# Parameter uncertainty analysis of a biokinetic model of caesium

W.B. Li, W. Klein, Eric Blanchardon, M. Puncher, R.W. Leggett, U. Oeh, B.

Breustedt, D. Nosske, M.A. Lopez

► To cite this version:

W.B. Li, W. Klein, Eric Blanchardon, M. Puncher, R.W. Leggett, et al. Parameter uncertainty analysis of a biokinetic model of caesium. Radiation Protection Dosimetry, 2014, 163 (1), pp.37-57. 10.1093/rpd/ncu055. hal-02611869

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# PARAMETER UNCERTAINTY ANALYSIS OF A BIOKINETIC MODEL OF CAESIUM

W.B. Li¹*, W. Klein², E. Blanchardon³, M. Puncher⁴, R.W. Leggett⁵, U. Oeh¹, B. Breustedt², D. Noßke⁶, M.A. Lopez⁷

¹HMGU-Research Unit Radiation Physics and Diagnostics, Helmholtz Zentrum München - German Research Center for Environmental Health (GmbH), D-85764 Neuherberg, Germany

²KIT-Institute of Radiation Research, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany

³IRSN-Internal Dose Assessment Lab., PRP-HOM/SDI/LEDI, BP-17 F-92262 Fontenay-aux-Roses Cedex, France

⁴HPA Centre for Radiation, Chemical and Environmental Hazards, Chilton, Didcot OX11 0RQ, UK.

⁵Environmental Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

⁶BfS-Department of Radiation Protection and Health, Ingolstädter Landstr. 1, 85764 Oberschleißheim, Germany

⁷CIEMAT-Dosimetría Interna, Departamento de Medio Ambiente, Avda Complutense 40, 28040 Madrid, Spain

*Received month date year, amended month date year, accepted month date year*

Parameter uncertainties for the biokinetic model of caesium developed by Leggett and co-workers were inventoried and evaluated. The methods of parameter uncertainty analysis were used to assess the uncertainties of model predictions with the assumptions of model parameter uncertainties and distributions. Furthermore, the importance of individual model parameters was assessed by means of sensitivity analysis. The calculated uncertainties of model predictions were compared with human data of Cs measured in blood and in the whole body. It was found that propagating the derived uncertainties in model parameter values reproduced the range of bioassay data observed in human subjects at different times after intake. The maximum ranges, expressed as uncertainty factors (defined as a square root of ratio between 97.5th and 2.5th percentiles) of blood clearance, whole body retention and urinary excretion of Cs predicted at earlier time after intake were respectively: 1.5, 1.0 and 2.5 at the first day; 1.8, 1.1 and 2.4 at day 10; 1.8, 2.0 and 1.8 at day 100; for the late times (1000 days) after intake, the uncertainty factors were increased to 43, 24 and 31, respectively. The model parameters of transfer rates between kidneys and blood, muscle and blood, and the rate of transfer from kidneys to urinary bladder content are most influential to the blood clearance and to the whole body retention of Cs. For the urinary excretion, the parameters of transfer rates from urinary bladder content to urine and from kidneys to urinary bladder content impact mostly.

## INTRODUCTION

Biokinetic models are essential for calculations of internal radiation dose and risk estimation for occupational workers, members of the public and patients exposed to radionuclides; they therefore underpin regulatory policy decisions in radiation protection. In addition, biokinetic models serve as tools for interpretation of in-vitro bioassay analysis (e.g. measurement in blood, urine, faeces, hair and nails), and in-vivo monitoring (e.g. whole body and specific organ monitoring).

Usually, models incorporating human physiological parameters are needed to estimate the content and behaviour of radionuclides in the human body. Human organs and tissues are simplified in the model as separate compartments; the translocations of materials between the organs and tissues are described by transfer rates, commonly designated as model

parameters. These parameters are derived mostly from experimental data obtained from studies of laboratory animals and from a follow-up of accidental human contamination cases. Because of the physiological variability of individuals, the uncertainty in the extrapolation of animal data to humans, the uncertainties in measurement techniques, the sparsity or inconsistency in reported data and the choice of parameter values, the values of parameters in biokinetic models are often subject to significant uncertainties(1-5).

A quantitative analysis of the uncertainties in the model inputs and outputs, including bioassay predictions and doses, can help to guide regulators in setting appropriate limits on intakes(6). The information on parameter sensitivity can help to guide national and international organizations and stakeholders on decisions regarding where to invest scientific efforts for updating the models(7). With the information both on the uncertainty and sensitivity of the model parameters and model predictions, policy decision

<footnote>*Corresponding author: wli@helmholtz-muenchen.de</footnote>
makers and monitoring service personnel can make proper decisions and adapt monitoring programs in order to provide uncertainties of doses for occupational workers<sup>(8)</sup>. Moreover, quantifying uncertainties on doses that result from sources of uncertainty in the biokinetic models can be used to assess the reliability of dose coefficients as a radiation protection device; that is, supporting their use in ensuring exposures comply with regulatory dose limits and constraints<sup>(9)</sup>. Finally, the uncertainty on the assessed doses is an important input to the study of the resulting health risk.

For an overall reliability analysis and quality control of the biokinetic models, it is therefore useful to develop a strategy to quantify the uncertainties and identify the model parameters that strongly influence the model outputs pertinent to the assessment of internal doses and in the monitoring of internal exposures. By doing so, the model developers and users can help prepare for unexpected situations (i.e., retrospective internal dose estimation after an accident) and apply the information to help in the experimental design to acquire the relevant parameters.

In the framework of the EURADOS network on internal dosimetry<sup>(10, 11)</sup>, a task group on uncertainties of internal dose assessment has been established. This group aims to review the basis of the biokinetic models, including uncertainties in model structure and parameter values, bioassay measurements, conditions of exposure and methods for dose assessment in order to evaluate uncertainties in doses and other model outputs using different methods, e.g. frequentist<sup>(12-14)</sup> and Bayesian methods<sup>(15-17)</sup>. In this paper, the biokinetic model of Cs developed by Leggett and coworkers<sup>(18)</sup> was analysed by means of a global uncertainty and sensitivity analysis. This model formed the basis of the slightly modified model for radiocaesium<sup>(19)</sup> and the new proposed ICRP model in its coming publication - Occupational Intakes of Radionuclides Part 3. A local sensitivity analysis was performed as well for comparisons. First, the parameter uncertainties were evaluated from biokinetic data available in the literature. The uncertainties of predicted retention and excretion of Cs in humans were calculated and the most influential model parameters that affect model predictions were identified. The benefits of the uncertainty and sensitivity analysis, and some limitations or difficulties in the approach with respect to situations where experimental data are sparse, are discussed.

## MATERIALS AND METHODS

In this section uncertainties in the values of parameters in the Cs biokinetic model, described by Leggett et al

(2003), are evaluated. An uncertainty analysis method is used to calculate the uncertainty in model predictions. Furthermore, a global and a local sensitivity analysis are used to identify the most influential parameters affecting blood clearance, whole body retention and urinary excretion of Cs.

## Some concepts used in the present study

In the present study, several concepts are used to quantify the uncertainty of model parameters and the model predictions:

**Standard uncertainty:** The concept of uncertainty defined by ISO associated with the result of measurement<sup>(20)</sup> was applied. The standard uncertainty refers to a standard deviation (SD) of a quantity, say, a model parameter.

**Uncertainty factor (UF):** A subjective confidence interval [A, B] was determined for a model feature, the true but unknown value was judged with reasonably high confidence, say 95%, to lie between A (2.5th percentile) and B (97.5th percentile). The associated UF is defined<sup>(21)</sup> as $(B/A)^{1/2}$. Another definition<sup>(5)</sup> of UF is the maximum value of C/A and B/C, where the value of C might be selected as the 50th percentile of a distribution. The first definition of $UF=(B/A)^{1/2}$ was used in the present study.

**Partial rank correlation coefficient (PRCC):** A measure of the rank correlation between one defined output variable with an input variable, under the condition that the indirect influence on this defined output variable due to other further input variables is somehow eliminated<sup>(22, 23)</sup>. PRCC takes a value from [-1, 1], a value near ±1 indicates a strong influence; a value near 0 shows a weak influence.

**Coefficient of variation (CV)<sup>(24)</sup>:** A positive random variable; it is the standard deviation divided by the mean.

**Confidence interval (CI)<sup>(24)</sup>:** Interval estimator ($T_0$, $T_1$) for the parameter $\theta$ with the statistics $T_0$ and $T_1$ as interval limits and for which it holds that $P(T_0 < \theta < T_1) \ge 1 - \alpha$, where $\alpha$ is generally a small number.

## Caesium model structure and parameter

The biokinetic model structure for Cs proposed by Leggett et al.<sup>(18)</sup> is shown in Figure 1. There are 24 compartments and 50 parameters in the model. The retention and excretion of Cs in the compartmental model are described by a system of first-order linear ordinary differential equations.

## Parameter uncertainty analysis

The principal method used for evaluating the uncertainty of model parameters in the present work is
to follow the steps involved in the derivation of the parameters of the biokinetic model for Cs. The uncertainties of the individual components which constitute the model parameter are then determined. The combined uncertainty of each parameter, which is derived from the different components, is thereby calculated by the law of uncertainty propagation(20).

Most of the model parameters of Cs are derived by addition, like p=p₁+p₂ or multiplication, like p=p₁×p₂ or division, like p=p₁/p₂ of several uncertain components or factors. According to the law of uncertainty propagation(20, 25):

$$ \sigma^2 = \sigma_1^2 + \sigma_2^2 + 2\sigma_{12} \quad (1) $$

for addition,

$$ (\sigma/p)^2 = (\sigma_1/p_1)^2 + (\sigma_2/p_2)^2 + 2\sigma_{12}^2/p_1p_2 \quad (2) $$

for multiplication,

$$ (\sigma/p)^2 = (\sigma_1/p_1)^2 + (\sigma_2/p_2)^2 - 2\sigma_{12}^2/p_1p_2 \quad (3) $$

for division,

$$ \sigma_{12} = \sigma_1 \sigma_2 r_{12} \quad (4) $$

for covariance and correlation,

where p and pᵢ are respectively model parameter and component or factor i, from which the model parameter is derived; σ and σᵢ are standard deviation of model parameter and component or factor i, respectively. σᵢⱼ and rᵢⱼ are covariance and correlation of components or factors i and j. The correlation rᵢⱼ normally takes a value from [-1, 1], with a negative value it indicates a negative correlation between components i and j.

