Simplification of Complex Physiologically Based Pharmacokinetic Models of Monoclonal Antibodies

Abstract

Monoclonal antibodies (mAbs) exhibit biexponential profiles in plasma that are commonly described with a standard two-compartment model with elimination from the central compartment. These models adequately describe mAb plasma PK. However, these models ignore elimination from the peripheral compartment. This may lead to underestimation of the volume of distribution of the peripheral compartment and thus over-predicts concentration in the peripheral compartment. We developed a simple and physiologically relevant model that incorporates information on binding and dissociation rates between mAb and FcRn receptor, mAb uptake, reflection, and catabolic degradation. We employed a previously published PBPK model and, with assumptions regarding rates of processes controlling mAb disposition, reduced the complex PBPK model to a simpler circular model with central, peripheral, and lymph compartments specifying elimination from both central and peripheral. We successfully applied the model to describe the PK of an investigational mAb. Our model presents an improvement over standard two-compartmental models in predicting whole-body average tissue concentrations while adequately describing plasma PK with minimal complexity and physiologically more meaningful parameters.

Appendices

Appendices

Appendix 1: Original Model Equations

Notes:

• 1. A superscript J is used to denote endogenous IgG or mAb in most equations. If the equations describing the mAb and IgG are different two independent systems of equations are used.

• 2. Dependent variables are described in amounts, not concentrations, as in (13).

• 3. The original model (13) described binding to FcRn receptor as the change from initial FcRn concentration in different tissues and the initial conditions for this change is set to 0. In this work, FcRn receptor amounts were expressed as  Rfree_{Organ ‐ X}.

• 4. The original model used two estimated coefficients to reflect the non-FcRn dependent tissue uptake of mAb (13). In our study, simulations were performed without these coefficients.

I. plasma

Endogenous IgG

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{plasma}}^{\mathrm{endo}}}{\operatorname{d}t}\\ {}={K}_0\times {V}_{\mathrm{Plasma}}+\frac{Q_{\mathrm{Lung}}-{L}_{\mathrm{Lung}}}{V_{\mathrm{VAS}}^{\mathrm{Lung}}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{endo}}-\frac{L_{\mathrm{GIT}}+{L}_{\mathrm{Spleen}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{endo}}\\ {}-\frac{Q_{\mathrm{Liver}}+{Q}_{\mathrm{Heart}}+{Q}_{\mathrm{Kidney}}+{Q}_{\mathrm{Skin}}+{Q}_{\mathrm{Muscle}\ }}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{endo}}+\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}^{\mathrm{endo}}\times {F}_{\mathrm{return}\ }\end{array}\hfill \\ {}\hfill \mathrm{IC}={A}_{\mathrm{plasma}}^{\mathrm{endo}}\mathrm{SS}\hfill \end{array} $$

(3)

Exogenous mAb

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{plasma}}}{\operatorname{d}t}\\ {}=\frac{Q_{\mathrm{Lung}}-{L}_{\mathrm{Lung}}}{V_{\mathrm{VAS}}^{\mathrm{Lung}}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}-\frac{L_{\mathrm{GIT}}+{L}_{\mathrm{Spleen}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}\\ {}-\frac{Q_{\mathrm{Liver}}+{Q}_{\mathrm{Heart}}+{Q}_{\mathrm{Kidney}}+{Q}_{\mathrm{Skin}}+{Q}_{\mathrm{Muscle}\ }}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}+\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}\times {F}_{\mathrm{return}\ }\end{array}\hfill \\ {}\hfill \mathrm{IC}={\mathrm{Dose}}_{\mathrm{mAb}}\hfill \end{array} $$

(4)

II. Organs

A. Vascular Space

Equations for the vascular compartment in liver and lung are different from other organs.

i. lung

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{V}\mathrm{AS}\ \mathrm{Lung}}^J}{\operatorname{d}t}\\ {}=\frac{Q_{\mathrm{Liver}}-{L}_{\mathrm{Liver}}}{V_{\mathrm{V}\mathrm{AS}}^{\mathrm{Liver}}}\times {A}_{\mathrm{V}\mathrm{AS}\ \mathrm{Liver}}^J+\frac{Q_{\mathrm{Heart}}-{L}_{\mathrm{Heart}}}{V_{\mathrm{V}\mathrm{AS}}^{\mathrm{Heart}}}\times {A}_{\mathrm{V}\mathrm{AS}\ \mathrm{Heart}}^{\mathrm{J}}+\frac{Q_{\mathrm{Kidney}}-{L}_{\mathrm{Kidney}}}{V_{\mathrm{V}\mathrm{AS}}^{\mathrm{Kidney}}}\times {A}_{\mathrm{V}\mathrm{AS}\ \mathrm{Kidney}}^{\mathrm{J}}\\ {}+\frac{Q_{\mathrm{Skin}}-{L}_{\mathrm{Skin}}}{V_{\mathrm{V}\mathrm{AS}}^{\mathrm{Skin}}}\times {A}_{\mathrm{V}\mathrm{AS}\ \mathrm{Skin}}^{\mathrm{J}}+\frac{Q_{\mathrm{Muscle}}-{L}_{\mathrm{Muscle}}}{V_{\mathrm{V}\mathrm{AS}}^{\mathrm{Muscle}}}\times {A}_{\mathrm{V}\mathrm{AS}\ \mathrm{Muscle}}^{\mathrm{J}}-\frac{Q_{\mathrm{Lung}}-{L}_{\mathrm{Lung}}}{V_{\mathrm{V}\mathrm{AS}}^{\mathrm{Lung}}}\times {A}_{\mathrm{V}\mathrm{AS}\ \mathrm{Lung}}^{\mathrm{J}}-\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Lung}}}{{\mathrm{V}}_{\mathrm{V}\mathrm{AS}}^{\mathrm{Lung}}}\\ {}\times {A}_{\mathrm{V}\mathrm{AS}\ \mathrm{Lung}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{V}\mathrm{AS}}\right)\times \frac{L_{\mathrm{Lung}}}{V_{\mathrm{V}\mathrm{AS}}^{\mathrm{Lung}}}\times {A}_{\mathrm{V}\mathrm{AS}\ \mathrm{Lung}}^{\mathrm{J}}+ FR\times \frac{C{L}_{\mathrm{Uptake}}^{\mathrm{Lung}}}{V_{\mathrm{END}}^{\mathrm{Lung}}}\times {\mathrm{ARCPX}}_{\mathrm{Lung}\hbox{-} 5}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{V}\mathrm{AS}\ \mathrm{Lung}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{V}\mathrm{AS}\ \mathrm{Lung}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{V}\mathrm{AS}\ \mathrm{Lung}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(5)

ii. liver

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{J}}}{\operatorname{d}t}\\ {}=\frac{Q_{\mathrm{GIT}}-{L}_{\mathrm{GIT}}}{V_{\mathrm{VAS}}^{\mathrm{GIT}}}\times {A}_{\mathrm{VAS}\ \mathrm{GIT}}^J+\frac{Q_{\mathrm{Spleen}}-{L}_{\mathrm{Spleen}}}{V_{\mathrm{VAS}}^{\mathrm{Spleen}}}\times {A}_{\mathrm{VAS}\ \mathrm{Spleen}}^J+\frac{L_{\mathrm{GIT}}+{L}_{\mathrm{Spleen}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^J\\ {}+\frac{Q_{\mathrm{Liver}}-{Q}_{\mathrm{GIT}}-{Q}_{\mathrm{Spleen}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^J-\frac{Q_{\mathrm{Liver}}-{L}_{\mathrm{Liver}}}{V_{\mathrm{VAS}}^{\mathrm{Liver}}}\times {A}_{\mathrm{VAS}\ \mathrm{Liver}}^J-\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Liver}}}{V_{\mathrm{VAS}}^{\mathrm{Liver}}}\times {A}_{\mathrm{VAS}\ \mathrm{Liver}}^J-\left(1-{\sigma}_{\mathrm{VAS}}\right)\\ {}\times \frac{L_{\mathrm{Liver}}}{V_{\mathrm{VAS}}^{\mathrm{Liver}}}\times {A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{J}}+\mathrm{FR}\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Liver}}}{V_{\mathrm{END}}^{\mathrm{Liver}}}\times {\mathrm{ARCPX}}_{\mathrm{Liver}\hbox{-} 5}^J\end{array}\hfill \\ {}\hfill {A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(6)

iii. Other Organs (heart, kidney, skin, muscle, GIT, spleen)

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}}{\operatorname{d}t}\\ {}=\frac{Q_{\mathrm{Organ}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{J}}-\frac{Q_{\mathrm{Organ}}-{L}_{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}-\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{VAS}}\right)\times \frac{L_{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\\ {}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+\mathrm{FR}\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{endo}}\operatorname{ss};{A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(7)

B. Interstitial Space (Applies for all organs including liver and lung, all termed “Organ”)

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}}{\operatorname{d}t}\\ {}=\left(1-{\sigma}_{\mathrm{VAS}}\right)\times \frac{L_{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{LY}}\right)\times \frac{L_{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}+\left(1-\mathrm{FR}\right)\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\\ {}\times {\mathrm{A}\mathrm{RCPX}}_{\mathrm{Organ}\hbox{-} 5}^J-\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{endo}}\mathrm{IC}={\mathrm{A}}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(8)

C. Endosomal Space (Applies for all organs including liver and lung, all termed “Organ”)

i. Subcompartment 1

Endogenous IgG

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}}{\operatorname{d}t}\\ {}=\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{endo}}+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{endo}}-k\frac{k_{\mathrm{on}\hbox{-} 1}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}\right.\\ {}\left.-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{exo}}\right)+{k}_{\mathrm{off}\hbox{-} 1}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}-\frac{1}{\tau}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}\mathrm{ss}\hfill \end{array} $$

(9)

Exogenous mAb

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{exo}}}{\operatorname{d}t}\\ {}=\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{exo}}+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{exo}}-\frac{k_{\mathrm{on}\hbox{-} 1}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{exo}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{exo}}\right.\\ {}\left.-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}\right)+{k}_{\mathrm{off}\hbox{-} 1}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{exo}}-\frac{1}{\tau}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{exo}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(10)

