Physiological Modeling to Understand the Impact of Enzymes and Transporters on Drug and Metabolite Data and Bioavailability Estimates

Abstract

Purpose

To obtain mathematical solutions that correlate drug and metabolite exposure and systemic bioavailability (F _sys) with physiological determinants, transporters and enzymes.

Methods

A series of physiologically-based pharmacokinetic (PBPK) models that included renal excretion and sequential metabolism within the intestine and/or liver as metabolite formation organs were developed. The area under the curve for drug (AUC) and formed metabolite (AUC{mi,P}) were solved by matrix inversion.

Results

The PBPK models revealed that AUC{mi,P} was dependent on dispositional parameters (transport and elimination) for the drug and metabolite. The solution was unique for each metabolite formation organ and was dependent on the type of drug and metabolite elimination organs. The AUC ratio of the formed metabolite after oral and intravenous drug dosing was useful for determination of the fraction absorbed (F _abs) and not the systemic bioavailability (F _sys) when either intestine or liver was the only drug elimination organ.

Conclusions

The AUC ratio of the formed metabolite after oral and intravenous drug dosing differed from that for drug and would not provide F _sys. However, the AUC ratio of the formed metabolite for oral and intravenous drug dosing furnished the estimate of F _abs when intestine or liver was the only drug metabolic organ.

Appendix

Mass balance equations and the corresponding matrices for the physiologically based pharmacokinetic model (case 1 to case 4, as shown in Figs. 1, 2 and 3)

Q:

blood flow rate

V:

blood or tissue volume

P:

parent drug

Mi:

the primary metabolite of interest

SB:

systemic blood, used as subscripts

HP:

highly perfused organ, used as subscript

PP:

poorly perfused organ, used as subscript

Intb:

intestinal blood, used as subscript

Int:

intestinal tissue, used as subscript

Lumen:

intestinal lumen, used as subscript

LB:

liver blood, used as subscript

L:

liver tissue, used as subscript

PV:

portal vein, used as subscript

HA:

hepatic artery, used as subscript

CL_r, CL_r{mi}:

apparent renal clearances of the parent drug and the metabolite, Mi, respectively

${{CL}}_{\rm{d1}}^{\rm{I}}{,}{\kern 1pt} {{CL}}_{\rm{d2}}^{\rm{I}}{\kern 1pt} {\kern 1pt}$ :

basolateral influx and efflux clearances of enterocytes, respectively

CL_int,met1,I :

metabolic intrinsic clearance for formation of the Mi in the intestinal tissue

CL_int,met2,I :

metabolic intrinsic clearance for formation of other metabolites in the intestinal tissue

CL_int,sec,I :

secretory intrinsic clearance for drug in the intestinal tissue

k_a :

rate constant of drug absorption in the intestine

k_g :

rate constant of intestinal transit and degradation

${{CL}}_{\rm{d1}}^{\rm{H}}{,}{\kern 1pt} {{CL}}_{\rm{d2}}^{\rm{H}}{\kern 1pt} {\kern 1pt}$ :

basolateral influx and efflux clearances of the hepatocyte, respectively

CL_int,met1,H :

metabolic intrinsic clearance for formation of the metabolite of interest in liver

CL_int,met2,H :

metabolic intrinsic clearance for formation of other metabolites in the liver

CL_int,sec,I :

secretory intrinsic clearance of drug in the liver

{mi} and {mii}:

symbols used to qualify the parameters for primary metabolites formed in intestine and other primary metabolite formed in liver for case 3

• (1) Case 1 (see Fig. 1 for the model scheme)

In systemic blood (denoted by the subscript, SB),

$$ {{{V}}_{\rm{SB}}}\frac{{{\text{d}}{{\text{P}}_{\rm{SB}}}}}{{dt}} = {{{Q}}_{\rm{HP}}}\frac{{{{\text{P}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}}} + {{{Q}}_{\rm{PP}}}\frac{{{{\text{P}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}}} + \left( {{{{Q}}_{\rm{HA}}} + {{{Q}}_{\rm{PV}}}} \right){{{P}}_{\rm{LB}}} - \left( {{{C}}{{{L}}_{\rm{r}}} + {{{Q}}_{\rm{HP}}} + {{{Q}}_{\rm{PP}}} + {{{Q}}_{\rm{HA}}} + {{{Q}}_{\rm{PV}}}} \right){{{P}}_{\rm{SB}}} $$

(1)

$$ {{{V}}_{\rm{SB}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{SB}}}}}{{dt}} = {{{Q}}_{\rm{HP}}}\frac{{{\text{M}}{{\text{i}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}\left\{ {{mi}} \right\}}} + {{{Q}}_{\rm{PP}}}\frac{{{\text{M}}{{\text{i}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}\left\{ {{mi}} \right\}}} + \left( {{{{Q}}_{\rm{HA}}} + {{{Q}}_{\rm{PV}}}} \right){{M}}{{{i}}_{\rm{LB}}} - {\kern 1pt} \left( {{{C}}{{{L}}_{\rm{r}}}\left\{ {{mi}} \right\} + {{{Q}}_{\rm{HP}}} + {{{Q}}_{\rm{PP}}} + {{{Q}}_{\rm{HA}}} + {{{Q}}_{\rm{PV}}}} \right){{M}}{{{i}}_{\rm{SB}}} $$

(2)