A degree of correlation is introduced in the present study between uptakes of Cs from plasma by the fact that transfer rates from plasma to tissue are related to cardiac output. A similar correlation between the equilibrium contents of Cs in plasma and in organs or tissues are assumed, this is related to the extraction fraction of Cs in tissues. The correlation between model parameters is usually measured by the correlation coefficient rᵢⱼ. Because of lack of experimental data for estimation of the correlation coefficient, a value of 0.9 or -0.9 is assumed for taking into account an extremely strong positive and negative correlation.

For convenience, the 50 parameters of the systemic model of Cs were classified into four groups: (1) parameters affecting transfer of Cs from plasma to tissues, (2) parameters affecting transfer of Cs from tissues to plasma, (3) parameters affecting transfer of Cs through and from the gastrointestinal tract tissues, and (4) other parameters. Derivation of the uncertainty

of the value of each parameter is described in the following sections.

### Transfer rates from plasma to tissue

There are 19 parameters representing the transfer rates of Cs flowing from plasma to organs or tissues. Those transfer rates are derived from three factors: (1) cardiac output (CO) (in unit of plasma volumes per day, PVPD); (2) fraction of cardiac output (FCO) to the organ or tissue and (3) extraction fraction (denoted as E<sub>Cs</sub>) of Cs from blood in organs or tissues. The multiplication of CO×FCO (denoted as r<sub>blood→tissue</sub>=CO×FCO) represents the flow rate from blood to the organs or tissue:

$$ k_{\text{plasma→tissue}} = r_{\text{blood→tissue}} \times E_{\text{Cs}} = CO \times FCO \times E_{\text{Cs}} \quad (5) $$

where k<sub>plasma→tissue</sub> is the transfer rate from plasma to organ or tissue. It is assumed that there is no correlation among these three quantities: CO, FCO and E<sub>Cs</sub> for each organ or tissue.

The value of CO in unit of PVPD can be calculated as following from two quantities, the total blood volume (5300 ml) and the cardiac output (6500 ml min⁻¹): 6500 (ml min⁻¹)×24 (h/d)×60 (min/h)/5300 (ml per cardiac output) = 1766 d⁻¹.

The reference values and uncertainties of cardiac output for adult males (6500±1500 ml min⁻¹) and females (5900±1400 ml min⁻¹) were reported by Williams(26) and ICRP(27). In the same reference, the reference values and uncertainties of blood volume for adult female (3900±600 ml) and male (5300±900 ml) were also given. In the present work, the reference value and uncertainty (6500±1500 ml for cardiac output and 5300±900 ml for one blood volume) for adult males are used in the analysis for a reference model of the adult male. Consequently, the uncertainty of CO in unit of PVPD is calculated to be 505 volumes d⁻¹ according to Eq.(2). The uncertainties of FCO and E<sub>Cs</sub> for each organ and tissue are evaluated based on the data used for the derivation of parameters by Leggett et al. 2003(18). These ranges reflect inter-subject variability – the range of values in the population, ultimately the “aleatory” uncertainties reproduce variability in the human dataset.

The extraction fraction of Cs in tissue, E<sub>Cs</sub> is defined as the fraction of atoms extracted by the tissue in passage from arterial to venous plasma. The extraction fractions estimated in this work are intended as effective values over a period of minutes rather than ‘first-pass’ extraction fractions. As mentioned earlier, this fraction introduces a correlation of equilibrium contents of Cs in different tissues or organs. The extraction fraction is estimated by scaling the values reported in the previous
work on K and Rb by factors of 0.25 and 0.3(18). The range of E<sub>Rb</sub> for kidneys is assumed as 0.80-0.95 with mean value of 0.89(28). E<sub>K</sub> for kidneys is assumed as 0.8 to 1.0(29) with mean value of 0.9(28, 30), the lower value of 0.8 is assigned here in order to obtain a mean value of 0.9. The uncertainty of extraction fraction estimated in this way is applicable to GI tract tissue and heart(18). The upper or lower values are generally introduced to preserve the reported mean value with a 95% confidence interval for a parameter.

E<sub>Cs</sub> for liver is assumed to be 0.04 as minimum, 0.05 as mean and 0.06 as maximum(18). The mean value of 0.05 is calculated as one-fourth of the value for myocardium which is available. The max and min values of 0.06 and 0.04 are assumed in order to obtain the mean value of 0.05.

E<sub>Cs</sub> for brain is assumed to be 0.001 as minimum (this value is not given and it is assumed in order to obtain the mean value of 0.002), 0.002 as mean and 0.003 as maximum(18).

Data of E<sub>Cs</sub> for skin is limited in the literature. The SD is assumed to be the same as liver 0.004. The mean value of 0.05 was reported by Leggett et al. 2003(18).

The values of E<sub>Cs</sub> for all other tissues ranged from 0.15 to 0.2(18). The mean of 0.1 is assumed for all other tissues. We assumed 0.3 as a upper value and 0.01 as a lower value.

The rates of transfer of Cs to and from red blood cells (RBC) is assumed to be slower than the transfer rates of 6 d<sup>-1</sup>, which is evaluated for K in a range of 5.3 – 6.5 d<sup>-1</sup>(30). In terms of fractional transfer of the plasma content per unit time, transfer of Cs into RBC appears to be roughly 0.3 times that of K in humans, rabbits and rats (31-33). The factor of 0.3 is applied to humans and, a mean value of 1.8 d<sup>-1</sup> with a range from 1.59 d<sup>-1</sup> to 1.95 d<sup>-1</sup>, is obtained for transfer from plasma to RBC.

Applying Eq.(1), the uncertainty of model parameter values of k<sub>plasma→tissue</sub> can then be calculated based on combining (propagating) the uncertainties from the above three components (see Table 1).

### Transfer rates from tissues to plasma

There are 17 transfer rates which flow back to plasma from organs and tissues. The translocation of Cs in different organs is described by a system of first-order process (Figure 2):

$$dA(t)/dt=aP(t)-bA(t) \qquad (6)$$

where A(t) and P(t) are time-dependent content of Cs in organs X and Y.

At equilibrium dA(t)/dt=0, A(t) and P(t) can be reduced to constant A and P representing the fractions of whole body Cs at equilibrium. The transfer rates from tissue to plasma can be estimated by the transfer rate from plasma to tissue and the equilibrium contents of Cs in plasma and in tissue:

$$k_{\text{tissue→plasma}}=(C_{\text{tissue}}/C_{\text{plasma}})\times k_{\text{plasma→tissue}} \qquad (7)$$

where C<sub>tissue</sub> and C<sub>plasma</sub> are the equilibrium content of Cs in tissue and in plasma, respectively. The quantities of C<sub>tissue</sub> and C<sub>plasma</sub> are negatively correlated.

The uncertainty of the ratio of C<sub>tissue</sub>/C<sub>plasma</sub> is evaluated by the range of the data used for C<sub>tissue</sub> and C<sub>plasma</sub> which are presented in Table 5 of Williams and Leggett 1987(34). The uncertainty of model parameter k<sub>tissue→plasma</sub> is calculated from the uncertainty of the ratio of C<sub>tissue</sub>/C<sub>plasma</sub> and the uncertainty of k<sub>plasma→tissue</sub> according to Eq.(2) and Eq.(3).

The uncertainty of model parameters in this category is shown in Table 2 with the uncertainty contribution from each component.

### Transfer rates related to gastrointestinal (GI) tract

There are 10 parameters of transfer rates related to GI tract.

Parameter k<sub>GI tract tissue→liver</sub>: The uncertainty of k<sub>plasma→GI tract tissue</sub> is known (see Table 1). The uncertainty of total outflow rate from GI tract tissue can be evaluated from the uncertainties of k<sub>plasma→GI tract tissue</sub> and fractions of Cs in plasma and in GI tract tissues at equilibrium. This uncertainty is combined with the uncertainty of the extraction fraction 0.05 for liver to represent the uncertainty of the parameter k<sub>GI tract tissue→liver</sub>.

Parameters k<sub>GI tract tissue→St</sub>, k<sub>GI tract tissue→SI</sub> and k<sub>GI tract tissue→LI</sub>: In the model, the cell sloughing from GI tract walls to the contents of the GI tract is depicted as a set of transfer rates from GI tract tissue to stomach (St), small intestine (SI) and large intestine (LI). The transfer rates of GI tract tissue→St, GI tract tissue→SI and GI tract tissue→LI are calculated as daily flow of substances 40 g d<sup>-1</sup> (for St), 130 g d<sup>-1</sup> (for SI) and 80 g d<sup>-1</sup> (for LI) divided by the mass of the GI tract tissue (1200 g) (Table 1 in reference(28)). The uncertainty of flow rate is not explicitly given in Leggett and Williams 1988(28) and it is calculated with an assumption of CV=10%. Uncertainty of mass of GI tract tissue is taken from ICRP Publication 89(27) (1140
g for female and 1210 g for male) and ICRP
Publication 56<sup>(35)</sup> (1160 g for reference man).

Parameter $k_{\text{St→SI}}$: The absorption from the stomach to blood is assumed to be negligible, the outflow rate from stomach to the small intestines is assumed to be the same as for K and Rb, 40 d⁻¹ (28, 30). This value is larger than the ICRP Publication 30<sup>(36)</sup> value of 24 d⁻¹ but more consistent with the value of 48 d⁻¹ which is reported by ICRP Publication 100<sup>(37)</sup> for non-caloric liquid. In this paper, a uniform distribution for this parameter is assumed with lower and upper values of 14 and 48 d⁻¹, respectively (see ICRP Publication 100<sup>(37)</sup>), consequently, the mean value is 31 d⁻¹, which is lower than the value of 40 d⁻¹ as designated by Leggett et al.<sup>(18)</sup>

Parameter $k_{\text{SI→LI}}$: The total rate of removal of Cs from the small intestine contents is reported as 30 d⁻¹ which is the same as the values for K and Rb<sup>(18, 28, 30)</sup> without indication on uncertainty. It is assumed 1% or 2% of the outflow from small intestine contents goes to large intestine contents<sup>(18)</sup>. In this paper, a CV of 10% is assumed to the parameter, $k_{\text{SI→LI}}=0.3$ d⁻¹.

Parameter $k_{\text{SLIVER}}$: This transfer rate is derived from three factors: outflow from small intestine to plasma 99%, total removal rate from small intestine 30 d⁻¹ and the extraction fraction for liver 0.05: 0.99x0.05×30=1.485 d⁻¹. The absorption fraction of Cs to plasma is assumed to be a constant value of 99%. The uncertainty of the value 30 d⁻¹ is assumed to be represented by a CV=10% and the uncertainty on the fraction 0.05 is represented by a uniform distribution with minimum and maximum values 0.04 and 0.06, respectively.