Endogenous IgG-FcRn complex

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}}{\operatorname{d}t}\\ {}=\frac{k_{\mathrm{on}\hbox{-} 1}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{exo}}\right)-{k}_{\mathrm{off}\hbox{-} 1}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}\\ {}-\frac{1}{\tau}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}\end{array}\hfill \\ {}\hfill {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}\mathrm{IC}={\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}\mathrm{ss}\hfill \end{array} $$

(11)

Exogenous mAb-FcRn complex

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{exo}}}{\operatorname{d}t}\\ {}=\frac{k_{\mathrm{on}\hbox{-} 1}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{exo}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{exo}}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}\right)-{k}_{\mathrm{off}\hbox{-} 1}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{exo}}\\ {}-\frac{1}{\tau}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{exo}}\end{array}\hfill \\ {}\hfill {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(12)

ii. Subcompartments 2, 3 and 4 (i = 2, 3, 4)

Endogenous IgG

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}}{\operatorname{d}t}\\ {}=\frac{1}{\tau}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \left(\mathrm{i}\hbox{-} 1\right)}^{\mathrm{endo}}-\frac{k_{\mathrm{on}\hbox{-} \mathrm{i}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}}\right)\\ {}+{k}_{\mathrm{off}\hbox{-} \mathrm{i}}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}-\frac{1}{\tau}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}\operatorname{ss}\hfill \end{array} $$

(13)

Exogenous mAb

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}}}{\operatorname{d}t}\\ {}=\frac{1}{\tau}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \left(\mathrm{i}\hbox{-} 1\right)}^{\mathrm{exo}}-\frac{k_{\mathrm{on}\hbox{-} \mathrm{i}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}\right)\\ {}+{k}_{\mathrm{off}\hbox{-} \mathrm{i}}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}}-\frac{1}{\tau}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(14)

Endogenous IgG-FcRn complex

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}}{\operatorname{d}t}\\ {}=\frac{1}{\tau}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \left(\mathrm{i}\hbox{-} 1\right)}^{\mathrm{endo}}+\frac{k_{\mathrm{on}\hbox{-} \mathrm{i}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}}\right)\\ {}-{k}_{\mathrm{off}\hbox{-} \mathrm{i}}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}-\frac{1}{\tau}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}\end{array}\hfill \\ {}\hfill {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}\mathrm{IC}={\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}\mathrm{ss}\hfill \end{array} $$

(15)

Exogenous mAb-FcRn complex

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}}}{\operatorname{d}t}\\ {}=\frac{1}{\tau}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \left(\mathrm{i}\hbox{-} 1\right)}^{\mathrm{exo}}+\frac{k_{\mathrm{on}\hbox{-} \mathrm{i}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}\right)\\ {}-{k}_{\mathrm{off}\hbox{-} \mathrm{i}}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}}-\frac{1}{\tau}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}}\end{array}\hfill \\ {}\hfill {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(16)

iii. Subcompartment 5

Endogenous IgG

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}}{\operatorname{d}t}\\ {}=\frac{1}{\tau}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 4}^{\mathrm{endo}}-\frac{k_{\mathrm{on}\hbox{-} 5}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 5}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\right)+{k}_{\mathrm{off}\hbox{-} 5}\\ {}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}-\frac{{\mathrm{CL}}_{\mathrm{Proteolysis}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}\mathrm{ss}\hfill \end{array} $$

(17)

Exogenous mAb

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\operatorname{END} Organ-5}^{\mathrm{exo}}}{\operatorname{d}t}\\ {}=\frac{1}{\tau}\times {A}_{\operatorname{END} Organ-4}^{\mathrm{exo}}-\frac{k_{\operatorname{on}-5}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\operatorname{END} Organ-5}^{\mathrm{exo}}\times \left({\operatorname{Rtotal}}_{Organ-5}-{\operatorname{ARCPX}}_{\operatorname{Organ}-5}^{\mathrm{exo}}-{\operatorname{ARCPX}}_{\operatorname{Organ}-5}^{\mathrm{endo}}\right)+{k}_{\operatorname{off}-5}\\ {}\times {\operatorname{ARCPX}}_{\operatorname{Organ}-5}^{\mathrm{exo}}-\frac{{\operatorname{CL}}_{\mathrm{Proteolysis}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\operatorname{END} Organ-5}^{\mathrm{exo}}\end{array}\hfill \\ {}\hfill {A}_{\operatorname{END} Organ-5}^{\mathrm{exo}}\operatorname{IC}=0\hfill \end{array} $$

(18)

Endogenous IgG-FcRn complex

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}}{\operatorname{d}t}\\ {}=\frac{1}{\tau}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 4}^{\mathrm{endo}}+\frac{k_{\mathrm{on}\hbox{-} 5}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 5}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\right)\\ {}-{k}_{\mathrm{off}\hbox{-} 5}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}-\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}\end{array}\hfill \\ {}\hfill {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}\mathrm{IC}={\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}\mathrm{ss}\hfill \end{array} $$

(19)

Exogenous mAb-FcRn complex

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}}{\operatorname{d}t}\\ {}=\frac{1}{\tau}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 4}^{\mathrm{exo}}+\frac{k_{\mathrm{on}\hbox{-} 5}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 5}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\right)\\ {}-{k}_{\mathrm{off}\hbox{-} 5}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}-\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\end{array}\hfill \\ {}\hfill {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(20)

III. Lymph Node

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{LN}}^{\mathrm{J}}}{\operatorname{d}t}\\ {}=\left(1-{\sigma}_{\mathrm{LY}}\right)\times \frac{L_{\mathrm{Lung}}}{V_{\mathrm{ISF}}^{\mathrm{Lung}}}\times {A}_{\mathrm{ISF}\ \mathrm{Lung}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\times \frac{L_{\mathrm{Liver}}}{V_{\mathrm{ISF}}^{\mathrm{Liver}}}\times {A}_{\mathrm{ISF}\ \mathrm{Liver}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\times \frac{L_{\mathrm{GIT}}}{V_{\mathrm{ISF}}^{\mathrm{GIT}}}\times {A}_{\mathrm{ISF}\ \mathrm{GIT}}^{\mathrm{J}}+\left(1\right.\\ {}\left.-{\sigma}_{\mathrm{LY}}\right)\times \frac{L_{\mathrm{Spleen}}}{V_{\mathrm{ISF}}^{\mathrm{Spleen}}}\times {A}_{\mathrm{ISF}\ \mathrm{Spleen}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\times \frac{L_{\mathrm{Heart}}}{V_{\mathrm{ISF}}^{\mathrm{Heart}}}\times {A}_{\mathrm{ISF}\ \mathrm{Heart}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\times \frac{L_{\mathrm{Kidney}}}{V_{\mathrm{ISF}}^{\mathrm{Kidney}}}\times {A}_{\mathrm{ISF}\ \mathrm{Kidney}}^{\mathrm{J}}+\left(1\right.\\ {}\left.-{\sigma}_{\mathrm{LY}}\right)\times \frac{L_{\mathrm{Skin}}}{V_{\mathrm{ISF}}^{\mathrm{Skin}}}\times {A}_{\mathrm{ISF}\ \mathrm{Skin}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\times \frac{L_{\mathrm{Muscle}}}{V_{\mathrm{ISF}}^{\mathrm{Muscle}}}\times {A}_{\mathrm{ISF}\ \mathrm{Muscle}}^{\mathrm{J}}-\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{LN}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{LN}}^{\mathrm{endo}}\operatorname{ss};{A}_{\mathrm{LN}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(21)

Appendix 2: Fast Endosomal Transfer and Rapid Binding Assumptions

Transfer of mAb/IgG between endosomal compartments is fast relative to the time scale of other processes such as changes in plasma and lymph as well as the rate of binding of mAb/IgG to FcRn. The assumption that the endosomal transfer rate is faster than FcRn binding process holds true for earlier compartments where FcRn binding is considered negligible (in the first endosomal subcompartment) or low (in intermediate compartments). As the drug is transferred to later compartments, rate of binding increases and thus rapid binding assumption is applied to the last compartment where the majority of binding to FcRn takes place. Physiologically these 2 processes (transfer and binding) might interfere with each other. However, since these 2 assumptions (i.e., fast endosomal transfer and rapid binding) are applied sequentially, the 2 assumptions do not contradict each other, at least from a theoretical aspect.

Assumption: Endosomal transfer of mAb/IgG between endosomal compartments is very fast relative to the time scale of other processes:

$$ \tau \to 0 $$

(22)

It can be shown that input in to the first endosomal compartment will be transferred to the terminal compartment as τ → 0 (Please refer to Appendix 9),

$$ \frac{1}{\tau}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 4}\to \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}+\frac{C{L}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}\;\mathrm{as}\kern0.37em \tau \to 0 $$

(23)

Thus, equation for free mAb/IgG in the terminal compartment can be expressed as:

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}}{\operatorname{d}t}=\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{exo}}+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{exo}}-\frac{k_{\mathrm{on}\hbox{-} 5}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 5}\right.\hfill \\ {}\left.-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\right)+{k}_{\mathrm{off}\hbox{-} 5}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}-\frac{{\mathrm{CL}}_{\mathrm{Proteolysis}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\hfill \end{array} $$

(24)

and the equation for complex mAb/IgG in the terminal compartment can be expressed as:

$$ \begin{array}{l}\frac{\operatorname{d}{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}}{\operatorname{d}t}=\frac{k_{\mathrm{on}\hbox{-} 5}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 5}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\right)-{k}_{\mathrm{off}\hbox{-} 5}\hfill \\ {}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}-\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {\operatorname{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\hfill \end{array} $$

(25)

Assumption: Free drug, the target and the complex are at rapid equilibrium meaning that the binding and dissociation rates are much faster than other processes (18).

