A Procedure to Estimate Dose and Time of Exposure to Ionizing Radiation from the γ-H2AX Assay ^†

Abstract

Accurately estimating the radiation dose received by an individual is essential for evaluating potential damage caused by exposure to ionizing radiation. Most retrospective dosimetry methods require the time since exposure to be known and rely on calibration curves specific to that time point. In this work, we introduce a novel method tailored to the γ-H2AX assay, which is a protein-based biomarker for radiation exposure, that enables the estimation of both the radiation dose and the time of exposure within a plausible post-exposure interval. Specifically, we extend calibration curves available at two distinct time points by incorporating the biological decay of foci, resulting in a model that captures the joint dependence of foci count on both dose and time. We demonstrate the applicability of this approach using both real-world and simulated data.

1. Introduction

Exposure to potentially harmful ionizing radiation is a common yet often underappreciated public health concern. Individuals may be exposed in everyday contexts, including occupational settings—such as among nuclear workers, medical staff, and radiographers—and as patients undergoing diagnostic or therapeutic procedures for conditions like cancer, orthopedic injuries, or cardiovascular disease. Less frequently, exposure may occur on a larger scale, for example, during nuclear accidents. Under either risk scenario, following a potential (over-)exposure to ionizing radiation in a radiological accident or incident, one needs retrospective dosimetry techniques to estimate the dose of the exposure to inform medical intervention and/or triage of patients for treatment. This is particularly important during mass casualty incidents to inform on appropriate medical interventions and to identify individuals most in need of treatment due to higher dose exposures. Current approaches for retrospective dosimetry can be categorized into physical (such as based on ceramic chips in mobile phones [1]) and biological dosimetry, also known as biodosimetry, which is commonly considered to include the use of physiological, chemical or biological markers of exposure of human tissues to ionizing radiation for the purpose of reconstructing doses to individuals or populations [2]. The work presented here applies to a biological dosimetry technique.

Exposure to ionizing radiation induces DNA double-strand breaks (DSBs), which can lead to chromosome aberrations if not correctly repaired during DNA replication [3]. The dicentric chromosome assay (DCA) is the well-established gold standard for accurate dose estimation, relying on the quantification of dicentric chromosomes in peripheral blood lymphocytes [4,5,6,7]. These abnormal chromosomes arise predominantly from ionizing radiation and serve as a highly specific biomarker for radiation exposure, which has been shown to be robust to inter-individual [8] and inter-laboratory variations [9]. The method is the recommended by the IAEA [10] which also provides detailed protocols for sample processing, culture, metaphase preparation, and statistical analysis tailored to emergency and routine biodosimetry. The assay has been used to provide evidence in occupational adjudication and regulatory decisions [11] and is considered robust for use in legal proceedings.

However, there are also some drawbacks of the dicentric assay such as being costly [12], and rather too slow for emergency scenarios, traditionally requiring four to five days [13] after the collection of samples to produce the dose estimation. The time taken can be reduced using automatic or semi-automatic evaluation systems; however, the dose estimate provided is less accurate than that from manual scoring, and currently more appropriate for triage than exact dose estimates [14,15]. There has also been recent progress in fully automizing the evaluation pipeline including the statistical analysis leading to the dose estimation [16]; the degree of adoption of such methods by dosimetry labs is yet less clear.

Despite the undisputed advantages and high degree of validation and harmonization of the dicentric assay, continued research into methods which can reliably quantify the degree of exposure within the first one or two days following the incident appears highly desirable. Promising candidates for such alternative biomarkers include protein-based assays that detect the presence of DNA damage response proteins at damage sites, such as γ-H2AX and 53BP1 [17]. These assays quantify the cellular response to ionizing radiation by measuring the accumulation of such proteins at sites of DNA double-strand breaks, providing a sensitive indicator of damage recognition and repair activity. In this work, we focus on γ-H2AX as a radiation biomarker.

When a cell is exposed to ionizing radiation and DSBs occur, a type of protein, the H2AX histone, is expressed to coordinate repair of DNA [18] and, in this process, is phosphorylated to become γ-H2AX [19]. The clusters of γ-H2AX proteins that form at the location of each DSB break are referred to as foci when visualized under fluorescent microscopy following immunofluorescent staining, and can be detected and counted minutes after ionizing radiation exposure [20]. The γ-H2AX foci reach the highest concentration at around 1 h after exposure, and then slowly decay until almost disappearing at around 96 h after exposure [21,22].

The γ-H2AX assay which uses γ-H2AX foci as the biomarker is still a relatively new method in the biodosimetrical toolbox, but has great potential as it is able to produce a dose estimate much faster than the DCA, which is particularly useful for triage in emergency response scenarios where many individuals may be affected. Common approaches utilizing this biomarker involve collecting a blood sample [22,23] from healthy, unexposed individuals and exposing them ex vivo to design doses of ionizing radiation (e.g., 0.5/1/2/4 Gy), then counting the number of γ-H2AX foci in a fixed number of lymphocyte cells in each sample at specified time points post-exposure, such as 1, 2, 4, 24 h [13,24]. We will hence refer to the mean count (which corresponds to the average or the expectation depending on the context) of γ-H2AX foci in a sample at a time after exposure as ‘yield’, and denote it as y. Linear calibration curves are fitted that describe the functional relationship between yield and design doses at each time after exposure.

These approaches, while having advantages over DCA such as allowing the dose estimation to take place much sooner (within 4–5 h) after exposure [25], are, however, still limited by their time dependence. For instance, if samples from a potentially irradiated individual were taken say 6 h after exposure, the lab would have either to resort to sub-optimal calibration curves at non-matching time points, or perhaps take another sample at a later time for which a calibration curve is available. Ref. [23] described this “a logistic problem difficult to solve”. Dose–time surface models have been posed as a solution to this problem, however, still require time since exposure to be known in order to provide an accurate dose estimate [23].

The situation would of course be even more difficult if this individual is unsure about the time of the possible exposure. To address such issues, we present here a new method that allows us to estimate ionizing radiation dose without knowing exact time since exposure, as well as estimate the time since exposure if needed. To do this, we take existing calibration curves and generalize them using the decay mechanism of γ-H2AX foci to build a model that describes the functional relationship between yield, dose, and time since exposure. To apply this model, we simply take γ-H2AX foci count measurements twice, i.e., at different time points, to infer dose and time since exposure.

