This section presents the details of the GDR calculation process.

4.1 Overview of Design

This section describes the methods for the surface and volumetric calculations used in the CPM calculator. The Monte Carlo N-Particle (MCNP) v6.2 radiation transport code was utilized to model the CPM calculator surface contamination by simulating the average energy deposition pulse height tally (Werner (ed.), 2017). A detector response function methodology (Asano et al., 2022), combining capabilities in MCNP with hybrid radiation transport capabilities implemented in the MC code Shift (Pandaya et al., 2016), was used to derive the Volume CPM CFs in a computationally feasible manner. The latter methodology is herein referred to as the "MCNP/Shift method". Since Shift currently does not support a pulse height tally, the average cell flux in Shift was used to approximate the MCNP-equivalent pulse height for cases involving buried source attenuation. Previous benchmarking of Shift capabilities indicated that radiation transport of photons through contaminated volumetric depths, as required in the volume CPM function, superseded capabilities of MCNP in default mode by a speed factor of 10 for simulations of average cell flux. The ability to conduct rapid, high-fidelity radiation transport simulations was paramount for this initiative, given the initially desired 23,120 discrete simulations for a series of contamination depths, 19 monoenergetic photon energies, 17 photon emitting radionuclides, and four NaI(Tl) scintillation detector models outlined in the table shown below. The small amount of thallium doping material in the detector crystals was omitted in the MCNP and Shift models for simplicity; this has a negligible effect on the energy deposition-based GDR, since the total mass attenuation coefficient is largely dependent on the sodium and iodide material concentrations.

Material	Monoenergetic Energies	Radionuclide Energies	Contamination Depths	Detector Distances	Number of Detectors	Total Runs
Concrete	19	17	7	5	4	5,320
Drywall	19	17	4	5	4	3,040
Glass	19	17	4	5	4	3,040
Soil	19	17	7	5	4	5,320
Steel	19	17	7	5	4	5,320
Wood	19	17	5	5	4	3,800
Total						23,120

Parameters modeled and simulated in MCNP and Shift included the following: plane radii, discrete photon energies, Superfund radionuclides (Am- 241, Co-60, Ba-137m/Cs-137, Pu-238, Pu-239, Pu-240, Ra-226, Ra-228, Tc-99, Th-228, Th-230, Th-232, U-234, U-235, U-238/Pa-234m), and source material compositions. All Four NaI(Tl) scintillation detectors (0.5" × 1", 1" × 1", 2" × 2", 3" × 3") and source-detector distances of 0.5 cm, 1 cm, 2.54 cm, 10 cm, 30 cm were modeled. Contaminated media of soil, concrete, wood, steel, drywall, and glass were evaluated for depths (source thicknesses) of ground plane (surface), 1 cm, 2 cm, 3 cm, 5 cm, 15 cm, and infinite. Nineteen discrete energies between 20 keV and 3 MeV were evaluated (0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1.0, 1.5, 2.0, and 3.0 MeV). Additionally, the list of radionuclides has been extended to include all radionuclides from ICRP 107 that satisfy the photon emission criteria previously described. Notably, both Rn-220 and Rn-222 lack the criteria for photon emission and have therefore been eliminated. In order to eliminate the computational runtime of simulating the photon emitting radionuclide spectra in MCNP or Shift, an interpolation method was employed to approximate the radionuclide CFs.

The table below presents the source thicknesses that were initially considered for each media. For some material and discrete energy combinations, the infinite thickness may be less than the discrete thickness value considered. Hence, for cases where the discrete simulation thickness is greater than four mean free paths, the source depth was not evaluated. Over the four detector sizes evaluated, 156 of the 23,120 possible combinations were not evaluated.

Material	Ground Plane	1cm	2cm	3cm	5cm	15cm	Infinite
Soil	Yes	Yes	Yes	Yes	Yes	Yes	Yes
Concrete	Yes	Yes	Yes	Yes	Yes	Yes	Yes
Wood	Yes	Yes	Yes	Yes	Yes	No	No
Steel	Yes	Yes	Yes	Yes	Yes	Yes	No
Glass	Yes	Yes	Yes	Yes	No	No	No
Drywall	Yes	Yes	Yes	Yes	No	No	No

Only primary photon source emissions were considered, due to Shift's capability to conduct neutral particle and not charged particle transport. Therefore, beta emissions from the radionuclide of interest as well as pure beta-emitters (H-3 and Sr-90) and their associated bremsstrahlung emissions cannot be run using Shift and were therefore not addressed. A flow chart is provided below to summarize the approach.

