamaged ones have three possible outcomes: being killed, being SLD, or maintaining their undamaged cell conditions. The SLDCs have two first possible outcomes. The probability of these outcomes depends on the radiosensitivity.

Till date, the TCP has been generally determined through experiments/observations that allow for deriving its values and phenomenological models. Actually, the experiments/observations are few or none; for this reason, the development of virtual simulations is a very interesting option for the determination of TCP.

Nowadays, the radiosensitivity studies function of d are widely described using the LQ S(d) model. S is the probabilistic complement of K; i.e., probabilistically S = 1K. Using this model, the radiosensitivity for K is determined as K(d) = 1 − LQ S(d). This cellular characteristic and others, such as radiosensitivity for cell sub-lethal damage, cell repair, volume and cell density, are used for simulating a fractioned radiation treatment, where all possible results of interactions of radiation with whatever type of cell in an irradiated tumor are analyzed.

Based on the descriptions in Table 1, the TCPsim methodology could replace the current ones employing analytical models. Also, TCPsim will represent a huge contribution due to one potential innovation being that rather than evaluating TCP by analytical calculation, it is calculated based on its own probabilistic definition, and could be considered as an extension of the MC methods, where macrocellular outcomes of the radiation interactions with three possible types of tumor cells are analyzed, instead of the DNA damages.

Table 1.

Comparison between the analytical TCP models (ATCPMs) and the TCPsim methodology.

Aspect	ATCPMs	TCPsim methodology	Remarks
Probabilistic foundation	Some of them are Poisson-based TCP models as in Mesbahi and Oladghaffari6 and Rana et al.,7 or function of BED as in Sood et al.8	Based on the own probabilistic definition of the TCP and computational simulations of radiation interactions with the tumor cells of the region with Dmin.	As described in Frometa-Castillo et al.,9 the Poisson-based TCP model is an incoherent derivation.
Complexity	Mathematically high	Computationally low.
Numerical and graphical information about the tumor DVH	It is used	Only uses the value of the region with Dmin.
Independent variable	EUD	dmin and n, where dmin = Dmin/n.	EUD has a mathematically complex expression for its calculation.
Use of the LQ models	Involve the LQ S(n,D_i) formalism	Involves the LQ S(dmin) model.	D_i: All tumor DVH valuesLQ S(n,D) = [LQ S(d)]ⁿ; where D = nd.*
The isoeffective dose in 2 Gy fraction (LQED2)	It is required its use due to EUD is function of D_i	It is not required for use.	The CSs use dmin.

ATCPM: analytical TCP models; TCP: tumor control probability; BED: biologically effective dose; DVH: Dose-volume histogram; EUD: equivalent uniform dose; LQED: isoeffective dose in 2 Gy fraction.

In Allen Li et al.,10 the TCP formulation is based on the Poisson distribution as follows:

TCP = \exp (- SN)

(1)

where N is the initial number of clonogens, the average number of surviving clonogens is given by SN, and S is the overall surviving fraction after a course of radiation therapy.

Equation (1) is known as the Poisson-based TCP model. As described in Frometa-Castillo et al.,9 this TCP model represents an incoherent derivation. Also, this work will help the medical physics community and others to understand the following: (1) how the binomial distribution (BD) and Poisson distribution (PD) were derived; (2) what really are the BD and PD; (3) how one should use the BD; (4) the PD is not a new probabilistic function (PF), but the own BD with simplifications valid for some values of BD parameters, and with changes in the variables and parameters; and (5) given the essential condition for a PF being not satisfied in the PD for some values of its parameter, we can also say that, The PD is not a PF. For these reasons, the PD(x;µ) could be replaced with BD(x;Xmax,p), where Xmax is the possible outcome of a stochastic process, Xmax = n and p = µ/Xmax. The BD(x;Xmax,p) would represent a unification in only one expression of the current BD(k;n,p) and PD(x;µ) expressions.

While D expresses a mathematical accumulation of dose, our CSs show that when a living tissue is fractionally irradiated, there is accumulation of RBEfs.

The new radiobiological simulator developed in this study, compared to its previous one by Frometa-Castillo et al.,11 includes the following improvements: (1) cell repair as a temporal-cellular factor; (2) LQ S(d) parameters (α and α/β) for determining the radiosensitivity K; (3) a tumor non-homogenously irradiated is considered; (4) a better explanation of the MC applied to radiotherapy; and (5) major graphical information.