Parameter $k_{\text{LIVER→SI}}$: Biliary secretion of Cs is depicted as a transfer from liver to small intestine<sup>(18)</sup>. However, this value is derived from the total transfer rate from the liver multiplying the loss of 5% from liver to bile. The transfer rate from liver to small intestine contents in bile is then 0.05×2.32 d⁻¹=0.116 d⁻¹, where 2.32 d⁻¹ is the total outflow from liver, which is calculated by multiplying the total inflow rate of 23.197 d⁻¹ to the liver by the ratio of Cs in plasma and in liver at equilibrium<sup>(18)</sup>. The transfer rate of 23.197 d⁻¹ is the sum of the transfer rate, $k_{\text{plasma→liver}}=19.515$ d⁻¹ and other inflow into liver. The uncertainty of this parameter $k_{\text{liver→SI}}$ is evaluated by the uncertainties of $k_{\text{plasma→liver}}$ and the loss fraction of 5% (with a CV=10%) from liver to bile.

Parameter $k_{\text{Pancreas→liver}}$: This parameter is derived from the parameter of $k_{\text{plasma→pancreas}}=1.766$ d⁻¹ and the fraction uptake of 0.05 by liver from the portal vein<sup>(18)</sup>. The uncertainty of this parameter can easily be

evaluated by combining the uncertainties on these two
components.

Parameter $k_{\text{Spleen→liver}}$: Similar to the parameter of $k_{\text{pancreas→liver}}$, the uncertainty of this parameter $k_{\text{liver}}$ is derived from the uncertainties of parameter of $k_{\text{plasma→liver}}=5.298$ d⁻¹ and the fraction uptake of 0.05 by liver from the portal vein<sup>(18)</sup>.

The derived uncertainties and some further discussion
are provided in Table 3.

Other transfer rates

There are 4 parameters which are not included in the
above discussion: these are the transfer rates: (1) from
kidneys to UBC, (2) from UBC to urine, (3) from large
intestine to faeces and (4) from skin to sweat. Those
parameters are evaluated in the following.

The uncertainty on the transfer rate from kidneys to UBC is derived from two factors: the flow fraction from kidneys to UBC and the total outflow rate from kidneys to all destinations: 0.05×33.554 d⁻¹=1.678 d⁻¹, where 33.554 d⁻¹ is obtained from the transfer rate from plasma to kidneys and the fraction of Cs in plasma and in kidneys at equilibrium. 0.05 is the removal fraction from kidney to UBC. The uncertainty of each component was propagated to the uncertainty of this parameter, $k_{\text{kidneys→UBC}}$.

The rate $k_{\text{UBC→urine}}$ from UBC to urine is assumed to be 12 d⁻¹, in accordance with ICRP Publication 67<sup>(38)</sup>. The uncertainty is assumed to be represented by a triangular distribution with minimum of 6, vertex of 12 and maximum value of 22 as assumed by Li et al.<sup>(12)</sup> for zirconium and by Apostoaei<sup>(39)</sup> for strontium. The modelling uncertainty by using first order kinetics for the urinary bladder voiding instead of a realistic filling and voiding kinetics is less than 0.1% and can be neglected.

The mean transfer rate $k_{\text{LI→faeces}}$, from large intestine to faeces is assumed to be 0.5 d⁻¹ in the model<sup>(18)</sup>; however, the transfer rate of 1.8 d⁻¹ from upper large intestine to lower large intestine and the transfer rate of 1.0 d⁻¹ from lower large intestine to faeces are recommended by ICRP<sup>(36)</sup>; the rate from large intestine to faeces therefor can be calculated as about 0.6 d⁻¹. In the new alimentary human tract model<sup>(37)</sup> the transfer rate from large intestine to faeces can be estimated from the rates from right to left colon, and left colon to rectosigmoid and rectosigmoid to faeces: 0.67 d⁻¹. In the present analysis, a triangular distribution with minimum value and vertex of 0.5 d⁻¹, and maximum value of 1 d⁻¹, is assumed. This distribution gives a mean value of 0.67 d⁻¹ and the assumption of such a
triangular distribution means the modal value of 0.5 d⁻¹ is sampled with the highest frequency.

The uncertainty of excretion of Cs in skin to sweat is calculated by two uncertainty factors: the outflow fraction from skin to sweat and the total outflow of transfer rate from skin⁽¹⁸⁾: 0.018×0.883 d⁻¹=0.0159 d⁻¹. The transfer rate 0.883 d⁻¹ is calculated from the transfer rate from plasma to skin and the fraction of Cs in plasma and in skin at equilibrium. The outflow fraction 0.018 is the removal fraction from skin to sweat. The uncertainty of each component was propagated to the uncertainty of this parameter, *k*<sub>skin→sweat</sub>.

The uncertainties and the 95% CI of those 4 parameters are summarized in Table 4. The assigned distributions are also shown.

## Distributions of model parameters

The experimental data available are not sufficient to determine the shape of the probability distribution for the model parameters. Possibility may be an alternative to deal with certain types of uncertainty in case of the lack of knowledge and it is applied in modelling the imprecision in prospective dosimetry of internal exposure to uranium⁽⁴⁰⁾.

In the present study, the uncertainties reflect inter-subject variability in these processes (regarded as “aleatory” uncertainty) rather than “epistemic” uncertainty, i.e. values that are fixed but unknown. Generally, a normal or lognormal distribution is assumed for most transfer rates. Uniform, triangle distributions are assigned to some parameters by taking into account the justification for the choice of these distributions by other authors.

The methods used for the evaluation of uncertainties of all parameters are summarised in Table A.1 in the Annex. In that table, the mean values given by Leggett and co-workers⁽¹⁸⁾, the 95% confidence intervals, the assigned distributions and some notes for derivations of the parameters are presented.

## Parameter sensitivity analysis

The goal of sensitivity analysis is to quantify the effects of parameter variations on calculated results. The principal method used to evaluate the model parameter sensitivity is sampling-based⁽⁴¹, ⁴²⁾. In this method, the sampling of the model inputs and calculated model outputs are analysed by a regression algorithm based on the rank transformation of the sampled data, the PRCC as a sensitivity measure index was applied in this work to identify the mostly

influential parameter⁽¹³⁾. As a second method, a local sensitivity analysis was applied. The local approach generates easily interpreted time-dependent data in terms of the variation of a particular input parameter leads to a variation of model output.

## Quality assurance of simulations

To reach a high quality assurance of simulations and modelling, two independent programs⁽¹², ¹⁴⁾ are used in the computations using the same uncertainties of the input model parameters evaluated in this work.

Although different mean values were reported for the following model parameters: *k*<sub>St→SI</sub>, *k*<sub>UBC→urine</sub> and *k*<sub>LI→faeces</sub>; only the latter two parameters concern the excretions of Cs to urine and to faeces, and the influence to other compartments in the model by taking different mean values were examined.

Furthermore, the influences of the normal and lognormal distributions of model parameters on uncertainties of model predictions were explored as well.

# RESULTS

## Uncertainty of model parameters as inputs

The model parameters and their uncertainties described in Tables 1-4, are summarised in Table 5. The 95% CI values and distribution are used for sampling the computer inputs.

## Uncertainty of model prediction

In this section, the uncertainties of model predictions in the compartments are presented following injection for blood, whole body and daily urinary excretion; this is because these bioassay quantities are normally measured in experiments and radiation protection practice, and used to estimate the intake and internal doses.

The 50th and 2.5th and 97.5th percentiles of the model predictions for blood clearance are plotted in Figure 3a, and compared with the measured data of Cs in human blood. The UF of blood clearance calculated by the 97.5th and 2.5th percentiles is less than 2.0 in the first 100 days after intake. The predicted UF increases to 17 at day 500 and to 44 at day 1000 (the data is not shown in the Figure).

The 50th and 2.5th and 97.5th percentiles of model predictions for whole body retention is presented in Figure 4a, and compared to the measured whole body
data of Cs in humans. According to the ICRP
Publication 78<sup>(43)</sup>, in the in vivo monitoring, the whole
body content of Cs is defined as the total amount of Cs
in the following organs and tissues: blood, heart, liver,
kidneys, muscle (skeleton muscle), GI tract tissue,
content in urinary bladder, spleen, pancreas, brain, red
marrow, skeleton (other skeleton), skin, lungs, adipose
tissue, Other 1, Other 2, RBC. During the first 100
days, the UF of whole-body retention of Cs is less than
a factor of 2 with the maximum value of 2 at day 100.
The UF increases to a factor of 20 at day 500 and a
factor of 23 at day 1000. The greater of the UF can be
attributed to fact that model predicts very small values
at these times. The median value closely agrees with
the observed human data, the 95% range of values
possibly overstates the range of the human data;
however, the uncertainty of the each measurement
point in whole body content of Cs is not available for
comparison to the calculated uncertainty.

The 50th and 2.5th and 97.5th percentiles of daily urinary excretion is shown in Figure 5a. The UF of urinary excretion is very similar to that of blood clearance. A UF of 12 at day 500 and a factor of 29 at day 1000 were revealed, smaller than that of blood at the same time point. However, unfortunately, there is no measurement on the urinary excretion data for comparison to the calculated uncertainty.

In summary, for the retentions in blood and in whole-body and the urinary excretion curve, the UF is generally smaller than 2 during the time course from day 1 to day 100. The UF is dramatically increased to 17 – 23 at day 500, and is gradually approaching 29 – 44 at day 1000.

Sensitivity of model parameter

The sensitivity of model parameter on the retention in
blood, in whole-body and in daily urine excretion is
presented in Figures 3b, 4b and 5b, respectively
together with percentiles of model prediction.

For blood clearance (Figure 3b), it is found that, in the
first 50 days after intake, the parameters k<sub>muscle→blood</sub>
and k<sub>blood→muscle</sub> are most influential. At day 100 after
intake, the parameters k<sub>kidneys→UBC</sub> and k<sub>blood→kidneys</sub>
have a large effect on blood retention. However, the
parameter k<sub>kidneys→blood</sub> has a strong and consistent
influence on blood retention.

Throughout the whole-body retention course (Figure
4b), the parameters *k*<sub>kidneys→blood</sub> and *k*<sub>kidneys→UBC</sub> have
the greatest impact.

For daily urinary excretion, the parameter $k_{\text{UBC-urine}}$ shows an important role. Parameters $k_{\text{kidneys-UBC}}$ and

$k_{\text{kidneys} \to \text{blood}}$ impact the excretion curve in the first 100 days after intake positively and negatively, respectively; after 400-500 days, the effect is reversed.

