Theoretical Examination Seeking Tangible Physical Meanings of Slopes and Intercepts of Plasma Concentration–Time Relationships in Minimal Physiologically Based Pharmacokinetic Models

Abstract

In minimal physiologically based pharmacokinetic (mPBPK) models, physiological (e.g., cardiac output) and anatomical (e.g., blood/tissue volumes) variables are utilized in the domain of differential equations (DEs) for mechanistic understanding of the plasma concentration–time relationships ${C}_{p}(t)$. Although fundamental biopharmaceutical variables in terms of distribution (e.g., ${K}_{p}$ and ${f}_{d}$) and elimination kinetics (e.g., $CL$) in mPBPK provide greater insights in comparison to classical compartment models, an absence of kinetic elucidation of slopes and intercepts in light of such DE model parameters hinders more intuitive appreciation of ${C}_{p}(t)$. Therefore, this study seeks the tangible physical meanings of slopes and intercepts of the plasma concentration–time relationships in one- and two-tissue mPBPK models (i.e., m2CM and m3CM), with respect to time parameters that are readily understandable in PK analyses, i.e., the mean residence ($MRT$) and transit ($MTT$) times. Utilizing the explicit equations (EEs) for the slopes, intercepts, and areas of each exponential phase in the m2CM and m3CM, we theoretically and numerically examined the limiting/boundary conditions of such kinetic properties, based on the ratio of the longest tissue $MTT$ to the $MRT$ in the body (i.e., ${K}_{det}={MTT}_{max}/MR{T}_{B}$) that is useful for dissecting complex PBPK systems. The kinetic contribution of the area of each exponential phase to the total drug exposure was assessed to identify the elimination phase between the terminal and non-terminal phases of the ${C}_{p}\left(t\right)$ in the m2CM and m3CM. This assessment provides improved understanding of the complexities inherent in all PBPK profiles and models.

Graphical Abstract

Appendices

Appendix 1

The mathematically attainable values of ${F}_{1}$ (as the minimum) and ${F}_{2}$ (as the maximum) are obtained at the condition of ${MTT}_{T}={MRT}_{B}$. Melding the relationships of (i) $MR{T}_{B}/MR{T}_{c}=a$ and (ii) ${MTT}_{T}/{MRT}_{B}=1$ in Eqs. 14a and 14b, the ${F}_{1}$ and ${F}_{2}$ terms can be rearranged and expressed as:

$${F}_{1}=\frac{1}{2}+\frac{1-1/a}{2\left(\sqrt{1-1/a}\right)}$$

(52)

$${F}_{2}=\frac{1}{2}-\frac{1-1/a}{2\left(\sqrt{1-1/a}\right)}$$

(53)

Since the denominators and numerators of the second terms in Eqs. 52 and 53 both converge to 0 when $a\to 1$, L'Hospital's rule can apply as:

$$\underset{a\to 1}{\mathrm{lim}}{F}_{1}=\underset{a\to 1}{\mathrm{lim}}\left[\frac{1}{2}+\frac{1/{a}^{2}\cdot \sqrt{1-1/a}}{1/{a}^{2}}\right]=\frac{1}{2}$$

(54)

$$\underset{a\to 1}{\mathrm{lim}}{F}_{2}=\underset{a\to 1}{\mathrm{lim}}\left[\frac{1}{2}-\frac{1/{a}^{2}\cdot \sqrt{1-1/a}}{1/{a}^{2}}\right]=\frac{1}{2}$$

(55)

Therefore, the minimum value of ${F}_{1}$ and the maximum value of ${F}_{2}$ are both 0.5 at the condition of ${MTT}_{T}={MRT}_{B}$.

