Generalized methods for predicting biological response to mixed radiation types and calculating equieffective doses (EQDX)

Abstract

1.1. Background

Predicting biological responses to mixed radiation types is of considerable importance when combining radiation therapies that use multiple radiation types and delivery regimens. These may include the use of both low- and high-linear energy transfer (LET) radiations. A number of theoretical models have been developed to address this issue. However, model predictions do not consistently match published experimental data for mixed radiation exposures. Furthermore, the models are often computationally intensive. Accordingly, there is a need for efficient analytical models that can predict responses to mixtures of low- and high-LET radiations. Additionally, a general formalism to calculate equieffective dose (EQDX) for mixed radiations is needed.

1.2. Purpose

To develop a computationally efficient analytical model that can predict responses to complex mixtures of low- and high-LET radiations as a function of either absorbed dose or equieffective dose (EQDX).

1.3. Methods

The Zaider-Rossi model (ZRM) was modified by replacing the geometric mean of the quadratic coefficients in the interaction term with the arithmetic mean. This modified ZRM model (mZRM) was then further generalized to any number of radiation types and its validity was tested against published experimental observations. Comparisons between the predictions of the ZRM and mZRM, and other models, were made using two and three radiation types. In addition, a generalized formalism for calculating EQDX for mixed radiations was developed within the context of mZRM and validated with published experimental results.

1.4. Results

The predictions of biological responses to mixed-LET radiations calculated with the mZRM are in better agreement with experimental observations than ZRM, especially when high- and low-LET radiations are mixed. In these situations, the ZRM overestimated the surviving fraction. Furthermore, the EQDX calculated with mZRM are in better agreement with experimental observations.

1.5. Conclusion

The mZRM is a computationally efficient model that can be used to predict biological response to mixed radiations that have low- and high-LET characteristics. Importantly, interaction terms are retained in the calculation of EQDX for mixed radiation exposures within the mZRM framework. The mZRM has application in a wide range of radiation therapies, including radiopharmaceutical therapy.

2. Introduction

The effects of mixtures of different types of radiation on biological response are of great importance for multiple areas including radiopharmaceutical therapy, ion beam therapy, radiation safety, aviation, and space exploration. Such mixtures can include radiations of low and high linear-energy-transfer (LET). Even though the exact mechanisms underlying the biological response to mixed radiations are not well understood, various theoretical formulations have been proposed by both experimental and theoretical groups over the years. Studies have been done on the effects of sequential irradiation with high- and low-LET radiation ^1–3 as well as with simultaneous irradiation ^4–9. The debate of whether the effects of each radiation type in a mixed-LET radiation are independent or dependent, and if dependent, whether the effects are synergistic or not, has been ongoing for decades. Barendsen et al. in 1960 ¹⁰ studied the combined effect of alpha and X-ray irradiation and concluded that the two effects were independent in nature. McNally et al. ² conducted a similar study and observed a dependent interaction between alpha and X-ray irradiation of V79 cells. Bird et al. ¹ also studied the combined effect of high- and low-LET irradiation of V79 cells and concluded that the effect was synergistic within an absorbed dose range of 10–15 Gy.

The underlying mechanisms of a dependent action between the different radiation types of a mixed-radiation field are complicated and efforts have been made to predict their biological effect. Some of the widely used models are the Zaider-Rossi Model (ZRM) ¹¹, the Lesion Additivity Model (LAM) ¹², the Local Effect Model (LEM) ^13–16, and the Microdosimetric Kinetic Model (MKM) ^17,18. These models account for the interaction of the radiation effects caused by the different radiations that comprise the mixture. The ZRM uses an analytical approach to predict cell survival following exposure to a mixture of radiations with different LETs. It is based on the theory of dual radiation action and linear quadratic (LQ) model. It uses LQ model parameters from individual cell survival curves to predict cell survival following exposure to mixed radiations. The LAM is based on the principle of lesion addition and, without focusing on the exact nature of the radiation-induced damage produced by different radiation types, it relies on the response curves to the individual radiations in the mixture and numerically solving differential equations to obtain the response curve for the radiation mixture. The LEM works on the concept of “local dose” with the assumption that identical local doses produce similar biological responses irrespective of the radiation type. This model involves calculation of the microscopic local absorbed-dose-distribution in the cell nucleus using Monte Carlo (MC) track structure calculations and uses this knowledge to simulate DNA double strand breaks (DSB) in the cell nucleus. Finally, the MKM model was developed by Hawkins ^17,18, and subsequently modified by Bertolet et al. ¹⁹, to predict the combined effect of low- and high-LET radiations under both sequential and simultaneous irradiation. This model requires knowledge of the nuclear radii of the cells being irradiated and uses the concept of a domain radius in which two sublethal lesions combine to form a lethal lesion.

In the present work we seek to identify an analytical and computationally efficient model that can be integrated into our MIRDcell software application ²⁰ to predict the surviving fraction of a large cell population following exposure to mixed radiation fields that arise from one or more radiopharmaceuticals. MIRDcell V3.13, and earlier versions, use a non-interacting model (NIM) that can overestimate the probability of cell survival. In the present work, we consider the NIM and some of the aforementioned models for implementation in MIRDcell with an eye toward computationally efficient accuracy. The LEM model was not considered because its DNA-centric nature is not suited for integration into MIRDcell V3.13. The ZRM ¹¹ is most efficient but it has shortcomings that we outline and present a modified version, mZRM, to overcome some of said shortcomings. We also generalize the mZRM for a mixture of any number of radiation types having pairwise interactions. This will be most beneficial in the context of radiopharmaceutical therapy with alpha particle emitters which can involve parent/progeny radionuclides that emit multiple radiation types ($\alpha$ particles, $\beta$ particles, Auger electrons, conversion electrons, X rays, $\gamma$ rays). Finally, recognizing that the International Commission on Radiation Units and Measurements (ICRU) has recommended the equieffective dose (EQDX) to be the standard quantity for biological response modeling for ionizing radiations ^21,22, we derive formulae for calculating EQDX for combinations of multiple radiation types.

3. Methods

3.1. Bioeffect modeling in absorbed dose space

3.1.1. Non-Interacting Model (NIM, existing).

The NIM implements the linear-quadratic (LQ) model and treats the biological consequences of the absorbed doses, from $m$ radiation types, independently for each radiation type $i$ ^20,23:

$$E_{mix}^{NIM}=\sum _{i=1}^m\left({\alpha }_i{D}_i+{\beta }_i{D}_i^2\right),$$ 1

$$P_{mix}^{NIM}\left(D\right)=e^{-E_{mix}^{ZRM}},$$ 2

where $P$ is the survival probability, and $D_i$, ${\alpha }_i$ and ${\beta }_i$ are the absorbed dose and LQ parameters for acute irradiation with radiation type $i$.