We explain the components of this modeling approach in Section 2. We demonstrate the efficiency of the proposed approach through simulation studies in Section 3. We provide two case studies with real data from two different laboratories in Section 4. We discuss limitations of the proposed approach in Section 5 before we finish this exposition with a Conclusion in Section 6. This article is a revised and expanded version of a paper entitled “Estimating dose and time of exposure from a protein-based radiation biomarker”, which was presented at the International Workshop on Statistical Modelling, Durham, UK, 15–19 July 2024 [26].

2. A Model Involving Dose and Time

The idea of our approach in its essence is to generalize the traditional, “fixed-time” calibration curves so that they become a function of dose and time. The decay mechanism of γ-H2AX foci is used to achieve this. We denote D as the dose of ionizing radiation received in Gy, t as the time after ionizing radiation exposure in hours and $y_t$ as the yield (the averaged foci count per cell) at time t hours after ionizing radiation exposure. The number of cells being used for foci counting in each sample is denoted as n. We proceed with explaining the components of the approach and the construction of the model.

2.1. Calibration Curves

Recall that, as mentioned in the Introduction, past literature describes the functional relationship between yield and dose at each time after exposure by fitting a linear calibration curve [13,24]. That is, one has, for fixed and known time since exposure t,

$$y_t=a_t+b_{t}D,$$

(1)

where $a_t$ and $b_t$ are parameters to be determined, as the basic calibration curve model.

We assume for our methodology that two such calibration curves, at times $t=t_1$ and $t=t_2\ne t_1$, are available. These curves will usually have been estimated from laboratory data, as described in the Introduction, by a generalized linear modeling technique such as a ‘quasi-Poisson’ model with identity link [24]. A technical note may here be in order: Since we are interested in modeling the yield (which is an average), but (quasi-)Poisson models operate on counts, a model of type

$$n{y}_t=a_{t}n+b_{t}nD$$

is usually fitted, where then $nD$ and n act as covariates [27]. The model can be fitted for instance using the glm function in R with option family = quasipoisson (link = “identity”). It is important to emphasize that the calibration data sets that give rise to the two calibration curves should be obtained under identical conditions (radiation type, software filter settings, etc.).

2.2. Decay Mechanism

As mentioned in the Introduction, the number of γ-H2AX foci decreases between 1 h and 96 h after exposure [21,22]. We refer to this as γ-H2AX foci decay. A previous study [22] shows that the yield follows

$$y_t=D(11.92e^{-0.3495t}+3.552e^{-0.01843t})$$

(2)

between 1 h and 96 h after exposure. There are several pieces of information in this equation: The general character of this mechanism, which takes biexponential form, and then specific numerical values given inside the equation. At this moment of the process, we put these specific numerical values aside, but assert that the general shape of this decay is a physical property and should presumably hold true across a wide range of individuals and labs. Therefore, we extract the form of the decay mechanism to obtain

$$y_t=D(A{e}^{-ut}+B{e}^{-vt})$$

(3)

that we refer to as the basic decay mechanism model. Specifically, $A,B$ and $u,v$ are positive parameters. All four of these parameters will be determined later.

2.3. Combining Calibration Curves with the Decay Mechanism

Now, let us reconsider model (1), where $b_t$ is closely related to the decay mechanism as it is multiplied by D. Therefore, we assume that $b_t$ changes according to the basic decay mechanism model (3). However, $a_t$, i.e., the background yield of foci when no exposure has taken place, should have very little to do with the decay mechanism since ideally we should be seeing a zero yield when $D=0$, and so changes in $a_t$ across time are likely driven by other factors in play in the environment such as laboratory conditions, individual differences [20], etc. The time dependency of $a_t$ was also noted by the authors of [23], who attributed it to ice-keeping procedures after lymphocyte isolation. Here, we make the assumption that $a_t$ changes linearly with t (in practice, this relationship may or may not be linear but this does not really matter—we are simply using the most simple way of linking the two calibration curves so that the combined curve is consistent with both of them). Hence, the combined calibration curve takes the form of

$$y_t=a+bt+D(A{e}^{-ut}+B{e}^{-vt})$$

(4)

to which we refer as the ‘dose–time model’ henceforth.

2.4. Determining Model Parameters

To acquire the values of the parameters $a,b,A,B,u,v$, we make a conceptual distinction between the parameters $u,v$, which can reasonably be considered to be determined by the nature of the physical decay mechanism, and the parameters $a,b,A,B$ which relate to the magnitude of foci observed and hence are likely to be lab-specific (e.g., depending on the filter settings of the foci detection software or the manual scoring criteria defined by the lab [25]). Following this rationale, we can bring back the parameters u and v from the source Equation (2) and fix them to $u=0.3495$ and $v=0.01843$ from (2).

We solve for parameters $a,b,A,B$ by equating the model with the relative terms of two calibration curves. Note again that these can be any two calibration curves as long as they are acquired with the same laboratory conditions at different times.

Specifically, supposing that we have calibration curves at times $t_1$ and $t_2$, that is,

$$y_{t_1}=a_{t_1}+b_{t_1}D$$

and

$$y_{t_2}=a_{t_2}+b_{t_2}D.$$

Then, by matching with the shape of (4), we would equate terms, yielding a system of four equations for the four parameters $a,b,A,B$:

$$a+b{t}_1=a_{t_1}$$

$$a+b{t}_2=a_{t_2}$$

$$A{e}^{-0.3495t_1}+B{e}^{-0.01843t_1}=b_{t_1}$$

$$A{e}^{-0.3495t_2}+B{e}^{-0.01843t_2}=b_{t_2}$$

These equations can be analytically solved (see Appendix A), and the resulting values for $a,b,A,B$ can be plugged back into (4) along with the determined values of u and v.

