A whole-body physiologically based pharmacokinetics (PBPK) model was constructed (Figure 1). It included several major organ compartments, including lungs, blood, heart, liver, spleen, intestine, kidney, and tumor. All other tissues were lumped into the Remainder compartment. Drug transport in each compartment was assumed to be homogeneously distributed and perfusion-limited modeling was utilized [19,20].

Following the mass balance through all the compartments, a general differential equation (1) was used to quantify drug concentration in a compartment:

Where C_compartment / M is drug concentration, V_compartment / L is volume, and Q_compartment / L h^-1is blood flow rate in the particular organ or compartment. C_flow-in represents the drug concentration in the blood flowing into the compartment and C_flow-out represents that flowing out. In the following equations, subscripts are used to denote respective organs, bl for blood, lu for lung, li for liver (in particular, Q_li denotes hepatic portal vein flow rate and Q_he Q_he hepatic artery flow rate), ht for heart, kd for kidney, tu for tumor, sp for spleen, and rm for remainder. Assuming drug diffusion in the tissue is not the rate-limiting step, the drug concentration in each organ can be described by drug partitioning between the blood and organ as quantified by the partition coefficient, K_compartment, which is unitless. PTX elimination mainly occurs in liver [21], which is assumed to be a linear process described by CL_liv . Each compartment is then described as by the following equations (2) to (9).

The multiple terms in the lung equation account for the venous flow-in from major organs and the arterial flow-out (Figure 1). The liver equation considers the flow-in from the spleen and hepatic clearance of the drug. Table 1 lists the physiological parameters used in the equations; the blood flow rates were obtained from the literature [22,23] and the organ volume were measured in our animal studies used to evaluate PTX NCs. Organ volumes were determined by organ weights with the tissue density of 1.18 g/ml [24]. Partition coefficients and the liver clearance rate were determined by data fitting of the PBPK model against our experimental values using nonlinear least squares (NLS)[25]. The PK parameters in the above questions, including partition coefficients, are adjusted and optimized during the NLS process to minimize the resident errors between the experimental data and model-predicted values of drug concentrations. The model was fitted with a tolerance of 10^-8. All calculations were done by MATLAB (version R2018a [26]).

A transit compartmental PD model was coupled with the drug exposure kinetics in the tumor that is calculated by PBPK (Figure 1) [27-29]. In the early stage of cell proliferation, tumor growth is treated exponentially. Angiogenesis and nutrient depletion will impede the process at the later stage, resulting in the linear growth of tumor [30]. To account for the lapse between drug exposure and cell apoptosis, three transit compartments are utilized in the PD model (Equation (10)):

where w₁ / g describes the tumor weight with the initial exponential growth characterized by the rate, k₁, followed by the linear growth characterized by the rate, k₂, and w_b is the weight cutoff above which the tumor growth switches from the exponential to a linear model. w₀ represents the initial tumor weight, and k_d denotes the tumor cell elimination rate by the drug with c denoting the drug concentration, which is calculated by the PK model. Moreover, compartments w₂ and w₃ are introduced to compensate for the lapse of tumor growth upon drug exposure, Equations (11) to (15):

where w is the total tumor weight and is the transit parameter to define the treatment lapse. In this study, NLS was first applied to fit the model against experimental data of tumor weight of mice treated with saline and obtain the tumor growth factors (k values and ).

To quantitatively assess the relative importance of each PBPK parameter that was obtained from NLS fitting on the PK and biodistribution of drug treatment, a sensitivity analysis was conducted. According to our PBPK model, drug distribution and accumulation mainly depend on blood flow rate, tissue partition coefficient, and renal clearance. Derivative-based local sensitivity was performed by applying the one-at-a-time (OAT) technique to evaluate the effect of one parameter on drug concentrations in organs while keeping other parameters fixed [31,32]. Specifically, sensitivity is calculated by taking the derivative of a particular output (e.g. drug concentration in the liver) over the parameter of interest (e.g. clearance), Equation (16):

As the output (i.e. concentration) is a function of time, a calculated sensitivity is then integrated over the treatment time. All the sensitivity values are assembled into a final sensitivity matrix for reporting with the output properties (i.e. concentrations in all respective organs) and PBPK parameters marked.