4.2 Design Testing and Benchmarking

Both MCNP and SHIFT software packages were considered for the CPM Calculator design. Each was tested and benchmarked against the other for repeatability and speed.

MCNP is a stochastic method that is widely used to solve radiation transport problems in reactor physics, shielding, dosimetry, radiation detection and measurements, and various other applications of interest. The Monte Carlo (MC) method enables detailed, explicit geometric, energy, and angular representations and hence is considered the most accurate method available for solving complex radiation transport problems (Wagner et al., 2011). MCNP functions by simulating the individual average particle behavior in a given environment based on sampling the probability density of particles in phase space. The "behavior" of a particle may be described by the types of interactions (neutron capture, photon scatter, photon leakage, etc.) or "events" that are sampled from probability distributions containing transport data based on the selection of a pseudorandom number. In MCNP, the lifetime of an individual particle from birth until termination (due to absorption, escape from the problem boundary) is described as its random "history."

At the end of the simulation, the final average quantities of interest, known as tallies (such as flux, detector pulse, dose, etc.), are estimated from the distribution of all histories and have an associated uncertainty (relative error) that is proportional to one over the square root of the total number of histories for statistically well-behaved simulations. Therefore, the uncertainty may be decreased by increasing the number of histories simulated in the problem to coincide with the "law of large numbers," where the sample mean approaches the population mean as the sample size increases. While MCNP is considered to be the most accurate method available for solving complex radiation transport problems, it possesses the inherent disadvantage of being computationally expensive, as the required number of histories to reach proper convergence (less than 5% for detector pulse height) is increased for complex problems (LANL, 2003).

SHIFT is a massively parallel Monte Carlo radiation transport package that implements hybrid methods (deterministic coupled with stochastic) to obtain a faster and more accurate solution to problems of interest, as opposed to purely stochastic Monte Carlo. The hybrid methodology utilized in this study is known as the CADIS (Consistent Adjoint Driven Importance Sampling) method, which is a tally optimization method that relies on the discrete ordinates deterministic solver, Denovo, as part of the Exnihilo code suite to generate an adjoint flux and associated importance map for particles simulated in a given environment. Once the adjoint flux and importance map is generated, particles are more likely to be born in regions of higher importance in the simulation environment. The CADIS method is known to be 10-100,000 times more efficient than analog (no variance reduction) Monte Carlo based on tally figure of merit (FOM) comparisons (Biondo et al., 2017). The hybrid capabilities of SHIFT are particularly useful for speeding up the transport of sources that undergo significant attenuation in large problem geometries as well as lowering the variance of tally results when compared to analog Monte Carlo. SHIFT is currently capable of neutral particle transport for only photons and neutrons and may be used to simulate select tallies that mirror those included in MCNP, such as average flux in a volumetric cell (F4 in MCNP).

A benchmark of SHIFT capabilities indicated that radiation transport of photons through contaminated volumetric depths, as required in the Volume CPM function, supersedes capabilities of MCNP by a speed factor of 10 for simulations of track length estimate of cell flux. The ability to conduct rapid, high-fidelity radiation transport simulations is paramount for this initiative, for 23,120 discrete simulations for a series of contamination depths, 19 monoenergetic photon energies, 17 photon emitting radionuclides, and four NaI(Tl) detector models, summarized Table 3. Photon emitters were alone considered from the list due to SHIFT's capability to conduct neutral particle and not charged particle transport. Any primary or secondary charged particle transport requires the use of Monte Carlo code (e.g., MCNP). Therefore, beta emissions from the list of radionuclides as well as pure beta- emitters (H-3 and Sr-90) and their associated radiative components cannot be run using SHIFT. In addition, both Rn-220 and Rn-222 lack the criteria for photon emission and have also been eliminated as photon emitters. These criteria reduce the list of Superfund Common Radionuclides to the photon emitting radionuclides.

Note that Bremsstrahlung beta (+/-) particle transport and annihilations from positron emissions, for the listed radionuclides, were not considered. Should photons from these actions be expected, further analysis would be necessary to accurately predict the GDR.