2.5. Estimating Dose and Time

Let us firstly consider the ‘trivial’ case in which the exposure time, say T, is known. In this case, we take one sample from the individual, yielding a patient data yield, $y_T$. The only unknown in the model would be D, which we can easily solve for as follows:

$$D={\displaystyle \frac{y_T-a-bT}{A{e}^{-uT}+B{e}^{-vT}}}.$$

However, the matter of interest in this paper is the event where the time of exposure is unknown. In this scenario, we take two samples at two different times. It is important that the laboratory conditions under which these samples are scored are still as similar as possible to the conditions under which the calibration data were obtained. We denote $T_1$ and $T_2$ as the time in hours after ionizing radiation exposure when we take the first and second sample, respectively, and define $\Delta =T_2-T_1$. Hence, we would not know $T_1$ or $T_2$, but we would know $\Delta$. It is also emphasized that $T_1$ and $T_2$ will usually not be the same as $t_1$ and $t_2$ introduced earlier, therefore, the explicit change in notation. From the dose–time model (4), we then have the two following equations:

$$y_{T_1}=a+b{T}_1+D(A{e}^{-u{T}_1}+B{e}^{-v{T}_1})$$

and

$$y_{T_1+\Delta }=a+b(T_1+\Delta )+D(A{e}^{-u(T_1+\Delta )}+B{e}^{-v(T_1+\Delta )})$$

where we have two unknowns, namely $T_1$ and D, and two equations in order to estimate them. However, this time we can not as easily find an explicit expression of $T_1$ and D. Therefore, we employ numerical methods to acquire solutions from this system of two equations. Specifically, for the applications in this paper, we consider using the Nsolve function of Mathematica, which is a solver for algebraic systems which provides all (real or complex) solutions of that system, and the nleqslv solver available in the nleqslv package [28] in R 4.4.3, which is a numerical solver of non-linear systems which just finds a single root depending on the starting point, which hence needs to be supplied. We suggest to use $T_1=1$ and $D=0$ generally for this purpose. Both solvers tend to work well and give the same unique real solutions, with some exceptions which are indicated later in the paper.

2.6. Example

Calibration. To illustrate how to obtain the parameters involved in the proposed model (4), we consider an example from [24]. Here, a pair of calibration curves was obtained from laboratory data with 7 design doses spanning the range from 0 Gy to 4 Gy, with samples taken 1 h and 24 h after exposure. All yields in this experiment were computed from samples consisting of $n=500$ cells. Specifically, the reported calibration curves were, 1 h after exposure:

$$y_1=0.131+12.559D$$

and 24 h after exposure:

$$y_{24}=0.179+1.937D.$$

Determine model parameters. Equating the relative terms as illustrated at the end of Section 2.4, we have

$$a+b{t}_1=0.131$$

$$a+b{t}_2=0.179$$

$$A{e}^{-0.3495t_1}+B{e}^{-0.01843t_1}=12.559$$

$$A{e}^{-0.3495t_2}+B{e}^{-0.01843t_2}=1.937$$

where $t_1=1$, $t_2=24$. This can be easily solved for $a,b,A,B$ to get $a=0.129$, $b=0.002087$, $A=13.6222$, $B=3.0095$ so that we have

$$y_t=0.129+0.002087t+D(13.6222e^{-0.3495t}+3.0095e^{-0.01843t})$$

as the dose–time model.

Estimation of dose and time. Given a new patient sample, with yields $y_{T_1}$ and $y_{T_2}$, collected at two time points where the time difference between them is known, we can fit the data to the model. For this example, we consider a pair of yields available in [24], where $y_{T_1}=5.02$ and $y_{T_2}=2.92$, with $n=200$ each, where the time difference between $T_1$ and $T_2$ is 23 h. From the dose–time model derived above, we obtain a pair of equations as follows:

$$5.02=0.129+0.002087T_1+D(13.6222e^{-0.3495T_1}+3.0095e^{-0.01843T_1}),$$

$$2.92=0.129+0.002087(T_1+23)+D(13.6222e^{-0.3495(T_1+23)}+3.0095e^{-0.01843(T_1+23)}).$$

By using the R solver nleqslv, we obtain the estimated time ${\widehat{T}}_1=9.895097$ and the estimated dose $\widehat{D}=1.658466$ (with the true dose being 1.5 Gy). Notably the time estimate is rather poor here, but this will not be of much importance in practice. What is important is that the methodology has enabled a very good dose estimate. We will assess the uncertainty associated with dose and time estimates more thoroughly in the next section.

3. Simulation Studies

In this section, we will test the proposed methodology with simulated data. Specifically, we setup a simulation model where we generate N = 100 samples in each experiment with each sample containing n ‘cells’ and, therefore, n simulated foci counts, where the settings n = 200, 500, 1000, and 2000 will be considered.

3.1. The Simulation Model

When simulating foci counts, it is important to carefully choose the distribution of the generated data. While there is indication from early research on the γ-H2AX biomarker that, at least conceptually, the foci counts should follow a Poisson distribution under idealized conditions [13,22], in practice, foci counts produced from laboratory experiments or collected from patient samples almost always show overdispersion [23]. This has to do with artifacts and heterogeneities relating to the samples, the laboratories, and the measurement process [24]. The overdispersion of (raw, non-aggregated) foci cell counts has typically been reported to reside between 1 and 3, sometimes up to 10 [23,29]. For the purposes of calibration curve estimation and dose estimation, one can deal with overdispersion quite easily by using quasi-Poisson models [24]. However, quasi-Poisson models are not associated with a probability distribution, so one cannot simulate from such a model. Hence, we will generate overdispersed count data from a Negative Binomial model.

For the purposes of this simulation, define the ‘design yield’ $y_{D,t}$ as the yield at time t hours after ‘exposure’ to design dose D Gy of ionizing radiation. To acquire the design yields, we will use the calibration curves $y_1$ and $y_{24}$ given in Section 2.6. Therefore, we substitute $D=1$ and $D=4$ into these curves, giving us the design yields $y_{1,1}=12.6898,$$y_{1,24}=2.1159,y_{4,1}=50.3661,y_{4,24}=7.9264$ for design doses $D=1$ and $D=4$ at time $t_1=1$ and $t_2=24$, respectively. We then define the random variables

$$C_{D,t}\stackrel{\mathrm{iid}}{\sim }NegBin(r_{D,t},p_{D,t}),$$

where $NegBin(r_{D,t},p_{D,t})$ describes the distribution of the number of failures in a sequence of trials with success probability $p_{D,t}$ before $r_{D,t}$ successes occur. We will generate from this distribution $n=500$ counts $C_{D,t}^{\left(1\right)},C_{D,t}^{\left(2\right)},$$\dots ,$$C_{D,t}^{\left(n\right)}$ for each design yield.