The TCPsim methodology, as the MC method, uses virtual and CSs; solves problems that are very difficult or impossible to solve with analytical methods; could replace the analytical methods in the future; and can treat the uncertainties of its parameters. In our methodology, as in other calculation methods, the precise determination of its parameters is one of the major difficulties. While MC uses PMs associated with the physical effects occurring during the tracking of a particle/photon in a simulated material, the TCPsim employs radiobiological PMs and others such as cell kill and the probability of meeting one of the three kinds of cells of an irradiated living tissue. Despite the differences, given the similarities, the TCPsim could be considered as an extension of the MC.

1.1. The LQ S(d) model

Current radiosensitive studies are described using the LQ S(d) model, and is shown in previous works12–14 as follows:

LQ S (d) = \exp (- α d - β d^{2})

(2)

Equation (2) represents the cell survival probability; i.e., the mean value of the ratio of the survived cells and their total, when a determined living tissue characterized with parameters α and α/β is homogenously is irradiated with one fraction of dose d. For this reason, this equation can be considered for whatever healthy cell of a determined tissue as the probability of becoming a survived cell after the irradiation.

The parameters α and α/β that characterize the radiosensitivity for S in a dose d; are used in our simulator for obtaining the radiosensitivity for K, given that probabilistically K(d) = 1 − LQ S(d).

It is important to establish the difference between the LQ S(d) model and LQ S(n,D) formalism. The LQ S(d) correlates S in function of dose d; while the LQ S(n,D) = [LQ S(d)]ⁿ was obtained for n fractions of an irradiation with dose per fraction d, under the following conditions: (1) 100% cell repair (or 0% cell sub-lethal damage) during the interfractions and (2) no presence of killed cells.

The BED expression is a derivation of the LQ S(n,D) and was mathematically created for evaluating the as mentioned biological dose, and is an unreal physical quantity. Indeed, as a result of interactions of radiation with the living tissues, the RBEfs are produced. RBEf is defined for the number of killed and SLDCs. More details about the BED and RBEf are explained in Frometa-Castillo et al.15

LQED2 is a derivation of the BED, and establishes equivalence between a fractioned treatment with 2 Gy per fraction and others with a dose d_i for n fractions.

Our simulator is based on the RBEf evaluations done with CSs. Nowadays, the LQ S(n,D) formalism is widely used for evaluating RBEfs. Table 2 shows comparisons between this formalism and CSs.

Table 2.

Comparison between the LQ S(n,D) formalism and CSs as methods evaluating RBEfs in an irradiation of n fractions.

No.	Aspect	LQ S(n,D)	CSs	Remarks
1	Probabilistic foundation	It is not wholly appropriate its probabilistic derivation	Strong	See Points 3,4, and 9 of the LQ S(n,D).
2	Complexity	Mathematically low	Computationally low
3	Sub-lethally damaged cells (SLDCs)	It is not considered	It is considered	In the LQ S(n,D) derivation is assumed SL = 0%, i.e., all SLC is repaired during interfraction periods.
4	Killed cells	It is not considered	It is considered	In the LQ S(n,D) derivation is missed the presence of KCs.
5	Metrics evaluated	S	Individually treated the UD, SL, and K	S is a mixed metric, where S = UD + SL.
6	BED	It is one of its derivations	It is not used	The CSs evaluates RBEfs. As described in Frometa-Castillo et al.,15 the BED is only a mathematical derivation.
7	LQED2	It is required its use due to S is function of the total absorbed dose (D)	It is not required its use	The CSs use the dose per fraction (d).
8	Relationship with the DNA	It is related	None	The CSs do macrocellular analysis.
9	CR	It is assumed CR = 100% in its derivation	CR <= 100%

SLDC: sub-lethally damaged cell; UD: mean undamaged cell; BED: biologically effective dose; LQED: isoeffective dose in 2 Gy fraction; CR: probability of the cell repair for the SLDCs.

As shown in Table 1, the current analytical TCP models have some incoherencies, so the most appropriate is the validation of our simulator with observational real data. This kind of validation is very difficult due to real data being little or none.

