Studies of strontium biokinetics in humans

Radiat Environ Biophys (2002) 41:281–287

The wrong version of Appendix was published. The correct Appendix appears here.

Appendix

Derivation of the f₁ equation:

$$ {\rm f}_{\rm 1} = {{z_{86po} } \over {z_{84iv} }} \cdot {{m_{84iv} } \over {m_{86po} }} $$

$$ {{z_{86po} } \over {z_{84iv} }} = {{\left( {{{z_{84} } \over {z_{88} }}} \right) \cdot \left( {h_{86} \cdot k_{88} - k_{86} } \right) + \left( {{{z_{86} } \over {z_{88} }}} \right) \cdot \left( {1 - h_{84} \cdot k_{88} } \right) + h_{84} \cdot k_{86} - h_{86} } \over {\left( {{{z_{84} } \over {z_{88} }}} \right) \cdot \left( {1 - h_{86} \cdot l_{88} } \right) + \left( {{{z_{86} } \over {z_{88} }}} \right) \cdot \left( {h_{84} \cdot l_{88} - l_{84} } \right) + h_{86} \cdot l_{84} - h_{84} }} $$

where ${{z_{84} } \over {z_{88} }}$, ${{z_{86} } \over {z_{88} }}$ are the ratios of strontium isotopes in blood and urine, measured by TI-MS

h₈₄, h₈₆: ratios of isotopes ⁸⁴Sr, and ⁸⁶Sr in relation to natural ⁸⁸Sr

k₈₆, k₈₈: ratios of isotopes ⁸⁶Sr, and ⁸⁸Sr in relation to ⁸⁴Sr in the intravenous tracer

l₈₄, l₈₈: ratios of isotopes ⁸⁴Sr, and ⁸⁸Sr in relation to ⁸⁶Sr in the oral tracer

m₈₄, m₈₆: amounts of the isotopes ⁸⁴Sr and ⁸⁶Sr as administered tracers, multiplied with the enrichment factor (enrichment of ⁸⁴Sr: 76.4%, of ⁸⁶Sr: 95.70%, see Table 2) iv=intravenous, po=per oral.

For calculation of factors h_84,86, k_86,88, l_84,88: take the corresponding data given in Table 2 (e.g.: h₈₄=ratio of 0.56 to 82.58).

Considering the composition of the measured values at masses z₈₄, z₈₆ and z₈₈:

$$ z_{84} = z_{84nat} + z_{84iv} + z_{84po} $$

(1)

$$ z_{86} = z_{86nat} + z_{86iv} + z_{86po} $$

(2)

$$ z_{88} = z_{88nat} + z_{88iv} + z_{88po} $$

(3)

It is:

$$ z_{84nat} = h_{84} \cdot z{}_{88nat} $$

(4)

$$ h_{84} = {{z_{84nat} } \over {z_{88nat} }} $$

(4a)

$$ z_{86nat} = h_{86} \cdot z_{88nat} $$

(5)

$$ h_{86} = {{z_{86nat} } \over {z_{88nat} }} $$

(5a)

$$ z_{86iv} = k_{86} \cdot z_{84iv} $$

(6)

$$ k_{86} = {{z_{86iv} } \over {z_{84iv} }} $$

(6a)

$$ z_{88iv} = k_{88} \cdot z_{84iv} $$

(7)

$$ k_{88} = {{z_{88iv} } \over {z_{84iv} }} $$

(7a)

$$ z_{84po} = l_{84} \cdot z_{86po} $$

(8)

$$ l_{84} = {{z_{84po} } \over {z_{86po} }} $$

(8a)

$$ z_{88po} = l_{88} \cdot z_{86po} $$

(9)

$$ l_{88} = {{z_{88po} } \over {z_{86po} }} $$

(9a)

$$ {{\rm{The}}\;{\rm{ratio}}\;{\rm{of}}\;{{z_{{84}} } \over {z_{{88}} }}\;{\rm{is}}\;{\rm{set}}\;{\rm{equal}}\;{\rm{to}}\;{\rm{ratio}}\;{\rm{of}}\;{\rm{Eq}}{\rm{.}}\;{\left( {\rm{1}} \right)}\;{\rm{to}}\;{\rm{Eq}}{\rm{.}}\;{\left( {\rm{3}} \right)}} $$

(10)

$$ {{\rm{and}}\;{\rm{the}}\;{\rm{ratio}}\;{\rm{of}}\;{{z_{{86}} } \over {z_{{88}} }}\;{\rm{is}}\;{\rm{set}}\;{\rm{equal}}\;{\rm{to}}\;{\rm{ratio}}\;{\rm{of}}\;{\rm{Eq}}{\rm{.}}\;{\left( 2 \right)}\;{\rm{to}}\;{\rm{Eq}}{\rm{.}}\;{\left( 3 \right)}} $$

(11)

$$ {{\rm{Solve}}\;{\rm{Eq}}{\rm{.}}\;{\left( {{\rm{10}}} \right)}\;{\rm{for}}\;{\rm{z}}_{{{\rm{88nat}}}} } $$

(12)

and

$$ {{\rm{solve}}\;{\rm{Eq}}{\rm{.}}\;{\left( {{\rm{11}}} \right)}\;{\rm{for}}\;{\rm{z}}_{{{\rm{88nat}}}} \;{\left( {{\rm{taking}}\;{\rm{into}}\;{\rm{account}}\;{\rm{Eqs}}{\rm{.}}\;{\rm{4 - 9a}}} \right)}.} $$

(13)

Finally, set Eq. (12) equal to Eq. (13), and solve for ${{z_{86po} } \over {z_{84iv} }}$.