In highly perfused organs (denoted by the subscript, HP),

$$ {{{V}}_{\rm{HP}}}\frac{{{\text{d}}{{\text{P}}_{\rm{HP}}}}}{{dt}} = {{{Q}}_{\rm{HP}}}\left( {{{{P}}_{\rm{SB}}} - \frac{{{{\text{P}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}}}} \right) $$

(3)

$$ {{{V}}_{\rm{HP}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{HP}}}}}{{dt}} = {{{Q}}_{\rm{HP}}}\left( {{{M}}{{{i}}_{\rm{SB}}} - {\kern 1pt} \,\frac{{{\text{M}}{{\text{i}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}\left\{ {{mi}} \right\}}}} \right) $$

(4)

In poorly perfused organs (denoted by the subscript, PP),

$$ {{{V}}_{\rm{PP}}}\frac{{{\text{d}}{{\text{P}}_{\rm{PP}}}}}{{dt}} = {{{Q}}_{\rm{PP}}}\left( {{{{P}}_{\rm{SB}}} - \frac{{{{\text{P}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}}}} \right) $$

(5)

$$ {{{V}}_{\rm{PP}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{PP}}}}}{{dt}} = {{{Q}}_{\rm{PP}}}\left( {{{M}}{{{i}}_{\rm{SB}}} - \frac{{{\text{M}}{{\text{i}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}\left\{ {{mi}} \right\}}}} \right) $$

(6)

In intestinal blood (denoted by the subscript, Intb),

$$ {{{V}}_{\rm{Intb}}}\frac{{{\text{d}}{{\text{P}}_{\rm{Intb}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{{P}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{I}}{{{P}}_{\rm{Int}}} - \left( {{{{Q}}_{\rm{PV}}} + {{CL}}_{\rm{d1}}^{\rm{I}}} \right){{{P}}_{\rm{Intb}}} $$

(7)

$$ {{{V}}_{\rm{Intb}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{Intb}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{M}}{{{i}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{I}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{Int}}} - \left( {{{{Q}}_{\rm{PV}}} + {{CL}}_{\rm{d1}}^{\rm{I}}\left\{ {{mi}} \right\}} \right){{M}}{{{i}}_{\rm{Intb}}} $$

(8)

In intestinal tissue (denoted by the subscript, Int),

$$ {{{V}}_{\rm{Int}}}\frac{{{\text{d}}{{\text{P}}_{\rm{Int}}}}}{{dt}} = {{CL}}_{\rm{d1}}^{\rm{I}}{{{P}}_{\rm{Intb}}} + {{{k}}_{\rm{a}}}{{{P}}_{\rm{lumen}}}{{{V}}_{\rm{lumen}}} - {\kern 1pt} \left( {{{CL}}_{\rm{d2}}^{\rm{I}} + {{C}}{{{L}}_{{\rm{int,met1,I}}}} + {{C}}{{{L}}_{{\rm{int,met2,I}}}} + {{C}}{{{L}}_{{\rm{int,sec,I}}}}} \right){{{P}}_{\rm{Int}}} $$

(9)

$$ {{{V}}_{\rm{Int}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{Int}}}}}{{dt}} = {{CL}}_{\rm{d1}}^{\rm{I}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{Intb}}} + {{{k}}_{\rm{a}}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{lumen}}}{{{V}}_{\rm{lumen}}} + {{C}}{{{L}}_{{\rm{int,met1,I}}}}{{{P}}_{\rm{Int}}} - \left( {{{CL}}_{\rm{d2}}^{\rm{I}}\left\{ {{mi}} \right\} + {{C}}{{{L}}_{{\rm{int,met,I}}}}\left\{ {{mi}} \right\} + {{C}}{{{L}}_{{\rm{int,sec,I}}}}\left\{ {{mi}} \right\}} \right){{M}}{{{i}}_{\rm{Int}}} $$

(10)

In intestinal lumen (denoted by the subscript, lumen),

$$ {{{V}}_{\rm{lumen}}}\frac{{{\text{d}}{{\text{P}}_{\rm{lumen}}}}}{{dt}} = {{C}}{{{L}}_{{\rm{int,sec,I}}}}{{{P}}_{\rm{Int}}} - \left( {{{{k}}_{\rm{a}}} + {{{k}}_{\rm{g}}}} \right){{{P}}_{\rm{lumen}}}{{{V}}_{\rm{lumen}}} $$

(11)

$$ {{{V}}_{\rm{lumen}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{lumen}}}}}{{dt}} = {{C}}{{{L}}_{{\rm{int,sec,I}}}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{Int}}} - \left( {{{{k}}_{\rm{a}}}\left\{ {{mi}} \right\} + {{{k}}_{\rm{g}}}\left\{ {{mi}} \right\}} \right){{M}}{{{i}}_{\rm{lumen}}}{{{V}}_{\rm{lumen}}} $$

(12)

In liver blood (denoted by the subscript, LB),

$$ {{{V}}_{\rm{LB}}}\frac{{{\text{d}}{{\text{P}}_{\rm{LB}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{{P}}_{\rm{Intb}}} + {{{Q}}_{\rm{HA}}}{{{P}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{H}}{{{P}}_{\rm{L}}} - \left( {{{{Q}}_{\rm{PV}}} + {{{Q}}_{\rm{HA}}} + {{CL}}_{\rm{d1}}^{\rm{H}}} \right){{{P}}_{\rm{LB}}} $$

