Plutonium Biokinetics in Human Body A. Luciani - Kit-Bibliothek - FZK

3.1.1 MATHEMATICAL TOOLS
3.1.1.1 Solution of a compartmental model
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3.1 BIOKINETIC MODELLING
As briefly introduced in paragraph 1.3.2, the biokinetic models used in internal
dosimetry are based on systems of compartments describing the kinetic of the radionuclide of
interest in different organs or tissues. The amount of activity in the compartments is
calculated by solving the system of differential equations which mathematically describes the
compartmental model [56]. In case of a single acute intake the resulting system of differential
equations with its initial conditions is given by:
where:
qi(t) is the amount of activity in the i-th compartment at time t;
ri,j is the transfer rate from j-th to i-th compartment;
λR is the physical decay constant;
i
q0 is the initial activity amount in the i-th compartment.
as:
dq1 t
dt
dq2 t
dt
.......
dq i t
dt
.......
dq n t
dt
n
j 2
n,j 2
j 1
n,j i
j 1
r 1,jq j(t)
r 2, jq j (t)
r i,jq j(t)
n 1
rn,j q j (t)
j 2
q 1(t) r j,1
j 2
n,j2
equation 3.1.1
In the more compact matrix form, the same system of differential equations is written
dq(t)
dt
where:
q(t) is the vector of the amount of activity in the n compartments at time t;
R is the matrix of the transfer rates;
I is the unit matrix;
q 0 is the vector of initial conditions.
n
q 2(t) r j,2
j 1
n,ji
q i(t) r j,i
j 1
n 1
q n(t) r j,n
j 2
λ Rq 1(t)
λ Rq 2(t)
λ Rq i(t)
λ Rq n(t)
(R λ RI)q(t) q(0) q 0
1
q1(0) q0 2
q2(0) q0 ........
i
qi (0) q0 ......
n
qn(0) q0 equation 3.1.2