The TCP calculated with our simulator shows that our computationally simulated results have logic radiobiological behaviors and approaches compared with those calculated by other researchers; Table 1 illustrates the previously quoted incoherencies, and the advances of our methodology.

Till date, the TCP has been calculated using analytical models. Even in some MC works, the TCP is calculated with these models, and the simulations are used for determining distributions of the physical absorbed dose. Thus, we can say that our simulator is analogous to a previous one.

1.2. MC methods in radiotherapy

“MC methods represent a wide class of numerical computer simulation tecniques that utilizes statistical resampling to solve complex systems tha are not easily tractable analytically. The current applications of MC dosimetry methods to radiation therapy could be broadly into two areas of macrodosimetry and micro (nano) dosimetry. In the former is mainly related to scoring radition dose in longer volumes (milimeters), while the latter is related to scoring dose in very small areas at the cellular, sub-cellular or molecular level.”16 Let us call the first method as MCSR. In addition to MCSR, there is another application of MC: the propagation of uncertainties (MCPU), which is widely used for the anlysis of risk in investments in the industrial field.17 In El Naqa et al.,16 many examples of the MCPU are reported, where MC was used to study how inter-patient differences can affect TCP.

“MC techniques have been succesfully applied for calculating radiotherapy dose distribution, treatment plan evaluation and uncertainty estimates, modeling tumor growth and simulating particle track structure. However, their application for radiobiological modelling of response and treatment outcomes has been limited thus far.”16

The MC has been used for calculating the TCP, as is shown in Lucas et al.18 and Santos and Neves.19 Contrary to the procedures used by these MC works, our simulator does not use analytical TCP expressions, but its own probabilistic definition based on virtual simulations. Different from MC works, the simulator analyzes all cells in their possible stages and the final result of the interactions with radiation after each fraction; the cell repair is taken into account as a temporal process during the interfractions.

The MC techniques work with PMs associated with physical effects, such as photoelectrical, Compton, and pair production, but our CSs employ radiobiological PMs, such as cell kill and sub-lethal damage probability. Besides, the simulator makes macrocellular analysis, where all cells are in their possible stages and the final result of the interactions are with radiation after each fraction; the cell repair is taken into account as a temporal process during the interfractions.

As described in Muraro et al.,20 the MC methods have been used in the IR field for solving problems that are very difficult to be solved by analytical ways. Our CSs could be considered as an extension of the MC methods.

2. Materials and methods

After the first fraction of irradiation to a living tissue region with dose d, a number of killed cells and SLD are generated, and the number of undamaged cells is calculated as follows:

nudc = NTC - (nkc + nslc)

(3)

The nslc can be repaired during the interfractions, and its mean amount is determined as follows:

nrc = nsl c_{j} * CR

(4)

In the second and successive fractions, nkc increases by contributions from the SLD and undamaged cells. If nkc is equal to NTC after n fractions, then the tumor is controlled. As a result of virtually simulating a fractioned treatment, TCP is calculated as follows:

TPC = \frac{TCOK}{nvs}

(5)

As shown in the diagram in Figure 1, Figure 2 and Figure 3, for simulating a fractioned treatment, the following are considered:

The first fraction generates mean nkc killed cells, nslc SLDCs, and nudc undamaged cells.

For the second and successive fractions, the three kinds of cells are analyzed in their possible final outcomes in each fraction.

The MATLAB function rand is used for generating a random number gnum <= 1. The probability of meeting a killed cell (PMKC) is calculated as nkc/NTC; then if gnum < PMKC, the analyzed cell is dead, but if it has survived, it is as shown in Figure 2.

For a killed cell, the simulator will analyze a new cell; but for a survived cell, there are two possibilities: the cell is undamaged or SLD. The probability of meeting an SLDC is defined with a new gnum > nslc/(nslc+nudc).

For an undamaged cell, if a new gnum<probability for K, this cell will die, but if gnum <= K+probability for SL, this becomes an SLD.

For an SLDC there is a range of damage degrees. Two new random numbers, gnum1 and gnum2, are generated, and we define KSL = max(gnum1;1-gnum1). If gnum2 <= KSL, the cell will die, but it is kept as an SLD. The previous condition is associated with a major probability of killing the SLDC.

While the number of fractions increases, nkc increases and nudc decreases. The nslc increases after the first fractions, and can increase or decrease and finally decrease after the second or successive fractions.