(13)

$$ {{{V}}_{\rm{LB}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{LB}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{M}}{{{i}}_{\rm{Intb}}} + {{{Q}}_{\rm{HA}}}{{M}}{{{i}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{H}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{L}}} - \left( {{{{Q}}_{\rm{PV}}} + {{{Q}}_{\rm{HA}}} + {{CL}}_{\rm{d1}}^{\rm{H}}\left\{ {{mi}} \right\}} \right){{M}}{{{i}}_{\rm{LB}}} $$

(14)

In liver tissue (denoted by the subscript L),

$$ {{{V}}_{\rm{L}}}\frac{{{\text{d}}{{\text{P}}_{\rm{L}}}}}{{dt}} = {{CL}}_{\rm{d1}}^{\rm{H}}{{{P}}_{\rm{LB}}} - {\kern 1pt} {\kern 1pt} {{CL}}_{\rm{d2}}^{\rm{H}}{{{P}}_{\rm{L}}} $$

(15)

$$ {{{V}}_{\rm{L}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{L}}}}}{{dt}} = {{CL}}_{\rm{d1}}^{\rm{H}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{LB}}} - {\kern 1pt} {\kern 1pt} {{CL}}_{\rm{d2}}^{\rm{H}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{L}}} $$

(16)

• (2) Case 2 (see Fig. 2 for the model scheme)

In systemic blood (denoted by the subscript, SB),

$$ {{{V}}_{\rm{SB}}}\frac{{{\text{d}}{{\text{P}}_{\rm{SB}}}}}{{dt}} = {{{Q}}_{\rm{HP}}}\frac{{{{\text{P}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}}} + {{{Q}}_{\rm{PP}}}\frac{{{{\text{P}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}}} + \left( {{{{Q}}_{\rm{HA}}} + {{{Q}}_{\rm{PV}}}} \right){{{P}}_{\rm{LB}}} - \left( {{{C}}{{{L}}_{\rm{r}}} + {{{Q}}_{\rm{HP}}} + {{{Q}}_{\rm{PP}}} + {{{Q}}_{\rm{HA}}} + {{{Q}}_{\rm{PV}}}} \right){{{P}}_{\rm{SB}}} $$

(17)

$$ {{{V}}_{\rm{SB}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{SB}}}}}{{dt}} = {\kern 1pt} {\kern 1pt} {\kern 1pt} {{{Q}}_{\rm{HP}}}\frac{{{\text{M}}{{\text{i}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}\left\{ {{mi}} \right\}}} + {{{Q}}_{\rm{PP}}}\frac{{{\text{M}}{{\text{i}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}\left\{ {{mi}} \right\}}} + \left( {{{{Q}}_{\rm{HA}}} + {{{Q}}_{\rm{PV}}}} \right){{M}}{{{i}}_{\rm{LB}}} - \left( {{{C}}{{{L}}_{\rm{r}}}\left\{ {{mi}} \right\} + {{{Q}}_{\rm{HP}}} + {{{Q}}_{\rm{PP}}} + {{{Q}}_{\rm{HA}}} + {{{Q}}_{\rm{PV}}}} \right){{M}}{{{i}}_{\rm{SB}}} $$

(18)

In highly perfused organs (denoted by the subscript, HP),

$$ {{{V}}_{\rm{HP}}}\frac{{{\text{d}}{{\text{P}}_{\rm{HP}}}}}{{dt}} = {{{Q}}_{\rm{HP}}}\left( {{{{P}}_{\rm{SB}}} - \frac{{{{\text{P}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}}}} \right) $$

(19)

$$ {{{V}}_{\rm{HP}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{HP}}}}}{{dt}} = {{{Q}}_{\rm{HP}}}\left( {{{M}}{{{i}}_{\rm{SB}}} - \frac{{{\text{M}}{{\text{i}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}\left\{ {{mi}} \right\}}}} \right) $$

(20)

In poorly perfused organs (denoted by the subscript, PP),

$$ {{{V}}_{\rm{PP}}}\frac{{{\text{d}}{{\text{P}}_{\rm{PP}}}}}{{dt}} = {{{Q}}_{\rm{PP}}}\left( {{{{P}}_{\rm{SB}}} - \frac{{{{\text{P}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}}}} \right) $$

(21)

$$ {{{V}}_{\rm{PP}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{PP}}}}}{{dt}} = {{{Q}}_{\rm{PP}}}\left( {{{M}}{{{i}}_{\rm{SB}}} - \frac{{{\text{M}}{{\text{i}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}\left\{ {{mi}} \right\}}}} \right) $$

(22)

In intestinal blood (denoted by the subscript, Intb),

$$ {{{V}}_{\rm{Intb}}}\frac{{{\text{d}}{{\text{P}}_{\rm{Intb}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{{P}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{I}}{{{P}}_{\rm{Int}}} - \left( {{{{Q}}_{\rm{PV}}} + {{CL}}_{\rm{d1}}^{\rm{I}}} \right){{{P}}_{\rm{Intb}}} $$

(23)

$$ {{{V}}_{\rm{Intb}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{Intb}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{M}}{{{i}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{I}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{Int}}} - \left( {{{{Q}}_{\rm{PV}}} + {{CL}}_{\rm{d1}}^{\rm{I}}\left\{ {{mi}} \right\}} \right){{M}}{{{i}}_{\rm{Intb}}} $$

(24)