In summary, the parameters, that connect blood with
muscle and kidneys, seem to be important in the
biokinetic model of Cs developed by Leggett and co-
workers<sup>(18)</sup>.

Inference of parameter uncertainty and sensitivity

With respect to uncertainty of whole-body retention of Cs, the results of the parameter uncertainty and sensitivity analysis indicate that the most important parameters are $k_{\text{kidneys} \to \text{blood}}$ and $k_{\text{kidneys} \to \text{UBC}}$, and these contribute equally to the variation in retention during the time course.

Regarding blood clearance, although the parameters $k_{\text{muscle} \to \text{blood}}$ and $k_{\text{blood} \to \text{muscle}}$ have the biggest effect on the uncertainty of blood clearance at early times, the magnitude of the uncertainty on blood clearance is small. For the larger uncertainty (UF ca. 40) at later times, say at day 1000, the parameters $k_{\text{kidneys} \to \text{UBC}}$ and $k_{\text{kidneys} \to \text{blood}}$ show a stronger influence.

Quality assurance

The simulation with two independent programs showed
a very good agreement on the uncertainty of model
predictions. The influences of normal and lognormal
distributions of model parameters on the model
production in different compartments are shown in the
Appendix B (Figure B1).

The transfer rate from stomach to small intestine
changes the velocity of GI tract passage and thereby
recycling velocity as well. A comparison of the
predictions of the model parameters reported by
Leggett et al. with the changed parameter values show
only a negligible influence. Only for the daily faecal
excretion in the first ten days small changes can be
observed (Figures B2-B3)

The results of a local sensitivity analysis were
presented in Figure B4 in Appendix B. Comparison of
two methods, very similar parameter sensitivity is
observed.

DISCUSSION

In the present study, the uncertainty of model
parameters was analysed by taking into account the
experimental data that are available from the literature.
Strictly speaking, the term “uncertainty” here refers
primarily to inter-individual variability in Cs retention and excretion rather than to uncertainty on the population mean or median value. It therefore primarily represents “aleatory” rather than “epistemic” uncertainty, and as is evident from Figure 4a, reproduces the range of whole body retention observed among human subjects. Furthermore, a key source of epistemic uncertainty – the choice and structure of the model itself – was not included in the analysis as the model structure was assumed to be accurate. It should be noted however, that the distributions reflect some additional variance incurred by the fact that the measurement data, on which the distributions are based, contain an element of measurement uncertainty. This is consistent with the observation that the range of whole body retention, predicted by the uncertainty analysis, is over-dispersed with respect to the human data.

However, in radiation protection monitoring, it is an individual person who is monitored. In this situation, if the model does not fit the monitored data (in-vivo or in-vitro measurements), as might occur in situations where special monitoring has been initiated and many measurements are available, then model parameter values can be adjusted to best fit the data. If the uncertainties described here accurately reflect inter-subject variability in Cs retention and excretion, then the choice of values should typically be within the 95% ranges of values given in Table A1. In fact, these distributions could be used as prior distributions in a Bayesian analysis of a worker’s monitoring data. Under these circumstances, the distributions represent epistemic uncertainty on the fixed but unknown values of the worker in question. The posterior distributions of model parameters can then be compared with their respective priors to check the consistency of the approach: it would be expected that posterior values of parameters are “not too far” removed from the range of values indicated by their corresponding prior if the model and parameter distributions reflect accurately the retention of Cs for the population in question (adult males in this instance).

In individual monitoring programs of internal exposures, the content of ¹³⁷Cs in whole-body and in urine is measured for intake and dose assessment with regulatory purpose. For a single acute intake, monitoring interval of 180 day is recommended(⁴⁴, ⁴⁵). For calculation of an intake, the retention at the middle of the monitoring interval, i.e. 90 days before the measurement is assumed. The calculation of this work shows that the uncertainty factor of the model prediction in whole-body and in urine at 90 day is 2.0 and 1.8, respectively; at day 180 the corresponding UF is 3.5 and 1.9, respectively, the statistical inference of the analysis implies that the use of urinary data for

internal dose assessment is preferred to whole-body counter. The smaller uncertainty in urinary prediction implies that urine measurement is reliable for internal dose assessment. However, there is a larger variability in the urine measurements from day to day compared to whole body measurements, and the uncertainty associated with the measurement technique, notably in the sampling procedure and in the detector calibration, should also be taken into account when designing the monitoring program. As mentioned above, due to the lack of measured urinary excretion data on Cs, it is difficult to judge the validation of the calculated uncertainty of urinary excretion. In the practice, two measurements are suggested to be applied.

The present work shows that simultaneous uncertainty and sensitivity analysis can identify important model parameters that may guide further research effort in model calibration. The benefit of uncertainty and sensitivity analysis can broaden our knowledge about the impact of parameters at different time points, for example, if a larger uncertainty exists at a certain time, then sensitivity information can help to identify the parameter that is responsible.

It is acknowledged that there are some limitations in the analysis, especially the paucity of experimental data that makes it difficult to determine the standard uncertainty of the parameter. Furthermore, the assignment of a distribution to a parameter is generally difficult with scarce measurement data; in this case, the rationale for the choice of distribution for model parameters usually is assumed based on experience in the literature, sometimes this is subjective. These problems may cause an additional uncertainty to the analysis, which is difficult to quantify, namely “uncertainty” in uncertainty analysis; it sometimes is called “imprecision”(⁴⁰). Nevertheless, the chosen distributions and their parameters can be justified on the basis that the subsequent uncertainty analysis reproduces the median and range of observed human data reasonably well.

As noted earlier, the quantification of uncertainties in biokinetic models can also be used to assess the reliability of effective dose coefficients (Sv per Bq intake) as a protection device. To this end, Puncher and Harrison(⁴⁶) derived uncertainties in a slightly modified version of the revised Cs model (ICRP, in preparation) that is compatible with the ICRP Publication 60 and Publication 103 dosimetry framework (ICRP 1991, 2007). A local sensitivity analysis indicated that the effective dose per unit injection was most sensitive to the rates to and from muscle and kidneys to blood, and from kidney to bladder, which is consistent with the sensitivity analysis described here. For simplicity, Puncher and Harrison(⁴⁶) assumed that the rates to and
from blood to all soft tissues are 100% positively correlated through their relationship between cardiac output and the efficiency of tissues in extracting caesium from plasma via the plasma membrane bound K+/Na+-ATPase enzyme complex. In a slightly different approach to that adopted here, the authors chose a sampling regime that reproduced the variability in blood and whole body retention given in figures 3 and 4, respectively. In this regime the rates to and from all soft tissues were multiplied by a common factor which was sampled from a lognormal distribution with median of unity and GSD of 1.4. This gave similar UF values to those calculated here, and a geometric 95% range of effective dose per unit ingestion of around a factor of ten for adults, calculated using the ICRP Publication 60 schema (ICRP 1991).

The present work will be extended to propagate the derived uncertainties to calculate distributions of effective dose per unit ingestion. These will be compared with the results of Puncher and Harrison<sup>(46)</sup> to ascertain the sensitivity of the output to judgements and choices made by the respective analysts and permit an assessment of the robustness of the uncertainty analysis procedures and the affect of differences in model structure.

## CONCLUSION

In the framework of the EURADOS Internal Dosimetry Working Group, a global uncertainty analysis method is developed for the calculation of confidence intervals of the model predictions of a biokinetic model of Cs developed by Leggett and co-workers<sup>(18)</sup>. The uncertainties of the model parameters were evaluated by analysis of the experimental data of Cs obtained from animal and humans. Using sampling technique uncertainty distribution of each compartment is calculated by biokinetic modelling. Furthermore, sensitivity analysis was performed to identify the important model parameter to the model prediction.

A relative small uncertainty factor, say less than 2, is revealed during the first 100 days after intake. However, a larger uncertainty factor, up to a maximum factor of 43 in whole-body compartment, was found in the later time, say, after day 500. Taking the blood clearance, whole body retention and the urinary excretion into account, transfer rates of muscle to and from blood, transfer rates of kidneys to and from blood and the rate of kidney to UBC are mostly influential parameters in the investigated biokinetic model.

The described approach pertains to adult male workers but could be readily extended to adult females as required.

## FUNDING

This work was supported by EURADOS Association (www.eurados.org) and was partially supported by the German Federal Ministry of Education and Research through the Competence Network of Radiation Research – Radiation and Environment II (KVSF II) project.

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LEGEND OF FIGURES

Figure 1. Biokinetic model of Cs developed by Leggett and co-workers(18).

Figure 2. A system of first-order process.

Figure 3. Uncertainty (a) and sensitivity (b) of model prediction in blood clearance of Cs.

Figure 4. Uncertainty (a) and sensitivity (b) of model prediction in whole body retention of Cs.

Figure 5. Uncertainty (a) and sensitivity (b) of model prediction in daily urinary excretion of Cs.

Figure B1. Comparison of calculated uncertainties on blood clearance, whole body retention and urinary excretion.

Figure 2B. Influence of change of parameters on blood clearance and whole body retention.

Figure 3B. Influence of change of parameters on small intestine, urinary and faecal excretions.

