2.1.

Macroscopic Model for Photodynamic Therapy

The macroscopic PDT model is briefly described in this section, as well as the definitions of the five parameters to be optimized during the process of fitting in the in vivo experimental results. The intention of this work is to investigate the oxygen supply term in the model using a microscopic model instead of fitting experimental results to derive parameters. More detailed descriptions of the model and the fitting routine can be found elsewhere.¹³

In the macroscopic model,¹³ spatial distribution of $\varphi$ in the tumor is calculated via Eq. (2) based on the diffusion approximation. Temporal and spatial distributions of PS ($S_0$), $\left[{}^3{\mathrm{O}}_2\right]$ and $\left[{}^1{\mathrm{O}}_2\right]$ concentrations are obtained by solving a set of coupled time-dependent differential Eqs. (3) to (5). The cumulative concentration of ${[{{}^1\mathrm{O}}_2]}_{\mathrm{rx}}$ can then be derived by the integration of Eq. (5) over time ($t$). In Eq. (4), the symbol ${\mathrm{\Gamma }}_{\mathrm{s}}$ denotes the rates at which ${}^3{\mathrm{O}}_2$ is supplied to the surrounding tissue, which is the term to be examined in this study.

The parameters ${{\mu }_{\mathrm{s}}}^{\prime }$ and ${\mu }_{\mathrm{a}}$ represent optical scattering and absorption coefficients, respectively. $\delta$ is the low concentration correction parameter and $S$ describes the light source. The mathematical definitions of all parameters are given in Tables 1 and 2. Some reported values of the photochemical parameters for Photofrin are shown in Table 1.¹³

Table 1

Photodynamic therapy (PDT) photochemical parameters used for calculations for Photofrin.13

Table 2

Definitions of the photochemical parameters.

2.2.

Microscopic Model for Photodynamic Therapy

A microscopic PDT model was developed based on the Krogh cylinder model.¹⁹ In the model, the tumor has uniformly spaced cylindrical blood capillaries (with a radius of $R_{\mathrm{c}}$) in parallel with the linear light source. The inter-capillary distance between two adjacent capillaries is assumed to be large enough so that each capillary can supply oxygen only to its immediate concentric surrounding tissue. The oxygen concentration in the oxygen supply term as expressed in Eq. (1) will be an average value over the entire tissue element volume. Due to the values of both inter-capillary distance ($R_{\mathrm{t}}$) and capillary length ($l_{\mathrm{z}}$) used in this study, it is reasonable to assume that $\varphi$ within the small tissue element in Fig. 1 is constant. Figure 1 shows the schematic of the cylindrical Krogh model. Note that a three-dimensional Krogh model can be simplified as a 2-D cylindrical symmetric model given the above assumptions.

Fig. 1

(a) A schematic of the Krogh cylinder model. (b) three-dimensional (3-D) mesh plot of the Krogh cylinder model. The light fluence rate within the Krogh model is considered to be a constant because the spatial scale of light transport ($\sim 1$ to 10 mm) is much larger than the spatial scale of the Krogh model ($<0.4\mathrm{mm}=400\mu \mathrm{m}$). The incident direction of light is randomly distributed.

Before introducing the governing equations for oxygen and its carrier, oxyhemoglobin (HbO), some basic physiological assumptions are discussed first. Oxygen is normally present in the blood in two forms: chemically bound to hemoglobin forming HbO, and free molecules dissolved in the plasma. Most oxygen is bound to form HbO, which is contained in the red blood cell (RBC). However, there is still a small fraction of oxygen dissolved in the liquid media such as blood plasma and RBC water. The concentration of these free oxygen molecules can be represented using a quantity called oxygen partial pressure ($P$). The relation between them can be defined using Henry’s law in Eq. (6), where $\alpha$ is the oxygen solubility coefficient:

