Mouse PBPK Model Formulation

The system-wide distribution of Doc was described using a PBPK model, as shown in Figure 4.1. IV administration is included in the model as an infusion into the venous compartment, which subsequently distributes throughout the rest of the body. In PBPK models, tissue compartments are typically treated as either perfusion-limited (one compartment), as in Equation (4.1),

dC_i

or diffusion-limited (two compartments), as in Equations (4.2)-(4.3):

dC_iv For the perfusion-limited system (Equation (4.1)), C_i is the drug concentration in tissue i, Fi is the blood flow rate into tissue i, Vi is the volume of tissue i, and Cin is the inlet drug

RBCs RBCs

Lung

Brain

Gut Spleen

Liver IV Dose

Heart

Kidney

Tumor

Muscle

Fat

VenousBlood ArterialBlood

Other

Figure 4.1: PBPK model for the distribution of Doc in plasma, various mouse tissues, and subcutaneously implanted tumor xenografts.

concentration for tissue i, which comes from: (i) the venous blood supply in the case of the lungs; (ii) from the spleen, gut, and arterial compartment for the liver; or (iii) from the arterial compartment for all other body tissues. Diffusion-limited tissues (Equations (4.2)-(4.3)) have both a vascular (v) and extra-vascular tissue (e) space (volumes V_iv, and V_ie, respectively), with separate concentrations, C_iv and C_ie, intratissue transfer rates, k_ive and k_iev, and an unbound drug fraction (f_unb) approximated as 15% for mice. Due to the long Doc retention times observed in tissues, an effect of intracellular Doc binding and transport, all modeled tissues had improved fits using diffusion-limited compartments. However, a number of these tissue models were still incapable of simultaneously fitting rapid post-dose dynamics and Doc retention. To improve model fits, an additional subcompartment was added to each tissue. The resulting model for tissues that do not eliminate drug is as follows:

dCiv

Here, C_ib is the drug concentration in the additional “bound” subcompartment, and k_bindout and kbindin are retention rates within the tissue. The binding rate parameters were kept the same for all investigated tissues to limit the overall number of model parameters. For drug eliminating tissues, primarily the liver in the case of Doc, Equations (4.5)-(4.6) add a clearance term, as follows:

Here, kcl_li is the rate of drug clearance from the liver. Note, that clearance was allowed to occur in both the extravascular space in addition to the bound subcompartment, and the same parameter was utlized in both tissue spaces due to data limitations regarding

liver metabolism. Finally, venous plasma and red blood cell (RBC) concentrations were represented by the following equations:

dC_ven

Here, C_ven and C_rbc are the concentrations in venous plasma and circulating red blood cells, respectively, f_hem is the hematocrit fraction in mice (f_hem = 0.45, [1]), F_j and C_j are the flow rates and extravascular concentrations, respectively, from tissue j, and u represents IV administration of Doc. Transition rates between plasma and RBCs are given by k_plasrbc and k_rbcplas, respectively. RBCs were represented as a side compartment to the venous plasma concentration as shown in Figure 4.1. For the arterial concentration, Equations (4.9)-(4.10) are left unchanged except for the removal of the drug input term, u, and the use of “art” in place of “ven”. The overall set of ordinary differential equations governing the PBPK model can be found in Appendix B.

To reduce the number of parameters requiring estimation, tissue and tumor masses from the mouse PK study were adjusted based on tissue densities obtained from the literature [1] and incorporated as the appropriate tissue volumes. Blood flow rates, vascular tissue fractions f_i, and masses for tissues not entirely extracted (fat and muscle) were also obtained from the literature [1]. These parameters are summarized in Table 4.1. Tumor blood flow rate was calculated based on the following equation for ovarian xenografts taken from the literature [166]:

log (tumor blood flow) = −0.808 log (tumor wet weight) − 0.436 (4.11)

Here, the tumor wet weight and tumor blood flow are in units of g and mL

g min, respectively.

Vascular and extra-vascular tissue volumes were calculated as V_iv = V_if_i and V_ie = V_i− V_iv. Similarly, the tissue volume and blood flow rate for the other compartment was calculated as V_o = V_tot−P

i6=oV_i and F_o = F_tot−P

i6=o,lF_i. The remaining parameters (tissue-specific exchange rates, plasma-RBC exchange rates, tissue binding rates, and liver clearance rate)

were estimated by minimizing the weighted sum squared error between model predictions and mouse PK data, as follows:

minρ (Y_act− Y_pred(ρ))^T W₁^TW₁(Y_act− Y_pred(ρ)) (4.12)

Here, Y_act and Y_pred are vectors of the actual PK study data and model predictions, respectively, W₁ is a diagonal matrix of the inverse standard deviations from the PK study, and ρ are the system parameters. The estimated parameters are summarized in Table 4.1.

Parameter estimation and structure selection was accomplished by sequentially adding tissues while minimizing the weighted sum of squares between the model predictions and collected experimental data. The inverse square of the variance at individual points was used as the weights (W₁^TW₁), and simulations were performed in MATLAB ( c 2007, The MathWorks, Natick, MA) using the fmincon function. The steps are summarized below:

1. A compartmental model was fit to the Doc plasma data.

2. Parameters for the lung compartment, assuming a perfusion-limited structure, were estimated, and then the structure was reevaluated as limited and as diffusion-limited with internal binding to assess which structure provided the most accurate representation of lung Doc PK.

3. RBC binding rates were estimated using the estimated plasma Doc profile as the input.

4. A PBPK model with those compartments deemed essential (RBC, lung, liver, gut, spleen, tumor, and “other”) was constructed, with each tissue evaluated first as perfusion-limited. Tissues were then evaluated individually as limited then diffusion-limited with internal binding.

5. Additional tissues were added and evaluated sequentially, in the following order: brain, kidney, heart, muscle, and fat.