Appendix 2

The mathematical expressions for ${F}_{AUC1}$ and ${F}_{AUC2}$ can be found at ${MRT}_{B}/MR{T}_{c}=\infty$. From Eqs. 5a and 5b, the products ${MRT}_{c}{\lambda }_{1}$ and ${MRT}_{c}{\lambda }_{2}$ can be expressed as:

$$MR{T}_{c}{\lambda }_{1}=\frac{1}{2}\left(1+\frac{MR{T}_{B}}{MT{T}_{T}}\right)\left(1+\sqrt{1-4\frac{MT{T}_{T}MR{T}_{c}}{{\left(MT{T}_{T}+MR{T}_{B}\right)}^{2}}}\right)$$

(56)

$$MR{T}_{c}{\lambda }_{2}=\frac{1}{2}\left(1+\frac{MR{T}_{B}}{MT{T}_{T}}\right)\left(1-\sqrt{1-4\frac{MT{T}_{T}MR{T}_{c}}{{\left(MT{T}_{T}+MR{T}_{B}\right)}^{2}}}\right)$$

(57)

When ${MRT}_{B}/MR{T}_{c}$ goes to infinity, Eqs. 55 and 56 converge to:

$$MR{T}_{c}{\lambda }_{1}\to 1+\frac{MR{T}_{B}}{MT{T}_{T}}$$

(58)

$$MR{T}_{c}{\lambda }_{2}\to 0$$

(59)

Since ${MRT}_{c}\ne 0$ in typical m2CM structures, Eqs. 18a and 18b for ${F}_{AUC1}$ and ${F}_{AUC2}$ can be rearranged for the case of ${MRT}_{B}/MR{T}_{c}=\infty$ as:

$${F}_{AUC1}=\frac{1-MR{T}_{c}{\lambda }_{2}}{MR{T}_{c}{\lambda }_{1}-MR{T}_{c}{\lambda }_{2}}=\frac{{MTT}_{T}}{MT{T}_{T}+MR{T}_{B}}$$

(60)

$${F}_{AUC2}=\frac{MR{T}_{c}{\lambda }_{1}-1}{MR{T}_{c}{\lambda }_{1}-MR{T}_{c}{\lambda }_{2}}=\frac{MR{T}_{B}}{MT{T}_{T}+MR{T}_{B}}$$

(61)

which are also graphically illustrated as red dashed curves in Fig. 3c.

In addition, we herein show that the equal contribution of the initial and terminal phases to the overall $AUC$ in the 2CM (i.e., ${F}_{AUC1}={F}_{AUC2}=0.5$) is achieved iff ${MRT}_{B}={MTT}_{T}$, when ${MRT}_{B}>MR{T}_{c}$ (i.e., ${R}_{T}MT{T}_{T}\ne 0$). Equation 4b is recalled and rearranged as:

$${\lambda }_{1}+{\lambda }_{2}=\frac{1}{MT{T}_{c}}+\frac{1}{MT{T}_{T}}=\frac{1}{{MRT}_{c}}\left(1+\frac{MR{T}_{B}}{MT{T}_{T}}\right)$$

(62)

When the contributions of the initial and terminal phases to the total drug exposure are equal to each other (i.e., ${F}_{AUC1}={F}_{AUC2}$), the sum of two slopes ${\lambda }_{1}$ and ${\lambda }_{2}$ can be obtained from Eqs. 18a and 18b as:

$${\lambda }_{1}+{\lambda }_{2}=\frac{2}{MR{T}_{c}}$$

(63)

that can be obtained only when ${\lambda }_{1}\ne {\lambda }_{2}$ (i.e., except for the multiple root case in Eq. 4a). As described in the main text, it is noteworthy that the condition ${R}_{T}MT{T}_{T}\ne 0$ can lead to two different real roots ${\lambda }_{1}$ and ${\lambda }_{2}$. Considering that a typical m2CM structure has a non-zero $MR{T}_{c}$, the right-hand sides of Eqs. 62 and 63 can be equated and rearranged, which leads to the relationship ${MRT}_{B}={MTT}_{T}$.

Conversely, melding the relationship ${MRT}_{B}={MTT}_{T}$ in Eqs. 5a and 5b results in:

$${\lambda }_{1}=\frac{1}{MR{T}_{c}}\left(1+\sqrt{1-\frac{MR{T}_{c}}{MR{T}_{B}}}\right)$$

(64)

$${\lambda }_{2}=\frac{1}{MR{T}_{c}}\left(1-\sqrt{1-\frac{MR{T}_{c}}{MR{T}_{B}}}\right)$$