When $P$ decreases in the surrounding environment, HbO will release oxygen and vice versa. The percentage of hemoglobin that is saturated with oxygen is usually referred to as hemoglobin oxygen saturation (${\mathrm{SaO}}_2$). The relationship between $P$ and ${\mathrm{SaO}}_2$ is described by Hill’s oxygen dissociation curve. A mathematic expression for ${}^3{\mathrm{O}}_2$ dissociation is Hill’s equation, as shown in Eq. (7):¹⁶

Oxygen molecules can diffuse freely from RBC into the blood stream due to negligible resistance in the membrane.¹⁹ Therefore, the first assumption is that released oxygen from HbO can instantaneously be present in blood plasma (i.e., ignoring the diffusion from RBC). The second assumption is that RBCs are uniformly distributed in the blood.

Given the above assumptions, the time-dependent governing equations for ${}^3{\mathrm{O}}_2$ and HbO transport inside the capillary are given in Eqs. (8) and (9). First, note that the concentration of ${}^3{\mathrm{O}}_2$ is hereafter expressed using the partial pressure of ${}^3{\mathrm{O}}_2$ ($P$) based on Eq. (6) because of the continuity boundary conditions that will be discussed later. Second, Sa in Eq. (9) is the hemoglobin oxygen saturation describing the percentage of HbO concentration to total hemoglobin concentration.

The product of Sa and $C_{\mathrm{H}}$ (total hemoglobin concentration in capillary) is the HbO concentration. The first terms on the right hand sides of Eqs. (8) and (9) are the diffusion terms of ${}^3{\mathrm{O}}_2$ and hemoglobin, respectively; the second terms describe the convection processes. The third term (${\mathrm{\Gamma }}_{\mathrm{rec}}$) is the so-called “reaction” term representing the ${}^3{\mathrm{O}}_2$ loading/unloading from deoxyhemoglobin/oxyhemoglobin. The parameters $D_{\mathrm{c}}$ and $D_{\mathrm{H}}$ represent the diffusion coefficients of ${}^3{\mathrm{O}}_2$ and hemoglobin in the capillary, respectively. ${\alpha }_{\mathrm{c}}$ is the solubility of ${}^3{\mathrm{O}}_2$ in plasma and $v$ is the blood velocity in the capillary.

By manipulating Eqs. (7) to (9), one can derive the main governing Eq. (10) for $P$ in the capillary, where $K$ and $M$ are defined in Eqs. (11) and (12), respectively.

The boundary conditions in the microscopic model are summarized in Eqs. (13) to (16). The bottom end of the capillary (i.e., $z=0$) is the entrance of blood flow, which is assumed to have a constant ${}^3{\mathrm{O}}_2$ partial pressure $P_{\mathrm{ts}}$. On the boundary between capillary and tissue, both ${}^3{\mathrm{O}}_2$ flux and $P$ are continuous as shown in Eqs. (14) and (15). Other boundaries are considered as insulation.

The parameters ${\alpha }_{\mathrm{t}}$ and $D_{\mathrm{t}}$ are the solubility and diffusion coefficients of oxygen in tissue, respectively.

The governing equation for $\left[{}^3{\mathrm{O}}_2\right]$ in tissue during PDT in the microscopic model is given by Eq. (17), which has the same terms on the left-hand side as Eq. (4) to describe PDT consumption of oxygen. The right-hand side of the equation contains both ${}^3{\mathrm{O}}_2$ diffusion (the first term) and the metabolic consumption (the second term). However, only a general oxygen supply term ${\mathrm{\Gamma }}_{\mathrm{s}}$ is used in the macroscopic model [as shown in Eq. (4)].

The parameter $q_0$ represents the maximum metabolic ${}^3{\mathrm{O}}_2$ consumption rate and $P_{\mathrm{m}}$ is the half-maximum oxygen consumption.

Table 3 presents the magnitude of the physiological parameters based on the literature values for the normal and tumor tissues by either measurements (in both living animals and fixed tissues) or theoretical studies.^16,19–25 The normal capillaries appear as fine, nearly parallel vessels that are served by orderly branching arterial and venous trees.²¹ In contrast, the tumor vessels are disorganized, leaving large, irregular avascular spaces.²¹ We tried to cover the existing wide range data sets representing both normal and cancer tissues in our study.