The cell repair is a temporal-cellular process and the number of repaired cells is determined by means of Equation (4) after each fraction.

If nkc = NTC after n fractions, there is tumor control, then TCP is calculated using Equation (5).

Figure 1.

Diagram of the TCPsim application.

Figure 2.

Illustration that shows the probability conditions of the type of cell (killed, SLD, or healthy or undamaged) for being analyzed.

Figure 3.

Illustration that shows the probabilities of transformation of the different types of cells.

3. Results

3.1. Description of the code

The “TCPsim” MATLAB application developed in this study is available at https://gitlab.com/tfrometa/TCPsim.

3.2. Reproducibility

During application, the input values (IVs) appear in yellow, the outcome of the “TCP” appears in green, and the “Radiosensitivity of tumor for cell survival, the LQ S(dmin)” and “Starting/Finishing time” in cyan. One should press the “Enter” key placed on each field for introducing the IVs into the application.

The steps for executing the simulation are as follows:

Introduce for a tumor non-homogeneously irradiated: dmin in Gy, α in Gy⁻¹, α/β in Gy, SL in %, CR in %, Vol in mm³ (it is suggested to always use a value ≥1 mm³), and Cden in cells/cm³;

Introduce for the virtual simulations: nvs, and n;

Press the “For calculating” button for obtaining the simulated TCP.

4. Discussion

Being responsible for radiation treatments, radiation oncologists with the collaboration of medical physicists choose/estimate the values of the TCPsim parameters: α, α/β, CR, SL, and Cden. It is suggested to always use values of Vol ≥ 1 mm3, and nvs ≥ 30. Further research is required for obtaining better CR values or ranges for each phase of a fractionated radiation treatment. The current radiosensitivity studies are described with S, which is probabilistically related to K, SL, and UD as shown in Equation (6), and S = SL + UD. Due to little available information about the SL, in many cases the SL values should be assumed taking into account that SL ≤ S.

In order to obtain better characterizations of the TCPsim parameters of different radiations and tumor tissues, a review of the most updated studies, like Van Leeuwen et al.,21 should be done.

Figure 4 illustrates a tumor non-homogenously irradiated with dose per fraction d_i and minimum dose dmin. In the region with the minimum dose of a tumor heterogeneously irradiated, there is the highest value S, as shown by the LQ S(d) model of Equation (2); i.e., there is the lowest value of probability of killing the tumor cells. For this reason, the TCP should be calculated by analyzing the results of interactions in this region.

Figure 4.

Illustration of a non-homogeneous distribution of dose di in a tumor with minimum dose dmin, and explanations of why TCP should be determined with the dmin.

The mean outcomes of the radiation interactions with the cells are probabilistically related as follows:

K + S L + U D = 1 (100 %)

(6)

S = SL + UD

(7)

K + S = 1 (100 %)

(8)

K = 1 - S

(9)

As Equation (8) shows, the survived cells involve SLD and undamaged cells. Nowadays, the descriptions of SL are little or none. For this reason, the values of the radiosensitivity for SL are taken as less than S, and have been assumed in our simulations.

Table 2 contains the results obtained with our TCPsim application, which have acceptable approaches with the TCP reported in their respective references.

Table 3 shows the simulated mean TCP results in percent for tumor characterized with α and α/β, respectively and Table 4 shows the relationship of TCP with dmin and n.

Table 3.

Simulated mean TCP results and TCP values reported in their respective references.

No.	d [Gy]	α [Gy⁻¹]	α/β [Gy]	ρ10*⁷ [cell/cm³]	SL [%]	CR [%]	n	References	TCP reported[%]	TCP simulated[%]	Obs.
1	2	0.262	7	1	10	40	35	Kristin et al.22	70	74	TCP reported for α/β = 10 Gy
2	3.1	0.307	10	1	25	40	20	Nahum and Uzan23	80	86
3	18	0.06	5	0.01	1	10	3	Ohri et al.24	85	83

CR: probability of the cell repair for the sub-lethally damaged cells; TCP: tumor control probability.

Table 4.