Thus,

$$ \begin{array}{l}\frac{k_{\mathrm{on}\hbox{-} 5}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 5}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\right)-{k}_{\mathrm{off}\hbox{-} 5}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\hfill \\ {}=0\hfill \end{array} $$

(26)

And

$$ {k}_{\mathrm{off}\hbox{-} 5}/\frac{k_{\mathrm{on}\hbox{-} 5}}{V_{\mathrm{END}}^{\mathrm{Organ}}}={\mathrm{KD}}_{\mathrm{END}} $$

(27)

Solve for ARCPX ^exo_{Organ ‐ 5}

$$ {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}=\frac{A_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 5}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}\right)}{{\mathrm{KD}}_{\mathrm{END}}+{A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}} $$

(28)

Similarly,

$$ {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}=\frac{A_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 5}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\right)}{{\mathrm{KD}}_{\mathrm{END}}+{A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{endo}}} $$

(29)

Substitute ARCPX ^endo_{Organ ‐ 5} in Eq. 27 to solve for ARCPX ^exo_{Organ ‐ 5} in terms of the free amounts of endogenous (A ^endo_{END Organ ‐ 5} ) IgG and exogenous mAb (A ^exo_{END Organ ‐ 5} )

$$ {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}=\frac{A_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\times {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 5}\times {\mathrm{KD}}_{\mathrm{END}}\ }{\left[{\mathrm{KD}}_{\mathrm{END}}+{A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\right]\times \left[{\mathrm{KD}}_{\mathrm{END}}+{A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}\right]-{A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{exo}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{endo}}} $$

(30)

and A _{END Organ ‐ i} and A ^endo_{END Organ ‐ i} are negligible compared to KD_END (A ^exo_{END Organ ‐ 5}  < < KD_END and A ^endo_{END Organ ‐ 5}  < < KD_END) (Please refer to Supplementary Figure 2)

Thus, Eq. 29 for ARCPX ^J_{Organ ‐ 5} can be simplified to

$$ {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{J}}=\frac{A_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{J}}\times {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 5}}{{\mathrm{KD}}_{\mathrm{END}}\ } $$

(31)

Equations derived for ARCPX ^J_{Organ ‐ 5} showed that the IgG/mAb-FcRn complex is proportional to the free IgG/mAb. In the following section, equations for free IgG/mAb (Eq. 27) and complex (Eq. 28) are added to get A ^J_{Total END Organ ‐ 5}

$$ {\mathrm{ARCPX}}_{\mathrm{Total}\ \mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{J}}={A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{J}}+{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{J}} $$

(32)

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{Total}\ \mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{J}}}{\operatorname{d}t}\hfill \\ {}=\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}-\frac{{\mathrm{CL}}_{\mathrm{Proteolysis}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{J}}-\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\hfill \\ {}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{J}}\hfill \end{array} $$

(33)

Assumption: total mAb/IgG in compartment 5 is in equilibrium, i.e.

$$ \frac{\operatorname{d}{A}_{\mathrm{Total}\ \mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{J}}}{\operatorname{d}t}=0 $$

(34)

Consequently,

$$ \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}-\frac{{\mathrm{CL}}_{\mathrm{Proteolysis}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{J}}-\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{J}}=0 $$

(35)

Substituting ARCPX ^J_{Organ ‐ 5} by Eq. 31 allows one to solve Eq. 43 for A _{END Organ ‐ 5}

$$ {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 5}^{\mathrm{J}}=\frac{\left(\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}\right)}{\left(\frac{{\mathrm{CL}}_{\mathrm{Proteolysis}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times \frac{{\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 5}}{{\mathrm{KD}}_{\mathrm{END}}\ }\right)} $$

(36)

Substituting A ^J_{END Organ ‐ 5} from Eqs. 38 and 34 allows solving for ARCPX ^J_{Organ ‐ 5} in terms of vascular and interstitial amounts

$$ {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{J}}={A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}\times \frac{\frac{{\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 5}}{{\mathrm{KD}}_{\mathrm{END}}}\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}}{\frac{{\mathrm{CL}}_{\mathrm{Proteolysis}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times \frac{{\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 5}}{{\mathrm{KD}}_{\mathrm{END}}}}+{A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}\times \frac{\frac{{\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 5}}{{\mathrm{KD}}_{\mathrm{END}}}\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}}{\frac{{\mathrm{CL}}_{\mathrm{Proteolysis}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times \frac{{\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 5}}{{\mathrm{KD}}_{\mathrm{END}}\ }} $$

(37)

Let the terms multiplying A ^J_VAS Organ and A ^J_ISF Organ be k ₁ and k ₂, respectively. Thus,

$$ {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 5}^{\mathrm{J}}={A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}\times {k}_1+{A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}\times {k}_2 $$

(38)

Substitute ARCPX_{Organ ‐ 5} in the equation for vascular and interstitial compartments (Please refer to Fig. 3):

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}}{\operatorname{d}t}=\frac{Q_{\mathrm{Organ}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{J}}-\frac{Q_{\mathrm{Organ}}-{L}_{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{VAS}}\right)\times \frac{L_{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}-\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times \hfill \\ {}{A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+\mathrm{FR}\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_1\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+\mathrm{FR}\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_2\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}\hfill \end{array} $$

(39)

Introduce the input term from the vascular to the interstitial compartment: (This term appears in the interstitial compartment)

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}}{\operatorname{d}t}=\frac{Q_{\mathrm{Organ}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{J}}-\frac{Q_{\mathrm{Organ}}-{L}_{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{VAS}}\right)\times \frac{L_{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}-\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times \hfill \\ {}{A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+\mathrm{FR}\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_1\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+\mathrm{FR}\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_2\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}-\left(1-\mathrm{FR}\right)\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_1\times \hfill \\ {}{A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+\left(1-\mathrm{FR}\right)\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_1\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}\hfill \end{array} $$

(40)

which can be simplified to:

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Organ}}}{\operatorname{d}t}=\frac{Q_{\mathrm{Organ}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{J}}-\frac{Q_{\mathrm{Organ}}-{L}_{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{VAS}}\right)\times \frac{L_{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}-\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times \hfill \\ {}{A}_{\mathrm{VAS}\;\mathrm{Organ}}^{\mathrm{J}}+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_1\times {A}_{\mathrm{VAS}\;\mathrm{Organ}}^{\mathrm{J}}+\mathrm{FR}\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_2\times {A}_{\mathrm{ISF}\;\mathrm{Organ}}^{\mathrm{J}}-\left(1-\mathrm{FR}\right)\times \frac{C{L}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_1\times \hfill \\ {}{A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}\hfill \end{array} $$

(41)

A similar substitution of ARCPX ^J_{Organ ‐ 5} in the equation for the interstitial compartment yields:

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}}{\operatorname{d}t}=\left(1-{\sigma}_{\mathrm{VAS}}\right)\times \frac{L_{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\operatorname{VAS} Organ}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{LY}}\right)\times \frac{L_{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\operatorname{ISF} Organ}^{\mathrm{J}}-\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\operatorname{ISF} Organ}^{\mathrm{J}}+\hfill \\ {}\left(1-\operatorname{FR}\right)\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_1\times {A}_{\operatorname{VAS} Organ}^{\mathrm{J}}+\left(1-\operatorname{FR}\right)\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_2\times {A}_{\operatorname{ISF} Organ}^{\mathrm{J}}\hfill \end{array} $$

(42)

Introduce the input term from the interstitial to the vascular compartment: (This term appears in the vascular compartment and should be accounted for)

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\operatorname{ISF} Organ}^J}{\operatorname{d}t}=\left(1-{\sigma}_{\mathrm{VAS}}\right)\times \frac{L_{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\operatorname{VAS} Organ}^J-\left(1-{\sigma}_{\mathrm{LY}}\right)\times \frac{L_{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\operatorname{ISF} Organ}^J-\frac{C{L}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\operatorname{ISF} Organ}^J+\hfill \\ {}\left(1-\mathrm{FR}\right)\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_1\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+\left(1-\mathrm{FR}\right)\times \frac{C{L}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_2\times {A}_{\operatorname{ISF} Organ}^{\mathrm{J}}+ FR\times \frac{{\operatorname{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_2\times \hfill \\ {}{A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}-\mathrm{FR}\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_2\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}\hfill \end{array} $$

(43)

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{ISF}\ \mathrm{Organ}}}{\operatorname{d}t}\hfill \\ {}=\left(1-{\sigma}_{\mathrm{VAS}}\right)\times \frac{L_{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\operatorname{VAS} Organ}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{LY}}\right)\times \frac{L_{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\operatorname{ISF} Organ}^{\mathrm{J}}-\frac{{\operatorname{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\operatorname{ISF} Organ}^{\mathrm{J}}+\left(1\right.\hfill \\ {}\left.-\operatorname{FR}\right)\times \frac{{\operatorname{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_1\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_2\times {A}_{\operatorname{ISF} Organ}^{\mathrm{J}}-\operatorname{FR}\hfill \\ {}\times \frac{{\operatorname{CL}}_{Uptake}^{Organ}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_2\times {A}_{\operatorname{ISF} Organ}^{\mathrm{J}}\hfill \end{array} $$

(44)

Let $\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_1$ be a first-order input from the vascular compartment to itself (k ^Organ_{VAS ‐ VAS} )

Let $\mathrm{FR}\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_2$ be a first-order input from interstitial compartment to vascular compartment (k ^Organ_{ISF ‐ VAS} )

Let $\left(1-\mathrm{FR}\right)\times \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_1$ be a first-order input from vascular compartment to interstitial compartment (k ^Organ_{VAS ‐ ISF} )

Let $\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {k}_2$ be a first-order input from the interstitial compartment to itself (k ^Organ_{ISF ‐ ISF} ).