Table 3

Physiological parameters in the microscopic PDT model. The standard values are considered as the benchmark condition for the comparison.

2.3.

Simulation, Procedures, and Initial Conditions

The microscopic model was simulated by the finite element method (FEM) analysis, solver, and simulation software package COMSOL Multiphysics v4.3b (Comsol AB, Stockholm, Sweden), which was run on an iMAC OSX version 10.9.5 (Processor 3.1 GHz Intel Core 17 and Memory 16 GB 1600 MHz DDR3). LiveLink for MATLAB® was also used to work with COMSOL Multiphysics in combination with MATLAB R2013a (64-bit, Massachusetts).

The first step was to examine an expression for oxygen supply. For this purpose, the instantaneous total change rates of ${}^3{\mathrm{O}}_2$ in tissue [i.e., the sum of the two terms on the right-hand side of Eq. (17)] were calculated, and then plotted as a function of the instantaneous oxygen concentration at the corresponding time normalized to its value prior to PDT ($[{}^3{\mathrm{O}}_2]/{[{}^3{\mathrm{O}}_2]}_0$). These simulations were first performed for a range of $\varphi$ (25 to $150\mathrm{mW}/{\mathrm{cm}}^2$), and some typical treatment conditions for Photofrin-, mTHPC-, BPD-, and HPPH-mediated PDT. Different photochemical parameters $\xi$ ($3.7\times {10}^{-3}-76\times {10}^{-3}{\mathrm{cm}}^2{\mathrm{mW}}^{-1}{\mathrm{s}}^{-1}$), $\sigma$ ($1.55\times {10}^{-5}-7.6\times {10}^{-5}\mu {\mathrm{M}}^{-1}$) and $\beta$ (8.7 to $11.9\mu \mathrm{M}$) were used in this step. The results were analyzed together to see the $\varphi$, $\xi$, $\sigma$, and $\beta$ dependence.

The main objective of the second step was to find $g$ values in Photofrin-mediated PDT for a range of physiological microenvironments at $\varphi =150\mathrm{mW}/{\mathrm{cm}}^2$. The initial Photofrin concentration in the tissue was assumed to be $7\mu \mathrm{M}$ and total treatment time was chosen to be 100 min. The standard values of the PDT photochemical and physiological parameters, reported by the other studies, were used as shown in Tables 1 and 3.^16,19–25 The radius of the capillary ($R_{\mathrm{c}}=2.5$, 4.0 or $10\mu \mathrm{m}$), the length of the capillary ($l_{\mathrm{z}}=100$, 220 or $400\mu \mathrm{m}$), the density of blood vessels ($R_{\mathrm{t}}=18$, 30 or $60\mu \mathrm{m}$), the blood flow velocity ($v_{\mathrm{z}}=50$, 100 or $200\mu \mathrm{m}/\mathrm{s}$), and maximum ${{}^3\mathrm{O}}_2$ metabolic consumption rate ($q_0=0.9$, 2.4 or $6\mu \mathrm{M}/\mathrm{s}$) were set at three different values, which cover the existing data sets representing both normal and tumor vasculatures.^19–24 The magnitude of $P_{\mathrm{ts}}$ was set at two different values of 50 and 100 mmHg representing ${{}^3\mathrm{O}}_2$ partial pressure in both tumor and normal arteries.^19,22–25, ²⁷ For the simulations, the physiological parameters were first set at their standard values and then the magnitude of $R_{\mathrm{c}}$, $R_{\mathrm{t}}$, $v_{\mathrm{z}}$, $q_0$, $P_{\mathrm{ts}}$ and $l_{\mathrm{z}}$ were varied from their standard values. The varied parameters with their values are listed in Table 4 along with the fitted results in the next section.

Table 4

Physiological parameters varied for the range of microenvironment examined for g.

3.1.