4.2.1 Denovo Deterministic Solver Overview

A 3-D discrete ordinates transport solver utilized in Shift, known as Denovo, discretizes (conversion of continuous space to equivalent discrete space) the steady-state Boltzmann transport equation in space, angle, and energy to approximate the adjoint fluxes (importance functions that describe the expected tally contribution of particles) that are used to generate variance reduction parameters for accelerating fixed-source MC simulations (Wagner et al., 2011). The CPM Calculator photon sources are fixed, following the matrix equation:

as opposed to a k-eigenvalue sources, following the characteristic equation:

For brevity, all equations shown to demonstrate Shift's capabilities are derived. The steady-state Boltzmann transport equation solved in Denovo is given by (Evans et al., 2010):

Here, the state is defined by the angular flux ψ, and the independent variables are r = (x,y,z) (in centimeters); Ω = (θ,ψ) (in steradians); and E (in mega-electron-volts) representing space, angle, and energy, respectively (Evans et al., 2010). This equation may be used to describe how various particle types (i.e., neutrons, photons) travel through a medium. The variables on the left side of the above equation describe the net transport (leakage) and total interactions that the particle undergoes, respectively. The variables on the right side describe the total accumulated source density. The above equation is converted to a discrete angular form that accounts for anisotropic scattering of the source within a problem geometry (Evans et al., 2010), as shown here:

In simpler terms, the above equation represents the continuous steady-state Boltzmann transport equation in a discrete space, such that Denovo can solve for the flux component through discretization over the problem geometry "mesh" that is defined by the user with respect to space and angle. This equation accounts for group (g) scattering (with respect to energy and angle) as well as even and odd spherical harmonics functions (Y^e,Y^o) used to evaluate Legendre polynomials that expand the angular flux scattering cross section.

The above equation may be expressed in operator form (Evans et al., 2010):

The same principle applies, where "L" represents the differential transport operator describing the net transport and total interactions of the source (left-hand side of the above equations). The "M" operator is used to convert the harmonic moments to discrete angles; the "S" operator defines the group transfer cross sections for particles (scattering from one energy group to another): and q_e represents the external source term (for photons this is negligible in the current study). The above equation may be further expanded to depict what Denovo is solving in matrix form (Evans et al., 2010):

The goal is to solve for the angular flux groups, given by [ψ]_G. Shift couples SCALE continuous-energy (Shift MC solver) and multigroup physics (Shift MC solver and Denovo deterministic solver) cross section libraries for hybrid transport. The CPM Calculator photon sources discretized in Denovo are separable energy-space isotropic sources, which facilitate the hybrid methods. As opposed to traditional (Sn) iterative discretization methods, Denovo utilizes the Koch-Baker-Alcouffe (KBA) parallel sweep algorithm to perform a 2-D decomposition of a 3-D cartesian orthogonal structured spatial mesh (i.e., the simulation problem space) to solve the steady-state Boltzmann transport equation in space, angle, and energy (Evans et al., 2010). In other words, the KBA algorithm directly inverts the "L" term in the matrix form to obtain a more computationally efficient solution of the angular flux by "sweeping" through the 3-D cartesian orthogonal structured spatial mesh, as opposed to traditional source iteration techniques that may take longer converge. The defined 3-D mesh in Shift is decomposed into computational domain blocks defined by the user that scale with the number of processors, as shown below (Evans et al., 2010):

The I,J, and K terms represent the total, global number of cells in the x-, y-, and z-directions, respectively. The I_b,J_b, and K_b terms represent the size of each x-, y-, and z-domain on each processor, which are defined by the user. The number of "small blocks" previously mentioned that discrete ordinates methods visualize in the phase space to transport particles, denoted by I, J, and K, are automatically calculated. The KBA decomposition may be visually depicted by the following figure. In this example, the grid is decomposed on nine processors.

The red lines (thick gray lines) indicate computational blocks in the z direction. Each processor has I_b ×J_b×K_b cells, and each block has size I_b ×J_b×K_b cells(Evans et al., 2010).