3.1.2. Lesion Additivity Model (LAM, existing).

The LAM model¹² works on the assumption that the intermediate lesions, from multiple radiation types, are additive. Based on this, the cellular response to multiple radiation types can be predicted using the responses for individual radiations. However, to achieve this, the response to mixed radiations needs to be solved using numerical techniques. The usage of the model, along with the relevant equations, is shown in the Supplement.

3.1.3. Modified Microdosimetric Kinetic Model (mMKM, existing).

A modified version of the original MKM has been presented by Bertolet et al. ¹⁹. In their work, they have presented a model which takes the LET of the high-LET radiation and the target cell nuclear radius into account to predict the cellular response to a mixture of high- and low-LET radiations. This has been achieved by using a parameter which uses the microdosimetric distribution of the incident radiation and the size of the target nucleus. The equations used in the model are given in the Supplement. The model, as described by Bertolet et al, accommodates a mixture of only two radiations.

3.1.4. Zaider-Rossi Model (ZRM, existing).

In the ZRM ¹¹, the biological effects of a mixture of two radiations are addressed by considering sublesions that are produced due to the interaction of radiation with biological matter. The ZRM assumes that the sublesions interact pairwise to produce lesions and is based on the earlier work of Kellerer and Rossi ²⁴. Pfuhl et al. expanded the ZRM model to include any number of radiations in the mixture and, following their notation ²⁵, the survival probability for mixed radiation can be written using the linear quadratic (LQ) model:

$$\begin{gathered} E_{mix}^{ZRM}\left(D\right)={\alpha }_{mix}D+{\beta }_{mix}{D}^2, \\ {P}_{mix}^{ZRM}\left(D\right)=e^{-E_{mix}^{ZRM}}, \end{gathered}$$ 3

where $D=\sum _{i=1}^m{D}_i$, and ${\alpha }_{mix}$ and ${\beta }_{mix}$ are the LQ parameters of the mixed radiation:

$${\alpha }_{mix}=\sum _{i=1}^m{f_i\alpha }_i,$$ 4

$$\sqrt{{\beta }_{mix}}=\sum _{i=1}^m{f}_i\sqrt{{\beta }_i}.$$ 5

The quantity $f_i=\frac{D_i}{D}$ is the fraction of the total absorbed dose that is delivered by radiation type $i$. The $E_{mix}^{ZRM}$ of a mixed radiation of only two radiation types with doses $D_1$ and $D_2$ is written as:

$$E_{mix}^{ZRM}={\alpha }_1{D}_1+{\beta }_1{D}_1^2+{\alpha }_2{D}_2+{\beta }_2{D}_2^2+2\sqrt{{\beta }_1{\beta }_2}{D}_1{D}_2$$ 6

The last term in equation 6 represents the interaction occurring due to the dependent nature of the effects caused by the two radiation types. Zaider and Rossi describe this term as the interaction between the sublesions produced by the two radiation types. However, it is important to note here that, according to ZRM, if either one of the radiation types is characterized with an LQ parameter ${\beta }_i=0$, the interaction term drops out (i.e. no interaction term for a mixture of low- and high-LET radiations).

3.1.5. Modified ZRM (mZRM, new).

Given that the interaction terms in the ZRM model drop out when one or both ${\beta }_i$ parameters are very small, it is apparent that the ZRM may perform poorly when predicting the biological response to mixed radiations when one or more of the radiations produces a high-LET type survival curve (i.e. ${\beta }_i~0$). Here we investigate an empirical approach to the interaction term that retains interaction terms even when a ${\beta }_i=0$. This approach uses the arithmetic mean $\left(\frac{\left({\beta }_1+{\beta }_2\right)}{2}\right)$ instead of the geometric mean ($\sqrt{{\beta }_1{\beta }_2}$) for modeling responses to mixed radiations. We recognize that this approach does not have a mechanistic basis, yet we point out that none of the aforementioned models are truly mechanistic in that they do not account for the complex myriad of molecular responses induced by ionizing radiation. In that sense, they too are largely empirical. Following this argument, a modified ZRM (mZRM) with radiation types interacting pairwise can be written as:

$$E_{mix}^{mZRM}={\alpha }_{mix}^{mZRM}D+{\beta }_{mix}^{mZRM}{D}^2,$$ 7

where ${\alpha }_{mix}^{mZRM}$ is given by equation 4, and

$${\beta }_{mix}^{mZRM}=\frac{1}{D^2}\left[\sum _{i=1}^m{\beta }_i{D}_i^2+\sum _{i<j}{\left({\beta }_i+{\beta }_j\right)D}_i{D}_j\right]$$ 8

where $i,j=\mathrm{1,2}\dots m$ are the indices denoting the radiation type and $D_1,D_2,\dots \dots ,D_m$ are the absorbed doses from each radiation type.

A further generalized version of equation 8 which accounts for dose rate effects is given in the Appendix. Furthermore, the expansion of equation 8 for three radiation types is also given in the Appendix.

3.2. Bioeffect modeling in EQDX space

3.2.1. Conventional Formulae for EQDX (existing).

The EQDX formalism has been used for bioeffect modeling, especially in external beam therapy, to estimate the effect of a given radiation treatment plan (test plan) relative to a predetermined reference treatment plan ²¹. The usual choice for the reference treatment plan has been acute 2-Gy fractions of megavoltage X-rays because of its relevance to the clinic. As per ICRU recommendations ²¹, the EQDX of a test treatment plan with fraction size $d$ and a total absorbed dose from low-LET radiation, $D$, can be written, following the Withers formula ²⁶, as:

$${\mathrm{E}\mathrm{Q}\mathrm{D}X}_{\frac{\alpha }{\beta }}=D\frac{d+\frac{{\alpha }_R}{{\beta }_R}}{X+\frac{{\alpha }_R}{{\beta }_R}}.$$ 9

where, $\alpha ={\alpha }_R$ and $\beta ={\beta }_R$ are the LQ parameters for acute irradiation, $X$ is the fraction size of the reference radiation treatment plan, and ${\alpha }_R$ and ${\beta }_R$ are the LQ parameters of the reference plan. Also, when $D$ is delivered in a single fraction, the above formula is written as:

$${\mathrm{E}\mathrm{Q}\mathrm{D}X}_{\frac{\alpha }{\beta }}=D\frac{D+\frac{{\alpha }_R}{{\beta }_R}}{X+\frac{{\alpha }_R}{{\beta }_R}}$$ 10

It is important to note that, in both equations 9 and 10 above, the LQ parameters for the test radiation are assumed to be that of the reference radiation.