The final equations for vascular and interstitial compartments can be written as:

A. Vascular Space

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}}{\operatorname{d}t}\hfill \\ {}=\frac{Q_{\mathrm{Organ}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}-\frac{Q_{\mathrm{Organ}}-{L}_{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{VAS}}\right)\times \frac{L_{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}-\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\hfill \\ {}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{VAS}}^{\mathrm{Organ}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Organ}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Organ}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}\hfill \end{array} $$

(45)

B. Interstitial Space

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}}{\operatorname{d}t}\hfill \\ {}=\left(1-{\sigma}_{\mathrm{VAS}}\right)\times \frac{L_{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{LY}}\right)\times \frac{L_{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}-\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}\hfill \\ {}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Organ}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{ISF}}^{\mathrm{Organ}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}-{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Organ}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}\hfill \end{array} $$

(46)

Appendix 3: Simplified Model Equations After Rapid Binding and Fast Endosomal Transfer Assumptions (Step 1)

I. plasma

Endogenous IgG

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{plasma}}^{\mathrm{endo}}}{\operatorname{d}t}\\ {}={K}_0\times {V}_{\mathrm{Plasma}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{endo}}-\frac{L_{\mathrm{GIT}}+{L}_{\mathrm{Spleen}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{endo}}-\frac{Q_{\mathrm{Liver}}+{Q}_{\mathrm{Heart}}+{Q}_{\mathrm{Kidney}}+{Q}_{\mathrm{Skin}}+{Q}_{\mathrm{Muscle}\ }}{V_{\mathrm{Plasma}}}\times \\ {}{A}_{\mathrm{Plasma}}^{\mathrm{endo}}+\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}^{\mathrm{endo}}\times {F}_{\mathrm{return}\ }\end{array}\hfill \\ {}\hfill \mathrm{IC}={A}_{\mathrm{plasma}}^{\mathrm{endo}}\mathrm{SS}\hfill \end{array} $$

(47)

Exogenous mAb

$$ \begin{array}{ll}\frac{\operatorname{d}{A}_{\mathrm{plasma}}}{\operatorname{d}t}=\hfill & {k}_{VAS- circul.}^{\mathrm{Lung}}\times {A}_{\operatorname{VAS} Lung}-\frac{L_{\mathrm{GIT}}+{L}_{\mathrm{Spleen}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}\hfill \\ {}\hfill & -\frac{Q_{\mathrm{Liver}}+{Q}_{\mathrm{Heart}}+{Q}_{\mathrm{Kidney}}+{Q}_{\mathrm{Skin}}+{Q}_{\mathrm{Muscle}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}+\frac{1}{{\operatorname{Tau}}_{LN}}\times {A}_{\mathrm{LN}}\times {F}_{\mathrm{return}}\hfill \\ {}\hfill & \operatorname{IC}={\operatorname{Dose}}_{\mathrm{mAb}}\hfill \end{array} $$

(48)

II. Organs

A. Vascular Space

Equations for the vascular compartment in liver and lung are different from other organs.

i. lung

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}}{\operatorname{d}t}\\ {}={k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Liver}}\times {A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Heart}}\times {A}_{\mathrm{VAS}\ \mathrm{Heart}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Kidney}}\times {A}_{\mathrm{VAS}\ \mathrm{Kidney}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Skin}}\times {A}_{\mathrm{VAS}\ \mathrm{Skin}}^{\mathrm{J}}\\ {}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Muscle}}\times {A}_{\mathrm{VAS}\ \mathrm{Muscle}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{EL}}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Lung}}\\ {}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{VAS}}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Lung}}\times {A}_{\mathrm{ISF}\ \mathrm{Lung}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(49)

ii. liver

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{J}}}{\operatorname{d}t}={k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{GIT}}\times {A}_{\mathrm{VAS}\ \mathrm{GIT}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Spleen}}\times {A}_{\mathrm{VAS}\ \mathrm{Spleen}}^{\mathrm{J}}+\frac{L_{\mathrm{GIT}}+{L}_{\mathrm{Spleen}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{J}}+\frac{Q_{\mathrm{Liver}}-\left({Q}_{\mathrm{GIT}}+{Q}_{\mathrm{Spleen}}\right)}{V_{\mathrm{Plasma}}}\ast {A}_{Plasma}^{endo}-\\ {}{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Liver}}\times {A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{EL}}^{\mathrm{Liver}}\times {A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Liver}}\times {A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{VAS}}^{\mathrm{Liver}}\times \\ {}{A}_{\mathrm{VAS}\ \mathrm{Liver}}^J+{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Liver}}\times {A}_{\mathrm{ISF}\ \mathrm{Liver}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Liver}}\times {A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(50)

iii. Other Organs (heart, kidney, skin, muscle, GIT, spleen)

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}}{\operatorname{d}t}\\ {}=\frac{Q_{\mathrm{Organ}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Organ}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{EL}}^{\mathrm{Organ}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Organ}}\\ {}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{VAS}}^{\mathrm{Organ}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Organ}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Organ}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(51)

B. Interstitial Space (Applies for all organs including liver and lung, all termed “Organ”)

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}}{\operatorname{d}t}\\ {}=\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Organ}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Organ}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}-{k}_{\mathrm{ISF}\hbox{-} \mathrm{EL}}^{\mathrm{Organ}}\\ {}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Organ}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{ISF}}^{\mathrm{Organ}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}-{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Organ}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(52)

III. Lymph Node

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{LN}}^{\mathrm{J}}}{\operatorname{d}t}\\ {}=\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Lung}}\times {A}_{\mathrm{ISF}\ \mathrm{Lung}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Liver}}\times {A}_{\mathrm{ISF}\ \mathrm{Liver}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{GIT}}\\ {}\times {A}_{\mathrm{ISF}\ \mathrm{GIT}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Spleen}}\times {A}_{\mathrm{ISF}\ \mathrm{Spleen}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Heart}}\times {A}_{\mathrm{ISF}\ \mathrm{Heart}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\\ {}\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Kidney}}\times {A}_{\mathrm{ISF}\ \mathrm{Kidney}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Skin}}\times {A}_{\mathrm{ISF}\ \mathrm{Skin}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Muscle}}\times {A}_{\mathrm{ISF}\ \mathrm{Muscle}}^{\mathrm{J}}\\ {}-\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{LN}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{LN}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{LN}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(53)

Appendix 4: Model Equations for Lumping of GIT and Spleen (Splanchnic) and Heart, Kidney, Skin, and Muscle (Visceral)

This is Step #2 in the model development. Please refer to Fig. 3 for model structure and Table I for model parameters.

I. plasma

Endogenous IgG

For the lumped compartment, the lymphatic can be described as the sum of flow for each individual organ as:

$$ \begin{array}{l}{L}_{\mathrm{Splanchnic}}={L}_{\mathrm{GIT}}+{L}_{\mathrm{Spleen}}\hfill \\ {}{L}_{\mathrm{Visceral}}={L}_{\mathrm{Heart}}+{L}_{\mathrm{Kidney}}+{L}_{\mathrm{Skin}}+{L}_{\mathrm{Muscle}\ }\hfill \end{array} $$

Similarly, the plasma flow for the lumped compartment can be described as:

$$ \begin{array}{l}{Q}_{\mathrm{Splanchnic}}={Q}_{\mathrm{GIT}}+{Q}_{\mathrm{Spleen}}\hfill \\ {}{Q}_{\mathrm{Visceral}}={Q}_{\mathrm{Heart}}+{Q}_{\mathrm{Kidney}}+{Q}_{\mathrm{Skin}}+{Q}_{\mathrm{Muscle}\ }\hfill \end{array} $$

Then the equation for endogenous IgG in plasma becomes:

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{plasma}}^{\mathrm{endo}}}{\operatorname{d}t}={K}_0\times {V}_{\mathrm{Plasma}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{endo}}-\frac{L_{\mathrm{Splanchnic}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{endo}}-\frac{Q_{\mathrm{Liver}}+{Q}_{\mathrm{Visceral}\ }}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{endo}}+\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times \\ {}{A}_{\mathrm{LN}}^{\mathrm{endo}}\times {F}_{\mathrm{return}\ }\end{array}\hfill \\ {}\hfill IC={A}_{\mathrm{plasma}}^{\mathrm{endo}}\mathrm{SS}\hfill \end{array} $$

(54)

Exogenous mAb

$$ \begin{array}{c}\hfill \frac{\operatorname{d}{A}_{\mathrm{plasma}}}{\operatorname{d}t}={k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}-\frac{L_{\mathrm{Splanchnic}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}-\frac{Q_{\mathrm{Liver}}+{Q}_{\mathrm{Visceral}\ }}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}+\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}\times {F}_{\mathrm{return}\ }\hfill \\ {}\hfill \mathrm{IC}={\mathrm{Dose}}_{\mathrm{mAb}}\hfill \end{array} $$

(55)

II. Organs

A. Vascular Space

Equations for the vascular compartment in liver and lung are different from other organs.

i. lung

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}}{\operatorname{d}t}\\ {}={k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Liver}}\times {A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Visceral}}\times {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{EL}}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}\\ {}\begin{array}{c}\hfill -\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{VAS}}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Lung}}\times {A}_{\mathrm{ISF}\ \mathrm{Lung}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}\hfill \\ {}\hfill {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array}\end{array} $$

(56)

ii. Liver

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{J}}}{\operatorname{d}t}={k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{J}}+\frac{L_{\mathrm{Splanchnic}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{J}}+\frac{Q_{\mathrm{Liver}}-\left({Q}_{\mathrm{Splanchnic}}\right)}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Liver}}\times \\ {}{A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{EL}}^{\mathrm{Liver}}\times {A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Liver}}\times {A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{VAS}}^{\mathrm{Liver}}\times {A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{J}}+\\ {}\ {k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Liver}}\times {A}_{\mathrm{ISF}\ \mathrm{Liver}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Liver}}\times {A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{VAS}\ \mathrm{Liver}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(57)

iii. splanchnic (GIT and spleen lumped)

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{J}}}{\operatorname{d}t}=\frac{Q_{\mathrm{Splanchnic}}\ }{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{EL}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{J}}-\left(1-\right.\\ {}\left.{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{VAS}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{J}}-\\ {}{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{J}}\\ {}{A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{exo}}\mathrm{IC}=0\end{array} $$

(58)

iv. visceral (heart, kidney, skin, muscle lumped)

$$ \begin{array}{c}\hfill \frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}}{\operatorname{d}t}=\frac{Q_{\mathrm{Visceral}}\ }{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Visceral}}\times {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{EL}}^{\mathrm{Visceral}}\times {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{VAS}}\right)\times \hfill \\ {}\hfill {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Visceral}}\times {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{VAS}}^{\mathrm{Visceral}}\times {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Visceral}}\times {A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Visceral}}\times {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}\hfill \\ {}\hfill {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(59)

B. Interstitial Space

i. lung, liver

(All organs but splanchnic and visceral)