In intestinal tissue (denoted by the subscript, Int),

$$ {{{V}}_{\rm{Int}}}\frac{{{\text{d}}{{\text{P}}_{\rm{Int}}}}}{{dt}} = {{CL}}_{\rm{d1}}^{\rm{I}}{{{P}}_{\rm{Intb}}} + {{{k}}_{\rm{a}}}{{{P}}_{\rm{lumen}}}{{{V}}_{\rm{lumen}}} - {{CL}}_{\rm{d2}}^{\rm{I}}{{{P}}_{\rm{Int}}} $$

(25)

$$ {{{V}}_{\rm{Int}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{Int}}}}}{{dt}} = {{CL}}_{\rm{d1}}^{\rm{I}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{Intb}}} - {{CL}}_{\rm{d2}}^{\rm{I}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{Int}}} $$

(26)

In intestinal lumen (denoted by the subscript, lumen),

$$ {{{V}}_{\rm{lumen}}}\frac{{{\text{d}}{{\text{P}}_{\rm{lumen}}}}}{{dt}} = - \left( {{{{k}}_{\rm{a}}} + {{{k}}_{\rm{g}}}} \right){{{P}}_{\rm{lumen}}}{{{V}}_{\rm{lumen}}} $$

(27)

In liver blood (denoted by the subscript, LB),

$$ {{{V}}_{\rm{LB}}}\frac{{{\text{d}}{{\text{P}}_{\rm{LB}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{{P}}_{\rm{Intb}}} + {{{Q}}_{\rm{HA}}}{{{P}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{H}}{{{P}}_{\rm{L}}} - \left( {{{{Q}}_{\rm{PV}}} + {{{Q}}_{\rm{HA}}} + {{CL}}_{\rm{d1}}^{\rm{H}}} \right){{{P}}_{\rm{LB}}} $$

(28)

$$ {{{V}}_{\rm{LB}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{LB}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{M}}{{{i}}_{\rm{Intb}}} + {{{Q}}_{\rm{HA}}}{{M}}{{{i}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{H}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{L}}} - \left( {{{{Q}}_{\rm{PV}}} + {{{Q}}_{\rm{HA}}} + {{CL}}_{\rm{d1}}^{\rm{H}}\left\{ {{mi}} \right\}} \right){{M}}{{{i}}_{\rm{LB}}} $$

(29)

In liver tissue (denoted by the subscript, L),

$$ {{{V}}_{\rm{L}}}\frac{{{\text{d}}{{\text{P}}_{\rm{L}}}}}{{dt}} = {{CL}}_{\rm{d1}}^{\rm{H}}{{{P}}_{\rm{LB}}} - \left( {{{CL}}_{\rm{d2}}^{\rm{H}} + {{C}}{{{L}}_{{\rm{int,met1,H}}}} + {{C}}{{{L}}_{{\rm{int,met2,H}}}} + {{C}}{{{L}}_{{\rm{int,sec,H}}}}} \right){{{P}}_{\rm{L}}} $$

(30)

$$ {{{V}}_{\rm{L}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{L}}}}}{{dt}} = {\kern 1pt} {\kern 1pt} {{CL}}_{\rm{d1}}^{\rm{H}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{LB}}} + {{C}}{{{L}}_{{\rm{int,met1,H}}}}{{{P}}_{\rm{L}}} - \left( {{{CL}}_{\rm{d2}}^{\rm{H}}\left\{ {{mi}} \right\} + {{C}}{{{L}}_{{\rm{int,met,H}}}}\left\{ {{mi}} \right\} + {{C}}{{{L}}_{{\rm{int,sec,H}}}}\left\{ {{mi}} \right\}} \right){{M}}{{{i}}_{\rm{L}}} $$

(31)

• (3) Case 3 (see Fig. 3 for the model scheme, different metabolites were formed in intestine (as metabolite mi) and liver (as metabolite mii))

In systemic blood (denoted by the subscript, SB),

$$ {{{V}}_{\rm{SB}}}\frac{{{\text{d}}{{\text{P}}_{\rm{SB}}}}}{{dt}} = {{{Q}}_{\rm{HP}}}\frac{{{{\text{P}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}}} + {{{Q}}_{\rm{PP}}}\frac{{{{\text{P}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}}} + \left( {{{{Q}}_{\rm{HA}}} + {{{Q}}_{\rm{PV}}}} \right){{{P}}_{\rm{LB}}} - \left( {{{C}}{{{L}}_{\rm{r}}} + {{{Q}}_{\rm{HP}}} + {{{Q}}_{\rm{PP}}} + {{{Q}}_{\rm{HA}}} + {{{Q}}_{\rm{PV}}}} \right){{{P}}_{\rm{SB}}} $$

(32)

$$ {{{V}}_{\rm{SB}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{SB}}}}}{{dt}} = {\kern 1pt} {\kern 1pt} {\kern 1pt} {{{Q}}_{\rm{HP}}}\frac{{{\text{M}}{{\text{i}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}\left\{ {{mi}} \right\}}} + {{{Q}}_{\rm{PP}}}\frac{{{\text{M}}{{\text{i}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}\left\{ {{mi}} \right\}}} + \left( {{{{Q}}_{\rm{HA}}} + {{{Q}}_{\rm{PV}}}} \right){{M}}{{{i}}_{\rm{LB}}} - \left( {{{C}}{{{L}}_{\rm{r}}}\left\{ {{mi}} \right\} + {{{Q}}_{\rm{HP}}} + {{{Q}}_{\rm{PP}}} + {{{Q}}_{\rm{HA}}} + {{{Q}}_{\rm{PV}}}} \right){{M}}{{{i}}_{\rm{SB}}} $$