When the test radiation produces a high-LET type response ($\beta =0$), EQDX is written as ^22,27:

$${\mathrm{E}\mathrm{Q}\mathrm{D}X}_{\frac{\alpha }{\beta }}=D_H\frac{\left(\frac{\kappa }{{\beta }_R}\right)}{\left(X+\frac{{\alpha }_R}{{\beta }_R}\right)}.$$ 11

Here, and $D_H$ is the absorbed dose of the high-LET test radiation, and $\kappa$ is the LQ-model linear-parameter (i.e. $\alpha$) that characterizes the biological response to acute irradiation with high-LET radiation.

When the test radiation is a variable, low dose-rate, low-LET radiation, the EQDX can be written as a function of $D$, the LQ parameters of the reference radiation, and the Lea-Catcheside time-dose $G$-factor ($G$).

$${\mathrm{E}\mathrm{Q}\mathrm{D}X}_{\frac{\alpha }{\beta }}=D\left(t\right)\left(\frac{G\left(t\right)D\left(t\right)+\frac{{\alpha }_R}{{\beta }_R}}{X+\frac{{\alpha }_R}{{\beta }_R}}\right).$$ 12

The $G$-factor can also be written as a function of the relative effectiveness (RE) ^28,29. The RE for an exponentially increasing and decreasing dose rate has been derived by Howell et al. ³⁰. An analytical form for the $G$-factor is given in equation 9 in ³¹ for an exponentially increasing and decreasing dose rate function. The $G$-factor accounts for the DNA repair that takes place during low dose rate, low-LET radiation. It should be used with caution, however, as Solanki et al. ³¹ have demonstrated that functional form for the $G$-factor derived in their work failed to satisfactorily model their experimental results for the exponentially increasing and decreasing dose rates.

3.2.2. Generalized EQDX formula for all radiation types (new).

One of the features of equations 9 and 10 is that, for low-LET radiations, the LQ parameters for acute delivery of the reference radiation are used for both the reference treatment plan and the test treatment plans. When applied to other low-LET test radiation treatment plans, the test and reference radiations are assumed to have the same LQ parameters. As a result, these equations would produce the same EQDX value for two test radiations even when they have dissimilar LQ parameters. In some circumstances, this may be a less than desirable outcome. In contrast, equations 11 and 12 have a parameter that is characteristic of the test radiation ($\kappa$ in equation 11 and $G\left(t\right)$ in equation 12). Therefore, treatment plans with different high-LET radiations (i.e. different $\kappa$) will appropriately produce different values of EQDX. The same is true for low-dose-rate treatment plans with different $G\left(t\right)$, although it still assumes that the LQ parameters are the same. Therefore, to set the stage for modeling of mixed radiation treatments such as those in radiopharmaceutical therapy and space radiation, a generalized form for EQDX that accounts for the LQ characteristics of the test radiation treatment plan needs to be formulated.

Consider a test radiation with LQ parameters ${\alpha }_1$ and ${\beta }_1$, and a reference radiation with LQ parameters ${\alpha }_R$ and ${\beta }_R$ for a particular cell line. Further assume that the reference radiation is given in fractions of $X$ Gy. The survival probability of the cells to an acute bolus of the test radiation is written as:

$$P_1=e^{-\left({\alpha }_1{D}_1+{\beta }_1{D}_1^2\right)}.$$ 13

Furthermore, when the reference radiation is delivered as a fractionated dose regimen with fraction size $X$, assuming total recovery between fractions, the survival probability can be written as ^32,33:

$$P_R=e^{-n_{R}X\left({\alpha }_R+{\beta }_{R}X\right)},$$ 14

where $n_R$ is the number of fractions and $n_{R}X=D_R$ is the total absorbed dose from the reference radiation treatment plan. For equieffect,

$${\alpha }_1{D}_1+{\beta }_1{D}_1^2=n_{R}X\left({\alpha }_R+{\beta }_{R}X\right)=D_R({\alpha }_R+{\beta }_{R}X).$$ 15

Hence,

$${\mathrm{E}\mathrm{Q}\mathrm{D}X}_{{\alpha }_R,{\beta }_R,{\alpha }_1,{\beta }_1}=D_R=\frac{{\alpha }_1{D}_1+{\beta }_1{D}_1^2}{\left({\alpha }_R+{\beta }_{R}X\right)}.$$ 16

Equation 16 is independent of whether the effect of the test radiation is high-LET or low-LET.

Although the generalized formalism given in equation 16 is applicable to all types of treatment plans (acute high-LET or low-LET; low-dose rate low-LET; and fractionated), further explanation is needed to show how this is applied to a fractionated plan. Since the survival probability for a fractionated plan can be written as:

$$P=e^{-n_{f}d\left({\alpha }_1+{\beta }_{1}d\right)},$$ 17

where $n_f$ and $d$ are the number of fractions and the fraction size, respectively, and ${\alpha }_1$, ${\beta }_1$ are the LQ parameters for an acute bolus of radiation. equation 17 can be rearranged as:

$$P=e^{-D_1\left({\alpha }_1+{\beta }_{1}d\right)}=e^{-{\alpha }_1{D}_1-{\beta }_1{D}_1^2/n_f},$$ 18

where

$$D_1=n_f\times d.$$ 19

Equation 18 can be re-written in terms of a new set of LQ parameters as:

$$P=e^{-{\alpha }_f{D}_1-{\beta }_f{D}_1^2}.$$ 20

where

$${\alpha }_f={\alpha }_{1}and{\beta }_f=\frac{{\beta }_1}{n_f}.$$ 21

In the limit $d\to 0$ (small fraction sizes with a given total absorbed dose), $n_f\to \infty$ and ${\beta }_f\to 0$. Therefore, $\underset{n_f\to \infty }{\lim }P=e^{-{\alpha }_1{D}_1}$. Similarly in the limit $d\to D_1$ (acute bolus), $n_f\to 1$ and ${\beta }_f\to {\beta }_1$. Therefore, $\underset{n_f\to 1}{\lim }P=e^{-{\alpha }_1{D}_1-{\beta }_1{D}_1^2}$, which is expected for an acute bolus of radiation. Therefore, for a fractionated plan with fixed fraction size, the LQ parameter set given by equation 21 can be used as needed in the equations that follow when calculating the EQDX for a combination of multiple radiation types.