Linear Correlation and $g$ Calculation for Different Photochemical Parameters and Fluence Rates in a Capillary with Standard Physiological Parameters

Figures 2(a) and 2(b) show the volume-averaged oxygen supply rate and ${{}^3\mathrm{O}}_2$ concentration over the whole Krogh cylinder as a function of time, calculated using the microscopic model for standard values presented in Table 3 and $\varphi =25$, 50, 75, 100, and $150\mathrm{mW}/{\mathrm{cm}}^2$. Based on our calculations, the initial volume-averaged oxygen concentration ${[{{}^3\mathrm{O}}_2]}_0$, prior to Photofrin-mediated PDT (with $\xi =3.7\times {10}^{-3}{\mathrm{cm}}^2{\mathrm{mW}}^{-1}{\mathrm{s}}^{-1}$, $\sigma =7.6\times {10}^{-5}\mu {\mathrm{M}}^{-1}$, $\beta =11.9\mu \mathrm{M}$, $\delta =33\mu \mathrm{M}$), was around $39.41\mu \mathrm{M}$. The corresponding volume-averaged oxygen supply rate versus $[{{}^3\mathrm{O}}_2]/{[{{}^3\mathrm{O}}_2]}_0$ is presented in Fig. 2(c), which is “zero” prior to PDT and shows a linear correlation independent of $\varphi$. A linear fit to these data (has an intercept at 1 on $x$-axis) results in a slope of about $4.9\pm 0.1$, which represents the $g$ value in $\mu \mathrm{M}/\mathrm{s}$ with the standard deviation of the mean (STDM). The simulations were also performed for different PSs: mTHPC (with $\xi =30\times {10}^{-3}{\mathrm{cm}}^2{\mathrm{mW}}^{-1}{\mathrm{s}}^{-1}$, $\sigma =2.97\times {10}^{-5}\mu {\mathrm{M}}^{-1}$, $\beta =8.7\mu \mathrm{M}$, $\delta =33\mu \mathrm{M}$), BPD (with $\xi =51\times {10}^{-3}{\mathrm{cm}}^2{\mathrm{mW}}^{-1}{\mathrm{s}}^{-1}$, $\sigma =1.7\times {10}^{-5}\mu {\mathrm{M}}^{-1}$, $\beta =11.9\mu \mathrm{M}$, $\delta =33\mu \mathrm{M}$) and HPPH (with $\xi =76\times {10}^{-3}{\mathrm{cm}}^2{\mathrm{mW}}^{-1}{\mathrm{s}}^{-1}$, $\sigma =1.55\times {10}^{-5}\mu {\mathrm{M}}^{-1}$, $\beta =11.9\mu \mathrm{M}$, $\delta =33\mu \mathrm{M}$) at $\varphi =150\mathrm{mW}/{\mathrm{cm}}^2$. The volume-averaged oxygen supply rate versus $[{{}^3\mathrm{O}}_2]/{[{{}^3\mathrm{O}}_2]}_0$ is presented in Fig. 2(d) and shows a linear correlation independent of $\xi$, $\sigma$ and $\beta$. The linear fit to these data results in a $g$ value of about $5.1\pm 0.2\mu \mathrm{M}/\mathrm{s}$ ($g$ is presented with STDM). While the $g$ value is extremely sensitive to change in the physiological parameters and ${{}^3\mathrm{O}}_2$ concentration, the results show that $g$ is independent of $\varphi$ and PS photochemical parameters.

