Appendix 1: Analysis of the two-compartment model with first-order absorption
Since this is the most complex model compared with the other three previously discussed models, the number of solutions to its main Eq. (22) is analysed slightly differently. It was found that the standard Laplace transform
$$ Z_n=\int\limits_0^{\infty} {e^{-nt} C(t)\hbox{d}t} $$
(45)
simplifies the analysis in this case, therefore values Z
n
are used here instead of previously used moments S
n
Eq. (4).
For the model in question we have
$$ Z_0=\frac{C_0}{k_{10}}, $$
(46)
$$ Z_n=C_0\frac{k_{\rm a}(k_{21}+n)}{(k_{\rm a}+n)(\alpha+n)(\beta+n)}. $$
(47)
In this paper we are going to demonstrate that the exponential rate constants of Eq. (22) can be explicitly defined in terms of the roots of cubic equation.
Considering a symmetric form Eq. (21) of the Eq. (22) where all three exponential rate constants λ1, λ2 and λ3 and coefficients A
1, A
2, A
3 (A
1 + A
2 + A
3 = 0) are equivalent, the cubic equation, three roots of which are our exponential rate constants, is constructed as
$$ x^3+a_2 x^2+a_1 x+a_0=0=(x-\lambda_1)(x-\lambda_2)(x-\lambda_3), $$
(48)
the expanded form of which defines how the coefficients a
0, a
1 and a
2 should look like:
$$ x^3-(\lambda_1+\lambda_2+\lambda_3)x^2+ (\lambda_1\lambda_2+\lambda_2\lambda_3+\lambda_3\lambda_1)x -\lambda_1\lambda_2\lambda_3 = 0. $$
(49)
Next, by calculating three “neighbour” moments Z
s
, Z
s+1 and Z
s+2 according to Eq. (45), after several rearrangements we have a linear combination of the coefficients a
0,a
1 and a
2 in a form
$$ c_0(s) a_0 +c_1(s) a_1 +c_2(s) a_2 = b(s), $$
(50)
where
$$ c_0(s) = Z_s-2 Z_{s+1}+Z_{s+2}, $$
(51)
$$ c_1(s)=-sZ_s+2(s+1)Z_{s+1}-(s+2)Z_{s+2}, $$
(52)
$$ c_2(s)=s^2Z_s-2(s+1)^2Z_{s+1} +(s+2)^2 Z_{s+2}, $$
(53)
$$ b(s)=s^3Z_s - 2(s+1)^3 Z_{s+1}+(s+2)^3 Z_{s+2}. $$
(54)
Then a substitution of three suitable values of s into Eqs. (50)–(54) gives a standard linear system of three equations with three unknowns (a
0, a
1 and a
2). For example, with s = 0, 1, 2 we have a 3 × 3 system
$$ \left[ \begin{array}{ccc} c_0(0) & c_1(0) & c_2(0) \\ c_0(1) & c_1(1) & c_2(1)\\ c_0(2) & c_1(2) & c_2(2) \end{array}\right]\,\left[ \begin{array}{c} a_0 \\ a_1 \\ a_2 \end{array}\right] = \left[\begin{array}{c} b(0) \\ b(1) \\ b(2) \end{array}\right], $$
(55)
solution of which expresses coefficients a
0, a
1, a
2 in terms of exact values of five moments—
\(Z_0,{\dots},Z_4\):
$$ a_0=24 ((3 Z_{1} (Z_{2}-Z_{3})+Z_{2} Z_{3}) Z_{4}-Z_{1} Z_{2} Z_{3})/d, $$
(56)
$$ \begin{aligned} a_1&=(((2 Z_{2}-18 Z_{3}+24 Z_{4}) Z_{1} +54 Z_{2} Z_{3}+72 Z_{3} Z_{4}-128 Z_{2} Z_{4}) Z_{0} \\ &\quad+(126 Z_{2} Z_{4}-114 Z_{3} Z_{4}-44 Z_{2} Z_{3}) Z_{1}+26 Z_{2} Z_{3} Z_{4})/d, \end{aligned} $$
(57)
$$ \begin{aligned} a_2&=(((3 Z_{2}-24 Z_{3}+30 Z_{4}) Z_{1} +45 Z_{2} Z_{3}+42 Z_{3} Z_{4}-96 Z_{2} Z_{4}) Z_{0} \\ &\quad+(63 Z_{2} Z_{4}-48 Z_{3} Z_{4}-24 Z_{2} Z_{3}) Z_{1}+9 Z_{2} Z_{3} Z_{4})/d, \end{aligned} $$
(58)
$$ \begin{aligned} \hbox{where}\, d&=((Z_{2}+6 (Z_{4}-Z_{3})) Z_{1}+9 Z_{2} Z_{3}+6 Z_{3} Z_{4}-16 Z_{2} Z_{4}) Z_{0} \\ &\quad +(9 Z_{2} Z_{4}-4 Z_{2} Z_{3}-6 Z_{3} Z_{4}) Z_{1}+Z_{2} Z_{3} Z_{4}. \end{aligned} $$
(59)
Since all the values of a
0, a
1, a
2 are known at this point, the roots λ1, λ2, λ3 of the cubic equation Eq. (48) can be found by means of any standard method.