3.2.3. EQDX formalism for mixed radiations of two types (new).

When multiple radiation types are involved, it is important to formulate a methodology of combining their effect in the EQDX space. That is, one should be able to convert the absorbed dose from each radiation type to EQDX individually, and then combine their EQDXs in a proper way to get the same effect as in the absorbed dose space. The total EQDX for two radiation types within the ZRM, ${\mathrm{E}\mathrm{Q}\mathrm{D}X}^{\mathrm{Z}\mathrm{R}\mathrm{M}}\left(total\right)$, can be expressed in the following way.

Consider two radiation types acting simultaneously on a particular cell with absorbed doses $D_1$ and $D_2$. Then the EQDXs for each of the two radiation types are:

$$\mathrm{E}\mathrm{Q}\mathrm{D}X1=\frac{{\alpha }_1{D}_1+{\beta }_1{D}_1^2}{\left({\alpha }_R+{\beta }_{R}X\right)}$$ 22

and

$$\mathrm{E}\mathrm{Q}\mathrm{D}X2=\frac{{\alpha }_2{D}_2+{\beta }_2{D}_2^2}{\left({\alpha }_R+{\beta }_{R}X\right)}$$ 23

Adding equations 22 and 23:

$$\mathrm{E}\mathrm{Q}\mathrm{D}X1+\mathrm{E}\mathrm{Q}\mathrm{D}X2=\frac{{\alpha }_1{D}_1+{\beta }_1{D}_1^2}{\left({\alpha }_R+{\beta }_{R}X\right)}+\frac{{\alpha }_2{D}_2+{\beta }_2{D}_2^2}{\left({\alpha }_R+{\beta }_{R}X\right)}$$ 24

For ${\alpha }_1={\alpha }_2=\alpha$ and ${\beta }_1={\beta }_2=\beta$, the above equation reduces to:

$$\mathrm{E}\mathrm{Q}\mathrm{D}X1+\mathrm{E}\mathrm{Q}\mathrm{D}X2=\frac{\alpha \left(D_1+D_2\right)+\beta \left(D_1^2+D_2^2\right)}{\left({\alpha }_R+{\beta }_{R}X\right)}$$ 25

However, if the EQDX for the total absorbed dose from both radiation types $\left(D_1+D_2\right)$, is used, one would get:

$$\mathrm{E}\mathrm{Q}\mathrm{D}X\left(total\right)=\frac{\alpha \left(D_1+D_2\right)+\beta {\left(D_1+D_2\right)}^2}{\left({\alpha }_R+{\beta }_{R}X\right)}$$ 26

From equations 25 and 26, it can be seen that $\mathrm{E}\mathrm{Q}\mathrm{D}X1+\mathrm{E}\mathrm{Q}\mathrm{D}X2$ is missing a term when compared to $\mathrm{E}\mathrm{Q}\mathrm{D}X\left(total\right)$. The additional term in $\mathrm{E}\mathrm{Q}\mathrm{D}X\left(total\right)$ is:

$$\frac{2D_1{D}_2\beta }{\left({\alpha }_R+{\beta }_{R}X\right)}$$ 27

When ${\beta }_1\ne {\beta }_2$, for the model to be self-consistent, $\beta$ can be taken of as the geometric mean of ${\beta }_{1}and{\beta }_2:\sqrt{{\beta }_1{\beta }_2}$. Therefore, the total EQDX for the ZRM can be written as:

$${\mathrm{E}\mathrm{Q}\mathrm{D}X}^{ZRM}\left(total\right)=\mathrm{E}\mathrm{Q}\mathrm{D}X1+\mathrm{E}\mathrm{Q}\mathrm{D}X2+\frac{2D_1{D}_2\sqrt{{\beta }_1{\beta }_2}}{\left({\alpha }_R+{\beta }_{R}X\right)}$$ 28

In the case of mZRM when ${\beta }_1\ne {\beta }_2,\beta$ is taken of as the arithmetic mean of ${\beta }_1$ and ${\beta }_2\left[\frac{{\beta }_1{+\beta }_2}{2}\right]$. In this case, the total EQDX can be written as:

Hence,

$${\mathrm{E}\mathrm{Q}\mathrm{D}X}^{mZRM}\left(total\right)=\mathrm{E}\mathrm{Q}\mathrm{D}X1+\mathrm{E}\mathrm{Q}\mathrm{D}X2+\frac{D_1{D}_2\left({\beta }_1{+\beta }_2\right)}{\left({\alpha }_R+{\beta }_{R}X\right)}$$ 29

Equation 29 can be generalized for $m$ radiation types as:

$$\begin{gathered} {\mathrm{E}\mathrm{Q}\mathrm{D}X}_{{\alpha }_R,{\beta }_R,{\alpha }_i,{\beta }_i}^{mZRM}\left(total\right) \\ =\frac{1}{\left({\alpha }_R+{\beta }_{R}X\right)}\left[\sum _{i=1}^m({\alpha }_i{D}_i+G_i{\beta }_i{D}_i^2)+\sum _{i<j}{\left({G_i\beta }_i+{G_j\beta }_j\right)D}_i{D}_j\right] \end{gathered}$$ 30

where the index $i$, $j$ represent each radiation type. For any radiation type/delivery that does not have a dose-rate effect, one should set the value of $G_i=1$. In equation 30, $D_1$, $D_2,\dots \dots ,D_m$, are the absorbed doses from each radiation type and are given as a single bolus.