Fig. 2

(a) Oxygen supply rate [right-hand side of Eq. (17)] and (b) oxygen concentration, defined as the volumetric average around each vessel versus time. The plots are for $\varphi =25$, 50, 75, 100, and $150\mathrm{mW}{\mathrm{cm}}^{-2}$. (c) The mean oxygen supply rate versus normalized mean oxygen for different $\varphi$. A linear fit to the spectra (as shown with red lines) results an average slope of about $4.99\pm 0.10$, which is $g(\mu \mathrm{M}/\mathrm{s})\pm \mathrm{STDM}$. (d) The mean oxygen supply rate versus normalized oxygen for different photosensitizers (PSs), Photofrin ($\xi =3.7\times {10}^{-3}{\mathrm{cm}}^2{\mathrm{mW}}^{-1}{\mathrm{s}}^{-1}$, $\sigma =7.6\times {10}^{-5}\mu {\mathrm{M}}^{-1}$, $\beta =11.9\mu \mathrm{M}$, $\delta =33\mu \mathrm{M}$), mTHPC ($\xi =30\times {10}^{-3}{\mathrm{cm}}^2{\mathrm{mW}}^{-1}{\mathrm{s}}^{-1}$, $\sigma =2.97\times {10}^{-5}\mu {\mathrm{M}}^{-1}$, $\beta =8.7\mu \mathrm{M}$, $\delta =33\mu \mathrm{M}$), BPD ($\xi =51\times {10}^{-3}{\mathrm{cm}}^2{\mathrm{mW}}^{-1}{\mathrm{s}}^{-1}$, $\sigma =1.7\times {10}^{-5}\mu {\mathrm{M}}^{-1}$, $\beta =11.9\mu \mathrm{M}$, $\delta =33\mu \mathrm{M}$) and HPPH ($\xi =76\times {10}^{-3}{\mathrm{cm}}^2{\mathrm{mW}}^{-1}{\mathrm{s}}^{-1}$, $\sigma =1.55\times {10}^{-5}\mu {\mathrm{M}}^{-1}$, $\beta =11.9\mu \mathrm{M}$, $\delta =33\mu \mathrm{M}$). A linear fit to the spectra (as shown with red lines) results $g=5.12\pm 0.17\mu \mathrm{M}/\mathrm{s}$. The data are plotted for the physiological parameters with standard values presented in Table 3. Normalized mean oxygen is defined as the volumetric average of ${{}^3\mathrm{O}}_2$ around each vessel divided by ${[{{}^3\mathrm{O}}_2]}_0$, which is the initial mean oxygen before PDT.

3.2.

Representative g Value Tests on More Than One Varying Parameter

The $g$ values were calculated for Photofrin-mediated PDT and a range of physiological parameters at $\varphi =150\mathrm{mW}/{\mathrm{cm}}^2$. In our model, the blood vessel network forms uniformly distributed Krogh cylinders and the spacing between vascular cylinders $R_{\mathrm{t}}$ varies between 18 and $60\mu \mathrm{m}$. The cylindrical blood capillary has $R_{\mathrm{c}}$ in the range of 2.5 to $10\mu \mathrm{m}$ and $l_{\mathrm{z}}$ in the range of 100 to $400\mu \mathrm{m}$. For both tumor and normal vasculatures, the maximum ${{}^3\mathrm{O}}_2$ metabolic consumption rate ($q_0$) and blood velocity in capillary ($v_{\mathrm{z}}$) are in the ranges of 0.9 to $6\mu \mathrm{M}/\mathrm{s}$ and 50 to $200\mu \mathrm{m}/\mathrm{s}$, respectively (see Table 3). The oxygen pressure at the aortal entrance of the blood vessel ($P_{\mathrm{ts}}$) is assumed to be in the range of 50 or 100 mmHg.^19,25,27 The linear correlation between the volume-averaged oxygen supply rate and $[{{}^3\mathrm{O}}_2]/{[{{}^3\mathrm{O}}_2]}_0$ as well as their respective $g$ values is presented in Figs. 3Fig. 4 to 5 and Table 4. The linear fits to the data result in $g$ values in the ranges of 0.9 to $286.1\mu \mathrm{M}/\mathrm{s}$ for $P_{\mathrm{ts}}=100\mathrm{mmHg}$ and $0.6-606.9\mu \mathrm{M}/\mathrm{s}$ for $P_{\mathrm{ts}}=50\mathrm{mmHg}$.