Finally, the PBPK model was scaled to humans, and PK simulation was conducted. We assumed that organ partitioning of PTX NCs is similar between mice and humans. To do this, physiological data was replaced by human values taken from literature [22,33], shown in Table 2. The clearance of the drug was also replaced with that in humans [34]. Note that the tumor data was of breast tumor, and its plasma flow was determined by macroscopic approaches [35].

In our previous studies with paclitaxel (PTX) nanocrystals, PK and PD kinetics were determined in tumor-bearing mice [10,11]. The murine tumor model was established by subcutaneous injection of 3×10⁶ HT-29 cells under the right arm and 3×10⁶ KB cells under the left arm of female nude outbred mice. Taxol, pure PTX nanocrystals and PTX nanocrystal surface-coated with F68 were administered intravenously once at an equivalent dose of 20 mg/kg via the tail vein. Biodistribution and pharmacokinetics were measured. By fitting the data with our PBPK model, partition coefficients in the major organs and the liver clearance were obtained (Table 3). The fit lines are shown in Figure 2.

Among all compartments, the partition coefficient of the liver is the highest for all three delivery forms. Compared to the conventional Taxol delivery system, the partition coefficients for the nanocrystal formulation increased as such: liver by 5-fold, spleen by 8-fold, and lung by 2-fold. This is likely due to the MPS sequestration, with the liver being the major organ for removing nanoparticles from blood circulation. For Taxol, the kidney had the second highest partition coefficient of 0.75 (unitless) after the liver; the number decreased to 0.41 for the nanocrystal formulation. With the increased liver uptake and decreased renal clearance, the liver could be considered a major depot site for drug nanocrystals. Moreover, surface treatment by F68 reduced the liver partition coefficient from 10.99 to 6.92, which was still 3.3 folds greater than Taxol's. F68 coating minimized the liver uptake and, compared with the other two formulations, significantly lowered clearance, suggesting prolonged blood circulation. The surface treatment also eased the drug tissue affinity in the spleen, heart, kidney, and lung, helping to lighten toxic effects in these organs and enhancing drug circulation. In the tumor, the partition coefficients of Taxol, PTX-NC, and F68-PTX-NC are 0.27, 0.28, and 0.65, respectively. The surface treatment increased the tumor partition (by 2.5 folds), supporting our experimental observation in animals [10,11].

While keeping the partition coefficients and parameters fixed, we ran PK simulations with several different values of the tumor partition coefficient. Table 4 shows the simulation results with the partition coefficient ranging from the smallest value of 0.27, which was of Taxol, to a more significant, hypothetical number of 0.87 for all three formulations. Both maximal concentration (C_max) and area under curve (AUC) values in tumor became larger as the tumor partition coefficient (K_partittion) increased. The C_max increased about two times and AUC about three to four times due to the increase in tumor partition. Thus, given the same dose, the higher the tumor partition coefficient, the more drug accumulation in the organ, resulting in better treatment efficacy. F68-PTX-NC showed the highest PK performance in the tumor compartment because of decreased tissue affinities with the surface treatment among the three delivery systems.

Experimental measurements of tumor growth from our earlier studies were used to establish the PD model. Values of the control treatment group were first fitted in order to obtain the growth rate parameters of the tumor (k₀ and k₁ in Eq. 11). Figure 3 shows the experimental data and predicted trends in tumor growth up to 23 days of individual mice injected with saline solution. The tumor growth model was able to describe the experimental data well. The average tumor linear growth factor k₁ and exponential growth factor k₂ were 0.17 (± 0.11) and 0.51 (± 0.56) grams per day, respectively. The growth factors were then applied to analyze k_d and of the drug treatment by Taxol and PTX NCs.

The tumor treatment parameters were obtained by fitting the PD model against the experimental data with the availability of k₁ and k₂. The transit parameter τ that characterizes the kinetics of cell damage are 0.93, 0.96, and 0.95 day^-1 for Taxol, PTX-NC, and F68-PTX-NC, respectively. The parameter to quantify drug efficacy, k_d, is 0.0039, 0.0037, and 0.0038 mL ng^-1 day^-1 of each respective treatment. The minor differences in the parameters among the drug delivery systems indicate similar anti-tumor activities. This could be understandable as the delivery systems containing the same drug, PTX, and the nanocrystal delivery systems do not alter the pharmacodynamics behavior. It also suggests that a nanocrystal has no antitumor effect; it is released or dissolved, whereby free drug molecules elicit the antitumor function.