Once the $\mathrm{E}\mathrm{Q}\mathrm{D}X\left(total\right)$ is calculated from either the ZRM or mZRM, the survival probability can be calculated as:

$$P_{\mathrm{E}\mathrm{Q}\mathrm{D}X}=e^{-\mathrm{E}\mathrm{Q}\mathrm{D}X\left(total\right)\left({\alpha }_R+{\beta }_{R}X\right)}$$ 31

It can be seen that, for two low-LET radiation types, and when ${\alpha }_1={\alpha }_2={\alpha }_R$ and $G_1{\beta }_1=G_2{\beta }_2=G_R{\beta }_R$, equations 28 and 29 and reduce to:

$$\begin{gathered} \text{EQD}{X}^{ZRM,mZRM}\left(total,2low-LETradiations\right)= \\ \text{EQD}X1+\text{EQD}X2+\frac{2D_1{D}_2{G}_R{\beta }_R}{\left({\alpha }_R+{\beta }_{R}X\right)} \end{gathered}$$ 32

where

$$\mathrm{E}\mathrm{Q}\mathrm{D}X1=\frac{{\alpha }_R{D}_1+G_R{\beta }_R{D}_1^2}{\left({\alpha }_R+{\beta }_{R}X\right)}$$

and

$$\mathrm{E}\mathrm{Q}\mathrm{D}X2=\frac{{\alpha }_R{D}_2+G_R{\beta }_R{D}_2^2}{\left({\alpha }_R+{\beta }_{R}X\right)}$$

4. Results

4.1. Comparison of LAM and mMKM with mZRM and ZRM

The LAM was compared with ZRM and mZRM for predicting cellular response to a simultaneous irradiation from a mixture of two radiation types using the LQ parameters obtained from the PIDE database³⁴. The mixture contained a high-LET type ($\alpha =1.27{\mathrm{G}\mathrm{y}}^{-1}$, $\beta =0{\mathrm{G}\mathrm{y}}^{-2}$) and a low-LET type ($\alpha =0.18{\mathrm{G}\mathrm{y}}^{-1}$, $\beta =0.156{\mathrm{G}\mathrm{y}}^{-2}$), which were identical to the values used for Figure 1C below. The predictions of LAM are closer to those of mZRM than to ZRM (Supplemental Figure 3). However, since the equations involved in LAM need to be solved numerically, its computation times are much longer than those of mZRM or ZRM. The computational time for LAM was about 100 times longer than that for mZRM (Supplemental Section 2.2). The same dataset was used to compare the response predictions from mMKM with that of mZRM. The mMKM model published in Bertolet et al. was used for this comparison and both their “independent” and “combined” versions were compared against mZRM (Supplemental Figure 4). Both versions of the mMKM deviate from expected predictions (i.e. between the low- and high-LET curves) and predictions of mZRM for the mixture of high- and low-LET radiations used here. Furthermore, the mMKM only accommodates a mixture of two radiations and requires knowledge of an additional parameter which depends on the LET of the high-LET radiation and the nuclear radius of the target cells.

Figure 1:

Surviving fraction of cells versus absorbed dose for mixed radiations: A) Experimental data from Fig. 2A in ¹² includes the response of V79 cells to 225-kVp X rays (●) and 31-MeV/u ${}_{10}{}^{20}{Ne}^{10+}$ ions (●) individually, and a 1: 1 mixture of the two radiations (●). B) Experimental data from Fig. 3G in ²⁵ includes the response of HSG cells to 150 kVp X-rays (●) and 48.1 MeV/u ${}_6{}^{12}\mathrm{C}^{6+}$ ions (●) individually, and a 1:2 (${}_6{}^{12}\mathrm{C}^{6+}$:X-ray) mixture of the two radiations (●). In both A and B, the solid blue (───) and red lines (───) are LQ fits to the experimental data. Solid lines represent the predictions for mixed radiations by the mZRM (───), ZRM (───), and NIM (───). C) Cell survival curves based on LQ parameters reported for He cells exposed to ¹³⁷Cs gamma rays (───) and 126-MeV ${}_6{}^{12}\mathrm{C}^{6+}$ ions (LET ~ 22 keV/$\mu$m, ───), and the predicted SF for a 1:1 mixture of the two by the mZRM (───), ZRM (───), and NIM (───). The LQ parameters were obtained from ³⁶ as listed in the PIDE database ³⁴. Statistical significance: A) ZRM: p-value=0.036, mZRM: p-value=0.033. Note that the error bars of the data points are not plotted here as accurate extraction of them using WebPlotDigitizer was difficult.

In view of the increased computation times required by the LAM (albeit minor differences in calculated survival probabilities compared to mZRM) and the lower accuracy of the mMKM for predicting the response to mixed radiations relative to mZRM, the more detailed comparisons below were only conducted for the NIM, ZRM, and mZRM.

4.2. Comparison of predicted effect of mixed radiations in absorbed dose space

The ZRM, mZRM, and NIM were further tested against several experimental datasets available in literature for mixed radiation effects. The data were extracted from the figures in the publications described below using a software tool WebPlotDigitizer ³⁵ and analyzed using an in-house code written in Python. Because the interaction term in the ZRM contains the geometric mean of ${\beta }_1$ and ${\beta }_2,\sqrt{{\beta }_1{\beta }_2}$, this term becomes negligible if either ${\beta }_1$ or ${\beta }_2$ becomes very small. Therefore, the models were tested against experimental datasets in literature where biological responses were reported for combinations of low- and high-LET radiation types. The LQ fitting of individual responses to each radiation type was performed by minimizing least-squares after weighting the data by 1/SF. The statistical significance for predicting the cellular response to mixed radiation for ZRM and mZRM was analyzed by performing a one-way ANOVA test on the fractional residuals from each model. A p-value less 0.05 was considered to be statistically significant.

4.2.1. X ray and neon ion data.

Data were extracted from Fig. 2A in ¹² which shows the response of V79 cells to 225-kVp X rays and 31-MeV/u ${}_{10}{}^{20}{\mathrm{N}\mathrm{e}}^{10+}$ ions/fragments delivered sequentially. These are the experimental data of Ngo et al. ³ where it was stated that the time between irradiations was less than 30 min and the cells were kept at ~4°C between irradiations. Figure 1A plots their surviving fraction data for X rays and ${}_{10}{}^{20}{\mathrm{N}\mathrm{e}}^{10+}$ ions along with their data for a 1:1 mixture of the two. Weighted (1/SF) LQ fits to the X ray (${\alpha }_{Xray}=0.195{\mathrm{G}\mathrm{y}}^{-1}$, ${\beta }_{Xray}=0.013{\mathrm{G}\mathrm{y}}^{-2}$) and ${}_{10}{}^{20}{\mathrm{N}\mathrm{e}}^{10+}$ ion (${\alpha }_{Ne}=0.835{\mathrm{G}\mathrm{y}}^{-1}$, ${\beta }_{Ne}=0{\mathrm{G}\mathrm{y}}^{-2}$) data are also shown along with the ZRM, mZRM, and NIM predictions for the mixture. The ZRM and NIM fall together. The mZRM deviates slightly from them, as well as from the experimental observations, as the total absorbed dose (Figure 1A abscissa) increases. Notably, the deviation of the model from the experimental data is a bit more substantial in the cases of ZRM and NIM. This can be attributed to the fact that the dose-response curve for ${}_{10}{}^{20}{\mathrm{N}\mathrm{e}}^{10+}$ ions is of high-LET type (no shoulder, $\beta ~0$) thereby causing the interaction term of the ZRM to be zero. The fractional absolute residuals of the responses to mixed radiations are shown in the Supplemental Figure 1.