Note: great care should be taken when defining the Denovo spatial mesh; a coarse mesh may result in unreliable importance maps, even when source biasing is implemented. The Denovo deterministic solver also utilizes a ray tracing method to generate track-length contributions (i.e., mean free path determinations) within the geometry of the problem; increasing the number of rays fired per computational domain block (mesh face) is particularly useful for large problems with small tally volumes. This is especially prevalent with the 0.5x1 and 1x1-inch detector CPM Calculator simulations.

4.2.2 CADIS Hybrid Method Overview

The hybrid capabilities of Shift are particularly useful for speeding up the transport of sources that undergo significant attenuation in large problem geometries and lowering the uncertainty associated with tally results when compared to analog (no variance reduction) MC. Shift is currently capable of neutral particle transport only for photons and neutrons and may be used to simulate select tallies that mirror those included in MCNP, such as average flux in a volumetric cell (F4 tally in MCNP). The hybrid methodology utilized in this study is known as the CADIS (Consistent Adjoint Driven Importance Sampling) hybrid method, which is a single tally region optimization method that relies on the discrete ordinates deterministic solver, Denovo, to generate an adjoint flux and associated importance map for particles simulated in a given environment. Once the adjoint flux and importance map are generated, particles are more likely to be born in regions of higher importance in the simulation environment. Omnibus code is used to couple the Denovo deterministic and Shift MC solver as part of the Exnihilo code suite to accelerate the Shift transport simulations. The CADIS hybrid method is known to be nearly 100,000 times more efficient than analog MC based on tally figure of merit (FOM) comparisons (Wagner et al., 2011).

In the CADIS method, particles are biased (increasing the number of sampled particles in order to decrease uncertainty). The biased source distribution is utilized such that particles are more likely to contribute to the total detector response and fall within the initial statistical weights (quantifies the tally contribution of particles in a simulation environment) generated from Denovo. The biased source distribution relationship is represented by the equations below (Wagner et al., 2011).

In the CADIS hybrid method, a weight window technique is applied by generating detailed space- and energy-dependent importance parameters (particle weights) and applying geometric splitting/roulette and energy splitting/roulette, while simultaneously controlling weight variations as a particle travels throughout an environment (Wagner et al., 2011). The deterministic Denovo solver generates the adjoint flux and associated weight window map; the Shift MC solver employs the stochastic splitting/roulette method in the phase space to determine the final contribution of particles in a tally region. The initial statistical weight of a particle is, w₀(,r→,E), is defined by the following equation (Wagner et al., 2011).

Upon initiation of the Shift MC solver, if a particle falls below a given weight in the importance map (3-D distribution of w₀(r→,E)), then the particle will be split (source biasing) or rouletted (terminated) in order to sufficiently increase its weight. This hybrid technique greatly increases the rate at which the Shift MC solver final solution converges and thus significantly increases computational efficiency.

4.3 Detector Details

Four sodium iodine (NaI) detector sizes are included in the CPM Calculator (0.5" × 1", 1" × 1", 2" × 2", 3" × 3"). Energy response curves (simulated) were created for each one using the pulse height energy deposition tally in MCNP (surface contamination) and MCNP/Shift method (volumetric contamination). The terms "pulse height", "response", and "detector response" are used interchangeably in the method to denote detector count rate. It is important to note that only the distribution of energy deposition pulses in the detector crystals are simulated; therefore, other factors that comprise the true detector response such as detector resolution, dead time, background, and pulse pile-up are unaccounted for. Nevertheless, the high-fidelity simulations of the energy deposition spectra result in conservative, yet scientifically rigorous screening value estimations of the detector count rate used to produce the GDR corresponding to a desired FAC or TAC.

The model assumes the detector is stationary and time is not considered. Any movement of the detector was not taken into account.

Schematics, images, and manufacturer instrument response curves are given below.

• Ludlum model 44-62 0.5x1 inch NaI detector

• Ludlum model 44-62 0.5x1 inch NaI detector schematic

• Ludlum model 44-2 1x1 inch NaI detector

• Ludlum model 44-2 1x1 inch NaI detector schematic

• Canberra model 802-2 2x2 inch NaI detector

• Canberra model 802-2 2x2 inch NaI detector schematic

• Ludlum model 44-10 2x2 inch NaI detector

• Ludlum model 44-10 2x2 inch NaI detector schematic

• Ludlum model 44-20 3x3 inch NaI detector

• Ludlum model 44-20 3x3 inch NaI detector schematic

4.4 Material Details

In this project, source data was obtained from a report published by Pacific Northwest National Laboratory, PIET-43741-TM-963 PNNL-15870, Rev. 1, Compendium of Material Composition Data for Radiation Transport Modeling, which is summarized in MCNP syntax in the table below (McConn et al., 2011). The exception is for the material definition of soil, which was taken from the Environmental Protection Agency's Federal Guidance Report 15 silty soil composition (Veinot et al., 2017).