Figure 4B. Parameter sensitivity on blood clearance, whole body retention and urinary excretion by a local sensitivity analysis.
<caption>PARAMETER UNCERTAINTY OF CS MODEL</caption>

<figure><img src="image_2.png" /><figcaption>Figure 1</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 2</figcaption></figure>
PARAMETER UNCERTAINTY OF CS MODEL

<figure><img src="image_4.png" /><figcaption>Figure 3</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 4</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 5</figcaption></figure>
<table><caption>Table 1. Uncertainties of parameters of transfer rate from plasma to organs and tissues</caption><thead><tr><th rowspan="2">No.</th><th rowspan="2"></th><th rowspan="2">CO±σ</th><th rowspan="2">FCO (range)<sup>a</sup></th><th rowspan="2">σ</th><th rowspan="2">E<sub>Cs</sub></th><th rowspan="2">σ</th><th>Transfer</th><th rowspan="2">σ</th><th rowspan="2">95% CI</th><th rowspan="2">Distribution</th></tr><tr><th>rate<sup>d</sup></th></tr><tr><th colspan="2"></th><th colspan="2">d<sup>-1</sup></th><th colspan="2">d<sup>-1</sup></th><th colspan="2">d<sup>-1</sup></th><th colspan="2"></th></tr></thead><tbody><tr><td>1</td><td>Plasma to Heart tissue</td><td>1766±505</td><td>0.04 (0.03 – 0.06)</td><td>0.00485</td><td>0.2</td><td>0.0054</td><td>14.128</td><td>4.335</td><td>5.631</td><td>22.625</td><td>Normal</td></tr><tr><td>2</td><td>Liver</td><td></td><td>0.065 (0.055 – 0.085)</td><td>0.00485</td><td>0.05</td><td>0.0032</td><td>19.515</td><td>5.8795</td><td>7.991</td><td>31.039</td><td>Normal</td></tr><tr><td>3</td><td>Kidneys</td><td></td><td>0.19 (0.16 – 0.21)</td><td>0.00809</td><td>0.2</td><td>0.0054</td><td>67.108</td><td>19.499</td><td>28.890</td><td>105.326</td><td>Normal</td></tr><tr><td>4</td><td>Muscle</td><td></td><td>0.17 (0.10 – 0.21)</td><td>0.0178</td><td>0.1</td><td>0.0307</td><td>30.022</td><td>13.068</td><td>4.408</td><td>55.636</td><td>Normal</td></tr><tr><td>5</td><td>GI tract tissue</td><td></td><td>0.15 (0.103 – 0.21)</td><td>0.0173</td><td></td><td>0.0054</td><td>52.980</td><td>16.308</td><td>21.016</td><td>84.944</td><td>Normal</td></tr><tr><td>6</td><td>Stomach</td><td></td><td>0.01 (0.005 – 0.02)</td><td>0.00243</td><td>0.2</td><td>0.0307</td><td>4.516</td><td>2.0891</td><td>0.4214</td><td>8.6106</td><td>Normal</td></tr><tr><td>7</td><td>Small intestine</td><td></td><td>0.1 (0.07 – 0.13)</td><td>0.009709</td><td></td><td>0.0307</td><td>1.048</td><td>0.4516</td><td>0.1628</td><td>1.933</td><td>Normal</td></tr><tr><td>8</td><td>Large intestine</td><td></td><td>0.04 (0.028 – 0.06)</td><td>0.005178</td><td></td><td>0.0307</td><td>0.02</td><td>0.0087</td><td>0.0029</td><td>0.0371</td><td>Normal</td></tr><tr><td>9</td><td>Spleen</td><td></td><td>0.03 (0.025 – 0.05)</td><td>0.00405</td><td>0.1</td><td>0.0307</td><td>5.298</td><td>2.2967</td><td>0.7964</td><td>9.7996</td><td>Normal</td></tr><tr><td>10</td><td>Pancreas</td><td></td><td>0.01 (0.006 – 0.019)</td><td>0.002104</td><td>0.1</td><td>0.0307</td><td>1.766</td><td>0.7988</td><td>0.2003</td><td>3.3317</td><td>Normal</td></tr><tr><td>11</td><td>Brain</td><td></td><td>0.12 (0.1 – 0.15)</td><td>0.008091</td><td>0.002</td><td>0.0003</td><td>0.424</td><td>0.142</td><td>0.1457</td><td>0.7023</td><td>Normal</td></tr><tr><td>12</td><td>Red marrow</td><td></td><td>0.03 (0.015 – 0.06)<sup>b</sup></td><td>0.007282</td><td>0.1</td><td>0.0307</td><td>5.298</td><td>2.4508</td><td>0.4943</td><td>10.1017</td><td>Normal</td></tr><tr><td>13</td><td>Other skeleton</td><td></td><td>0.02 (0.01 – 0.04)<sup>b</sup></td><td>0.00485</td><td>0.1</td><td>0.0307</td><td>3.532</td><td>1.6339</td><td>0.3296</td><td>6.7344</td><td>Normal</td></tr><tr><td>14</td><td>Skin</td><td></td><td>0.05 (0.033 – 0.07)</td><td>0.005988</td><td>0.05</td><td>0.0032</td><td>4.415</td><td>1.3925</td><td>1.6857</td><td>7.1443</td><td>Normal</td></tr><tr><td>15</td><td>Lungs</td><td></td><td>0.025 (0.01 – 0.055)</td><td>0.007282</td><td>0.1</td><td>0.0307</td><td>4.415</td><td>2.1011</td><td>0.2968</td><td>8.53324</td><td>Normal</td></tr><tr><td>16</td><td>Adipose tissue</td><td></td><td>0.05 (0.035 – 0.09)</td><td>0.0089</td><td>0.1</td><td>0.0307</td><td>8.830</td><td>3.9149</td><td>1.1568416.5032</td><td></td><td>Normal</td></tr><tr><td>17</td><td>Other 1</td><td></td><td>0.04998<sup>c</sup></td><td>0.0114</td><td>0.1</td><td>0.0307</td><td>8.826</td><td>4.2169</td><td>0.5608</td><td>17.0912</td><td>Normal</td></tr><tr><td>18</td><td>Other 2</td><td></td><td>0.00002<sup>c</sup></td><td>4.5602E-06</td><td>0.1</td><td>0.0307</td><td>0.00353</td><td>0.0017</td><td>0.00022</td><td>0.0068</td><td>Normal</td></tr><tr><td>19</td><td>RBC</td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.0583</td><td>1.686</td><td>1.914</td><td>Normal</td></tr><tr><td colspan="2">All</td><td></td><td>1.00</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

<footnote><sup>a</sup> From Table 1 of reference<sup>(18)</sup></footnote>

<footnote><sup>b</sup> Data are not given in Table 1 of reference<sup>(18)</sup>, only given for total, assumed half of and two times of the mean value as for total value.</footnote>

<footnote><sup>c</sup> The FCO value of “Other” is the sum of thyroid, lymph nodes, testes, adrenal, urinary bladder and all other tissues in the Table 1 of Leggett and co-workers<sup>(18)</sup>, this value is distributed to Other1 and Other2 by 99.96% and 0.04%, respectively.</footnote>

<footnote><sup>d</sup> The transfer rate given by Leggett and co-workers<sup>(18)</sup></footnote>
PARAMETER UNCERTAINTY OF CS MODEL

<table><caption>Table 2. Uncertainties of parameters of transfer rate from organs and tissues to plasma</caption><thead><tr><th rowspan="2">No.</th><th rowspan="2"></th><th colspan="3">Absolute value in tissue, g Cs 10<sup>-8</sup> per g</th><th colspan="2">Equilibrium content of Cs<sup>a</sup></th><th colspan="2">Transfer rate</th><th colspan="2">95% CI</th><th rowspan="2">Distribution</th></tr><tr><th>min.</th><th>max.</th><th>mean</th><th>σ</th><th>mean</th><th>σ</th><th>d<sup>-1</sup></th><th>d<sup>-1</sup></th></tr></thead><tbody><tr><td>1</td><td>Brain to Plasma</td><td>0.08</td><td>1.3</td><td>0.69</td><td>0.1974</td><td>0.01</td><td>0.002861</td><td>0.0848</td><td>0.031</td><td>0.0235</td><td>0.1461</td><td>Normal</td></tr><tr><td>2</td><td>Heart tissue</td><td>1.3</td><td>1.4</td><td>1.35</td><td>0.0162</td><td>0.0035</td><td>4.2E-05</td><td>8.0731</td><td>2.7745</td><td>2.6352</td><td>13.511</td><td>Normal</td></tr><tr><td>3</td><td>RBC</td><td>0.48</td><td>24</td><td>12.24</td><td>3.8058</td><td>0.014</td><td>0.004353</td><td>0.2571</td><td>0.0464</td><td>0.1663</td><td>0.347995</td><td>Normal</td></tr><tr><td>4</td><td>Lungs</td><td>1</td><td>2.1</td><td>1.55</td><td>0.178</td><td>0.006</td><td>0.000689</td><td>1.4717</td><td>0.7102</td><td>0.0798</td><td>2.8636</td><td>Normal</td></tr><tr><td>5</td><td>Liver</td><td>1.1</td><td>1.5</td><td>1.3</td><td>0.065</td><td>0.02</td><td>0.000996</td><td>2.204</td><td>0.7169</td><td>0.7989</td><td>3.60913</td><td>Normal</td></tr><tr><td>6</td><td>Pancreas</td><td>-</td><td></td><td>0.125</td><td>0.1004</td><td>0.002</td><td>0.10039</td><td>1.678</td><td>0.7696</td><td>0.1695</td><td>3.1865</td><td>Normal</td></tr><tr><td>7</td><td>Spleen</td><td>-</td><td></td><td>0.125</td><td>0.1004</td><td>0.002</td><td>0.10039</td><td>5.033</td><td>2.215</td><td>0.6916</td><td>9.3744</td><td>Normal</td></tr><tr><td>8</td><td>Red marrow</td><td>0.9</td><td>9.8</td><td>5.35</td><td>1.4401</td><td>0.015</td><td>0.004038</td><td>0.7064</td><td>0.3415</td><td>0.0371</td><td>1.3757</td><td>Normal</td></tr><tr><td>9</td><td>Other skeleton</td><td>0.9</td><td>9.8</td><td>5.35</td><td>1.4401</td><td>0.055</td><td>0.014805</td><td>0.1284</td><td>0.0621</td><td>0.006752</td><td>0.2501</td><td>Normal</td></tr><tr><td>10</td><td>Other 1</td><td>0.1</td><td>1.8</td><td>0.95</td><td>0.2751</td><td>0.0304878</td><td>0.008828</td><td>0.692</td><td>0.3482</td><td>0.009434</td><td>1.37457</td><td>Normal</td></tr><tr><td>11</td><td>Other 2</td><td></td><td></td><td></td><td></td><td>0.0000122</td><td>3.53E-06</td><td>0.00141</td><td>0.0007</td><td>1.922E-05</td><td>0.0028</td><td>Normal</td></tr><tr><td>12</td><td>Muscle</td><td>1.5</td><td>3.4</td><td>2.45</td><td>0.3074</td><td>0.8</td><td>0.10039</td><td>0.0751</td><td>0.0332</td><td>0.01006</td><td>0.1401</td><td>Normal</td></tr><tr><td>13</td><td>Adipose tissue</td><td>-</td><td></td><td>0.125</td><td>0.1004</td><td>0.01</td><td>0.10039</td><td>1.766</td><td>0.7944</td><td>0.2091</td><td>3.3229</td><td>Normal</td></tr><tr><td>14</td><td>Skin</td><td>1.1</td><td>2.4</td><td>1.75</td><td>0.2104</td><td>0.01</td><td>0.001202</td><td>0.867</td><td>0.2816</td><td>0.31504</td><td>1.41896</td><td>Normal</td></tr><tr><td>15</td><td>Kidneys</td><td>0.9</td><td>1.3</td><td>1.1</td><td>0.0647</td><td>0.004</td><td>0.000235</td><td>31.876</td><td>9.9657</td><td>12.3432</td><td>51.4088</td><td>Normal</td></tr><tr><td>16</td><td>GI tract tissue</td><td>-</td><td></td><td>0.125</td><td>0.1004</td><td>0.012</td><td>0.10039</td><td>8.191</td><td>2.5967</td><td>3.10139</td><td>13.2806</td><td>Normal</td></tr><tr><td>17</td><td>SI</td><td>-</td><td></td><td>0.125</td><td>0.1004</td><td>0.004</td><td>0.10039</td><td>28.215</td><td>12.3462</td><td>4.0165</td><td>52.4135</td><td>Normal</td></tr><tr><td></td><td>Plasma</td><td>0.097</td><td>0.3</td><td>0.1985</td><td>0.0328</td><td>0.002</td><td>0.000331</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>All</td><td></td><td></td><td></td><td></td><td></td><td>1</td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