We inform the values $r_{D,t}$ and $p_{D,t}$ by the respective design yield $y_{D,t}$ and a dispersion parameter

$$\varphi ={\displaystyle \frac{Var\left(C_{D,t}\right)}{y_{D,t}}},$$

which, when $\varphi >1$, encapsulates the overdispersion of the γ-H2AX foci relative to the Poisson distribution. We will use $\varphi =2$ in this simulation study, also making the implicit assumption that $\varphi$ is constant across D and t. From general properties of the Negative Binomial distribution, we have the equations

$$y_{D,t}={\displaystyle \frac{r_{D,t}(1-p_{D,t})}{p_{D,t}}}$$

and

$$\varphi y_{D,t}=Var\left(C_{D,t}\right)={\displaystyle \frac{r_{D,t}(1-p_{D,t})}{p_{D,t}^2}},$$

which can be solved for $p_{D,t}$ and $r_{D,t}$ as

$$p_{D,t}={\displaystyle \frac{1}{\varphi }}$$

and

$$r_{D,t}={\displaystyle \frac{p_{D,t}}{1-p_{D,t}}}{y}_{D,t}={\displaystyle \frac{1}{\varphi -1}}{y}_{D,t}.$$

After simulating the cell-wise counts, we compute the sample yields

$${\overline{C}}_{D,t}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{C}_{D,t}^{\left(i\right)}$$

Since the data have been generated from calibration curves at $t_1$ and $t_2$, in this particular instance, we do actually have $T_1=t_1$ and $T_2=t_2$. Still, we will treat $T_1$ and $T_2$ as unknown in order to assess our methodology. To acquire the estimations of dose and time, we substitute into the same model from the example in Section 2.6 for each design dose, namely

$$\begin{array}{ccc}\hfill {\overline{C}}_{D,1}& =& 0.129+0.002087T_1+D(13.6222e^{-0.3495T_1}+3.0095e^{-0.01843T_1}),\hfill \\ \hfill {\overline{C}}_{D,24}& =& 0.129+0.002087(T_1+23)+D(13.6222e^{-0.3495(T_1+23)}+3.0095e^{-0.01843(T_1+23)}).\hfill \end{array}$$

For each of the two considered pairs $(D,T_1)=(1,1)$ and $(4,1)$, solving the equations yields an estimated dose $\widehat{D}$ and an estimated ${\widehat{T}}_1$. We repeat this procedure $N=100$ times to simulate 100 samples and acquire ${\widehat{D}}^{\left(1\right),\left(2\right),\dots ,\left(N\right)}$ and ${\widehat{T}}_1^{\left(1\right),\left(2\right),\dots ,\left(N\right)}$ for each pair $(D,T_1)$.

3.2. Outputs

We display the simulation results with Box-and-Whisker Plots. We will also display the bias of $\widehat{D}$ and $\widehat{T}\equiv {\widehat{T}}_1$ denoted by $Bias\left(\widehat{D}\right)$ and $Bias\left(\widehat{T}\right)$ that we define as

$$\begin{array}{ccc}\hfill Bias\left(\widehat{D}\right)& =& {\displaystyle \frac{1}{N}}\sum _{k=1}^N{\widehat{D}}^{\left(k\right)}-D,\hfill \\ \hfill Bias\left(\widehat{T}\right)& =& {\displaystyle \frac{1}{N}}\sum _{k=1}^N{\widehat{T}}^{\left(k\right)}-T_1,\hfill \end{array}$$

$k=1,2,\dots ,N$, as well as the variance of $\widehat{D}$ and $\widehat{T}$ denoted by $Var\left(\widehat{D}\right)$ and $Var\left(\widehat{T}\right)$ that we define as

$$\begin{array}{ccc}\hfill Var\left(\widehat{D}\right)& =& {\displaystyle \frac{1}{N-1}}\sum _{k=1}^N{\left({\widehat{D}}^{\left(k\right)}-{\displaystyle \frac{1}{N}}\sum _{k=1}^N{\widehat{D}}^{\left(k\right)}\right)}^2,\hfill \\ \hfill Var\left(\widehat{T}\right)& =& {\displaystyle \frac{1}{N-1}}\sum _{k=1}^N{\left({\widehat{T}}^{\left(k\right)}-{\displaystyle \frac{1}{N}}\sum _{k=1}^N{\widehat{T}}^{\left(k\right)}\right)}^2,\hfill \end{array}$$

$k=1,2,\dots ,N$.

3.3. Results

We display the results with Box-and-Whisker Plots in Figure 1. The plots are arranged, from left to right, as a function of the number of cells n, equivalent to the number of foci counts we generate in each sample. Overall, it appears from this plot that both dose and time are reasonably well estimated, with the boxplots of estimates being centered at or close to the true values, and the spread of the boxplots decreasing as n increases. This gives empirical evidence towards the consistency of the proposed methodology.

The results in terms of bias and variance are summarized in Figure 2 and Figure 3. We see from these results that the variance decreases as n increases, which is to be expected. The bias seems to fluctuate with no visible pattern. However, when considering the magnitude of the bias, it is generally quite small when compared to the values of D and $T_1$, indicating that the accuracy of the model is quite good across all considered values of n. It is interesting to observe the “parallelity” of the biases of time and dose: positive biases of the time estimates go along with positive biases of the dose estimates, which makes intuitive sense since a slightly increased dose at a slightly higher exposure time may be indistinguishable from a slightly smaller exposure dose at a slightly smaller exposure time.

4. Analyses with Real Data Sets

4.1. UKHSA Data

4.1.1. Defining Calibration and Testing Data

The proposed methodology is applied to a real data set produced by the Cytogenetics group at UKHSA. In an experiment, four batches of peripheral blood from a healthy volunteer were exposed ex vivo to 250 kV X-rays with design doses 0, 0.5, 1, 2, respectively. The X-ray set (model CP160/1, Ago X-ray Ltd., Martock, UK) was calibrated with reference to national standards with a half-value layer of Cu/Al filtration used for all exposures. Each exposure was monitored using a calibrated UNIDOS E electrometer and “in-beam” monitor ionization chamber (all from PTW, Freiburg, Germany). Staining was performed using a purified anti-H2A.X (phospho Ser139) antibody. For each batch, the foci count in $n=200$ lymphocyte cells were taken at time points 1 h, 2 h, 4 h, 24 h and 48 h post-exposure. The design time points were chosen to cover the most likely post-exposure time-points for samples to be taken in a real life scenario, and having sufficient information to develop calibration curves, noting in particular the complex DNA repair kinetics in the first two hours.

Table 1. Laboratory data from UKHSA: foci yields in batches of 200 cells (adapted from [26]).