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}}{\operatorname{d}t}=\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\operatorname{LY} VAS- ISF}^{\mathrm{Organ}}\times {A}_{\operatorname{VAS} Organ}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\operatorname{LY} ISF- LN}^{\mathrm{Organ}}\times {A}_{\operatorname{ISF} Organ}^{\mathrm{J}}-{k}_{\operatorname{ISF}- EL}^{\mathrm{Organ}}\times \\ {}{A}_{\operatorname{ISF} Organ}^{\mathrm{J}}+{k}_{\operatorname{VAS}- ISF}^{\mathrm{Organ}}\times {A}_{\operatorname{VAS} Organ}^{\mathrm{J}}+{k}_{\operatorname{ISF}- ISF}^{\mathrm{Organ}}\times {A}_{\operatorname{ISF} Organ}^{\mathrm{J}}-{k}_{\operatorname{ISF}- VAS}^{\mathrm{Organ}}\times {A}_{\operatorname{ISF} Organ}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\operatorname{ISF} Organ}^{\mathrm{endo}}\operatorname{IC}={A}_{\operatorname{ISF} Organ}^{\mathrm{endo}}\operatorname{ss};{A}_{\operatorname{ISF} Organ}^{\mathrm{exo}}\operatorname{IC}=0\hfill \end{array} $$

(60)

ii. splanchnic (GIT and spleen lumped)

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{d{A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{J}}}{ dt}=\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{J}}\\ {}-{k}_{\mathrm{ISF}\hbox{-} \mathrm{EL}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{ISF}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{J}}-{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Splanchnic}}\times \\ {}{A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(61)

iii. visceral (heart, kidney, skin, muscle lumped)

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{J}}}{\operatorname{d}t}=\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Visceral}}\times {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Visceral}}\times {A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{J}}-{k}_{\mathrm{ISF}\hbox{-} \mathrm{EL}}^{\mathrm{Visceral}}\times \\ {}{A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Visceral}}\times {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{ISF}}^{\mathrm{Visceral}}\times {A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{J}}-{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Visceral}}\times {A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(62)

III. Lymph Node

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{LN}}^{\mathrm{J}}}{\operatorname{d}t}\\ {}=\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Lung}}\times {A}_{\mathrm{ISF}\ \mathrm{Lung}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Liver}}\times {A}_{\mathrm{ISF}\ \mathrm{Liver}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\\ {}\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Visceral}}\times {A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{J}}-\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{LN}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{LN}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{LN}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(63)

Appendix 5: Model Equations after fast transfer from liver to lung (Step#3)

Liver compartment no longer exists and the appearance of $\frac{Q_{\mathrm{Liver}}}{V_{\mathrm{Plasma}}}$ is solely indicative of the transfer from plasma to lung. Finally, splanchnic is now positioned to be parallel to other organs, i.e., visceral.

I. plasma

Endogenous IgG

$$ \begin{array}{l}\begin{array}{l}\frac{d{A}_{\mathrm{plasma}}^{\mathrm{endo}}}{ dt}={K}_0\times {V}_{\mathrm{Plasma}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{endo}}-\frac{Q_{\mathrm{Splanchnic}}+{Q}_{\mathrm{Visceral}}+{Q}_{\mathrm{Liver}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{endo}}+\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}^{\mathrm{endo}}\times \\ {}\ {F}_{\mathrm{return}\ }\end{array}\hfill \\ {}\hfill \mathrm{IC}={A}_{\mathrm{plasma}}^{\mathrm{endo}}\mathrm{SS}\hfill \end{array} $$

(64)

Exogenous mAb

$$ \begin{array}{c}\hfill \frac{d{A}_{\mathrm{plasma}}}{ dt}={k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}-\frac{Q_{\mathrm{Splanchnic}}+{Q}_{\mathrm{Visceral}}+{Q}_{\mathrm{Liver}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}+\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}\times {F}_{\mathrm{return}\ }\hfill \\ {}\hfill \mathrm{IC}={\mathrm{Dose}}_{\mathrm{mAb}}\hfill \end{array} $$

(65)

II. Organs

A. Vascular Space

Equations for the vascular compartment in lung are different from other organs.

i. lung

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{d{A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}}{ dt}={k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Visceral}}\times {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}+\frac{Q_{\mathrm{Liver}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Lung}}\times \\ {}{A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{EL}}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{VAS}}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}+\\ {}{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Lung}}\times {A}_{\mathrm{ISF}\ \mathrm{Lung}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(66)

ii. splanchnic (GIT and spleen lumped)

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{endo}}}{\operatorname{d}t}=\frac{Q_{\mathrm{Splanchnic}}\ }{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{EL}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{J}}-\left(1-\right.\\ {}\left.{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{VAS}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{J}}-\\ {}{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Splanchnic}}\times {A}_{\operatorname{VAS}\kern0.50em Splanchnic}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\operatorname{VAS}\kern0.5em Splanchnic}^{\mathrm{endo}}\operatorname{IC}={A}_{\mathrm{VAS}\kern0.5em \mathrm{Splanchnic}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(67)

iii. visceral (heart, kidney, skin, muscle lumped)

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}}{\operatorname{d}t}=\frac{Q_{\mathrm{Visceral}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Visceral}}\times {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{EL}}^{\mathrm{Visceral}}\times {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{VAS}}\right)\times \\ {}{k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Visceral}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{VAS}}^{\mathrm{Visceral}}\times {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Visceral}}\times {A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Visceral}}\times {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(68)

B. Interstitial Space

i. lung

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{ISF}\ \mathrm{Lung}}^{\mathrm{J}}}{\operatorname{d}t}=\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Lung}}\times {A}_{\mathrm{ISF}\ \mathrm{Lung}}^{\mathrm{J}}-{k}_{\mathrm{ISF}\hbox{-} \mathrm{EL}}^{\mathrm{Lung}}\times \\ {}{A}_{\mathrm{ISF}\ \mathrm{Lung}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{ISF}}^{\mathrm{Lung}}\times {A}_{\mathrm{ISF}\ \mathrm{Lung}}^{\mathrm{J}}-{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Lung}}\times {A}_{\mathrm{ISF}\ \mathrm{Lung}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(69)

ii. splanchnic (GIT and spleen lumped)

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{J}}}{\operatorname{d}t}=\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{J}}-\\ {}{k}_{\mathrm{ISF}\hbox{-} \mathrm{EL}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{VAS}\ \mathrm{Splanchnic}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{ISF}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{J}}-{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Splanchnic}}\times \\ {}{A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(70)

iii. visceral (heart, kidney, skin, muscle lumped)

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\operatorname{ISF} Visceral}^{\mathrm{J}}}{\operatorname{d}t}=\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Visceral}}\times {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Visceral}}\times {A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{J}}-{k}_{\mathrm{ISF}\hbox{-} \mathrm{EL}}^{\mathrm{Visceral}}\times \\ {}{A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Visceral}}\times {A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{ISF}}^{\mathrm{Visceral}}\times {A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{J}}-{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Visceral}}\times {A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(71)

III. Lymph Node

Endogenous IgG

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{LN}}^{\mathrm{J}}}{\operatorname{d}t}=\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Lung}}\times {A}_{\mathrm{ISF}\ \mathrm{Lung}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Splanchnic}}\times {A}_{\mathrm{ISF}\ \mathrm{Splanchnic}}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\times \\ {}{k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Visceral}}\times {A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{J}}-\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{LN}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{LN}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{LN}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(72)

Appendix 6: Model Equations for Lumping Splanchnic and Visceral into Peripheral (Step#4)

I. plasma

Endogenous IgG

$$ \begin{array}{c}\hfill \frac{\operatorname{d}{A}_{\mathrm{plasma}}^{\mathrm{endo}}}{\operatorname{d}t}={K}_0\times {V}_{\mathrm{Plasma}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{endo}}-\frac{Q_{\mathrm{Peripheral}}+{Q}_{\mathrm{Liver}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{endo}}+\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}^{\mathrm{endo}}\times {F}_{\mathrm{return}\ }\hfill \\ {}\hfill \mathrm{IC}={A}_{\mathrm{plasma}}^{\mathrm{endo}}\mathrm{SS}\hfill \end{array} $$

(73)

Exogenous mAb

$$ \begin{array}{c}\hfill \frac{\operatorname{d}{A}_{\mathrm{plasma}}}{\operatorname{d}t}={k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Lung}}\times {A}_{\mathrm{VAS}\ \mathrm{Lung}}-\frac{Q_{\mathrm{Peripheral}}+{Q}_{\mathrm{Liver}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}+\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}\times {F}_{\mathrm{return}\ }\hfill \\ {}\hfill \mathrm{IC}={\mathrm{Dose}}_{\mathrm{mAb}}\hfill \end{array} $$

(74)

II. Organs

A. Vascular Space

Equations for the vascular compartment in lung are different from other organs.

i. lung

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Lung}}^{\mathrm{J}}}{\operatorname{d}t}={k}_{VAS- circul.}^{\mathrm{Peripheral}}\times {A}_{\operatorname{VAS} Peripheral}^{\mathrm{J}}+\frac{Q_{\mathrm{Liver}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{J}}-{k}_{VAS- circul.}^{\mathrm{Lung}}\times {A}_{\operatorname{VAS} Lung}^{\mathrm{J}}-{k}_{\operatorname{VAS}- EL}^{\mathrm{Lung}}\times \\ {}\ {A}_{\operatorname{VAS} Lung}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\operatorname{LY} VAS- ISF}^{\mathrm{Lung}}\times {A}_{\operatorname{VAS} Lung}^{\mathrm{J}}+{k}_{\operatorname{VAS}- VAS}^{\mathrm{Lung}}\times {A}_{\operatorname{VAS} Lung}^{\mathrm{J}}+{k}_{\operatorname{ISF}- VAS}^{\mathrm{Lung}}\times {A}_{\operatorname{ISF} Lung}^{\mathrm{J}}\\ {}-{k}_{\operatorname{VAS}- ISF}^{\mathrm{Lung}}\times {A}_{\operatorname{VAS} Lung}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\operatorname{VAS} Lung}^{\mathrm{endo}}\mathrm{IC}={A}_{\operatorname{VAS} Lung}^{\mathrm{endo}}\mathrm{ss};{A}_{\operatorname{VAS} Lung}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(75)

ii. Peripheral (splanchnic and visceral lumped)