Figure 2:

Surviving fraction of He cells exposed to ¹³⁷Cs gamma rays (───), 126-MeV ${}_6{}^{12}\mathrm{C}^{6+}$ ions (LET ~ 22 keV/$\mu$m, ───), 26-MeV ${}_6{}^{12}\mathrm{C}^{6+}$ ions (LET ~ 75 keV/$\mu$m, ───) and the predicted SF for a (A) 1:1:1 and (B) 2:1:1 mixture of the three from ZRM & NIM (gold line),and mZRM (black line). The LQ parameters were obtained from Suzuki et al. ³⁶ as listed in the PIDE database ³⁴.

4.2.2. X ray and carbon ion data.

Figure 1B shows the surviving fraction of HSG cells following irradiation with 150 kVp X rays and 48.1 MeV/u ${}_6{}^{12}\mathrm{C}^{6+}$ ions individually, and when they are mixed at 1:2 (${}_6{}^{12}\mathrm{C}^{6+}$:X-ray) ratio. The experimental data were extracted from Fig. 3G in ²⁵ and were fitted with a weighted (1/SF) LQ fit (${\alpha }_{Xray}=0.198{\mathrm{G}\mathrm{y}}^{-1}$, ${\beta }_{Xray}=0.028$, ${\alpha }_C=0.596{\mathrm{G}\mathrm{y}}^{-1}$, ${\beta }_C=0.047{\mathrm{G}\mathrm{y}}^{-2}$). Again, the ZRM and mZRM similarly predict the experimental observations for the mixed radiations. However, the NIM model that does not have interaction terms deviates from the experimental observations by a considerable amount.

Figure 3:

Surviving fraction of T-1 kidney cells versus absorbed dose for a mixture of $X$ rays and alpha particles. The experimental data, represented by (●,●,●), correspond to the data in Fig. 3 of Ref. ³⁷. They represent their response to 250-kVp X-rays, alpha particles (peak energy = 2.9 MeV, LET ~ 140 keV/$\mu$m), and a mixture of the two radiation types: A) 11% alpha, 89% $X$ rays; B) 45% alpha, 55% $X$ rays; C) 64% alpha, 36% $X$ rays). The solid blue (───) and red lines (───) are LQ fits to the experimental data. The black and gold lines are predictions for the mixed radiations by the mZRM (───), ZRM & NIM (───). Statistical significance: A) ZRM: p-value=0.036, mZRM: p-value=0.040. B) ZRM: p-value=0.016, mZRM: p-value=0.002. C) ZRM: p-value=0.051, mZRM: p-value=0.032.

4.2.3. LQ parameters from PIDE database.

To illustrate a situation where the difference between the ZRM, mZRM, and NIM is more prominent, surviving fractions for mixed radiations of two types were predicted using sets of LQ parameters from the PIDE database ³⁴. The LQ parameters for HE cells, extracted from the database, were the fit parameters for the original data published in ³⁶. The two radiation types selected were 662-keV gamma rays (from ¹³⁷Cs) and 126-MeV ${}_6{}^{12}\mathrm{C}^{6+}$ ions with an LET of 22 keV/$\mu$m. The LQ parameters were: ${\alpha }_{\gamma }=0.18{\mathrm{G}\mathrm{y}}^{-1}$, ${\beta }_{\gamma }=0.156{\mathrm{G}\mathrm{y}}^{-2}$, ${\alpha }_{ion}=1.27{\mathrm{G}\mathrm{y}}^{-1}$, ${\beta }_{ion}=0{\mathrm{G}\mathrm{y}}^{-2}$. The SF predictions from ZRM, mZRM, and NIM for a 1:1 mixture of the two radiation types, are shown in Figure 1C. Also shown is the product of the individual surviving fraction curves. The SF predictions of ZRM and NIM are about seven-fold higher than mZRM at an mZRM surviving fraction of 10⁻³ and the ZRM and NIM are identical to the product of individual SFs. It is important to note that the predictions of ZRM and NIM versus mZRM deviate as the total absorbed dose increases (Figure 1C).

Figure 2 illustrates the application of the mZRM to experimental data corresponding to the same two radiation types shown in Figure 1C (662-keV gamma rays and 126-MeV ${}_6{}^{12}\mathrm{C}^{6+}$ ions with an LET of 22 keV/$\mu$m) plus 26-MeV ${}_6{}^{12}\mathrm{C}^{6+}$ ions (LET ~ 75 keV/$\mu$m). The LQ parameters for the 26-MeV ${}_6{}^{12}\mathrm{C}^{6+}$ ions are ${\alpha }_{ion}=1.85{\mathrm{G}\mathrm{y}}^{-1}$, ${\beta }_{ion}=0{\mathrm{G}\mathrm{y}}^{-2}$. These values were obtained from the PIDE database ³⁴ and the original data was from Suzuki et al. ³⁶. The ZRM, mZRM and NIM have been used to predict the response to a 1:1:1 mixture (Figure 2A) and a 2:1:1mixture (Figure 2B) of the three radiation types. The ZRM and the NIM have identical predictions in both Figure 2A and Figure 2B because two (out of three) of the LQ $\beta$-parameters are zero. When Figure 1C and Figure 2A are compared, it can be seen that the separation between ZRM and mZRM predictions is reduced in Figure 2A. This is because, in Figure 1C, contribution from low-LET to high-LET is 1:1, whereas in Figure 2A it is 1:2.

4.2.4. X ray and alpha particle data.

Another example, using experimental data, which illustrates the differences among NIM, ZRM and mZRM in predicting the response of mixed-LET radiations with varying high-LET percentages, is shown in Figure 3. The experimental data were obtained from Raju et al. ³⁷. The results indicate that, for all three percentages of alpha particles in the mixture, mZRM predictions more closely approximate experimental observations. Furthermore, both NIM and ZRM have identical predictions for the mixture because one of the two radiation types in the mixture is high-LET (i.e. $\beta =0$). The fractional absolute residuals of the responses to mixed radiations corresponding to these three plots are given in Supplemental Figure 2.