(33)

$$ {{\text{V}}_{{\text{SB}}}}\frac{{{\text{dMi}}{{\text{i}}_{{\text{SB}}}}}}{{{\text{dt}}}} = {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\text{Q}}_{{\text{HP}}}}\frac{{{\text{Mi}}{{\text{i}}_{{\text{HP}}}}}}{{{{\text{K}}_{{\text{HP}}}}\left\{ {{\text{mii}}} \right\}}} + {{\text{Q}}_{{\text{PP}}}}\frac{{{\text{Mi}}{{\text{i}}_{{\text{PP}}}}}}{{{{\text{K}}_{{\text{PP}}}}\left\{ {{\text{mii}}} \right\}}} + \left( {{{\text{Q}}_{{\text{HA}}}} + {{\text{Q}}_{{\text{PV}}}}} \right){\text{Mi}}{{\text{i}}_{{\text{LB}}}} - \left( {{\text{C}}{{\text{L}}_{\text{r}}}\left\{ {{\text{mii}}} \right\} + {{\text{Q}}_{{\text{HP}}}} + {{\text{Q}}_{{\text{PP}}}} + {{\text{Q}}_{{\text{HA}}}} + {{\text{Q}}_{{\text{PV}}}}} \right){\text{Mi}}{{\text{i}}_{{\text{SB}}}} $$

(34)

In highly perfused organs (denoted by the subscript, HP),

$$ {{{V}}_{\rm{HP}}}\frac{{{\text{d}}{{\text{P}}_{\rm{HP}}}}}{{dt}} = {{{Q}}_{\rm{HP}}}\left( {{{{P}}_{\rm{SB}}} - \frac{{{{\text{P}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}}}} \right) $$

(35)

$$ {{{V}}_{\rm{HP}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{HP}}}}}{{dt}} = {{{Q}}_{\rm{HP}}}\left( {{{M}}{{{i}}_{\rm{SB}}} - \frac{{{\text{M}}{{\text{i}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}\left\{ {{mi}} \right\}}}} \right) $$

(36)

$$ {{{V}}_{\rm{HP}}}\frac{{{\text{dMi}}{{\text{i}}_{\rm{HP}}}}}{{dt}} = {{{Q}}_{\rm{HP}}}\left( {{{Mi}}{{{i}}_{\rm{SB}}} - \frac{{{\text{Mi}}{{\text{i}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}\left\{ {{mii}} \right\}}}} \right) $$

(37)

In poorly perfused organs (denoted by the subscript, PP),

$$ {{{V}}_{\rm{PP}}}\frac{{{\text{d}}{{\text{P}}_{\rm{PP}}}}}{{dt}} = {{{Q}}_{\rm{PP}}}\left( {{{{P}}_{\rm{SB}}} - \frac{{{{\text{P}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}}}} \right) $$

(38)

$$ {{{V}}_{\rm{PP}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{PP}}}}}{{dt}} = {{{Q}}_{\rm{PP}}}\left( {{{M}}{{{i}}_{\rm{SB}}} - \frac{{{\text{M}}{{\text{i}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}\left\{ {{mi}} \right\}}}} \right) $$

(39)

$$ {{{V}}_{\rm{PP}}}\frac{{{\text{dMi}}{{\text{i}}_{\rm{PP}}}}}{{dt}} = {{{Q}}_{\rm{PP}}}\left( {{{Mi}}{{{i}}_{\rm{SB}}} - \frac{{{\text{Mi}}{{\text{i}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}\left\{ {{mii}} \right\}}}} \right) $$

(40)

In intestinal blood (denoted by the subscript, Intb),

$$ {{{V}}_{\rm{Intb}}}\frac{{{\text{d}}{{\text{P}}_{\rm{Intb}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{{P}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{I}}{{{P}}_{\rm{Int}}} - \left( {{{{Q}}_{\rm{PV}}} + {{CL}}_{\rm{d1}}^{\rm{I}}} \right){{{P}}_{\rm{Intb}}} $$

(41)

$$ {{{V}}_{\rm{Intb}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{Intb}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{M}}{{{i}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{I}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{Int}}} - \left( {{{{Q}}_{\rm{PV}}} + {{CL}}_{\rm{d1}}^{\rm{I}}\left\{ {{mi}} \right\}} \right){{M}}{{{i}}_{\rm{Intb}}} $$

(42)

$$ {{{V}}_{\rm{Intb}}}\frac{{{\text{dMi}}{{\text{i}}_{\rm{Intb}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{Mi}}{{{i}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{I}}\left\{ {{mii}} \right\}{{Mi}}{{{i}}_{\rm{Int}}} - \left( {{{{Q}}_{\rm{PV}}} + {{CL}}_{\rm{d1}}^{\rm{I}}\left\{ {{mii}} \right\}} \right){{Mi}}{{{i}}_{\rm{Intb}}} $$

(43)