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{J}}}{\operatorname{d}t}=\frac{Q_{\mathrm{Peripheral}}\ }{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}\times {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{EL}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{J}}-\left(1-\right.\\ {}\left.\ {\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{VAS}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}^{\mathrm{J}}-\\ {}{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(76)

B. Interstitial Space

Peripheral

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{d{A}_{\operatorname{ISF} Peripheral}^{\mathrm{J}}}{ dt}=\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\operatorname{LY} VAS- ISF}^{\mathrm{Peripheral}}\times {A}_{\operatorname{VAS} Peripheral}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\operatorname{LY} ISF- LN}^{\mathrm{Peripheral}}\times {A}_{\operatorname{ISF} Peripheral}^{\mathrm{J}}-\\ {}{k}_{\operatorname{ISF}- EL}^{\mathrm{Peripheral}}\times {A}_{\operatorname{ISF} Peripheral}^{\mathrm{J}}+{k}_{\operatorname{VAS}- ISF}^{\mathrm{Peripheral}}\times {A}_{\operatorname{VAS} Peripheral}^{\mathrm{J}}+{k}_{\operatorname{ISF}- ISF}^{\mathrm{Peripheral}}\times {A}_{\operatorname{ISF} Peripheral}^{\mathrm{J}}-{k}_{\operatorname{ISF}- VAS}^{\mathrm{Peripheral}}\times \\ {}{A}_{\operatorname{ISF} Peripheral}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}^{\mathrm{endo}}\operatorname{IC}={A}_{\mathrm{ISF}\ Visceral}^{\mathrm{endo}}\operatorname{ss};{A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(77)

III. Lymph Node

Endogenous IgG

$$ \begin{array}{c}\hfill \frac{\operatorname{d}{A}_{\mathrm{LN}}^{\mathrm{J}}}{\operatorname{d}t}=\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\operatorname{LY} ISF- LN}^{\mathrm{Lung}}\times {A}_{\operatorname{ISF} Lung}^{\mathrm{J}}+\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\operatorname{LY}\kern0.5em ISF- LN}^{\mathrm{Peripheral}}\times {A}_{\operatorname{ISF} Peripheral}^{\mathrm{J}}-\frac{1}{{\operatorname{Tau}}_{LN}}\times {A}_{\mathrm{LN}}^{\mathrm{J}}\hfill \\ {}\hfill {A}_{\mathrm{LN}}^{\mathrm{endo}}\operatorname{IC}={A}_{\mathrm{LN}}^{\mathrm{endo}}\operatorname{ss};{A}_{\mathrm{LN}}^{\mathrm{exo}}\operatorname{IC}=0\hfill \end{array} $$

(78)

Appendix 7: Model Equations for Fast Transfer from Lung to Plasma (Step#5)

(Figure 4d)

I. plasma

Endogenous IgG

$$ \begin{array}{c}\hfill \frac{\operatorname{d}{A}_{\mathrm{plasma}}^{\mathrm{endo}}}{\operatorname{d}t}={K}_0\times {V}_{\mathrm{Plasma}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}\times {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{endo}}-\frac{Q_{\mathrm{Peripheral}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{endo}}+\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}^{\mathrm{endo}}\times {F}_{\mathrm{return}\ }\hfill \\ {}\hfill \mathrm{IC}={A}_{\mathrm{plasma}}^{\mathrm{endo}}\mathrm{SS}\hfill \end{array} $$

(79)

Exogenous mAb

$$ \begin{array}{c}\hfill \frac{\operatorname{d}{A}_{\mathrm{plasma}}}{\operatorname{d}t}={k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}\times {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}-\frac{Q_{\mathrm{Peripheral}}}{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}+\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}\times {F}_{\mathrm{return}\ }\hfill \\ {}\hfill \mathrm{IC}={\mathrm{Dose}}_{\mathrm{mAb}}\hfill \end{array} $$

(80)

II. Peripheral

A. Vascular Space

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{J}}}{\operatorname{d}t}=\frac{Q_{\mathrm{Peripheral}}\ }{V_{\mathrm{Plasma}}}\times {A}_{\mathrm{Plasma}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}\times {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{J}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{EL}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{J}}\\ {}-\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{VAS}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}^{\mathrm{J}}-\\ {}\ {k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{VAS}\ \mathrm{Visceral}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(81)

B. Interstitial Space

Peripheral

$$ \begin{array}{c}\hfill \begin{array}{l}\frac{d{A}_{\mathrm{ISF}\ \mathrm{Peripheral}}^{\mathrm{J}}}{ dt}=\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{J}}-\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}^{\mathrm{J}}-\\ {}{k}_{\mathrm{ISF}\hbox{-} \mathrm{EL}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}^{\mathrm{J}}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}^{\mathrm{J}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{ISF}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}^{\mathrm{J}}-{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Peripheral}}\times \\ {}{A}_{\mathrm{ISF}\ \mathrm{Peripheral}}^{\mathrm{J}}\end{array}\hfill \\ {}\hfill {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{ISF}\ \mathrm{Visceral}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(82)

III. Lymph Node

$$ \begin{array}{c}\hfill \frac{\operatorname{d}{A}_{\mathrm{LN}}^{\mathrm{J}}}{\operatorname{d}t}=\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}^{\mathrm{J}}-\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}^{\mathrm{J}}\hfill \\ {}\hfill {A}_{\mathrm{LN}}^{\mathrm{endo}}\mathrm{IC}={A}_{\mathrm{LN}}^{\mathrm{endo}}\mathrm{ss};{A}_{\mathrm{LN}}^{\mathrm{exo}}\mathrm{IC}=0\hfill \end{array} $$

(83)

Appendix 8: Rapid equilibrium between plasma and Vascular compartments forming Central compartment (Model Equations) (Step#6)

(Figure 5) (Supplementary Table 4)

In this step, rapid equilibrium from plasma to vascular compartment is assumed. The rate of transfer from plasma to vascular compartment of Peripheral organs $\left(\frac{Q_{\mathrm{Peripheral}}}{V_{\mathrm{Plasma}}}\right)$ is of the same order as transfer rate from the vascular compartment of Peripheral organs to plasma (k ^Peripheral_{VAS ‐ circul.} ). These two rates are assumed to be faster than rates governing transfer between vascular and interstitial spaces:

$$ \begin{array}{l}\begin{array}{cccccc}\hfill {k}_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}\gg \hfill & \hfill {k}_{\mathrm{VAS}\hbox{-} \mathrm{EL}}^{\mathrm{Peripheral}},\hfill & \hfill {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Peripheral}},\hfill & \hfill {k}_{\mathrm{VAS}\hbox{-} \mathrm{VAS}}^{\mathrm{Peripheral}},\hfill & \hfill {k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}.}^{\mathrm{Peripheral}},\hfill & \hfill {k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Peripheral}}\hfill \end{array}\\ {}\begin{array}{cccccc}\hfill \begin{array}{cccccccccc}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \frac{1}{k_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}}\sim 0\hfill & \hfill \hfill \end{array}\end{array} $$

(Table III):

Further, we assume that the vascular compartment is in rapid equilibrium such that the rate of change is negligible (i.e., $\frac{\mathrm{d}{A}_{\mathrm{VAS}\ \mathrm{Peripheral}}}{\mathrm{d}t}$ is much smaller than the right hand side of the differential equation)

Dividing Eqs. 114/115 by k ^Peripheral_{VAS ‐ circul.} leads to

$$ 0=\frac{\frac{Q_{\mathrm{Peripheral}}\ }{V_{\mathrm{Plasma}}}}{k_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}}\ast {A}_{\mathrm{Plasma}}-{A}_{\mathrm{VAS}\ \mathrm{Peripheral}}-0-0+0+0-0 $$

(84)

After rearrangement

$$ {A}_{\mathrm{Plasma}}=\frac{k_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}}{\frac{Q_{\mathrm{Peripheral}}\ }{V_{\mathrm{Plasma}}}}\times {A}_{\mathrm{VAS}\ \mathrm{Peripheral}} $$

(85)

Introduce a new variable for the total amount in vascular and plasma compartments (Central)

$$ {A}_{\mathrm{Central}}={A}_{\mathrm{Plasma}}+{A}_{\mathrm{VAS}\ \mathrm{Peripheral}} $$

(86)

Then

$$ {A}_{\mathrm{Plasma}}=\frac{A_{\mathrm{Central}}}{\ \left(1+\frac{\frac{Q_{\mathrm{Peripheral}}\ }{V_{\mathrm{Plasma}}}}{k_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}}\right)} $$

(87)

and

$$ {A}_{\mathrm{VAS}\ \mathrm{Peripheral}}=\frac{A_{\mathrm{Central}}}{\ \left(1+\frac{k_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}}{\frac{Q_{\mathrm{Peripheral}}\ }{V_{\mathrm{Plasma}}}}\right)} $$

(88)

Adding differential equations for vascular and plasma compartments to get the differential equation for A _Central results in

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{Central}}}{\operatorname{d}t}=-{k}_{\mathrm{VAS}\hbox{-} \mathrm{EL}}^{\mathrm{Peripheral}}\times \frac{A_{\mathrm{Central}}}{\ \left(1+\frac{k_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}}{\frac{Q_{\mathrm{Peripheral}}\ }{V_{\mathrm{Plasma}}}}\right)}-\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Peripheral}}\times \frac{A_{\mathrm{Central}}}{\ \left(1+\frac{k_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}}{\frac{Q_{\mathrm{Peripheral}}\ }{V_{\mathrm{Plasma}}}}\right)}+{k}_{\mathrm{VAS}\hbox{-} \mathrm{VAS}}^{\mathrm{Peripheral}}\times \\ {}\frac{A_{\mathrm{Central}}}{\left(1+\frac{k_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}}{\frac{Q_{\mathrm{Peripheral}}\ }{V_{\mathrm{Plasma}}}}\right)}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{VAS}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}-{k}_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Peripheral}}\times \frac{A_{\mathrm{Central}}}{\ \left(1+\frac{k_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}}{\frac{Q_{\mathrm{Peripheral}}\ }{V_{\mathrm{Plasma}}}}\right)}+\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}\times {F}_{\mathrm{return}\ }\end{array} $$