We scaled the PBPK model to humans and conducted PK simulations under hypothetical conditions. The same partition coefficients determined by data fitting of the murine model were utilized; physiological parameters (Table 2) and the clearance value of PTX (25.81) in humans were taken from the literature [34]. The simulation results of the tumor compartment are shown in Figure 4. The initial spikes of concentration were likely caused by the mathematical model originally generated by data fitting of experimental results obtained in mice. The remaining trends appear more trustworthy; thus, only the AUC should be used to interpret the simulation results. The AUC values of drug accumulation in tumors are 1.54, 1.16, and 3.52 μg/ml·h of Taxol, PTX-NC, and F68-PTX-NC, respectively. The surface treatment seems to at least double the drug accumulation in tumors compared with the solubilized formulation or pure NCs. Relative to the total injected dose (316 mg for 70 kg adult with body surface area 1.7 m²), the tumor accumulation was 0.29, 0.44, and 0.95 %. The trend is similar to that in mice (0.62, 0.26, and 2.92 %). The further increase in mice for F68-coated NC resulted from the decreased clearance. Comparison of other organs indicates different trends between mice and humans (Table 5). Each value in the tablet is relative to the total injected dose, equivalent to the accumulated concentration in blood (i.e. 100 %).

We further simulated PD in humans based on the tumor exposure predicted by the PBPK discussed above. Tumor growth parameters were taken from our mouse model. Data from human breast tumors were simulated with the blood flow rate and tumor volume taken from the literature (3.2 L/h and 0.00419 L; Table 2). Note that, in our tumor-bearing mice (as well as numerous tumor murine models reported in the literature), the xenografted tumor is significantly larger relative to body weight (6 %) and, therefore, not scalable in humans (0.007 %; see later discussion). Figure 5 shows tumor growth inhibition by these treatments was evident within the first few hours (3.6, 2.9, and 7.5 % with Taxol, PTX-NC, and F-68-PTX-NC, respectively). At day 5, human breast tumor growth without treatment increased by 41.2 %, while the increases were 31.7, 33.4, 26.3 % of Taxol, PTX-NC, and F68-PTX-NC, respectively. In comparison, the tumor growth during a 5-day span in mice without treatment was 270 and 46-79 % with the treatments. The results show a considerable difference in tumor growth between mice and humans without treatment. Even with the treatment, the difference is still outstanding. The simulation also indicates that surface coating of NCs can positively inhibit tumor growth, agreeing with the biodistribution simulation (Figure 4).

Sensitivity analyses were conducted to examine the impact of individual PK parameters on the biodistribution kinetics in mice and humans. Figure 6 shows the two sensitivity matrices calculated for mice and humans. The data shown here was generated based on simulations of Taxol (considering the clearance data taken from the study of Taxol), but the data of the other two NC formulations (not shown here) yield the same trend and conclusion.

The heat map indicates intercorrelations between PBPK parameters and drug concentrations in the organs and tumor. For the drug concentration in a particular organ (i.e. the y-axis), the figure ranks the significance of impact by the respective partition coefficient and hepatic clearance (x-axis). Each box's color marks the sensitivity values of a particular derivative of organ concentration over partition coefficient or clearance, integrated over a period of time (e.g. 3 h used in Figure 6). The sensitivity map calculated from the mice PK model (Figure 6A) indicates that the hepatic clearance of PTX is the dominant factor in determining drug depositions in all organs and the tumor. The plasma concentration of the drug is affected mainly by the hepatic clearance. The next organ affected by the clearance is the liver, followed by the lung. From the mathematic and modeling perspective, the drug concentration accumulated in an organ is positively affected by the blood flow rate to the organ and negatively by the organ volume (or weight). The lung has the highest blood flow rate, followed by the liver. This can be further illustrated by comparing drug accumulations in tumors and the liver. The tumor has a comparable organ volume to the liver or blood, but its flow rate is only a fraction of that in the liver or blood (about 0.9% or 0.2%; Table 1). Besides the phagocytosis of drug nanocrystals by the MPS (indicated by the liver partition coefficient), the much-dwarfed systemic circulation into the tumor significantly limits drug deposition. Nonetheless, despite the liver partition being the largest among all partition coefficients (Table 3), it is the systemic clearance by the liver that exaggerates the liver partition in determining local drug distributions to other organs and the tumor.