In intestinal tissue (denoted by the subscript, Int),

$$ {{{V}}_{\rm{Int}}}\frac{{{\text{d}}{{\text{P}}_{\rm{Int}}}}}{{dt}} = {{CL}}_{\rm{d1}}^{\rm{I}}{{{P}}_{\rm{Intb}}} + {{{k}}_{\rm{a}}}{{{P}}_{\rm{lumen}}}{{{V}}_{\rm{lumen}}} - \left( {{{CL}}_{\rm{d2}}^{\rm{I}} + {{C}}{{{L}}_{{\rm{int,met1,I}}}} + {{C}}{{{L}}_{{\rm{int,met2,I}}}} + {{C}}{{{L}}_{{\rm{int,sec,I}}}}} \right){{{P}}_{\rm{Int}}} $$

(44)

$$ {{{V}}_{\rm{Int}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{Int}}}}}{{dt}} = {{CL}}_{\rm{d1}}^{\rm{I}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{Intb}}} + {{{k}}_{\rm{a}}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{lumen}}}{{{V}}_{\rm{lumen}}} + {{C}}{{{L}}_{{\rm{int,met1,I}}}}{{{P}}_{\rm{Int}}} - \left( {{{CL}}_{\rm{d2}}^{\rm{I}}\left\{ {{mi}} \right\} + {{C}}{{{L}}_{{\rm{int,met,I}}}}\left\{ {{mi}} \right\} + {{C}}{{{L}}_{{\rm{int,sec,I}}}}\left\{ {{mi}} \right\}} \right){{M}}{{{i}}_{\rm{Int}}} $$

(45)

$$ {{{V}}_{\rm{Int}}}\frac{{{\text{dMi}}{{\text{i}}_{\rm{Int}}}}}{{dt}} = {{CL}}_{\rm{dl}}^{\rm{I}}\left\{ {{mii}} \right\}{{Mi}}{{{i}}_{\rm{Intb}}} - {{CL}}_{{\rm{d}}2}^{\rm{I}}\left\{ {{mii}} \right\}{{Mi}}{{{i}}_{{\rm Int} }} $$

(46)

In intestinal lumen (denoted by the subscript, lumen),

$$ {{{V}}_{\rm{lumen}}}\frac{{{\text{d}}{{\text{P}}_{\rm{lumen}}}}}{{dt}} = {{C}}{{{L}}_{{\rm{int,sec,I}}}}{{{P}}_{\rm{Int}}} - \left( {{{{k}}_{\rm{a}}} + {{{k}}_{\rm{g}}}} \right){{{P}}_{\rm{lumen}}}{{{V}}_{\rm{lumen}}} $$

(47)

$$ {{{V}}_{\rm{lumen}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{lumen}}}}}{{dt}} = {{C}}{{{L}}_{{\rm{int,sec,I}}}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{Int}}} - \left( {{{{k}}_{\rm{a}}}\left\{ {{mi}} \right\} + {{{k}}_{\rm{g}}}\left\{ {{mi}} \right\}} \right){{M}}{{{i}}_{\rm{lumen}}}{{{V}}_{\rm{lumen}}} $$

(48)

In liver blood (denoted by the subscript, LB),

$$ {{{V}}_{\rm{LB}}}\frac{{{\text{d}}{{\text{P}}_{\rm{LB}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{{P}}_{\rm{Intb}}} + {{{Q}}_{\rm{HA}}}{{{P}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{H}}{{{P}}_{\rm{L}}} - \left( {{{{Q}}_{\rm{PV}}} + {{{Q}}_{\rm{HA}}} + {{CL}}_{\rm{d1}}^{\rm{H}}} \right){{{P}}_{\rm{LB}}} $$

(49)

$$ {{{V}}_{\rm{LB}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{LB}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{M}}{{{i}}_{\rm{Intb}}} + {{{Q}}_{\rm{HA}}}{{M}}{{{i}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{H}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{L}}} - \left( {{{{Q}}_{\rm{PV}}} + {{{Q}}_{\rm{HA}}} + {{CL}}_{\rm{d1}}^{\rm{H}}\left\{ {{mi}} \right\}} \right){{M}}{{{i}}_{\rm{LB}}} $$

(50)

$$ {{{V}}_{\rm{LB}}}\frac{{{\text{dMi}}{{\text{i}}_{\rm{LB}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{Mi}}{{{i}}_{\rm{Intb}}} + {{{Q}}_{\rm{HA}}}{{Mi}}{{{i}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{H}}\left\{ {{mii}} \right\}{{Mi}}{{{i}}_{\rm{L}}} - \left( {{{{Q}}_{\rm{PV}}} + {{{Q}}_{\rm{HA}}} + {{CL}}_{\rm{d1}}^{\rm{H}}\left\{ {{mii}} \right\}} \right){{Mi}}{{{i}}_{\rm{LB}}} $$

(51)

In liver tissue (denoted by the subscript, L),

$$ {{{V}}_{\rm{L}}}\frac{{{\text{d}}{{\text{P}}_{\rm{L}}}}}{{dt}} = {{CL}}_{\rm{d1}}^{\rm{H}}{{{P}}_{\rm{LB}}} - {\kern 1pt} {\kern 1pt} \left( {{{CL}}_{\rm{d2}}^{\rm{H}} + {{C}}{{{L}}_{{\rm{int,met1,H}}}} + {{C}}{{{L}}_{{\rm{int,met2,H}}}} + {{C}}{{{L}}_{{\rm{int,sec,H}}}}} \right){{{P}}_{\rm{L}}} $$