When the exponential rate constants λ1, λ2, λ3 are known, the coefficients A
1, A
2 can be found as
$$ A_2=\frac{\lambda_2(\lambda_2+1) \left((Z_{0}-Z_{1})\lambda_1\lambda_3-Z_{1}(\lambda_1+\lambda_3+1) \right)} {(\lambda_2-\lambda_1)(\lambda_2-\lambda_3)}, $$
(60)
$$ A_1=\frac{\lambda_1\left(Z_{0}\lambda_2\lambda_3+A_2(\lambda_2-\lambda_3)\right)} {\lambda_2(\lambda_3-\lambda_1)}. $$
(61)
Appendix 2: Nomenclature
-
: -
One- and two-compartment iv models
-
: -
One- and two-compartment ev models with first-order absorption
-
a
0, a
1, a
2, d
:
-
Cubic coefficients in Eqs. (48), (56)–(59)
-
A
1, A
2, A
3
:
-
Coefficients of the PK curve
- α, β:
-
Hybrid rate constants
- AUC:
-
Area under the PK curve C(t)
- AUMC:
-
Area under the curve tC(t)
- AUMMC:
-
Area under the curve t
2
C(t)
-
: -
Initial concentration, Solutions I, II, III
-
c
0(s), c
1(s), c
2(s), b(s):
-
Parameters of Eq. (50)
-
: -
Concentration in the central compartment, Solutions I, II, III
-
: -
Concentration in the absorption compartment, Solutions I, II, III
-
: -
Concentration in the elimination compartment, Solutions I, II, III
-
: -
Concentration in the peripheral compartment, Solutions I,II, III
-
Cl
:
-
Total clearance
-
Cl
d
:
-
Intercompartmental distribution clearance
-
: -
Maximum plasma concentration
-
: -
Maximum plasma concentration in the central compartment
-
: -
Maximum plasma concentration in the peripheral compartment
-
k
10
:
-
Elimination rate constant
-
k
12
:
-
Transfer rate constant from the central (1) to the peripheral (2) compartment
-
k
21
:
-
Transfer rate constant from the peripheral (2) to the central (1) compartment
-
k
a
:
-
Absorption rate constant
- λ1, λ2, λ3
:
-
Exponent of the ith exponential term of a polyexponential equation
- MAT:
-
Mean absorption time
- MRT:
-
Mean residence time
- MRTC:
-
Mean residence time in the central compartment
- MRTP:
-
Mean residence time in the peripheral compartment
-
S
i
:
-
ith S-moment
-
t
:
-
Time after drug administration
-
: -
Time to reach 
-
: -
Time to reach 
-
: -
Time to reach 
-
: -
Volume of distribution
-
: -
Volume of the central compartment
-
: -
Volume of the peripheral compartment
- VRT:
-
Variance in residence time
-
x
:
-
Cubic variable Eq. (48)
-
Z
i
:
-
ith Z-moment