<footnote><sup>a</sup> Given as a fraction of total body content of Cs</footnote>

<footnote>- The values are not given in reference<sup>(34)</sup>, it is assumed to be the same as that of muscle</footnote>
<table><caption>Table 3. Uncertainty of parameter of transfer rates related to GI tract tissue</caption><thead><tr><th rowspan="2">No.</th><th rowspan="2"></th><th rowspan="2"></th><th rowspan="2">Transfer rate</th><th rowspan="2">σ</th><th colspan="2">95% CI</th><th rowspan="2">Distribution</th></tr><tr><th>d<sup>-1</sup></th><th>d<sup>-1</sup></th></tr></thead><tbody><tr><td>1</td><td>GI tract tissue</td><td>to Liver</td><td>0.431</td><td>0.145505</td><td>0.1458</td><td>0.7162</td><td>Normal</td></tr><tr><td>2</td><td>GI tract tissue</td><td>to Stomach</td><td>0.0333</td><td>0.004709</td><td>0.0241</td><td>0.0425</td><td>Normal</td></tr><tr><td>3</td><td>GI tract tissue</td><td>to Small intestine</td><td>0.108</td><td>0.015274</td><td>0.0781</td><td>0.1379</td><td>Normal</td></tr><tr><td>4</td><td>GI tract tissue</td><td>to Large intestine</td><td>0.0667</td><td>0.009433</td><td>0.0482</td><td>0.0852</td><td>Normal</td></tr><tr><td>5</td><td>Stomach</td><td>to Small intestine</td><td>40</td><td></td><td>12 – 48</td><td></td><td>Uniform</td></tr><tr><td>6</td><td>Small intestine</td><td>to Large intestine</td><td>0.3</td><td>0.03</td><td>0.2412</td><td>0.3588</td><td>Normal</td></tr><tr><td>7</td><td>Small intestine</td><td>to Liver</td><td>1.485</td><td>0.176892</td><td>1.1383</td><td>1.8317</td><td>Normal</td></tr><tr><td>8</td><td>Liver</td><td>to Small intestine</td><td>0.116</td><td>0.037544</td><td>0.0424</td><td>0.1896</td><td>Normal</td></tr><tr><td>9</td><td>Pancreas</td><td>to Liver</td><td>0.0883</td><td>0.026749</td><td>0.03587</td><td>0.14073</td><td>Normal</td></tr><tr><td>10</td><td>Spleen</td><td>to Liver</td><td>0.265</td><td>0.080279</td><td>0.10765</td><td>0.42235</td><td>Normal</td></tr></tbody></table>
PARAMETER UNCERTAINTY OF CS MODEL

<table><caption>Table 4. Uncertainty of parameter of other 4 transfer rates</caption><thead><tr><th rowspan="2">No.</th><th rowspan="2">Parameter</th><th rowspan="2"></th><th rowspan="2">Transfer rate</th><th rowspan="2">σ</th><th colspan="2">95% CI</th><th rowspan="2">Distribution</th></tr><tr><th>d<sup>-1</sup></th><th>d<sup>-1</sup></th></tr></thead><tbody><tr><td>1</td><td>Kidneys</td><td>to</td><td>UBC</td><td>1.678</td><td>0.5508</td><td>0.598</td><td>2.758</td><td>Normal</td></tr><tr><td>2</td><td>UBC</td><td>to</td><td>Urine</td><td>12</td><td>3.2998</td><td>6, 12, 22<sup>a</sup></td><td></td><td>Triangular</td></tr><tr><td>3</td><td>Large intestine</td><td>to</td><td>Faeces</td><td>0.5</td><td>0.3536</td><td>0.5, 0.5, 1<sup>a</sup></td><td></td><td>Triangular</td></tr><tr><td>4</td><td>Skin</td><td>to</td><td>Sweat</td><td>0.0159</td><td>0.0054</td><td>0.00531</td><td>0.0265</td><td>Normal</td></tr></tbody></table>

<footnote>ª The values which are assumed as a triangular distribution.</footnote>
<table><caption>Table 5. Mean valuesª, default valuesᵇ by Leggett et al. and 95% confidence interval of transfer rates with assigned distributions</caption><thead><tr><th>No.</th><th>Parameter</th><th>Mean<sup>a</sup> (d<sup>-1</sup>)</th><th>95% CI (d<sup>-1</sup>)</th><th>Distribution</th><th>Values<sup>b</sup> (d<sup>-1</sup>)</th></tr></thead><tbody><tr><td>1</td><td>Blood→Heart</td><td>14.128</td><td>5.631 22.625</td><td>Normal</td><td>14.128</td></tr><tr><td>2</td><td>Blood→Liver</td><td>19.515</td><td>7.991 31.039</td><td>Normal</td><td>19.515</td></tr><tr><td>3</td><td>Blood→Kidney</td><td>67.108</td><td>28.890 105.326</td><td>Normal</td><td>67.108</td></tr><tr><td>4</td><td>Blood→Muscle</td><td>30.022</td><td>4.409 55.635</td><td>Normal</td><td>30.022</td></tr><tr><td>5</td><td>Blood→GI Tract Tissue</td><td>52.980</td><td>21.016 84.944</td><td>Normal</td><td>52.980</td></tr><tr><td>6</td><td>Blood→Stomach</td><td>4.516</td><td>0.421 8.611</td><td>Normal</td><td>4.516</td></tr><tr><td>7</td><td>Blood→Small intestine</td><td>1.048</td><td>0.163 1.933</td><td>Normal</td><td>1.048</td></tr><tr><td>8</td><td>Blood→Large Intestine</td><td>0.020</td><td>0.003 0.037</td><td>Normal</td><td>0.020</td></tr><tr><td>9</td><td>Blood→Spleen</td><td>5.298</td><td>0.796 9.800</td><td>Normal</td><td>5.298</td></tr><tr><td>10</td><td>Blood→Pancreas</td><td>1.766</td><td>0.200 3.332</td><td>Normal</td><td>1.766</td></tr><tr><td>11</td><td>Blood→Brain</td><td>0.424</td><td>0.146 0.702</td><td>Normal</td><td>0.424</td></tr><tr><td>12</td><td>Blood→Red Marrow</td><td>5.298</td><td>0.494 10.102</td><td>Normal</td><td>5.298</td></tr><tr><td>13</td><td>Blood→Other Skeleton</td><td>3.532</td><td>0.330 6.734</td><td>Normal</td><td>3.532</td></tr><tr><td>14</td><td>Blood→Skin</td><td>4.415</td><td>1.686 7.144</td><td>Normal</td><td>4.415</td></tr><tr><td>15</td><td>Blood→Lungs</td><td>4.415</td><td>0.297 8.533</td><td>Normal</td><td>4.415</td></tr><tr><td>16</td><td>Blood→Adipose Tissue</td><td>8.830</td><td>1.157 16.503</td><td>Normal</td><td>8.830</td></tr><tr><td>17</td><td>Blood→Other 1</td><td>8.826</td><td>0.561 17.091</td><td>Normal</td><td>8.826</td></tr><tr><td>18</td><td>Blood→Other 2</td><td>0.004</td><td>0.0002 0.007</td><td>Normal</td><td>0.004</td></tr><tr><td>19</td><td>Blood→RBC</td><td>1.800</td><td>1.686 1.914</td><td>Normal</td><td>1.800</td></tr><tr><td>20</td><td>Heart→Blood</td><td>8.073</td><td>2.635 13.511</td><td>Normal</td><td>8.073</td></tr><tr><td>21</td><td>Liver→Blood</td><td>2.204</td><td>0.799 3.609</td><td>Normal</td><td>2.204</td></tr><tr><td>22</td><td>Liver→Small intestine</td><td>0.116</td><td>0.042 0.190</td><td>Normal</td><td>0.116</td></tr><tr><td>23</td><td>Kidney→UBC</td><td>1.678</td><td>0.598 2.758</td><td>Normal</td><td>1.678</td></tr><tr><td>24</td><td>Kidney→Blood</td><td>31.876</td><td>12.343 51.409</td><td>Normal</td><td>31.876</td></tr><tr><td>25</td><td>Muscle→Blood</td><td>0.075</td><td>0.010 0.140</td><td>Normal</td><td>0.075</td></tr><tr><td>26</td><td>GI Tract Tissue→Blood</td><td>8.191</td><td>3.101 13.281</td><td>Normal</td><td>8.191</td></tr><tr><td>27</td><td>GI Tract Tissue→Liver</td><td>0.431</td><td>0.146 0.716</td><td>Normal</td><td>0.431</td></tr><tr><td>28</td><td>GI Tract Tissue→Stomach</td><td>0.033</td><td>0.024 0.043</td><td>Normal</td><td>0.033</td></tr><tr><td>29</td><td>GI Tract Tissue→Small intestine</td><td>0.108</td><td>0.078 0.138</td><td>Normal</td><td>0.108</td></tr><tr><td>30</td><td>GI Tract Tissue→Large intestine</td><td>0.067</td><td>0.048 0.085</td><td>Normal</td><td>0.067</td></tr><tr><td>31</td><td>Spleen→Blood</td><td>5.033</td><td>0.692 9.374</td><td>Normal</td><td>5.033</td></tr><tr><td>32</td><td>Spleen→Liver</td><td>0.265</td><td>0.108 0.422</td><td>Normal</td><td>0.265</td></tr><tr><td>33</td><td>Pancreas→Blood</td><td>1.678</td><td>0.170 3.186</td><td>Normal</td><td>1.678</td></tr><tr><td>34</td><td>Pnacreas→Liver</td><td>0.088</td><td>0.036 0.141</td><td>Normal</td><td>0.088</td></tr><tr><td>35</td><td>Skin→Blood</td><td>0.867</td><td>0.315 1.419</td><td>Normal</td><td>0.867</td></tr><tr><td>36</td><td>Skin→Excreta</td><td>0.016</td><td>0.005 0.026</td><td>Normal</td><td>0.016</td></tr><tr><td>37</td><td>Brain→Blood</td><td>0.085</td><td>0.023 0.146</td><td>Normal</td><td>0.085</td></tr><tr><td>38</td><td>Red Marrow→Blood</td><td>0.706</td><td>0.037 1.375</td><td>Normal</td><td>0.706</td></tr><tr><td>39</td><td>Other Skeleton→Blood</td><td>0.128</td><td>0.0066 0.250</td><td>Normal</td><td>0.128</td></tr><tr><td>40</td><td>Lungs→Blood</td><td>1.472</td><td>0.080 2.864</td><td>Normal</td><td>1.472</td></tr><tr><td>41</td><td>Adipose Tissue→Blood</td><td>1.766</td><td>0.209 3.323</td><td>Normal</td><td>1.766</td></tr><tr><td>42</td><td>Other 1→Blood</td><td>0.692</td><td>0.009 1.375</td><td>Normal</td><td>0.692</td></tr><tr><td>43</td><td>Other 2→Blood</td><td>0.001</td><td>0.000019 0.0028</td><td>Normal</td><td>0.001</td></tr><tr><td>44</td><td>RBC→Blood</td><td>0.257</td><td>0.166 0.348</td><td>Normal</td><td>0.257</td></tr><tr><td>45</td><td>UBC→Urine</td><td>13.33</td><td>6, 12, 22<sup>c</sup></td><td>Triangular</td><td>12</td></tr><tr><td>46</td><td>Stomach→Small intestine</td><td>31</td><td>14, 48<sup>d</sup></td><td>Uniform</td><td>40</td></tr><tr><td>47</td><td>Small intestine→Blood</td><td>28.215</td><td>4.017 52.413</td><td>Normal</td><td>28.215</td></tr><tr><td>48</td><td>Small intestine→Large intestine</td><td>0.3</td><td>0.241 0.359</td><td>Normal</td><td>0.3</td></tr><tr><td>49</td><td>Small intestine→Liver</td><td>1.485</td><td>1.138 1.832</td><td>Normal</td><td>1.485</td></tr><tr><td>50</td><td>Large intestine→Feces</td><td>0.667</td><td>0.5, 0.5, 1<sup>c</sup></td><td>Triangular</td><td>0.5</td></tr></tbody></table>