Time Point (h)	Average No. Foci
	0 Gy	0.5 Gy	1 Gy	2 Gy
1	0.675	2.265	4.62	5.905
2	0.165	2.045	3.79	6.04
4	0.15	1.74	3.17	4.48
24	0.265	0.545	1.12	2.025
48	0.17	0.635	0.76	1.285

displays the average foci count (yield) at each design dose and time point. Inspecting this data set, we can immediately notice two anomalies, highlighted in Table 1 in bold face. The average foci count with design dose 0.5 Gy at time point 48 h is larger than that of 24 h. The average foci count with design dose 2 Gy at time 1 h is abnormally small, in fact, even smaller than that of 2 h. While both of these anomalies could be related to a large variance of foci yield due to the rather small number of cells considered, other explanations are possible. For instance, it is known that, for small doses as 0.5 Gy, the image filters used for foci detection may overinterpret spurious signals, leading to increased dispersion [29]. Multiple foci in close proximity, as observed particularly for higher doses such as 2 Gy, can be ambiguous in interpretation [30], and may lead to overlapping foci which negatively impacts their detection [22,31]. Whatever the reason for these anomalies, we remove the data with design dose 0.5 Gy at time 48 h and the data with design dose 2 Gy at time 1 h from further consideration.

Before proceeding with our analysis, we briefly verify whether the calibration curves used in the examples from the last section appear appropriate here. A quick verification using the applet from [24], with the data for 1 h at 0 Gy and 1 h at 1 Gy as the reference sample for the calibration curve at 1 h shows that the reference samples do not validate this calibration curve. Using the data for 24 h at 0 Gy and 24 h at 1 Gy as the reference sample for the calibration curve at 24 h, this calibration curve does not get validated unless a large dispersion of $\varphi \ge 30$ is specified which is however rather inconsistent with the data at hand, see also Section 4.1.2. Hence, overall, we conclude that the match of those calibration curves to this new data set would be rather poor.

The parameters involved in the calibration curve can be affected by intra-individual variation arising from multiple sources, including how the data is produced, collected, and the samples are stored. However, according to [32], the estimation of parameters is not largely affected by such intra-individual variation. There is also report of inter-individual variation, which is on a similar scale as intra-individual variation [33], and of inter-scorer variation [24]. To some extent, these variations are accounted for by the overdispersion that we have already considered in our modeling approach.

For such reasons, it is useful to have calibration curves which are as “similar” as possible (in terms of the laboratory, the scoring procedure, the scorer, the software, etc.) to the patient sample from which the dose is eventually estimated. For the sake of this analysis, we use the average foci count at times 2 h and 24 h as our “training data” (

Table 2. The “training data” which were used to create calibration curves at $t_1=2$ h and $t_2=24$ h.

Time Point (h)	Average No. Foci
	0 Gy	0.5 Gy	1 Gy	2 Gy
2	0.165	2.045	3.79	6.04
24	0.265	0.545	1.12	2.025

) in order to estimate two calibration curves. We will then use the remaining average foci counts at times 1 h, 4 h and 48 h as our “testing data” (

Table 3. The “testing data” from which exposure doses and times will be estimated (the calibration data and two anomalous observations have been removed).

Time Point (h)	Average No. Foci
	0 Gy	0.5 Gy	1 Gy	2 Gy
1	0.675	2.265	4.62
4	0.15	1.74	3.17	4.48
48	0.17		0.76	1.285

).

4.1.2. Generating Calibration Curves and the Model

For the calibration curve estimation, we go though the process as described in Section 2.1. We estimate calibration curves with the generalized linear model (glm) function in R by fitting a quasi-Poisson generalized linear model with identity link and dose as covariate. The resulting calibration curves, for 2 h and 24 h after exposure, respectively, are as follows:

$$y_2=0.1954+3.2167D,$$

$$y_{24}=0.2355+0.8609D,$$

with estimated dispersion parameters 13.13 and 2.85, respectively. Going through the same process as described in Section 2.4, we get the following dose–time model:

$$y_t=0.19176+0.0018227t+D(3.87598e^{-0.3495t}+1.33846e^{-0.01843t}).$$

(5)

4.1.3. Dose and Time Estimation

The estimation procedure from Section 2.5 is now applied onto each possible pairing of observations arising from the “testing” data from Table 3. For each design dose, we use every time pairing available to represent $T_1$ and $T_2=T_1+\Delta$, from which we then estimate D and $T_1$ by substituting each corresponding yield into the model (5). Specifically, these are three possible pairings $(T_1,T_2)$ for each of 0 Gy and 1 Gy, and one pairing for each of 0.5 Gy and 2 Gy. The value $\Delta$ is hence defined by the difference $T_2-T_1$ of each pair.

That is, for each of the resulting eight pairings, we solve the set of two equations

$$\begin{array}{ccc}\hfill y_{T_1}& =& 0.19176+0.0018227T_1+D(3.87598e^{-0.3495T_1}+1.33846e^{-0.01843T_1})\hfill \\ \hfill y_{T_1+\Delta }& =& 0.19176+0.0018227(T_1+\Delta )+D(3.87598e^{-0.3495(T_1+\Delta )}+1.33846e^{-0.01843(T_1+\Delta )}).\hfill \end{array}$$

After solving the eight problems using the Nsolve function of Mathematica, we display the dose and time estimations produced from the testing data (Table 3) in

Table 4. Dose estimations for the testing data (adapted from [26]).

${\mathit{T}}_1,{\mathit{T}}_2$	Δ	Design Dose (Gy)
		0	0.5	1	2
1, 4	3	0.00	1.12	1.90
1, 48	47	0.00		0.86
4, 48	44	0.00		0.84	1.81

and

Table 5. Time ($T_1$) estimations for the testing data.

${\mathit{T}}_1,{\mathit{T}}_2$	Δ	Design Dose (Gy)
		0	0.5	1	2
1, 4	3	−183	5.20	3.67
1, 48	47	−58.9		0.04
4, 48	44	−55.9		1.56	3.56

. All dose estimates appear useful for triage purposes. It is noted that these results are the same when using instead the the nleqslv package [28] in R, apart from the first column (for 0 Gy) of the time estimates, which then take the values −43.08, −34.48, and 10.38, respectively. However, it is clear that “time from exposure” cannot be accurately or even meaningfully estimated if there had been no exposure. The important result is that the dose can still be accurately estimated in this case.

