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Efficient parameter estimation in multiresponse models measuring radioactivity retention

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Abstract

After incorporation of radioactive substances, workers are routinely checked by bioassays (isotopic activity excreted via urine, measurements of radionuclides retained in the whole body or in the lungs, etc.). From the results, the isotopic activity incorporated by the worker is inferred, as well as the values of other parameters related to the metabolism of the incorporated substance, using the ’response function’. This function depends on several factors and it is usually obtained by solving a system of linear differential equations, resulting from the compartmental model which describes the human body (or a part of it). The possibility of using different types of bioassays from the same worker improves estimation of some of the parameters that characterize the solution of the system of equations, specially the unknown incorporated activity to the system. The transfer coefficients are usually considered to be known, using the values that are published in the corresponding International Commission of Radiological Protection (ICRP) publication. In the present study some practical cases will be presented, and optimal design criteria are developed that allow taking the bio-samples at the most informative times. The methodology presented here requires solving the models of element distribution in the human organism as a function of time, for which the recently updated models recommended by the ICRP have been used. Initially thought for workers in facilities dealing with radioactive substances, the study results, procedures and conclusions can be applied to other clinical or laboratory settings, and to the design of action protocols in case of environmental public exposure.

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Acknowledgements

This research was supported by the Spanish Ministry of Education, Culture and Sports, and the Regional Government of Castilla y León (Projects ‘MTM 2013-47879-C2-2-P’, ‘MTM2016-80539-C2-2-R’ and ‘SA130U14’, ‘SA080P17’ respectively).

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Authors and Affiliations

  1. Department of Statistics, Faculty of Science, University of Salamanca, Pl. de los Caídos s/n, 37008, Salamanca, Spain

    J. M. Rodríguez-Díaz & G. Sánchez-León

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  1. J. M. Rodríguez-Díaz
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  2. G. Sánchez-León
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Correspondence to J. M. Rodríguez-Díaz.

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Appendix

Appendix

Cobalt model

For inhalation both the HRTM and the cobalt models are interconnected (Figs. 7, 8, Table 5).

Fig. 7
figure 7

Human respiratory tract model (HRTM ICRP 130)

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Fig. 8
figure 8

Cobalt model (ICRP 134 2016), including the Human Alimentary Tract Model HATM (ICRP 100 2006)

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Table 5 Absorption (default) parameters for inhaled and ingested cobalt (ICRP 134)
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The information matrix of the one-point scenario can be written as:

$$\begin{aligned} M_1=\frac{1}{1-S_{12}^2}\left( \begin{array}{cc} m_{11} &{} I\; m_{12} \\ I\; m_{21} &{} I\; m_{22}\\ \end{array} \right) , \end{aligned}$$

with

$$\begin{aligned} m_{11} &= (w-1)^2 {f_1^M}(t)^2 -2 (w-1) (w {f_1^S}(t)+s (w-1) {f_2^M}(t)-s w {f_2^S}(t)) {f_1^M}(t) \\& \quad + w^2 {f_1^S}(t)^2+(-w {f_2^M}(t)+{f_2^M}(t)+w {f_2^S}(t))^2 \\& \quad - 2 s w {f_1^S}(t) (-w {f_2^M}(t)+{f_2^M}(t)+w {f_2^S}(t)) , \\ m_{12} &= (w-1) {f_1^M}(t)^2 +((1-2 w) {f_1^S}(t)-s (2 (w-1) {f_2^M}(t)-2 w {f_2^S}(t)+{f_2^S}(t))) {f_1^M}(t) \\& \quad + w {f_1^S}(t)^2+s {f_1^S}(t) ((2 w-1) {f_2^M}(t)-2 w {f_2^S}(t)) \\& \quad + ({f_2^M}(t)-{f_2^S}(t)) ((w-1) {f_2^M}(t)-w {f_2^S}(t)), \\ m_{21} &= (1-w) {f_1^M}(t)^2 +((2 w-1) {f_1^S}(t)+s (2 (w-1) {f_2^M}(t)-2 w {f_2^S}(t)+{f_2^S}(t))) {f_1^M}(t) \\& \quad - w {f_1^S}(t)^2-({f_2^M}(t)-{f_2^S}(t)) ((w-1) {f_2^M}(t)-w {f_2^S}(t)) \\& \quad + s {f_1^S}(t) (-2 w {f_2^M}(t)+{f_2^M}(t)+2 w {f_2^S}(t)) , \\ m_{22} &= {f_1^M}(t)^2-2 ({f_1^S}(t) +s ({f_2^M}(t)-{f_2^S}(t))) {f_1^M}(t) \\& \quad + {f_1^S}(t)^2+({f_2^M}(t)-{f_2^S}(t))^2+2 s {f_1^S}(t) ({f_2^M}(t)-{f_2^S}(t)) . \end{aligned}$$