Material	ZAID	Weight Fraction
Concrete (Regular)	1,000	-0.010000
	8,000	-0.532000
	11000	-0.029000
	13,000	-0.034000
	14,000	-0.337000
	20,000	-0.044000
	26,000	-0.014000
Soil (FGR 15)	1,000	-0.021
	6,000	-0.016
	8,000	-0.577
	13,000	-0.05
	14,000	-0.271
	19,000	-0.013
	20,000	-0.041
	26,000	-0.011
Steel (304 Stainless)	6,000	-0.000400
	14,000	-0.005000
	15,000	-0.000230
	16,000	-0.000150
	24,000	-0.190000
	25,000	-0.010000
	26,000	-0.70730
	28,000	-0.092500
Drywall (Gypsum/Plaster of Paris)	1,000	-0.023416
	8,000	-0.557572
	16,000	-0.186215
	20,000	-0.232797
Glass (Plate)	8,000	-0.459800
	11,000	-0.096441
	14,000	-0.336553
	20,000	-0.107205
Wood (Southern Pine)	1,000	-0.059642
	6,000	-0.497018
	7,000	-0.004970
	8,000	-0.427435
	12,000	-0.001988
	16,000	-0.004970
	19,000	-0.001988
	20,000	-0.001988

The following table presents densities, mass attenuation coefficients, and the resulting infinite four mean free path values for discrete photon emission energies (in the air above the contaminated source and surrounding the detector) to define contamination plane radius.

Energy (MeV)	Mass Attenuation Coefficient µ/ρ (cm2/g)	Linear Attenuation Coefficient µ (cm-1) -	Mean Free Path 1/µ (cm)	Plane Radius 4/µ (cm)
0.01	5.120	6.62E-03	151.05	604.22
0.02	0.778	1.01E-03	994.21	3976.84
0.03	0.354	4.57E-04	2185.97	8743.87
0.04	0.249	3.21E-04	3112.25	12449.02
0.05	0.208	2.69E-04	3718.25	14872.98
0.06	0.188	2.42E-04	4124.77	16499.10
0.07	0.177	2.29E-04	4389.09	17556.35
0.08	0.166	2.15E-04	4653.40	18613.60
0.1	0.154	1.99E-04	5018.79	20075.15
0.15	0.136	1.75E-04	5703.50	22814.02
0.2	0.123	1.59E-04	6272.47	25089.87
0.3	0.107	1.38E-04	7248.31	28993.26
0.4	0.095	1.23E-04	8099.23	32396.91
0.5	0.087	1.13E-04	8877.36	35509.42
0.6	0.081	1.04E-04	9601.43	38405.72
0.8	0.071	9.15E-05	10932.93	43731.71
1	0.064	8.22E-05	12164.13	48656.51
1.5	0.052	6.69E-05	14944.83	59779.34
2	0.044	5.75E-05	17391.39	69565.57
3	0.036	4.63E-05	21597.19	86388.74

4.5 Progeny and Chains

The CPM Calculator calculates the detector response for the primary radionuclide in 100-1000 years of secular equilibrium with its progeny. This is meaningful, especially in the common case of Cs-137 (the well-known 662 keV photon of Cs-137 is actually produced by its metastable progeny, Ba-137m). This feature can be deactivated by deselecting the check box beneath the radionuclide selection list.

4.6 Model Geometry and Physics

This section describes the setup of the contaminated sources and the detector geometry in MCNP/Shift.

4.6.1 Ground Plane

The geometry of the ground plane model is a 1 micron thick disc source above which a detector is suspended. The height (h) of the detector is the user's estimate of the distance in centimeters between the detector and the source of contamination. The maximum radius of the disc (R) is calculated such that the distance from the detector to the outer circumference of the circle is four mean free paths of the simulated photon energy in air (3 MeV in the example depiction below), a distance at which the photon is safely assumed to be attenuated.