4.1.4. Uncertainty Quantification with Bootstrap Resampling

We employ the bootstrap [34] to estimate the uncertainty of the estimation process. For each combination of time and design dose in consideration, we take 1000 bootstrap samples of its n foci counts (corresponding to the n scored cells). Hence, each bootstrap sample consists again of n counts that were drawn from the original data with replacement. We take the average of each bootstrap sample which we use as yields to produce dose estimations. We display the bootstrapped bias and variance for the dose estimates in

Table 6. (Bias, variance) of dose estimates obtained via the bootstrap. Values $<5\times {10}^{-4}$ are given as 0.

${\mathit{T}}_1,{\mathit{T}}_2$	Design Dose (Gy)
	0	0.5	1	2
1, 4	(0, 0)	(1.943, 1552.39)	(0.924, 0.062)
1, 48	(0, 0)		(−0.138, 0.026)
4, 48	(−0.001, 0)		(−0.156, 0.025)	(−0.190, 0.059)

and for the time estimates in

Table 7. (Bias, variance) of time ($T_1$) estimates obtained via the bootstrap. NA indicates that the concept of a true time of exposure is not well defined.

${\mathit{T}}_1,{\mathit{T}}_2$	Design Dose (Gy)
	0	0.5	1	2
1, 4	(NA, 1981.31)	(5.10, 214.62)	(2.779, 0.899)
1, 48	(NA, 351.86)		(−0.976, 0.555)
4, 48	(NA, 389.83)		(−2.456, 0.760)	(−0.411, 0.709)

.

It is worth commenting on the large bias and variance estimates in the case $D=0.5$, in Table 6 and Table 7. It might seem like both estimations for $\widehat{D}$ and ${\widehat{T}}_1$ have completely failed in this case, but this is not entirely true. Due to the high amount of bootstrap samples we are taking from this relatively small data set with only 200 elements, it is not implausible that a very small amount samples might contain very skewed data, and cause a very small amount of estimations to go completely astray and so to inflate the bias and variance terms. Bearing this in mind, we have since examined the bootstrap samples and removed 2 extremely large outliers. We present the results from the remaining 998 samples in

Table 8. Bias and variance of $\widehat{D}$ and ${\widehat{T}}_1$ from bootstrap simulated data with true $D=0.5$, $T_1=1$, $T_2=4$, with 2 outliers removed from the 1000 bootstrap samples.

	Bias	Variance
$\widehat{D}$	0.627	0.035
${\widehat{T}}_1$	4.474	3.750

. There is no evidence in the literature to back the removal of outliers from bootstrap samples, but doing so provides us with a more realistic assessment of the uncertainty in this particular instance. The conclusion here is that the bias is acceptable and the variance quite small after removal of outliers.

In summary, it appears we would most likely be able to make good estimations with appropriate calibration curves even when we have only a small number of cells available in the patient sample. While there remains some notable biases, these have been inherited from the data themselves. Overall, the model performs well, and the variance is reasonably low across all scenarios.

4.2. UAB Data

We consider a real data set that was analyzed in [35] and originally produced at the Universitat Autónoma de Barcelona (UAB) [23]. Ionizing radiation (6MV photon beams) was applied to peripheral mononucleated cells at designed dose levels of 0, 0.25, 0.5, 0.75, 1, 1.25, 2, 2.5, 2.75, and 3 Gy. γ-H2AX foci counts in 500 cells were obtained using a semi-automatic method at the following time points after radiation: 1 h, 1.5 h, 2 h, 3 h, 4 h, 6 h, 8 h, 10 h, and 24 h. Data were collected from three exercises (0, 1, and 2) involving two donors (1 and 2).

To construct the two calibration curves, we use the average foci count at 2 h and 24 h. The calibration data is presented in

Table 9. (Top): foci yields at 2 h and 24 h for dose levels of 0 Gy, 0.25 Gy, 0.5 Gy, 0.75 Gy, and 1 Gy; (bottom): foci yields at 2 h and 24 h for dose levels of 1 Gy, 1.25 Gy, 1.5 Gy, 2 Gy, 2.5 Gy, and 3 Gy.

Time Point	Average No. Foci
	0 Gy	0.25 Gy	0.5 Gy	0.5 Gy	0.5 Gy	0.75 Gy	0.75 Gy	1 Gy
2 h	0.802	4.104	5.954	5.274	5.412	6.296	6.634	7.730
24 h	0.710	0.830	0.856	1.062	1.100	0.954	0.934	1.890
Time Point	Average No. Foci
	1 Gy	1 Gy	1.25 Gy	1.25 Gy	1.5 Gy	2 Gy	2.5 Gy	3 Gy
2 h	6.008	6.188	9.274	9.582	10.624	17.54	20.126	21.290
24 h	1.202	1.208	1.372	1.382	2.116	3.004	4.148	6.246

. By fitting a quasi-Poisson model with an identity link, we obtain the following two calibration curves:

$$y_2=1.2767+6.8964D,$$

and

$$y_{24}=0.3637+1.3068D.$$

The two calibration curves are depicted along with the raw data in Figure 4. Then we obtain the dose–time model as follows:

$$y_t=1.3597-0.0415t+D(9.937195e^{-0.3495t}+2.030271e^{-0.01843}).$$

(6)

For dose estimation, we use the average foci count at time points 1.5 h, 3 h, 6 h, and 10 h for both donors 1 and 2 from exercise 2 at dose levels of 1, 2, and 3 Gy. The paired data can be found in

Table 10. Data from donor 1 and exercise 2 at dose levels of 1 Gy, 2 Gy, and 3 Gy. The two values given in each column are the pair of yields obtained at the respective two time points.

Time Point	Average No. Foci
	1 Gy	2 Gy	3 Gy
1.5 h, 3 h	10.206, 5.036	15.352, 12.150	17.380, 15.984
1.5 h, 6 h	10.206, 2.850	15.352, 6.168	17.380, 11.278
1.5 h, 10 h	10.206, 1.902	15.352, 4.036	17.380, 7.120
3 h, 6 h	5.036, 2.850	12.150, 6.168	15.984, 11.278
3 h, 10 h	5.036, 1.902	12.150, 4.036	15.984, 7.120
6 h, 10 h	2.850, 1.902	6.168, 4.036	11.278, 7.120

and

Table 11. Data from donor 2 and exercise 2 at dose levels of 1 Gy, 2 Gy, and 3 Gy.