The whole body model for solubilities S and M can be written as:

$$\begin{aligned} f_1^S &= -6755.78 {\text {e}}^{-2.100 t}+317654 {\text {e}}^{-2.020 t}-567862 {\text {e}}^{-2.010 t}+ 256966 {\text {e}}^{-2.000 t}+ 0.000353 {\text {e}}^{-1.200 t}- 4.742\times 10^{-6} {\text {e}}^{-1.003 t}- 7.880\times 10^{-6} {\text {e}}^{-1.00036 t}-0.0000158 {\text {e}}^{-0.668 t}- 3.782\times 10^{-7} {\text {e}}^{-0.462 t}+0.000602 {\text {e}}^{-0.331 t}+0.0152 {\text {e}}^{-0.200 t}- 9.842\times 10^{-12} {\text {e}}^{-0.0994 t}+0.000286 {\text {e}}^{-0.0769 t}+0.0000762 {\text {e}}^{-0.0129 t}+ 0.0357 {\text {e}}^{-0.00346 t}+ 0.000131 {\text {e}}^{-0.00217 t}+0.0000362 {\text {e}}^{-0.00127 t}- 0.0000223 {\text {e}}^{-0.000848 t}+0.0173 {\text {e}}^{-0.000460 t}+ 0.00104 {\text {e}}^{-0.000441 t} \\ f_1^M &= -6546.61 {\text {e}}^{-2.100 t}+307970 {\text {e}}^{-2.020 t}-550585 {\text {e}}^{-2.010 t}+ 249162 {\text {e}}^{-2.00036 t}+0.00688 {\text {e}}^{-1.200 t}-0.0000998 {\text {e}}^{-1.00336 t}- 0.000158 {\text {e}}^{-1.00036 t}-0.000315 {\text {e}}^{-0.668 t}-7.499\times 10^{-6} {\text {e}}^{-0.462 t}+ 0.0119 {\text {e}}^{-0.331 t}+0.0122 {\text {e}}^{-0.205 t}+ 2.0657\times 10^{-8} {\text {e}}^{-0.0994 t}+ 0.00538 {\text {e}}^{-0.0769 t}-0.00137 {\text {e}}^{-0.0129 t}+0.0302 {\text {e}}^{-0.00836 t}+ 0.0137 {\text {e}}^{-0.00536 t}+0.00306 {\text {e}}^{-0.00217 t}+0.00305 {\text {e}}^{-0.00127 t}+ 0.000743 {\text {e}}^{-0.000848 t}+0.000641 {\text {e}}^{-0.000441 t} \end{aligned}$$

The 24-urine excretion model is as follows:

$$\begin{aligned} f_2^S &= 0.00144 {\text {e}}^{-2.100 t}+0.0000496 {\text {e}}^{-1.200 t}+0.000128 {\text {e}}^{-1.003 t}+ 0.0000650 {\text {e}}^{-t}+0.0000126 {\text {e}}^{-0.668 t}- 2.220\times 10^{-7} {\text {e}}^{-0.462 t}+ 0.000123 {\text {e}}^{-0.331 t}+ 2.405\times 10^{-6} {\text {e}}^{-0.200 t}+0.0000174 {\text {e}}^{-0.0769 t}+ 7.689\times 10^{-7} {\text {e}}^{-0.0129 t}+ 2.520\times 10^{-6} {\text {e}}^{-0.00346 t}+ 1.908\times 10^{-7} {\text {e}}^{-0.00217 t}+ 2.633\times 10^{-8} {\text {e}}^{-0.00127 t}- 9.073\times 10^{-9} {\text {e}}^{-0.000848 t}+ 1.396\times 10^{-6} {\text {e}}^{-0.000460 t}+ 6.908\times 10^{-8} {\text {e}}^{-0.000441 t}\\ f_2^M &= 0.0284 {\text {e}}^{-2.100 t}+0.000998 {\text {e}}^{-1.200 t}+0.00257 {\text {e}}^{-1.00336 t}+ 0.00130 {\text {e}}^{-1.00036 t}+0.000250 {\text {e}}^{-0.668 t}- 4.404\times 10^{-6} {\text {e}}^{-0.462 t}+ 0.00242 {\text {e}}^{-0.331 t}+0.0000598 {\text {e}}^{-0.205 t}+ 1.736\times 10^{-9} {\text {e}}^{-0.0994 t}+ 0.000328 {\text {e}}^{-0.0769 t}-0.0000138 {\text {e}}^{-0.0129 t}+0.000125 {\text {e}}^{-0.00836 t}+ 0.0000552 {\text {e}}^{-0.00536 t}+ 4.466\times 10^{-6} {\text {e}}^{-0.00217 t}+ 2.233\times 10^{-6} {\text {e}}^{-0.00127 t}+ 2.919\times 10^{-7} {\text {e}}^{-0.000848 t}+ 4.198\times 10^{-8} {\text {e}}^{-0.000441 t} \end{aligned}$$