Fig. 3

Calculated mean oxygen supply rate [right side of Eq. 17] versus normalized mean oxygen, $[{{}^3\mathrm{O}}_2]/{[{{}^3\mathrm{O}}_2]}_0$. The data are plotted for artery oxygen partial pressure $P_{\mathrm{ts}}=100\mathrm{mmHg}$, capillary length $l_{\mathrm{z}}=220\mu \mathrm{m}$ as well as different blood flow $v_{\mathrm{z}}$ and maximum metabolic oxygen consumption rate $q_0$. Each plot contains nine combinations of cylindrical tissue radius $R_{\mathrm{t}}(18,30,60\mu \text{m})$ and capillary radius $R_{\mathrm{c}}(2.5,4,10\mu \text{m})$ as presented with different colors and symbols. The 3 plots for the left column are for $v_{\mathrm{z}}=50\mu \text{m}/\text{s}$ and $q_0=0.9,2.4,\text{and}6\mu \text{M}/\text{s}$, respectively; the 3 plots for the middle column are for $v_{\mathrm{z}}=\text{\hspace{0.17em}100\hspace{0.17em}}\mu \text{m}/\text{s}$ and $q_0=0.9,2.4,\text{and}6\mu \text{M}/\text{s}$, respectively; the 3 plots for the right column are for $v_{\mathrm{z}}=200\mu \text{m}/\text{s}$ and $q_0=0.9,2.4,\text{and}6\mu \text{M}/\text{s}$, respectively. The calculated $g$ values are in the range of 0.89–$286.09\mu \mathrm{M}/\mathrm{s}$.

Fig. 4

Calculated mean oxygen supply rate [right side of Eq. 17] versus normalized mean oxygen, $[{{}^3\mathrm{O}}_2]/{[{{}^3\mathrm{O}}_2]}_0$. The data are plotted for artery oxygen partial pressure $P_{\mathrm{ts}}=50\mathrm{mmHg}$, capillary length $l_{\mathrm{z}}=220\mu \mathrm{m}$ as well as different blood flow $v_{\mathrm{z}}$ and maximum metabolic oxygen consumption rate $q_0$. Each plot contains nine combinations of cylindrical tissue radius $R_{\mathrm{t}}(18,30,60\mu \text{m})$ and capillary radius $R_{\mathrm{c}}(2.5,4,10\mu \text{m})$ as presented with different colors and symbols. The 3 plots for the left column are for $v_{\mathrm{z}}=50\mu \text{m}/\text{s}$ and $q_0=0.9,2.4,\text{and}6\mu \text{M}/\text{s}$, respectively; the 3 plots for the middle column are for $v_{\mathrm{z}}=\text{\hspace{0.17em}100\hspace{0.17em}}\mu \text{m}/\text{s}$ and $q_0=0.9,2.4,\text{and}6\mu \text{M}/\text{s}$, respectively; the 3 plots for the right column are for $v_{\mathrm{z}}=200\mu \text{m}/\text{s}$ and $q_0=0.9,2.4,\text{and}6\mu \text{M}/\text{s}$, respectively. The calculated $g$ values are in the range of 0.59–$606.96\mu \mathrm{M}/\mathrm{s}$.

Fig. 5

Calculated mean oxygen supply rate [right side of Eq. 17] versus normalized mean oxygen, $[{{}^3\mathrm{O}}_2]/{[{{}^3\mathrm{O}}_2]}_0$. The data are plotted for blood flow $v_{\mathrm{z}}=100\mu \text{m}/\text{s}$, maximum metabolic oxygen consumption rate $q_0=2.4\mu \text{M}/\text{s}$ as well as different artery oxygen partial pressure $P_{\mathrm{ts}}$ and capillary length $l_{\mathrm{z}}$. Each plot contains 9 combinations of cylindrical tissue radius $R_{\mathrm{t}}(18,30,60\mu \text{m})$ and capillary radius $R_{\mathrm{c}}(2.5,4,10\mu \text{m})$ as shown with different colors and symbols. The 3 plots for the left column are for $P_{\mathrm{ts}}=100\mathrm{mmHg}$ and $l_{\mathrm{z}}=100,220,\text{and}400\mu \mathrm{m}$, respectively; the 3 plots for the right column are for $P_{\mathrm{ts}}=50\mathrm{mmHg}$ and $l_{\mathrm{z}}=100,220,\text{and}400\mu \mathrm{m}$, respectively. The $g$ values are in the ranges of 1.01–$277.33\mu \mathrm{M}/\mathrm{s}$ and 0.93–$642.25\mu \mathrm{M}/\mathrm{s}$ for $P_{\mathrm{ts}}=100\text{and}50\mathrm{mmHg}$, respectively.