(65)

which can be used for obtaining the mathematical expressions for ${F}_{AUC1}$ and ${F}_{AUC2}$ as:

$${F}_{AUC1}=\frac{\frac{1}{MR{T}_{c}}-{\lambda }_{2}}{{\lambda }_{1}-{\lambda }_{2}}=\frac{\frac{1}{MR{T}_{c}}\sqrt{1-\frac{MR{T}_{c}}{MR{T}_{B}}}}{\frac{2}{MR{T}_{c}}\sqrt{1-\frac{MR{T}_{c}}{MR{T}_{B}}}}=\frac{1}{2}$$

(66)

$${F}_{AUC2}=\frac{{\lambda }_{1}-\frac{1}{MR{T}_{c}}}{{\lambda }_{1}-{\lambda }_{2}}=\frac{\frac{1}{MR{T}_{c}}\sqrt{1-\frac{MR{T}_{c}}{MR{T}_{B}}}}{\frac{2}{MR{T}_{c}}\sqrt{1-\frac{MR{T}_{c}}{MR{T}_{B}}}}=\frac{1}{2}$$

(67)

which proves that, the condition ${MRT}_{B}={MTT}_{T}$ along with a non-zero $MR{T}_{c}$ in typical m2CM structures can lead to the relationship ${F}_{AUC1}={F}_{AUC2}=0.5$, only when ${MRT}_{B}>MR{T}_{c}$ (> 0). Collectively, under the condition of ${MRT}_{B}>MR{T}_{c}$, the relationship ${F}_{AUC1}={F}_{AUC2}=0.5$ is achieved iff ${MRT}_{B}={MTT}_{T}$, consistent with the graphical illustration of Fig. 3c for the m2CM.

Appendix 3

In this section, we obtain a necessary condition for the upper limit of ${F}_{AUC3}$ in the range of ${MRT}_{B}<MT{T}_{2}$ in the m3CM. Equation 44c can be rearranged with respect to ${\lambda }_{3}$, utilizing Eqs. 22d and 22e:

$${F}_{AUC3}=\frac{\left(1-MT{T}_{1}{\lambda }_{3}\right)\left(1-MT{T}_{2}{\lambda }_{3}\right)}{2-MT{T}_{1}{\lambda }_{3}-MT{T}_{2}{\lambda }_{3}-MR{T}_{B}{\lambda }_{3}+MT{T}_{1}{\lambda }_{3}MT{T}_{2}{\lambda }_{3}MR{T}_{c}{\lambda }_{3}}$$

(68)

Since an addition of one more tissue compartment that has a ‘shorter’ $MTT$ to the m2CM results in the cases of ${F}_{AUC3}>MR{T}_{B}/(MR{T}_{B}+MT{T}_{2})$ (i.e., Fig. 6c), we reasoned that a necessary condition for the possible ${F}_{AUC3}$ range can be obtained at $MT{T}_{1}\to 0$. Accordingly, Eq. 68 can be rearranged, when $MT{T}_{1}\to 0$, as:

$${F}_{AUC3}=\frac{1-{P}_{det}{K}_{det}}{1-{P}_{det}{K}_{det}+1-{P}_{det}}=\frac{1}{1+\frac{1/{P}_{det}-1}{1/{P}_{det}-{K}_{det}}}$$

(69)

where ${P}_{det}$ is $MR{T}_{B}{\lambda }_{3}$ and ${K}_{det}$ is $MT{T}_{2}/MR{T}_{B}$ in the m3CM. Based on InEq. 27b, the following inequalities for the case of ${K}_{det}>1$ can be obtained as:

$${K}_{det}-1<\frac{1}{{P}_{det}}-1<{K}_{det}$$

(70)

$$0<\frac{1}{{P}_{det}}-{K}_{det}<1$$

(71)

Therefore, the ${F}_{AUC3}$ values under the conditions of ${K}_{det}>1$ and ${MTT}_{1}\to 0$ are found to fall within:

$$0<{F}_{AUC3}<\frac{1}{{K}_{det}}$$

(72)

It is noteworthy that the ${F}_{AUC3}$ term being limited to $1/{K}_{det}$ is achievable when ${K}_{det}+1$ (the upper bound of $1/{P}_{det}$) is sufficiently close to ${K}_{det}$ (the lower bound of $1/{P}_{det}$) (i.e., ${K}_{det}\gg 1$).