Time Point	Average No. Foci
	1 Gy	2 Gy	3 Gy
1.5 h, 3 h	8.070, 4.310	12.800, 9.390	17.786, 16.016
1.5 h, 6 h	8.070, 2.592	12.800, 5.874	17.786, 12.004
1.5 h, 10 h	8.070, 1.950	12.800, 3.838	17.786, 7.960
3 h, 6 h	4.310, 2.592	9.390, 5.874	16.016, 12.004
3 h, 10 h	4.310, 1.950	9.390, 3.838	16.016, 7.960
6 h, 10 h	2.592, 1.950	5.874, 3.838	12.004, 7.960

. By fitting the differences between two time points, $\Delta$, and their corresponding yields to the model (6), at the respective time points $T_1$ and $T_2=t_1+\Delta$, we obtain estimates for the dose and $T_1$. The results are shown in

Table 12. Estimates from donor 1 and exercise 2 at dose levels of 1 Gy, 2 Gy, and 3 Gy. In each bracket, the first element represents the estimate of dose D, while the second element represents the estimate of $T_1$.

Time Point	Dose Level
	1 Gy	2 Gy	3 Gy
1.5 h, 3 h	(0.058, −7.411)	(3.735, 4.680)	(8.400, 10.245)
1.5 h, 6 h	(0.060, −7.540)	(1.487, 0.830)	(5.390, 6.043)
1.5 h, 10 h	(0.127, −5.404)	(1.455, 0.754)	(3.577, 3.803)
3 h, 6 h	(0.253, −0.629)	(0.960, 0.213)	(4.655, 5.682)
3 h, 10 h	(0.325, 0.190)	(1.328, 1.356)	(3.289, 3.832)
6 h, 10 h	(0.351, 3.641)	(1.442, 5.162)	(2.886, 5.115)

and

Table 13. Estimates from donor 2 and exercise 2 at dose levels of 1 Gy, 2 Gy, and 3 Gy. In each bracket, the first element represents the estimate of dose D, while the second element represents the estimate of $T_1$.

Time Point	Dose Level
	1 Gy	2 Gy	3 Gy
1.5 h, 3 h	(0.059, −6.579)	(1.838, 2.374)	(7.872, 9.073)
1.5 h, 6 h	(0.061, −6.681)	(1.722, 2.117)	(5.955, 6.556)
1.5 h, 10 h	(0.310, −1.894)	(1.449, 1.463)	(4.181, 4.444)
3 h, 6 h	(0.278, 0.390)	(1.692, 3.496)	(5.439, 6.711)
3 h, 10 h	(0.457, 2.142)	(1.425, 2.759)	(3.932, 4.711)
6 h, 10 h	(0.534, 6.533)	(1.339, 5.085)	(3.470, 5.793)

. For dose estimation, we use the average foci count at time points 1.5 h, 3 h, 6 h, and 10 h for both donors 1 and 2 from exercise 2 at dose levels of 1, 2, and 3 Gy. The paired data can be found in Table 10 and Table 11. By fitting the differences between two time points, $\Delta$, and their corresponding yields to the model (6), at the respective time points $T_1$ and $T_2=t_1+\Delta$, we obtain estimates for the dose and $T_1$. We produced the estimates using the R solver (nleqslv), with the results provided in Table 12 and Table 13. In considering these tables, recall that, while the methdology delivers time and dose estimates, the primary focus of dosimetry is always dose estimation. We observe that this method provides dose estimates that are close to the true dose levels, particularly for higher doses such as 2 Gy and 3 Gy.

We also carried out these estimations using NSolve in Mathematica. For reasons unknown to us, NSolve struggled with finding the estimates for some pairings, such as (1.5 h, 6 h) for a true dose of 1 Gy, even after adjustment of the output range. We speculate that this has to do with the b parameter (−0.0415 here) being both negative and close to zero.

We then compare these results with the traditional dose estimation approach, which relies on a single calibration curve at a known time point [24]. Since the calibration curves at 2 h and 24 h have been calculated, we can derive interpolated calibration curves between these time points by combining the two known curves with appropriate weights. For the time point 1.5 h, interpolation is not feasible, but following the most likely course of action that a dosimetry unit would take in this case, we just choose the closest calibration curve, which is 2 h. The results are shown in

Table 14. Estimates for donor 1 from exercise 2 at dose levels of 1 Gy, 2 Gy, and 3 Gy, obtained using the calibration curve at 2 h and the interpolated curves at 3 h and 6 h.

Time Point	Dose Level
	1 Gy	2 Gy	3 Gy
1.5 h, 3 h	1.2948	2.0410	2.3350
1.5h 6h	1.2948	2.0410	2.3350
1.5h 10h	1.2948	2.0410	2.3350
3h 6h	0.5722	1.6432	2.2204
3h 10h	0.5722	1.6432	2.2204
6h 10h	0.3917	1.0739	2.1245

and

Table 15. Estimates for donor 2 from exercise 2 at dose levels of 1 Gy, 2 Gy, and 3 Gy, obtained using the calibration curve at 2 h and the interpolated curves at 3 h and 6 h.

Time Point	Dose Level
	1 Gy	2 Gy	3 Gy
1.5 h, 3 h	0.9851	1.6709	2.3939
1.5 h, 6 h	0.9851	1.6709	2.3939
1.5 h, 10 h	0.9851	1.6709	2.3939
3 h, 6 h	0.4629	1.2278	2.2252
3 h, 10 h	0.4629	1.2278	2.2252
6 h, 10 h	0.3387	1.0135	2.2738

. By comparing Table 12 with Table 14 and Table 13 with Table 15, we observe that our method provides dose estimates that are more sensitive to time changes, particularly at 3 h, 6 h, and 10 h after radiation. It delivers estimates of comparable accuracy to the single calibration curve method, which is a good result considering that the proposed method does not assume the time of exposure is known.

5. Limitations

It is clear that the general difficulties and issues with the γ-H2AX biomarker carry over to the presented method.

Among these, one finds several peculiarities relating to this biomarker at very low and high doses, partially already alluded to in the introduction. At very low doses (<100 mGy), the number of gamma-H2AX foci in each cell is very small, typically just 0, 1, or 2, making it hard to distinguish an actual exposure from background irradiation, which typically resides between 0 and 1 foci per cell [13]. This is exacerbated by a tendency of detection software to detect spurious signals in this dose range [29]. For high doses (>3 Gy), foci begin overlapping which complicates their detection under both manual and automated scoring. Indeed, when using quadratic calibration curves for the modeling of γ-H2AX foci, it has been observed that the quadratic terms tends to be negative (unlike for the dicentric assay where it is positive), which is presumably due to omitted foci counts due to the overlap effect. However, it also been found that the inclusion of this quadratic term adds more variance to the dose estimation than it reduces the bias so that the recommendation is still to just use linear calibration curves in practice [24]. This has also been empirically observed in experiments directly comparing the dicentric assay with the γ-H2AX assay using data from the same donors in the same study [36,37]. Due to the mentioned saturation effect, predictions from such linear curves yielding very high dose estimates should be considered with caution.