(52)

$$ {{{V}}_{\rm{L}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{L}}}}}{{dt}} = {{CL}}_{\rm{d1}}^{\rm{H}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{LB}}} - {{CL}}_{\rm{d2}}^{\rm{H}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{L}}} $$

(53)

$$ {{{V}}_{\rm{L}}}\frac{{{\text{dMi}}{{\text{i}}_{\rm{L}}}}}{{dt}} = {\kern 1pt} {\kern 1pt} {\kern 1pt} {{CL}}_{\rm{d1}}^{\rm{H}}\left\{ {{mii}} \right\}{{Mi}}{{{i}}_{\rm{LB}}} + {{C}}{{{L}}_{{\rm{int,met1,H}}}}{{{P}}_{\rm{L}}} - \left( {{{CL}}_{\rm{d2}}^{\rm{H}}\left\{ {{mii}} \right\} + {{C}}{{{L}}_{{\rm{int,met,H}}}}\left\{ {{mii}} \right\} + {{C}}{{{L}}_{{\rm{int,sec,H}}}}\left\{ {{mii}} \right\}} \right){{Mi}}{{{i}}_{\rm{L}}} $$

(54)

• (4) Case 4 (see Fig. 3 for the model scheme, the same metabolite was formed in intestine and liver)

In systemic blood (denoted by the subscript, SB),

$$ {{{V}}_{\rm{SB}}}\frac{{{\text{d}}{{\text{P}}_{\rm{SB}}}}}{{dt}} = {{{Q}}_{\rm{HP}}}\frac{{{{\text{P}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}}} + {{{Q}}_{\rm{PP}}}\frac{{{{\text{P}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}}} + \left( {{{{Q}}_{\rm{HA}}} + {{{Q}}_{\rm{PV}}}} \right){{{P}}_{\rm{LB}}} - \left( {{{C}}{{{L}}_{\rm{r}}} + {{{Q}}_{\rm{HP}}} + {{{Q}}_{\rm{PP}}} + {{{Q}}_{\rm{HA}}} + {{{Q}}_{\rm{PV}}}} \right){{{P}}_{\rm{SB}}} $$

(55)

$$ {{{V}}_{\rm{SB}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{SB}}}}}{{dt}} = {\kern 1pt} {\kern 1pt} {{{Q}}_{\rm{HP}}}\frac{{{\text{M}}{{\text{i}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}\left\{ {{mi}} \right\}}} + {{{Q}}_{\rm{PP}}}\frac{{{\text{M}}{{\text{i}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}\left\{ {{mi}} \right\}}} + \left( {{{{Q}}_{\rm{HA}}} + {{{Q}}_{\rm{PV}}}} \right){{M}}{{{i}}_{\rm{LB}}} - \left( {{{C}}{{{L}}_{\rm{r}}}\left\{ {{mi}} \right\} + {{{Q}}_{\rm{HP}}} + {{{Q}}_{\rm{PP}}} + {{{Q}}_{\rm{HA}}} + {{{Q}}_{\rm{PV}}}} \right){{M}}{{{i}}_{\rm{SB}}} $$

(56)

In highly perfused organs (denoted by the subscript, HP),

$$ {{{V}}_{\rm{HP}}}\frac{{{\text{d}}{{\text{P}}_{\rm{HP}}}}}{{dt}} = {{{Q}}_{\rm{HP}}}\left( {{{{P}}_{\rm{SB}}} - \frac{{{{\text{P}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}}}} \right) $$

(57)

$$ {{{V}}_{\rm{HP}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{HP}}}}}{{dt}} = {{{Q}}_{\rm{HP}}}\left( {{{M}}{{{i}}_{\rm{SB}}} - \frac{{{\text{M}}{{\text{i}}_{\rm{HP}}}}}{{{{{K}}_{\rm{HP}}}\left\{ {{mi}} \right\}}}} \right) $$

(58)

In poorly perfused organs (denoted by the subscript, PP),

$$ {{{V}}_{\rm{PP}}}\frac{{{\text{d}}{{\text{P}}_{\rm{PP}}}}}{{dt}} = {{{Q}}_{\rm{PP}}}\left( {{{{P}}_{\rm{SB}}} - \frac{{{{\text{P}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}}}} \right) $$

(59)

$$ {{{V}}_{\rm{PP}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{PP}}}}}{{dt}} = {{{Q}}_{\rm{PP}}}\left( {{{M}}{{{i}}_{\rm{SB}}} - \frac{{{\text{M}}{{\text{i}}_{\rm{PP}}}}}{{{{{K}}_{\rm{PP}}}\left\{ {{mi}} \right\}}}} \right) $$

(60)

In intestinal blood (denoted by the subscript, Intb),

$$ {{{V}}_{\rm{Intb}}}\frac{{{\text{d}}{{\text{P}}_{\rm{Intb}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{{P}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{I}}{{{P}}_{\rm{Int}}} - \left( {{{{Q}}_{\rm{PV}}} + {{CL}}_{\rm{d1}}^{\rm{I}}} \right){{{P}}_{\rm{Intb}}} $$

(61)

$$ {{{V}}_{\rm{Intb}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{Intb}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{M}}{{{i}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{I}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{Int}}} - \left( {{{{Q}}_{\rm{PV}}} + {{CL}}_{\rm{d1}}^{\rm{I}}\left\{ {{mi}} \right\}} \right){{M}}{{{i}}_{\rm{Intb}}} $$