<footnote>ª The mean values which was evaluated in the present work.</footnote>

<footnote>ᵇ The values of transfer rates given by Leggett and co-workers<sup>(18)</sup>.</footnote>

<footnote>ᶜ The values which are assumed as a triangular distribution.</footnote>

<footnote>ᵈ The values which are assumed as a uniform distribution.</footnote>
Appendix A

<table><caption>Table A.1: Summary of evaluation of uncertainties of the transfer rates for a biokinetic model of Cs</caption><thead><tr><th>No.</th><th>Parameter</th><th>Mean (d<sup>-1</sup>)</th><th>Derivation of parameter values</th><th>Uncertainty, σ (d<sup>-1</sup>)</th><th>95% CI (d<sup>-1</sup>)</th><th>Distribution</th><th>Note</th></tr></thead><tbody><tr><td>1</td><td>Blood→Heart</td><td>14.128</td><td>Generally, the uncertainty of transfer rate from plasma to organ or tissue is calculated by the uncertainties of three quantities: the cardiac output (CO), fraction cardiac output (FCO) and extraction fraction of Cs in tissue (E<sub>Cs</sub>). The standard uncertainty of CO is evaluated to be 505 d<sup>-1</sup>. The uncertainties of FCO and E<sub>T</sub> are evaluated from the experimental data of K and Rb in animal and in humans, mostly, reported by Leggett and coworkers, for detail, see text.</td><td>4.335</td><td>5.631 22.625</td><td>Normal</td><td>k<sub>plasma→tissue</sub>=CO×FCO×E<sub>T</sub>,<br>Leggett et al. 2003</td></tr><tr><td>2</td><td>Blood→Liver</td><td>19.515</td><td>See parameter No. 1</td><td>5.8795</td><td>7.991 31.039</td><td>Normal</td><td>Leggett et al. 2003</td></tr><tr><td>3</td><td>Blood→Kidney</td><td>67.108</td><td>See parameter No. 1</td><td>19.499</td><td>28.890 105.326</td><td>Normal</td><td>Leggett and Williams 1988, Black<br>et al. 1956</td></tr><tr><td>4</td><td>Blood→Muscle</td><td>30.022</td><td>See parameter No. 1</td><td>13.068</td><td>4.409 55.635</td><td>Normal</td><td></td></tr><tr><td>5</td><td>Blood→GI Tract Tissue</td><td>52.98</td><td>See parameter No. 1</td><td>16.308</td><td>21.016 84.944</td><td>Normal</td><td></td></tr><tr><td>6</td><td>Blood→St</td><td>4.516</td><td>The contents of GI tract tissue are divided into fractions contained in St, SI and LI. The transfer rate for Cs is assumed to be the same as for K. The transfer rate from plasma to St is assumed to be attributed from saliva and gastric juices, the sum of 1.935 d-1 +2.581 d-1=4.516 d<sup>-1</sup>. The uncertainty evaluation is similar to parameter No.1.</td><td>2.0981</td><td>0.404 8.628</td><td>Normal</td><td>See Leggett and Williams, 1988</td></tr><tr><td>7</td><td>Blood→SI</td><td>1.048</td><td>Similar to parameter No.6, the transfer rate from plasma to SI is assumed to be resulted from secretions, pancreatic juices and Brunner's gland secretion, the sum of 0.645+0.387+0.016=1.048 d<sup>-1</sup>. The uncertainty evaluation is similar to parameter No.1.</td><td>0.4516</td><td>0.163 1.933</td><td>Normal</td><td></td></tr></tbody></table>
<table><tr><td>8</td><td>Blood→Large Intestine</td><td>0.02</td><td>Similar to parameter No. 6, Cs is transferred into LI through the wall of large intestine. The uncertainty evaluation is similar to parameter No.1.</td><td>0.0087</td><td>0.003</td><td>0.037</td><td>Normal</td><td></td></tr><tr><td>9</td><td>Blood→Spleen</td><td>5.298</td><td>See parameter No. 1</td><td>2.2967</td><td>0.796</td><td>9.800</td><td>Normal</td><td></td></tr><tr><td>10</td><td>Blood→Pancreas</td><td>1.766</td><td>See parameter No. 1</td><td>0.7988</td><td>0.200</td><td>3.332</td><td>Normal</td><td></td></tr><tr><td>11</td><td>Blood→Brain</td><td>0.424</td><td>See parameter No. 1</td><td>0.142</td><td>0.146</td><td>0.702</td><td>Normal</td><td>Leggett et al. 2003</td></tr><tr><td>12</td><td>Blood→Red Marrow</td><td>5.298</td><td>See parameter No. 1</td><td>2.4508</td><td>0.494</td><td>10.102</td><td>Normal</td><td></td></tr><tr><td>13</td><td>Blood→Other Skeleton</td><td>3.532</td><td>See parameter No. 1</td><td>1.6339</td><td>0.330</td><td>6.734</td><td>Normal</td><td></td></tr><tr><td>14</td><td>Blood→Skin</td><td>4.415</td><td>See parameter No. 1</td><td>1.3925</td><td>1.686</td><td>7.144</td><td>Normal</td><td></td></tr><tr><td>15</td><td>Blood→Lungs</td><td>4.415</td><td>See parameter No. 1</td><td>2.1011</td><td>0.297</td><td>8.533</td><td>Normal</td><td></td></tr><tr><td>16</td><td>Blood→Adipose Tissue</td><td>8.83</td><td>See parameter No. 1</td><td>3.9149</td><td>1.157</td><td>16.503</td><td>Normal</td><td></td></tr><tr><td>17</td><td>Blood→Other 1</td><td>8.826</td><td>See parameter No. 1</td><td>4.2169</td><td>0.561</td><td>17.091</td><td>Normal</td><td></td></tr><tr><td>18</td><td>Blood→Other 2</td><td>0.00353</td><td>See parameter No. 1</td><td>0.0017</td><td>0.000</td><td>0.007</td><td>Normal</td><td></td></tr><tr><td>19</td><td>Blood→RBC</td><td>1.8</td><td>30% of values for K and Rb and comparative data, i.e. 6 d<sup>-1</sup>. A CV of 10% is assigned for this parameter.</td><td>0.0583</td><td>1.686</td><td>1.914</td><td>Normal</td><td>Leggett and Williams, 1986, 1988</td></tr><tr><td>20</td><td>Heart→Blood</td><td>8.073</td><td>According to Eqs.2-3, the uncertainty is calculated from that of transfer rate from blood to heart and that of equilibrium contents of Cs in plasma and in heart.</td><td>2.774</td><td>2.635</td><td>13.511</td><td>Normal</td><td></td></tr><tr><td>21</td><td>Liver→Blood</td><td>2.204</td><td>According to Eqs.2-3, the uncertainty is calculated from that of transfer rate from blood to liver and that of equilibrium contents of Cs in plasma and in liver.</td><td>0.717</td><td>0.799</td><td>3.609</td><td>Normal</td><td></td></tr><tr><td>22</td><td>Liver→SI</td><td>0.116</td><td>The uncertainty is mainly related to that of transfer rate of plasma to liver and that of loss from liver to bile.</td><td>0.0375</td><td>0.042</td><td>0.190</td><td>Normal</td><td></td></tr></table>
PARAMETER UNCERTAINTY OF CS MODEL