(89)

Regroup rates and change parameter names to reflect new model structure

Let

$$ {k}_{\mathrm{EL}}^{\mathrm{Central}}=\frac{k_{\mathrm{VAS}\hbox{-} \mathrm{EL}}^{\mathrm{Peripheral}}}{\ \left(1+\frac{k_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}}{\frac{Q_{\mathrm{Peripheral}}\ }{V_{\mathrm{Plasma}}}}\right)} $$

be elimination rate from the central compartment

Let $\frac{k_{\mathrm{LY}\ \mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Peripheral}}}{\left(1+\frac{k_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}}{\frac{Q_{\mathrm{Peripheral}}\ }{V_{\mathrm{Plasma}}}}\right)}$ be transfer rate from the central to interstitial (i.e., peripheral) compartment via lymphatic flow

Let $\frac{k_{\mathrm{VAS}\hbox{-} \mathrm{VAS}}^{\mathrm{Peripheral}}}{\left(1+\frac{k_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}}{\frac{Q_{\mathrm{Peripheral}}\ }{V_{\mathrm{Plasma}}}}\right)}$ first-order input from central compartment to itself which reflects the impact of recycled drug from endosomal compartment

Let $\frac{k_{\mathrm{VAS}\hbox{-} \mathrm{ISF}}^{\mathrm{Peripheral}}}{\left(1+\frac{k_{\mathrm{VAS}\hbox{-} \mathrm{circul}.}^{\mathrm{Peripheral}}}{\frac{Q_{\mathrm{Peripheral}}\ }{V_{\mathrm{Plasma}}}}\right)}$ be first-order input from central to interstitial compartment

Let k ^Peripheral_{ISF ‐ VAS} be termed k ^Peripheral_{ISF ‐ Central} to reflect the updated model structure

Thus equations for Central, Peripheral and lymph node compartments are provided as follows,

Central Compartment

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{Central}}\ }{\operatorname{d}t}=-{k}_{\mathrm{EL}}^{\mathrm{Central}}\times {A}_{\mathrm{Central}}-\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\hbox{-} \mathrm{ISF}}^{\mathrm{Central}}\times {A}_{\mathrm{Central}}+{k}_{\mathrm{Central}}^{\mathrm{Central}}\times {A}_{\mathrm{Central}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{Central}}^{\mathrm{Peripheral}}\times \\ {}{A}_{\mathrm{ISF}\ \mathrm{Peripheral}}-{k}_{\mathrm{Central}\hbox{-} \mathrm{ISF}}^{\mathrm{Central}}\times {A}_{\mathrm{Central}}+\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}\times {F}_{\mathrm{return}\ }\end{array} $$

(90)

Peripheral Compartment

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{ISF}\ \mathrm{Peripheral}}}{\operatorname{d}t}=\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\hbox{-} \mathrm{ISF}}^{\mathrm{Central}}\times {A}_{\mathrm{Central}}-\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}-{k}_{\mathrm{ISF}\hbox{-} \mathrm{EL}}^{\mathrm{Peripheral}}\\ {}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}+{k}_{\mathrm{Central}\hbox{-} \mathrm{ISF}}^{\mathrm{Central}}\times {A}_{\mathrm{Central}}+{k}_{\mathrm{ISF}\hbox{-} \mathrm{ISF}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}-{k}_{\mathrm{ISF}\hbox{-} \mathrm{Central}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}\end{array} $$

(91)

Lymph Node

$$ \frac{d{A}_{\mathrm{LN}}}{\operatorname{d}t}=\left(1-{\sigma}_{LY}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}-\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}} $$

(92)

Rearrange equation for A _Central and A _{ISF Peripheral}

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{Central}}\ }{\operatorname{d}t}=-\left({k}_{\mathrm{EL}}^{\mathrm{Central}}-{k}_{\mathrm{Central}}^{\mathrm{Central}}\right)\times {A}_{\mathrm{Central}}-\left(\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\mathrm{LY}\hbox{-} \mathrm{ISF}}^{\mathrm{Central}}+{k}_{\mathrm{Central}\hbox{-} \mathrm{ISF}}^{\mathrm{Central}}\right)\times {A}_{\mathrm{Central}}+\\ {}{k}_{\mathrm{ISF}\hbox{-} \mathrm{Central}}^{\mathrm{Peripheral}}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}+\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}\times {A}_{\mathrm{LN}}\times {F}_{\mathrm{return}\ }\end{array} $$

(93)

$$ \begin{array}{l}\frac{\operatorname{d}{A}_{\mathrm{ISF}\ \mathrm{Peripheral}}}{\operatorname{d}t}=\\ {}\ \left(\left(1-{\sigma}_{\mathrm{VAS}}\right)\times {k}_{\operatorname{LY}- ISF}^{\mathrm{Central}}+{k}_{\operatorname{Central}- ISF}^{\mathrm{Central}}\right)\times {A}_{\mathrm{Central}}-\left(\left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\operatorname{LY}\kern0.5em ISF- LN}^{\mathrm{Peripheral}}+{k}_{\operatorname{ISF}- Central}^{\mathrm{Peripheral}}\right)\times \\ {}{A}_{\operatorname{ISF}\kern0.5em Peripheral}-\left({k}_{\operatorname{ISF}- EL}^{\mathrm{Peripheral}}-{k}_{\operatorname{ISF}- ISF}^{\mathrm{Peripheral}}\right)\times {A}_{\operatorname{ISF}\kern0.5em Peripheral}\end{array} $$

(94)

Subsequently, the following rearrangements and assumptions were made:

• 1- Rearrangement: Subtracting the rate of recycling of mAb from the endosomes back to the central compartment (k ^Central_Central , 5.8E-06 h⁻¹) which acts as an input to the central compartment from the central compartment elimination, (k ^Central_EL , 6.1E-06 h⁻¹) results in a slower elimination rate termed, k ^C_EL (3E-07 h⁻¹).

  $$ {k}_{\mathrm{EL}}^{\mathrm{C}\mathrm{entral}}-{k}_{\mathrm{C}\mathrm{entral}}^{\mathrm{C}\mathrm{entral}}={k}_{\mathrm{EL}}^{\mathrm{C}} $$

  (95)

  Similarly, subtracting the rate of recycling of mAb from the endosomes back to the peripheral compartment (k ^Peripheral_Peripheral , 2.73E−06 h⁻¹) from the peripheral compartment elimination, k ^Peripheral_EL (2.85E−06 h⁻¹) results in a slower peripheral elimination rate termed, k ^P_EL (1.2E−07 h⁻¹).

  $$ {k}_{\mathrm{EL}}^{\mathrm{P}\mathrm{eripheral}}-{k}_{\mathrm{P}\mathrm{eripheral}}^{\mathrm{P}\mathrm{eripheral}}={k}_{\mathrm{EL}}^{\mathrm{P}} $$

  (96)

• 2- Rearrangement: Transfer from central to peripheral compartments via lymphatic flow and the reflection of the fraction mAb-FcRn complex recycled are combined into one term that describes the sum of the two processes.

  $$ \left(1-{\sigma}_{\mathrm{V}}\right)\times {k}_{\mathrm{LY}\ \mathrm{Cen}\hbox{-} \mathrm{P}\mathrm{er}}+{k}_{\mathrm{C}\mathrm{en}\hbox{-} \mathrm{P}\mathrm{er}}={k}_{\mathrm{C}\hbox{-} \mathrm{P}} $$

  (97)

• 3- Assumption: Rate of return of mAb to the lymph node via lymphatic flow ((1 − σ _LY) × k ^Peripheral_{LY ISF ‐ LN} , 0.39 h⁻¹) is faster than the reflection of the fraction mAb-FcRn complex recycled , (k ^Peripheral_{ISF ‐ Central} , 1.95E-06 h⁻¹)

  $$ \left(1-{\sigma}_{\mathrm{LY}}\right)\times {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Peripheral}}\gg {k}_{\mathrm{ISF}\hbox{-} \mathrm{Central}}^{\mathrm{Peripheral}} $$

  (98)

• 4- Assumption: Rate of return from peripheral compartment to lymph node ((1 − σ _LY) × k ^Peripheral_{LY ISF ‐ LN} , 0.39 h⁻¹) is comparable to rate of return from lymph node to central compartment ($\frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}$, 0.11 h⁻¹)

  $$ \left(1-{\sigma}_{\mathrm{LY}}\right)\ast {k}_{\mathrm{LY}\ \mathrm{ISF}\hbox{-} \mathrm{LN}}^{\mathrm{Peripheral}}\sim \frac{1}{{\mathrm{Tau}}_{\mathrm{LN}}}={k}_{\mathrm{LN}} $$

  (99)

• 5- Assumption: 100% of the drug in the lymph node is subject to return to the central compartment (i.e., F _return = 1)

  Rewrite Eqs. 93, and 94

  $$ \frac{\operatorname{d}{A}_{\mathrm{C}\mathrm{entral}}\ }{\operatorname{d}t}=-{k}_{\mathrm{EL}}^{\mathrm{C}}\times {A}_{\mathrm{C}\mathrm{entral}}-{k}_{\mathrm{C}\mathrm{P}}\times {A}_{\mathrm{C}\mathrm{entral}}+{k}_{\mathrm{LN}}\times {A}_{\mathrm{LN}} $$

  (100)
  $$ \frac{\operatorname{d}{A}_{\mathrm{ISF}\ \mathrm{Peripheral}}}{\operatorname{d}t}={k}_{\mathrm{CP}}\times {A}_{\mathrm{Central}}-{k}_{\mathrm{LN}}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}-{k}_{\mathrm{EL}}^{\mathrm{P}}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}} $$

  (101)

  Further, k ^C_EL  ∼ k ^P_EL  = k _EL. Thus, Final model equations can be written as:

A. Central Compartment

$$ \begin{array}{ll}\frac{\operatorname{d}{A}_{\mathrm{Central}}\ }{\operatorname{d}t}=\hfill & -{k}_{\mathrm{EL}}\times {A}_{\mathrm{Central}}-{k}_{\mathrm{CP}}\times {A}_{\mathrm{Central}}+{k}_{\mathrm{LN}}\times {A}_{\mathrm{LN}}\hfill \\ {}\hfill & \mathrm{IC}={\mathrm{Dose}}_{\mathrm{mAb}}\hfill \end{array} $$