Simulated mean TCP results in percent for tumor characterized with α and α/β, respectively, 0.307 Gy⁻¹ and 10 Gy of Nahum and Uzan23; Cden = 10⁷ cell/cm³; SL = 25%–30%; and CR = 40%.

dmin [Gy]	n
	10	15	20	25	30
1	0	0	0	5	12
1.8	0	5	51	66	95
2	0	10	58	83	99
2.5	0	14	62	87	100
3.1	0	17	88	100	100
3.5	5	59	89	100	100

5. Conclusion

The simulator shows direct relationships of the TCP with the radiosensitivity of the tumor tissue cells, dmin and n, and an inverse relationship with the cell repair. These relationships are positive elements for the validation of our simulator. When dmin increases, the radiosensitivities for cell kill and/or cell sub-lethal damage increase, while the increase in n represents an increase in the probabilities of the interaction of radiation with undamaged or SLDCs.

The simulated TCP calculations only use dmin, since in the zone with a minimum of dose there is the maximum probability of cell survival; in other words, the minimum probability for cell kill.

Although other temporal-cellular processes can be included in the simulations, nowadays only the cell repair is considered.

The chain of effects of the IR in living tissues initially involves physical effects, later biological ones, and finally clinical effects. The MC has been useful in radiotherapy only for calculating the first effects, while our radiobiological simulator calculates clinical effects in tumors.

Our future TCPsim researches will be aimed to virtually simulate the most feasible possible real fractionated radiation oncology treatments; therefore the improvements in the TCPsim application will involve the uncertainties of many of its radiobiological parameters and variables, such as α and α/β, cell repair, and cell sub-lethal damage.

While our TCPsim methodologies have solid-logic probabilistic-radiobiological foundations, many of the current analytical TCP models have incoherent ones, such as Poisson-based TCP model, and TCP function of the BED. For this reason, it is not of sense to continuously use and develop these models.

Footnotes

Appendix 1 Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

ORCID iD

Terman Frometa-Castillo

Author biographies

Terman Frometa-Castillo is the owner of the SMp, LLC, and has a broad background in M&S, mainly in applications to radiotherapy. He is creator of the “universal” probabilistic function, the SMp(x), and principal developer and researcher of a new methodology of calculating probabilities using computational simulators.

Anil Pyakuryal is an adjunct professiorial lecturer of mathemaics and statistics at American University, and an adjunct instructor of physics and engineering at the University of District of Columbia.

Ganesh Narayanasamy has a PhD in applied physics from the University of Michigan. His research interests include dosimetric studies in radiation therapy, assessment of new and existing technologies in radiation therapy, and study of radiation biology. Dosimetric comparison between treatment planning systems has been his special research interest.

Amadeo Wals-Zurita is a specialist in radiation oncology with experience in lung cancer, head and neck cancer lymphomas, and brain cancer. He is a member of the SEOR (Spanish Radiation Oncology Society), SAC (Andalusian Cancer Society), Professor at UNIA (Universidad Internacional de Andalucia), and founding member of FROG (Freelancer Radiation Oncologists Group)

Raul Piseaux-Aillon is currently an associate researcher in the Northwestern University, Feinberg School of Medicine, and has more than 30 years of wide experience related to oncology therapies.

Asghar Mesbahi has been working in the medical physics sector since 2004, with a focus on radiation therapy physics and dosimetry. Most of his publications have been on application of the Monte Carlo method to solve the calculation issues in clinical radiation physics. Recently, he has started to work on radiobiological modeling of tumor and normal tissue response to radiation therapy, and is one the cofounders of the molecular medicine research center in Tabriz University Medical sciences. In addition, he established a medical radiation sciences research team in 2016; Currently, he is serving as the head of the team.

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Grills

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dmin [Gy]	n
	10	15	20	25	30
1	0	0	0	5	12
1.8	0	5	51	66	95
2	0	10	58	83	99
2.5	0	14	62	87	100
3.1	0	17	88	100	100
3.5	5	59	89	100	100

dmin [Gy]	n
	10	15	20	25	30
1	0	0	0	5	12
1.8	0	5	51	66	95
2	0	10	58	83	99
2.5	0	14	62	87	100
3.1	0	17	88	100	100
3.5	5	59	89	100	100

dmin [Gy]	n
	10	15	20	25	30
1	0	0	0	5	12
1.8	0	5	51	66	95
2	0	10	58	83	99
2.5	0	14	62	87	100
3.1	0	17	88	100	100
3.5	5	59	89	100	100