<table><tr><td>23</td><td>Kidneys→UBC</td><td>1.678</td><td>The uncertainty is calculated by that of the total outflow rate from kidneys and flow fraction from kidneys to UBC.</td><td>0.551</td><td>0.598</td><td>2.758</td><td>Normal</td><td></td></tr><tr><td>24</td><td>Kidneys→Blood</td><td>31.876</td><td>According to Eqs.2-3, the uncertainty is calculated from that of transfer rate from blood to kidneys and that of equilibrium contents of Cs in plasma and in kidneys.</td><td>9.966</td><td>12.343</td><td>51.409</td><td>Normal</td><td></td></tr><tr><td>25</td><td>Muscle→Blood</td><td>0.0751</td><td>According to Eqs.2-3, the uncertainty is calculated from that of transfer rate from blood to muscle and that of equilibrium contents of Cs in plasma and in muscle.</td><td>0.0332</td><td>0.010</td><td>0.140</td><td>Normal</td><td></td></tr><tr><td>26</td><td>GI Tract Tissue→Blood</td><td>8.191</td><td>According to Eqs.2-3, the uncertainty is calculated from that of transfer rate from blood to GI tract tissue and that of equilibrium contents of Cs in plasma and in GI tract tissue.</td><td>2.597</td><td>3.101</td><td>13.281</td><td>Normal</td><td></td></tr><tr><td>27</td><td>GI Tract Tissue→Liver</td><td>0.431</td><td>The uncertainty is calculated by that of the total outflow rate from GI tract tissue and the uncertainty of equilibrium Cs content in plasma and in liver</td><td>0.1455</td><td>0.146</td><td>0.716</td><td>Normal</td><td></td></tr><tr><td>28</td><td>GI Tract Tissue→St</td><td>0.0333</td><td>The transfer rates of GI tract tissue→St, GI tract tissue→SI and GI tract tissue→LI are calculated as daily flow of substances 40 g d-1 (for St), 130 g d-1 (for SI) and 80 g d-1 (for LI) divided by the mass of the GI tract (1200 g). So the uncertainty is calculated from that of daily flow rate and that of GI tract tissue mass.</td><td>0.00471</td><td>0.024</td><td>0.043</td><td>Normal</td><td>Leggett Williams, 1986; 1988; ICRP Publ. 88</td></tr><tr><td>29</td><td>GI Tract Tissue→SI</td><td>0.108</td><td>See parameter No. 28.</td><td>0.0153</td><td>0.078</td><td>0.138</td><td>Normal</td><td></td></tr><tr><td>30</td><td>GI Tract Tissue→LI</td><td>0.0667</td><td>See parameter No. 28.</td><td>0.00943</td><td>0.048</td><td>0.085</td><td>Normal</td><td></td></tr><tr><td>31</td><td>Spleen→Blood</td><td>5.033</td><td>According to Eq.2-3, the uncertainty is calculated from that of transfer rate from blood to spleen and that of equilibrium contents of Cs in plasma and in spleen.</td><td>2.215</td><td>0.692</td><td>9.374</td><td>Normal</td><td>Leggett et al. 2003</td></tr></table>
<table><tr><td>32</td><td>Spleen→Liver</td><td>0.265</td><td>The uncertainty is derived from uncertainty of plasma to spleen and the uncertainty of fraction uptake of liver from portal vein.</td><td>0.0803</td><td>0.108</td><td>0.422</td><td>Normal</td><td></td></tr><tr><td>33</td><td>Pancreas→Blood</td><td>1.678</td><td>According to Eqs.2-3, the uncertainty is calculated from that of transfer rate from blood to pancreas and that of equilibrium contents of Cs in plasma and in pancreas.</td><td>0.7696</td><td>0.170</td><td>3.186</td><td>Normal</td><td>Leggett et al. 2003</td></tr><tr><td>34</td><td>Pancreas→Liver</td><td>0.0883</td><td>The uncertainty is derived from uncertainty of 1766 d-1 and the uncertainty of fraction uptake of liver from portal vein.</td><td>0.0267</td><td>0.036</td><td>0.141</td><td>Normal</td><td></td></tr><tr><td>35</td><td>Skin→Blood</td><td>0.867</td><td>According to Eq.2-3, the uncertainty is calculated from that of transfer rate from blood to skin and that of equilibrium contents of Cs in plasma and in skin.</td><td>0.282</td><td>0.315</td><td>1.419</td><td>Normal</td><td>Leggett et al. 2003</td></tr><tr><td>36</td><td>Skin→Excreta</td><td>0.0159</td><td>The uncertainty is calculated by that of total outflow rate from skin and that of flow fraction from skin to sweat.</td><td>0.0054</td><td>0.005</td><td>0.026</td><td>Normal</td><td></td></tr><tr><td>37</td><td>Brain→Blood</td><td>0.0848</td><td>According to Eq.2-3, the uncertainty is calculated from that of transfer rate from blood to brain and that of equilibrium contents of Cs in plasma and in brain.</td><td>0.0313</td><td>0.023</td><td>0.146</td><td>Normal</td><td>Leggett et al. 2003</td></tr><tr><td>38</td><td>Red Marrow→Blood</td><td>0.706</td><td>According to Eq.2-3, the uncertainty is calculated from that of transfer rate from blood to red marrow and that of equilibrium contents of Cs in plasma and in other red marrow.</td><td>0.3414</td><td>0.037</td><td>1.375</td><td>Normal</td><td>Leggett et al. 2003</td></tr><tr><td>39</td><td>Other Skeleton→Blood</td><td>0.128</td><td>According to Eq.2-3, the uncertainty is calculated from that of transfer rate from blood to other skeleton and that of equilibrium contents of Cs in plasma and in other skeleton.</td><td>0.0621</td><td>0.006</td><td>0.250</td><td>Normal</td><td>Leggett et al. 2003</td></tr><tr><td>40</td><td>Lungs→Blood</td><td>1.472</td><td>According to Eq.2-3, the uncertainty is calculated from that of transfer rate from blood to lungs and that of equilibrium contents of Cs in plasma and in lungs.</td><td>0.710</td><td>0.080</td><td>2.864</td><td>Normal</td><td>Leggett et al. 2003</td></tr><tr><td>41</td><td>Adipose Tissue→Blood</td><td>1.766</td><td>According to Eq.2-3, the uncertainty is calculated from that of transfer rate from blood to adipose tissue and that of equilibrium contents of Cs in plasma and in other adipose tissue.</td><td>0.794</td><td>0.209</td><td>3.323</td><td>Normal</td><td>Leggett et al. 2003</td></tr></table>
PARAMETER UNCERTAINTY OF CS MODEL

<table><tr><td>42</td><td>Other 1→Blood</td><td>0.692</td><td>The uncertainty is contributed from that of transfer rate from blood to other 1 and that of equilibrium contents of Cs in plasma and in other 1.</td><td>0.348</td><td>0.009</td><td>1.375</td><td>Normal</td><td>Leggett et al. 2003</td></tr><tr><td>43</td><td>Other 2→Blood</td><td>0.00141</td><td>Similar to parameter No. 42</td><td>0.0007096</td><td>0.000</td><td>0.003</td><td>Normal</td><td>Leggett et al. 2003</td></tr><tr><td>44</td><td>RBC→Blood</td><td>0.257</td><td>The uncertainty is calculated from that of transfer rate of blood to RBC and that of equilibrium content of Cs in plasma and in RBC.</td><td>0.257</td><td>0.166</td><td>0.348</td><td>Normal</td><td>Leggett et al. 2003</td></tr><tr><td>45</td><td>UBC→Urine</td><td>12</td><td>This transfer rate is given by ICRP as 12 d-1. In the present study, it is assumed as a triangular distribution with a=6 d-1, b=12 d-1 and c=22 d-1.</td><td></td><td>6, 12, 22</td><td></td><td>Triangular</td><td>ICRP 56, Li et al, Apostoaei</td></tr><tr><td>46</td><td>St→SI</td><td>40</td><td>This transfer rate is assumed to be the same as K and Rb, 40 d<sup>-1</sup>. However, it is larger comparing to 24 d-1 given by ICRP Publication 30 and it is close to 48 d-1 for non-carolic liquid recommended by ICRP Publication 100. In the present study, A uniform distribution is assumed with minimum and maximum values 14 d-1 and 48 d-1, respectively.</td><td></td><td>14, 48</td><td></td><td>Uniform</td><td>Leggett et al. 2003; ICRP 30; ICRP 100</td></tr><tr><td>47</td><td>SI→Blood</td><td>28.215</td><td>The uncertainty is contributed from transfer rate of plasma to SI (parameter No. 7) and that of equilibrium content of Cs in plasma and in SI.</td><td>12.346</td><td>4.017</td><td>52.413</td><td>Normal</td><td>Leggett et al. 2003</td></tr><tr><td>48</td><td>SI→LI</td><td>0.3</td><td>This transfer rate is assumed as the same value as K and Rb, 30 d<sup>-1</sup>. No uncertainty is given. It is assumed in the present analysis a CV of 10%.</td><td>0.03</td><td>0.241</td><td>0.359</td><td>Normal</td><td>Leggett et al. 2003, Leggett and Williams, 1986, 1988</td></tr><tr><td>49</td><td>SI→Liver</td><td>1.485</td><td>Three uncertainties contribute to this transfer rate: uncertainty of total removal rate from small intestine (σ=3 d<sup>-1</sup> for 30 d<sup>-1</sup>), the uncertainty of extraction fraction for liver ((σ=0.01 for 0.05) and the uncertainty of outflow from small intestine to plasma</td><td>0.177</td><td>1.138</td><td>1.832</td><td>Normal</td><td>Leggett et al. 2003, Leggett and Williams, 1986, 1988</td></tr></table>
(assumed to be constant 99%).

<table><tr><td>50</td><td>LI→Feces</td><td>0.5</td><td>A mean value of 0.5 d<sup>-1</sup> was given in Leggett et al. 2003. However, comparing to the transfer rate from large intestine to feces, 0.6 d<sup>-1</sup> calculated by transfer rates from ULI to LLI (1.8 d<sup>-1</sup>) and LLI to feces (1.0 d<sup>-1</sup>) reported by ICRP Publication 30 and the transfer rate, 0.67 d<sup>-1</sup> calculated from the transfer rates from right colon to left colon and left colon to rectosigmoid, and from rectosigmoid to feces (2.0 d<sup>-1</sup>) for male by ICRP Publication 100, in the present study, a triangular distribution is assumed with 0.5, 0.5 and 1 to obtain a mean value of 0.67 d<sup>-1</sup> and mode value of 0.5 d<sup>-1</sup>.</td><td>0.118</td><td>0.5, 0.5, 1</td><td>Triangular</td><td>Leggett et al. 2003; ICRP 30, ICRP 100</td></tr></table>
Appendix B

<table><caption>Figure B1. Blood, urinary excretion and Whole body</caption><tr><td><img src="blood_excretion_image.png" alt="Blood retention of injected <sup>137</sup>Cs (to be provided)"></td><td>Urinary excretion (to be provided)</td><td><img src="whole_body_excretion_image.png" alt="Whole body retention of injected <sup>137</sup>Cs (to be provided)"></td></tr></table>
<figure><img src="image_7.png" /><figcaption>Figure 2B. Effects on the blood, whole body retention</figcaption></figure>
PARAMETER UNCERTAINTY OF CS MODEL

<figure><img src="image_8.png" /><figcaption>Figure 3B. Effects on the small intestine and urinary excretion faecal excretion</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 4B. Parameter sensitivity on blood clearance, whole body retention and urinary excretion.</figcaption></figure>