(102)

B. Peripheral Compartment

$$ \begin{array}{ll}\frac{\operatorname{d}{A}_{\mathrm{ISF}\ \mathrm{Peripheral}}}{\operatorname{d}t}=\hfill & {k}_{\mathrm{CP}}\times {A}_{\mathrm{Central}}-{k}_{\mathrm{LN}}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}-{k}_{\mathrm{EL}}\times {A}_{\mathrm{ISF}\ \mathrm{Peripheral}}\hfill \\ {}\hfill & \mathrm{IC}=0\hfill \end{array} $$

(103)

C. Lymph node Compartment

$$ \begin{array}{ll}\frac{\operatorname{d}{A}_{\mathrm{LN}}}{\operatorname{d}t}=\hfill & {k}_{\mathrm{LN}}\times \left({A}_{\mathrm{ISF}\ \mathrm{Peripheral}}-{A}_{\mathrm{LN}}\right)\hfill \\ {}\hfill & \mathrm{IC}=0\hfill \end{array} $$

(104)

Appendix 9: Derivation of the Output Function in Transit Compartment System with Fast Transfer

In this appendix, we show that the output from the last of N transit compartments equals to the input, if the transfer rate between transit compartments is fast compared to other processes. We assume that all model variables describing amounts are nonnegative. We also skip the proofs of the following statements:

$$ 0\le {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}},0\le {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}},0\le {\mathrm{ARCPX}}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}},0\le {\mathrm{ARCPX}}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}},i=1,2,3,4 $$

(105)

and

$$ {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{endo}}+{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{exo}}\le {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} \mathrm{i}},i=1,2,3,4 $$

(106)

and there exist positive constants M ^endo and M ^exo such that

$$ \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}\le {M}^{\mathrm{J}},\mathrm{J}=\mathrm{endo}\ \mathrm{or}\ \mathrm{exo} $$

(107)

First we will show that for i = 1, 2, 3, 4,  J = endo or exo

$$ \frac{1}{\tau}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{J}}\le {M}^{\mathrm{J}}+{k}_{\mathrm{off}\hbox{-} 1}\times {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1}+\dots +{k}_{\mathrm{off}\hbox{-} \mathrm{i}}\times {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} \mathrm{i}} $$

(108)

and

$$ \frac{1}{\tau}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{J}}\le \tau \times \frac{k_{\mathrm{on}\hbox{-} \mathrm{i}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}\times \left({M}^{\mathrm{J}}+{k}_{\mathrm{off}\hbox{-} 1}\times {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1}+\dots +{k}_{\mathrm{off}\hbox{-} \mathrm{i}}\times {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}\right) $$

(109)

Applying the integrating factor e ^t/τ to (9) and (10) one can obtain the following solution for  A ^J_{END Organ ‐ 1} :

$$ \begin{array}{l}\ {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}(t)={\displaystyle \underset{0}{\overset{t}{\int }}}\left(\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\ast {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}(z)+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\ast {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}(z)\right)\ast {e}^{\frac{z-t}{\tau }}\operatorname{d}z\\ {}-\frac{k_{\mathrm{on}\hbox{-} 1}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\ast {\displaystyle \underset{0}{\overset{t}{\int }}}{A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}\ast \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}(z)-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}(z)\right)\times {e}^{\frac{z-t}{\tau }}\operatorname{d}z\\ {}+{k}_{\mathrm{off}\hbox{-} 1}\times {\displaystyle \underset{0}{\overset{t}{\int }}}{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}(z)\times {e}^{\left(z-t\right)/\tau}\operatorname{d}z\end{array} $$

(110)

and

$$ \begin{array}{l}\ {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}(t)=\\ {}\frac{k_{\mathrm{on}\hbox{-} 1}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {\displaystyle \underset{0}{\overset{t}{\int }}}{A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}(z)-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}(z)\right)\times {e}^{\frac{z-t}{\tau }}\operatorname{d}z\\ {}-{k}_{\mathrm{off}\hbox{-} 1}\times {\displaystyle \underset{0}{\overset{t}{\int }}}{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}(z)\times {e}^{\left(z-t\right)/\tau}\operatorname{d}z\end{array} $$

(111)

(107) and (108) imply that

$$ {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}(t)\le {M}^{\mathrm{J}}\times {\displaystyle \underset{0}{\overset{t}{\int }}}{e}^{\frac{z-t}{\tau }}\operatorname{d}z+{k}_{\mathrm{off}\hbox{-} 1}\times {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1}\times {\displaystyle \underset{0}{\overset{t}{\int }}}{e}^{\frac{z-t}{\tau }}\operatorname{d}z\le \tau \times \left({M}^{\mathrm{J}}+{k}_{\mathrm{off}\hbox{-} 1}\times {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1}\right) $$

(112)

and

$$ {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}(t)\le \frac{k_{\mathrm{on}\hbox{-} 1}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1}\times {\displaystyle \underset{0}{\overset{t}{\int }}}{A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}\times {e}^{\frac{z-t}{\tau }}\operatorname{d}z $$

(113)

From (113) it follows

$$ {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}(t)\le {\tau}^2\times \left({M}^{\mathrm{J}}+{k}_{\mathrm{off}\hbox{-} 1}\times {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1}\right)\times \frac{k_{\mathrm{on}\hbox{-} 1}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1} $$

This proves (109) and (110) for i = 1. The proof of cases i = 2, 3, 4 is analogous.

In the next step we will show that for i = 1, 2, 3, 4,  J = endo or exo

$$ \frac{1}{\tau}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \mathrm{i}}^{\mathrm{J}}\to \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}\;\mathrm{as}\;\tau \to 0 $$

(114)

That will yield (23) for i = 4. Consider first the case i = 1. Multiplying (110) by 1/τ results in

$$ \begin{array}{l}\frac{1}{\tau}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}(t)={\displaystyle \underset{0}{\overset{t}{\int }}}\left(\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}(z)+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}(z)\right)\times \frac{1}{\tau }{e}^{\frac{z-t}{\tau }}\operatorname{d}z-\frac{k_{\mathrm{on}\hbox{-} 1}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times \\ {}{\displaystyle \underset{0}{\overset{t}{\int }}}\frac{1}{\tau}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}\times \left({\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1}-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}(z)-{\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}(z)\right)\times {e}^{\frac{z-t}{\tau }}\operatorname{d}z+{k}_{\mathrm{off}\hbox{-} 1}\times \\ {}{\displaystyle \underset{0}{\overset{t}{\int }}}\frac{1}{\tau}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}(z)\times {e}^{\left(z-t\right)/\tau}\operatorname{d}z\end{array} $$

(115)

From (107) and (112) the following bound applies to the integral

$$ \begin{array}{l}{\displaystyle \underset{0}{\overset{t}{\int }}}\frac{1}{\tau}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}\ast \left({\operatorname{Rtotal}}_{Organ-1}-{\operatorname{ARCPX}}_{Organ-1}^J(z)-{\operatorname{ARCPX}}_{Organ-1}^J(z)\right)\times {e}^{\frac{z-t}{\tau }}\operatorname{d}z\le \\ {}\left({M}^{\mathrm{J}}+{k}_{\mathrm{off}\hbox{-} 1}\times {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1}\right)\times {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1}\times \tau \end{array} $$

(116)

This means that the integral in (117) approaches 0 as τ → 0. Similarly, (110) implies that the last integral in (116) can be bound as follows:

$$ {\displaystyle \underset{0}{\overset{t}{\int }}}\frac{1}{\tau}\times {\mathrm{ARCPX}}_{\mathrm{Organ}\hbox{-} 1}^{\mathrm{J}}(z)\times {e}^{\left(z-t\right)/\tau}\operatorname{d}z\le {\tau}^2\times \frac{k_{\mathrm{on}\hbox{-} \mathrm{i}}}{V_{\mathrm{END}}^{\mathrm{Organ}}}\times {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} \mathrm{i}}\times \left({M}^{\mathrm{J}}+{k}_{\mathrm{off}\hbox{-} 1}\times {\mathrm{Rtotal}}_{\mathrm{Organ}\hbox{-} 1}\right) $$

(117)

which implies that this integral approaches 0 as τ → 0. To conclude that

$$ \begin{array}{l}{\displaystyle \underset{0}{\overset{t}{\int }}}\left(\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}(z)+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}(z)\right)\times \frac{1}{\tau }{e}^{\frac{z-t}{\tau }}\operatorname{d}z\to \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}(t)+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times \\ {}{A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}(t)\end{array} $$

(118)

we observe that

$$ \gamma (t)=\frac{1}{\tau }{e}^{\frac{-t}{\tau }} $$

is the gamma probability density function which possesses the approximate identity property (22,23)

$$ {\displaystyle \underset{0}{\overset{t}{\int }}}f(z)\gamma \left(t-z\right)\operatorname{d}z\to f(t)\;\mathrm{as}\;\tau \to 0 $$

(119)

for any continuous function f(t). This completes proof of (114) for i = 1. The proof of cases i = 2, 3, 4 is differs only in the part:

$$ {\displaystyle \underset{0}{\overset{t}{\int }}}\frac{1}{\tau}\times {A}_{\mathrm{END}\ \mathrm{Organ}\hbox{-} \left(\mathrm{i}\hbox{-} 1\right)}^{\mathrm{J}}(z)\times \frac{1}{\tau }{e}^{\frac{z-t}{\tau }} dz\to \frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{VAS}}^{\mathrm{Organ}}}\times {A}_{\mathrm{VAS}\ \mathrm{Organ}}^{\mathrm{J}}(t)+\frac{{\mathrm{CL}}_{\mathrm{Uptake}}^{\mathrm{Organ}}}{V_{\mathrm{ISF}}^{\mathrm{Organ}}}\times {A}_{\mathrm{ISF}\ \mathrm{Organ}}^{\mathrm{J}}(t)\;\mathrm{as}\;\tau \to 0 $$

(120)

However, by induction, (107) implies (108) which concludes Appendix 9.