Figure 6B shows the sensitivity matrix in humans. The same partition coefficients determined from the mice model were utilized but with human physiological conditions (Table 2). The clearance was taken from human experimental data for paclitaxel [34]. The overall result is different from that in mice. The partition coefficients of each organ and tumor have the most relevant impact on drug accumulation. On the other hand, the hepatic clearance is not significant. The liver partition coefficient is the most influential parameter for blood plasma instead of clearance. Ostensibly, the sensitivity data indicate that instead of systemic elimination, drug distribution to organs contributes the most to drug disposition. The variations in physiological and anatomical conditions primarily cause the difference in biodistribution between mice and humans. In humans, the cardiac flow rate (Q_lu) is about four times greater than that in the liver; the blood volume is also 3-4 times higher (Table 2). In mice, however, even though the liver blood flow rate is about four times smaller than that in blood, the liver volume is similar to the blood volume. Given the same drug partition coefficients used in both mice and human models, such differences in blood flow and organ volume explain drastically different PK behaviors of drug nanocrystals between the two biological systems. This also implies that hepatic uptake and clearance are more significant in rodents, including mice than in humans. From an evolutionary perspective, it could make more sense that humans prepare and cook their food while animals digest raw organics. A stronger MPS would protect the body from exogenous particles and toxins absorbed into the systemic circulation. Additionally, it is noted that the scales are significantly different in the sensitivity maps between mice and humans, as shown in Figure 6.

A sensitivity value is much higher in mice than humans at any time point of simulation – mathematically, a direct result of the relative ratio between blood flow rate over organ volume (or weight) being more significant in mice than humans.

Figure 7 shows the time-integrated sensitivity values in mouse tumors over 24 hours. Again, it is shown that the hepatic clearance has the most significant impact on the drug accumulation, four times greater than the next contribution, tumor blood flow rate, followed by blood flow rates through the hepatic artery, spleen, and liver. While the blood flow rate and organ volume are both small in the spleen (Table 1), drug transport and deposition in the spleen directly contribute to the drug concentration in the liver (Eq. 5). The Tumor partition coefficient ranks as the fourth factor to alter the drug accumulation. This suggests that even with the so-called active targeting moieties on the surface of nanoparticles, the formulation strategy would have a minimal effect in tumor accumulation. Moreover, given that tumor blood flow only accounts for 0.2% of the total blood output, any possible impact by the EPR effect is very limited in varying the drug accumulation in the tumor relative to the total amount of injected dose. In comparison, blood flow in the liver and kidney accounts for more than 22.5 and 16.25 % of the cardiac output, respectively (Table 1).

Our analyses suggest that in mice when given intravenously, drug nanocrystals are mainly taken up by the liver; tissue distribution and tumor accumulation of the drug are subsequently determined by how fast the drug is eliminated from the system. However, tissue distribution in humans is likely driven by the systemic circulation and the organ affinity of the drug (delivery system). In humans, the drug has a greater tendency to be deposited in tissues and gradually released back into the systemic circulation (blood and lymph). With this regard, having a drug delivery system circulating long enough in the blood (for example, by putting biocompatible surfactants on nanoparticle surfaces) could significantly enhance tissue distribution and tumor accumulation in mice. Conversely, having such surface treatment or not solely to improve the blood circulation might make little distinction in humans. Although the blood concentration of PTX doubled (Figure 4), this may result in higher tissue partitioning of the NC surface coating. From the systemic perspective, the PK difference implies that treatment efficacy and PK of a drug delivery system tested in murine models may not directly extend to humans. As indicated by a literature report, two different formulations of PTX were seen to have opposite rankings in anticancer effects in mice vs. human clinical trials [36].