4.3. Comparison of predicted effect of mixed radiations in EQDX space

4.3.1. Application of EQDX to radiations with shouldered SF curves ($\beta \ne 0$).

We validated equation 16 using experimental data in the literature. Figure 4 illustrates the surviving fraction plotted as a function of absorbed dose and EQD2 for the same data obtained from Fig. 3G in ²⁵ (original data is from Furusawa et al. ⁸). The absorbed doses plotted in Figure 4A were converted to EQD2 using the general formula given in equation 16. The general formula given in equation 16 has linearized the surviving fraction data in the EQD2 space even though both the reference radiation (X ray) and the test radiation (C ion) have low-LET type behavior with different $\beta$s. This linearization cannot be achieved if the conventional version of equation 16 equation 10 is used, as it cannot accommodate two different $\beta$s. Multiple low-LET type responses are typical in the context of radiopharmaceutical therapy where different radiation types are involved, and therefore, a formalism which can accommodate such responses is needed. equation 16 provides such a formalism.

Figure 4:

Surviving fraction as a function of A) Absorbed dose, and B) EQD2, for HSG cells irradiated with a single fraction of 150 kVp X-rays (blue circles) and 48.1 MeV/u ${}_6{}^{12}\mathrm{C}^{6+}$ ions (red circles). The EQD2 was calculated using the formula given in equation 16 with ${\alpha }_R=0.198{Gy}^{-1}$, ${\beta }_R=0.028{Gy}^{-2}$ and ${\alpha }_1={\alpha }_C=0.596{Gy}^{-1}$, ${\beta }_1={\beta }_C=0.047{Gy}^{-2}$. Experimental data were from figure 3G in ²⁵ (original data is from Furusawa et al. ⁸). Note that equation 16 linearizes the EQD2-response curves while the absorbed-dose-response curves are shouldered ($\beta \ne 0$).

4.3.2. Application of EQDX for mixed radiations.

Figure 5 illustrates an application of using total EQDX (equations 28 and 29) to calculate the surviving fraction following irradiation with mixed radiations of different LET. The experimental data are what are used in Figure 3 and represent the surviving fraction to a radiation mixture of X rays and alpha particles. equations 28 and 29 were used to calculate the total EQD2 from ZRM and mZRM respectively, and equation 31 was used to calculate the surviving probability (dots) for each. The solid lines for mZRM and ZRM & NIM are the same surviving probability predictions as in Figure 3, that is, they are calculated using absorbed dose rather than from EDQ2. The surviving probability predictions from total EQD2 (black and yellow dots) and from mixed absorbed dose (black and yellow solid lines) are exactly the same, highlighting the self-consistency of the model. It can be seen from all three plots in Figure 5 that the predictions from mZRM agree more closely with experimental observations than both ZRM and NIM. Furthermore, the NIM calculations demonstrate that the EQDX from each radiation type cannot just be added to obtain the total EQDX of the mixture, rather, the interaction terms for EQDX need to be considered. Finally, when calculating the EQDX interaction terms, mZRM results in better agreement with experimental observations compared to ZRM.

Figure 5:

Surviving fraction of T-1 kidney cells predicted from EQDX and plotted as a function of absorbed dose from mixtures of $\mathrm{X}$ rays and alpha particles. Experimental data are from Raju et al. ³⁷ and are the same those used in Figure 3. The surviving fractions resulting from irradiation with the mixed radiations are depicted for the different percentages of absorbed dose delivered with each radiation type: A) 11% alpha, 89% $\mathrm{X}$ rays; B) 45% alpha, 55% $\mathrm{X}$ rays; C) 64% alpha, 36% $\mathrm{X}$ rays. Black and yellow dots are SF calculations from EQDX and solid lines are those from mixed absorbed dose.

5. Discussion

The focus of this study was to identify a computationally efficient and accurate bioeffect model that is suited for predicting responses to mixed radiations and compatible with the MIRDcell software application ²⁰. After first eliminating competing models, our approach was twofold: 1) modify the ZRM model to overcome some of the limitations of the model in predicting the biologic response to a mixture of a high- and low-LET radiations, and 2) generalize the ZRM and mZRM models within the context of the EQDX. Our modification to the ZRM is analytical and empirical in nature, and is supported by analyses that demonstrate better matches between model and experimental data while satisfying required mathematical and biophysical boundary conditions. The mZRM model is not based on a theoretical and classical biophysical mechanism of cell inactivation. This approach contrasts the direct-effect-mechanism bioeffect models such as the LEMs, which relate DNA double strand break distributions to local energy deposition. Such models require a substantial amount of computation time and resources and only consider the classical direct and indirect actions of ionizing radiation on DNA in the cell nucleus. Therefore, only a subset of the complex array of biological and physical variables that affect cellular responses to ionizing radiation are considered. Other biological factors such as radiation-induced bystander effects are not considered in these models ^38,39.

Both ZRM and mZRM imply that the order in which different radiations are applied is inconsequential. Ngo et al. ⁴⁰ have shown that, although the sequence of mixed exposure to X rays and neon ions was immaterial at lower absorbed doses of X rays (5 Gy), the effect of the mixed radiation regimen was greater when higher absorbed doses of X rays (8 Gy) were given before the neon ions. Higgins et al. have also reported different survival behaviors for simultaneous and sequential irradiation from mixed-LET radiations ⁴¹. This time factor is not explicitly incorporated in our current version of mZRM; dose protraction effects such as repopulation are not explicitly modeled within mZRM. Dose protraction was included by Zaider and Rossi in the ZRM ¹¹. Furthermore, McNally et al. ⁴ has reported an oxygen effect on the biologic response of V79 cells simultaneously irradiated by mixed-LET radiations (X rays and neutrons). These oxygen effects are also not taken into account within the mZRM. Since both ZRM and mZRM are based on the LQ formalism, it is important to be cognizant of some of the shortcomings of the LQ model. The LQ model fails to explain the increased radiosensitivity at low (~ up to 0.5 Gy) absorbed doses ^42,43. An alternative model, which can predict the hypersensitivity at low absorbed doses has been discussed by Lind et al. ⁴⁴. Furthermore, one should be aware that the mZRM can only be used to predict the biologic response to mixed-LET radiations when responses to the individual radiations are known.