(62)

In intestinal tissue (denoted by the subscript, Int),

$$ {{{V}}_{\rm{Int}}}\frac{{{\text{d}}{{\text{P}}_{\rm{Int}}}}}{{dt}} = {{CL}}_{\rm{d1}}^{\rm{I}}{{{P}}_{\rm{Intb}}} + {{{k}}_{\rm{a}}}{{{P}}_{\rm{lumen}}}{{{V}}_{\rm{lumen}}} - \left( {{{CL}}_{\rm{d2}}^{\rm{I}} + {{C}}{{{L}}_{{\rm{int,met1,I}}}} + {{C}}{{{L}}_{{\rm{int,met2,I}}}} + {{C}}{{{L}}_{{\rm{int,sec,I}}}}} \right){{{P}}_{\rm{Int}}} $$

(63)

$$ {{{V}}_{\rm{Int}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{Int}}}}}{{dt}} = {{CL}}_{\rm{d1}}^{\rm{I}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{Intb}}} + {{{k}}_{\rm{a}}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{lumen}}}{{{V}}_{\rm{lumen}}} + {{C}}{{{L}}_{{\rm{int,met1,I}}}}{{{P}}_{\rm{Int}}} - \left( {{{CL}}_{\rm{d2}}^{\rm{I}}\left\{ {{mi}} \right\} + {{C}}{{{L}}_{{\rm{int,met,I}}}}\left\{ {{mi}} \right\} + {{C}}{{{L}}_{{\rm{int,sec,I}}}}\left\{ {{mi}} \right\}} \right){{M}}{{{i}}_{\rm{Int}}} $$

(64)

In intestinal lumen (denoted by the subscript, lumen),

$$ {{{V}}_{\rm{lumen}}}\frac{{{\text{d}}{{\text{P}}_{\rm{lumen}}}}}{{dt}} = {{C}}{{{L}}_{{\rm{int,sec,I}}}}{{{P}}_{\rm{Int}}} - \left( {{{{k}}_{\rm{a}}} + {{{k}}_{\rm{g}}}} \right){{{P}}_{\rm{lumen}}}{{{V}}_{\rm{lumen}}} $$

(65)

$$ {{{V}}_{\rm{lumen}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{lumen}}}}}{{dt}} = {{C}}{{{L}}_{{\rm{int,sec,I}}}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{Int}}} - \left( {{{{k}}_{\rm{a}}}\left\{ {{mi}} \right\} + {{{k}}_{\rm{g}}}\left\{ {{mi}} \right\}} \right){{M}}{{{i}}_{\rm{lumen}}}{{{V}}_{\rm{lumen}}} $$

(66)

In liver blood (denoted by the subscript, LB),

$$ {{{V}}_{\rm{LB}}}\frac{{{\text{d}}{{\text{P}}_{\rm{LB}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{{P}}_{\rm{Intb}}} + {{{Q}}_{\rm{HA}}}{{{P}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{H}}{{{P}}_{\rm{L}}} - \left( {{{{Q}}_{\rm{PV}}} + {{{Q}}_{\rm{HA}}} + {{CL}}_{\rm{d1}}^{\rm{H}}} \right){{{P}}_{\rm{LB}}} $$

(67)

$$ {{{V}}_{\rm{LB}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{LB}}}}}{{dt}} = {{{Q}}_{\rm{PV}}}{{M}}{{{i}}_{\rm{Intb}}} + {{{Q}}_{\rm{HA}}}{{M}}{{{i}}_{\rm{SB}}} + {{CL}}_{\rm{d2}}^{\rm{H}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{L}}} - \left( {{{{Q}}_{\rm{PV}}} + {{{Q}}_{\rm{HA}}} + {{CL}}_{\rm{d1}}^{\rm{H}}\left\{ {{mi}} \right\}} \right){{M}}{{{i}}_{\rm{LB}}} $$

(68)

In liver tissue (denoted by the subscript, L),

$$ {{{V}}_{\rm{L}}}\frac{{{\text{d}}{{\text{P}}_{\rm{L}}}}}{{dt}} = {{CL}}_{\rm{d1}}^{\rm{H}}{{{P}}_{\rm{LB}}} - \left( {{{CL}}_{\rm{d2}}^{\rm{H}} + {{C}}{{{L}}_{{\rm{int,met1,H}}}} + {{C}}{{{L}}_{{\rm{int,met2,H}}}} + {{C}}{{{L}}_{{\rm{int,sec,H}}}}} \right){{{P}}_{\rm{L}}} $$

(69)

$$ {{{V}}_{\rm{L}}}\frac{{{\text{dM}}{{\text{i}}_{\rm{L}}}}}{{dt}} = {\kern 1pt} {{CL}}_{\rm{d1}}^{\rm{H}}\left\{ {{mi}} \right\}{{M}}{{{i}}_{\rm{LB}}} + {{C}}{{{L}}_{{\rm{int,met1,H}}}}{{{P}}_{\rm{L}}} - \left( {{{CL}}_{\rm{d2}}^{\rm{H}}\left\{ {{mi}} \right\} + {{C}}{{{L}}_{{\rm{int,met,H}}}}\left\{ {{mi}} \right\} + {{C}}{{{L}}_{{\rm{int,sec,H}}}}\left\{ {{mi}} \right\}} \right){{M}}{{{i}}_{\rm{L}}} $$

(70)