Detector responses from ground plane sources were simulated in MCNP using high-performance computing (HPC) clusters. Simulations were run for 19 monoenergetic photon sources, each using 120 combinations of contamination material, detector configuration, and source-detector distance. In each simulation, a pulse height spectrum was estimated throughout the NaI crystal of the modeled detector. All final MCNP pulse height tally results were reported with relative errors of less than 5%.

4.6.2 Volumetric

The 6 different options for source material are soil, concrete, plate glass, wood, steel, and drywall. These materials are based on a uniformly contaminated cylindrical slab source of varying thickness. The average cell flux throughout a void-filled "coupling surface" mimicking the geometry of a detector suspended in air at a distance above the source is estimated, which would be converted to average pulse height in the corresponding full model detector in a post-processing code. An example of a Volume CPM model for a 3 MeV source contaminated in soil at a discrete depth (shown in red) is depicted in this diagram.

The total depth of a medium (red and brown regions) is always based on four mean free paths for a 3 MeV source in the medium of interest.

Average cell fluxes for the Shift volumetric contamination runs were simulated using high-performance computing (HPC) clusters. All final Shift average cell flux tally results were reported with relative errors of less than 5%.

4.7 MCNP and SHIFT Implementation

The MCNP pulse-height tally may be approximated by multiplying the energy-dependent fluence (or average cell flux tally result) by a user-provided detector response function that accounts for phase space dependencies such as energy and position of photons incident on the detectors. Since the detectors of interest have dimensions that are (approximately) isotropic, an angular detector response function is not needed to account for fluctuations in the direction of photons incident on the detectors. See the following equation.

Detector response functions (RFs) were generated by normalizing the MCNP estimated pulse height for surface contamination to the average cell flux estimated using Shift for the same modeling conditions (emission energy, contamination depth, contamination material, detector configuration, and source-detector distance). See the following equation.

The detector response for monoenergetic volumetric sources was calculated as the product of the energy-binned fluence and the energy dependent response function. See the following equation.

For a given radionuclide, the detector response for each photon emission energy was interpolated from the responses of the 19 monoenergetic photon sources for the same conditions (contamination depth, contamination material, detector configuration, and source-detector distance) using the piecewise cubic hermite interpolating polynomial (PCHIP) interpolation algorithm in a post-processing code. The interpolated response at each emission energy was multiplied by the corresponding decay yield of the emitted photon, and the final radionuclide-specific response for the given conditions was calculated as the sum of the product over all emission energies. It was assumed that photons with emission energies below 20 keV did not contribute to the final response and were, therefore, omitted from the calculation.

4.8 Calculating the Activity Concentration to CPM Conversion Factor

4.8.1 Ground Plane Source Contamination

The MCNP pulse heights were normalized, as presented in the equation below, to obtain the i^th monoenergetic or radionuclide ground plane source activity concentration to CPM conversion factor.

4.8.2 Volumetric Source Contamination

The MCNP/Shift responses (MCNP-equivalent pulse heights) were normalized, as presented in the equation below, to obtain the i^th monoenergetic or radionuclide volumetric source activity concentration to CPM conversion factor.

4.9 Converting the Field and Target Activity to CPM

Field Activity Concentration in CPM, cpm_FAC, and Target Activity Concentration in CPM, cpm_TAC, are found by multiplying the cpm per activity concentration conversion factor, CF, by the user's TAC in pCi/g for a radionuclide. If multiple primary radionuclides were selected, the FACs in pCi/g are also multiplied by the result. The equations for cpm_FACi and cpm_TACi for a radionuclide i may be seen below.

4.10 Calculating the Relative Fraction

The relative fraction, f, is the fraction of the total activity contributed by each radionuclide. The FACs are used to find the relative fractions of each radionuclide, which are then applied to the GDR. The equation for calculating the relative fraction for a radionuclide i may be seen below.

4.11 Calculating the Gross Detector Response

The GDR is the total calculated response of the detector in cpm for the desired remedial activity of the particular radionuclides in the soil. MARSSIM Equation 4-4 "Gross Activity DCGL" (U.S. EPA, 2000) is applied to find the GDR and can be seen in an edited form below.

where f is the relative fraction of each radionuclide and cpm_TAC is the TAC of each radionuclide in units of detector cpm.