The general EQDX formalism presented here requires knowledge of the LQ parameters for each radiation type. These are obtained using experimental techniques (i.e. colony forming assay); the PIDE database is a valuable resource for such information ³⁴. One of the main objectives of this work is to formalize the ZRM and mZRM within the EQDX context, that is to derive a methodology of calculating the EQDX for mixed irradiations within this framework. It is important to note that the term “radiation type” used within the analyses presented here, denotes a source of radiation with a unique LQ parameter set. “Radiation types” can apply to different radiation types with different LQ parameters (e.g. alpha, beta and gamma radiation), or to the same radiation type with different LQ parameters (e.g. Auger electrons). Due to their extremely short range and their high-LET type nature within that range, Auger electrons have a greater biological effect (i.e. larger linear LQ parameter) when the decay occurs in the cell nucleus as opposed to a decay in the cytoplasm. The LQ parameters that are obtained when an Auger electron emitter is localized in these two source regions are very different ⁴⁵. These differences are accounted for in a complex LQ parameter set used in our MIRDcell V3 bioeffect modeling software ²⁰.

When calculating EQDX values for combinations of multiple radiation types, the individual EQDX values are summed along with additional interaction terms. This is similar to the presence of interaction terms for mZRM in the absorbed-dose space. Bodey et al. ^46,47 have presented a formalism to calculate EQD0 (i.e. biologically effective dose (BED)) from a combination of radiopharmaceutical therapy and fractionated external beam therapy with X rays by considering the interaction between the DNA damage caused by each process. However, the model’s use of a single set of LQ parameters for both modalities makes it applicable to low-LET radiopharmaceutical therapy only. In fact, when $\beta \to 0$, all of the interaction terms go to zero. Furthermore, in the limit of time between fractions $t_f\to \infty$ (i.e. long relative to repair half-time), the interaction terms also go to zero. In contrast the generalized ${\mathrm{E}\mathrm{Q}\mathrm{D}X}^{mZRM}$ (equation 30) retains interaction terms that accommodate combinations of low-LET radiation schedules, combinations of low-LET and high-LET radiations, and combinations of high-LET radiations. Interestingly, the current recommendation for clinical practice to stay below the dose limits for organs at risk is to simply sum the individual EQDX values for each treatment regimen. That is, EQDX values from multiple radiation regimens are added together without interaction terms ⁴⁸. This was justified by assuming the different regimens are well spaced in time. Nonetheless, one should be cognizant of possible undesired effects when chronic irradiation regimens, such as those in radiopharmaceutical therapies, are combined with other regimens such as external beam therapies. Such situations could lead to instances where simultaneous irradiations occur and the interaction terms may be of significant importance when calculating EQDX for the mixed regimens.

6. Conclusion

The mZRM is presented here which can be used to predict the biological response to a mixture of multiple high- and low-LET radiation types. Such radiation mixtures are commonplace in radiopharmaceutical therapy, carbon ion therapy, and during deep space exploration. Furthermore, methods for the calculation of EDQX for mixed radiation types has been presented for both ZRM and mZRM, illustrating the importance of the interaction terms in the latter. When predicting the biological response to multiple high- and low-LET radiations, the mZRM results match the experimental observations more closely than those of ZRM and the NIM. Therefore, the mZRM is an attractive alternative to more computationally intensive dose-response models, particularly when the models are implemented within complex dosimetry environments that require CPU/GPU-intensive calculations (e.g. multicellular clusters).

8. Appendix

8.1. mZRM for mixed radiations with dose rate effects

The mZRM can be further generalized for dose rate effects by including Lea-Catcheside G-factors for each relevant radiation type in the LQ response. This accounts for the dose rate effect that can arise in chronic irradiation with low-LET radiations such as X rays and beta particles. In these instances, the ${\beta }_{mix}^{mZRM}$ is written as follows:

$${\beta }_{mix}^{mZRM}=\frac{1}{D^2}\left[\sum _{i=1}^m{G}_i{\beta }_i{D}_i^2+\sum _{i<j}\left({G_i\beta }_i+{G_j\beta }_j\right)D_i{D}_j\right],$$ 33

where $G_i$ is the G-Factor for the chronic irradiation regimen of the $i^{th}$ radiation type. G-factors for exponentially increasing and decreasing dose rates are provided by Solanki et al. ³¹, and a general form is provided by Hobbs et al. ²⁷. For any radiation type/delivery $i$ that does not have a dose rate effect, $G_i=1$.

8.2. ZRM and mZRM expansions for multiple radiation types

8.2.1. ZRM for 3 radiations.

$$\sqrt{{\beta }_{mix}}=\sum _i{f}_i\sqrt{{\beta }_i}=\frac{D_1}{D}\sqrt{{\beta }_1}+\frac{D_2}{D}\sqrt{{\beta }_2}+\frac{D_3}{D}\sqrt{{\beta }_3},$$ 34

$${\beta }_{mix}^{ZRM}=\frac{1}{D^2}\left({{\beta }_{1}D}_1^2+{{\beta }_{2}D}_2^2+{{\beta }_{3}D}_3^2+{2D}_1{D}_2\sqrt{{\beta }_1{\beta }_2}+{2D}_1{D}_3\sqrt{{\beta }_1{\beta }_3}+{2D}_2{D}_3\sqrt{{\beta }_2{\beta }_3}\right),$$ 35

8.2.2. mZRM for 3 radiations.

For three radiation types, m=3. Therefore, from equation 8,

$${\beta }_{mix}^{mZRM}=\frac{1}{D^2}\left({{\beta }_{1}D}_1^2+{{\beta }_{2}D}_2^2+{{\beta }_{3}D}_3^2+D_1{D}_2\left({\beta }_1+{\beta }_2\right)+D_1{D}_3\left({\beta }_1+{\beta }_3\right)+D_2{D}_3\left({\beta }_2+{\beta }_3\right)\right)$$ 36

In a special case where ${\beta }_1={\beta }_2={\beta }_3=\beta$, equation 36 simplifies to:

$${\beta }_{mix}^{mZRM}=\frac{1}{D^2}\left(\beta \left(D_1^2+D_2^2+D_3^2\right)+2\beta D_1{D}_2+{2\beta D}_1{D}_3+{2\beta D}_2{D}_3\right)$$ 37

$${\beta }_{mix}^{mZRM}=\frac{\beta }{D^2}\left(D_1^2+D_2^2+D_3^2+2D_1{D}_2+{2D}_1{D}_3+{2D}_2{D}_3\right)$$ 38

$${\beta }_{mix}^{mZRM}=\frac{\beta }{D^2}{\left(D_1+D_2+D_3\right)}^2$$ 39

$${\beta }_{mix}^{mZRM}=\beta$$ 40