Posing further difficulty with the γ-H2AX assay, it has sometimes been stated that there is substantial inter-individual variation, presumably relating to differing radiosensitivity of the concerned individuals [38]. Specifically, it has been demonstrated that both control and exposure yields may depend on factors such as age or occupation [39]. In a study involving 28 individuals [40], it was shown that, while inter-individual variation in γ-H2AX foci yield is present, it is less prevalent in exposed than in background foci. Furthermore, this study showed that, after irradiation, the inter-individual variation is on a similar level as the intra-individual variation, which was similarly observed in [24,37]. In another study, inter-individual variation at 4 h post-radiation was found insignificant using flow cytometry methods [38]. It can be concluded that, jointly, the intra- and inter-individual variation just add to the general overdispersion of the γ-H2AX assay, which is why quasi-Poisson or related models are usually employed for the analysis of this biomarker. In doing so, it needs to be acknowledged that a possible dependency of the overdispersion on dose is ignored [29]. Since we are not using the estimated dispersion for uncertainty quantification or further inference, this is, however, not a particular area of concern in this present study.

There is a plethora of other experimental factors which impact the calibration and the reliability of the γ-H2AX assay. These include the type and quality of radiation [41], laboratory settings [32], scoring mechanisms [13], the technician performing the scoring [38], as well as image and microscope settings [20]. Further differences can incur due to shipment conditions [42] which is of particular importance for the γ-H2AX assay due to the quick foci decay. This point also relates to the temperature at exposure [37] and shipment [13], and to the type of blood actually shipped (whole blood or isolated lymphocytes) [32]. Note that irrespective of the type of blood shipped, the scored cells are almost always lymphocytes, so further discussion on dependencies of foci yield on cell type [43] is not of concern for us.

In summary, there is still a large number of possible biases and uncertainties associated with this assay. These biases and uncertainties will enter the estimation of the calibration curves which are taken as starting point of our analyses. In this work, no attempt has been made to account for any of these biases and uncertainties, other than those arising due to uncertain or unknown time of exposure. Hence, the conclusions in the paper should be seen as conditional on the presence of such issues which continue to be intrinsic to the γ-H2AX assay. Or, in other words, any bias and uncertainties which are inherent to the original calibration curves will, by our methodology, be carried forward into our dose–time model and hence may impact the accuracy of the final dose estimates. Having stated all this, despite these potential shortcomings, the γ-H2AX is assay has established itself as an alternative biomarker for triage situations, and the simulations and real data analyses carried out in this paper do lend support to its applicability for this purpose. For accurate clinical dose estimation, other assays such as the DCA (minimum detection of 100 mGy) or centromere stained micronuclei (detection limit of 50 mGy) may continue to be more appropriate [8].

A further limitation of this study lies in the considered samples sizes for collecting foci yields. We have considered here samples sizes between $n=200$ and $n=1000$, which would be the samples sizes typically used in clinical experiments to establish calibration curves and to validate the general functioning of the biomarker, and this was the mindset with which we have approached this study. For rapid emergency triage, smaller sample sizes of 50 or less scored cells are however preferable [32]. Hence the performance of the proposed methodology under full ‘emergency triage’ conditions should be further assessed.

The final limitation that needs to be commented on is the requirement to select the decay parameters u and v. Here, we have made the decision to keep these parameters as entirely constant, with values extracted from the literature. We believe this is to some extent justified as foci decay is a purely physical property which should have little to do with lab procedures, scorers or scoring systems, etc. This reasoning appears to be in line with the earlier literature on the γ-H2AX assay which indicated that ‘a single decay’ would be appropriate for describing foci loss until 96h after exposure [22]. However, statements of this type relate to samples from single individuals and, hence, do not serve to mitigate more recent evidence of inter-individual variation of the foci decay kinetics [44]. It has also been reported that the speed of foci signal decay is reduced for very low doses [45]. The choice of constant and fixed values for u and v is hence a limitation of the proposed methodology. An approach for possible estimation of u and v from calibration data is discussed in the Conclusions.

6. Conclusions

The proposed methodology provides a way to greatly reduce the post-exposure time dependency needed for accurate dose estimation of previous methods of ionizing radiation dose estimation that uses γ-H2AX foci as the biomarker. An intuitive way to describe our model is that it is a generalization of linear calibration curves through non-linear interpolation using the decay mechanism.

A notable feature of this methodology that deserves some discussion is that, for the proposed approach to function, we assume that there are two calibration curves available to us, which are then used to infer the parameters $a,b,A,B$. Recall, at this point, the dose–time model $y_t=a+bt+D(A{e}^{-ut}+B{e}^{-vt})$, where we decided to take u and v from past research and keep them as they are in the modeling process. However, if we had three or four calibration curves instead of two, we would in theory be able to infer u or/and v using Nsolve instead of taking them as given. However, attempts at this were unsuccessful. As long as there are at least three total variables, including at least one of u or v, Mathematica will run until problems where RAM fills up or crashes in the process. Future research with ample resources can attempt to generate all four of these parameters and test if this would increase the precision and accuracy of the model.

The overall performance of the models on real data is acceptably good, considering that the most likely use of the γ-H2AX foci assay is to serve as a triage biomarker [46]. Hence, this biomarker may often be used to make a quick assessment of whether or not some radiation exposure has occurred, specifically when time since exposure is unknown, as is the case in many emergency response scenarios. This may then be followed up by a more precise biomarker (with less time dependency) such as dicentric chromosomes (see e.g., [8]), or if needed, immediate medical treatment.

In this project, we aimed to build a model that would allow us to determine ionizing radiation dose and exposure time using a sequence of biomarker measurements. We tested this model using both simulated and real data. We conclude from the results that the model performs well with reasonable discrepancies that can be explained. A study with similar aims as this project is provided in [35]. This study used MCMC techniques instead of the decay mechanism to form a generalized relationship between dose, time of exposure, and yield.