Uranium model

$$\begin{aligned} f_{12} &= {\text {e}}^{-310.76 t} (2.362\times 10^{-6} {\text {e}}^{209.76 t} + 1.219\times 10^{-7} {\text {e}}^{210.759 t}- 0.0000152 {\text {e}}^{272.81 t} + 0.0000635 {\text {e}}^{290.19 t}- 0.000284 {\text {e}}^{298.76 t} + 0.000170 {\text {e}}^{299.76 t}- 0.0000388 {\text {e}}^{300.759 t}- 0.000374 {\text {e}}^{304.756 t} + 0.000133 {\text {e}}^{305.382 t}+ 0.000452 {\text {e}}^{308.66 t} + 0.000206 {\text {e}}^{309.56 t} + 0.000787 {\text {e}}^{309.757 t} + 0.000401 {\text {e}}^{309.76 t}+ 0.0000194 {\text {e}}^{310.416 t}+ 0.0000144 {\text {e}}^{310.559 t} + 0.0000592 {\text {e}}^{310.621 t} + 0.0000148 {\text {e}}^{310.661 t} + 0.000384 {\text {e}}^{310.663 t} + 0.000116 {\text {e}}^{310.728 t} - 6.678\times 10^{-6} {\text {e}}^{310.747 t}+ 0.00249 {\text {e}}^{310.756 t} + 0.000629 {\text {e}}^{310.759 t} + 0.000989 {\text {e}}^{310.759 t} + 4.326\times 10^{-6} {\text {e}}^{310.76 t}+ 1.790\times 10^{-6} {\text {e}}^{310.76 t}+ 8.012\times 10^{-6} {\text {e}}^{310.76 t} + 3.786\times 10^{-7} {\text {e}}^{310.76 t} )\\ f_{13} &= {\text {e}}^{-310.76 t} (0.000477 {\text {e}}^{309.56 t} + 0.00136 {\text {e}}^{309.757 t} + 0.000689 {\text {e}}^{309.76 t} + 7.971\times 10^{-} {\text {e}}^{310.416 t}+ 3.196\times 10^{-6} {\text {e}}^{310.559 t}+ 8.823\times 10^{-6} {\text {e}}^{310.621 t} + 1.545\times 10^{-6} {\text {e}}^{310.661 t} + 0.0000392 {\text {e}}^{310.663 t}+ 3.808\times 10^{-6} {\text {e}}^{310.728 t} - 9.2\times 10^{-8} {\text {e}}^{310.747 t} + 0.0000173 {\text {e}}^{310.756 t} + 3.307\times 10^{-6} {\text {e}}^{310.759 t} + 5.187\times 10^{-6} {\text {e}}^{310.759 t} + 2.247\times 10^{-8} {\text {e}}^{310.76 t} + 9.123\times 10^{-9} {\text {e}}^{310.76 t} + 4.039\times 10^{-8} {\text {e}}^{310.76 t} + 1.897\times 10^{-9} {\text {e}}^{310.76 t} ) \\ f_{22} &= 0.000267 {\text {e}}^{-1.2 t}+0.0016 {\text {e}}^{-1. t}+0.0517 {\text {e}}^{-0.201 t}+ 5.024 {\text {e}}^{-0.0035 t}+3.284 {\text {e}}^{-0.0005 t} \\ f_{23} &= 0.00032 {\text {e}}^{-1.2 t}+0.0016 {\text {e}}^{-1. t}+0.0104 {\text {e}}^{-0.201 t}+ 0.0349 {\text {e}}^{-0.0035 t}+ 0.0173 {\text {e}}^{-0.0005 t} \end{aligned}$$

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Rodríguez-Díaz, J.M., Sánchez-León, G. Efficient parameter estimation in multiresponse models measuring radioactivity retention. Radiat Environ Biophys 58, 167–182 (2019). https://doi.org/10.1007/s00411-019-00780-7

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