3.3.

Formulation of $g$ Directly from Blood Vessel Physiological Parameters

The convective oxygen delivery ($Q$) is related to the product of blood flow ($v_{\mathrm{z}}\pi R_{\mathrm{c}}^2$) and oxygen concentration by Fick’s principle:²⁸

Oxygen continuously diffuses from the plasma to the tissue (with the volume of $\pi R_{\mathrm{t}}^2{l}_{\mathrm{z}}$) where it is consumed. If one assumes that all the capillaries perfusing the tissue are identical and that the oxygen consumption is uniform within the small tissue element shown in Fig. 1(a), then the amount of ${{}^3\mathrm{O}}_2$ removed from the volume of blood per unit length along the capillary ($\pi R_{\mathrm{c}}^2$) is constant. The amounts of oxygen moving across the capillary wall is proportional to that consumed by tissue which is supplied by the capillary:²⁸

Figures 3 to 5 describing the changes of $g$ with $v_{\mathrm{z}}$, $R_{\mathrm{c}}$, $R_{\mathrm{t}}$, and $l_{\mathrm{z}}$ show close agreement with Fick’s principles. While with the same conditions of $R_{\mathrm{t}}$ and $R_{\mathrm{c}}$, the resulting $g$ values decreased roughly by $1/l_{\mathrm{z}}$, $g$ linearly increased with $v_{\mathrm{z}}$; $g$ showed a nonlinear relationship with $R_{\mathrm{c}}$, $R_{\mathrm{t}}$, and $q_0$. On the basis of the reduced chi-squared ($0.96\le {\chi }^2\le 1$), the $g$ values versus $R_{\mathrm{c}}$ [Fig. 6(a)], $R_{\mathrm{t}}$ [Fig. 6(b)], $v_{\mathrm{z}}$ [Figs. 6(c)], and $l_{\mathrm{z}}$ [Figs. 6(d)] were best fitted with a second-order polynomial, second power decay, linear, and first power decay curves, respectively. The fitting equations were used to obtain an empirical Eqs. (20) and (21) that can calculate $g$ directly from $v_{\mathrm{z}}$, $R_{\mathrm{c}}$, $R_{\mathrm{t}}$, $l_{\mathrm{z}}$, and $q_0$ for two $P_{\mathrm{ts}}$ conditions:

Fig. 6

$g$ versus (a) $R_{\mathrm{c}}$, (b) $R_{\mathrm{t}}$, (c) $v_{\mathrm{z}}$, (d) $l_{\mathrm{z}}$, and (e) $q_0$ for $P_{\mathrm{ts}}=100\mathrm{mmHg}$ (in the left column) and $P_{\mathrm{ts}}=50\mathrm{mmHg}$ (in the right column). The vasculature conditions assumed to have the standard values presented in Table 3. The actual $g$ values (filled-set symbols) are compared with those calculated by using Eqs. (20) and (21) (empty-set symbols). On the basis of the reduced chi-squared ($0.96\le {\chi }^2\le 1$), the $g$ values versus $R_{\mathrm{c}}$, $R_{\mathrm{t}}$, $v_{\mathrm{z}}$, and $l_{\mathrm{z}}$ were best fitted with second-order polynomial, second power decay, linear and first power decay curves, respectively.

The relationship between $g$ and $q_0$ as well as the constant values were obtained and optimized manually based on the relative errors of the actual $g$ and those obtained by Eqs. (20) and (21). For the range of microenvironments shown in Table 3, Eqs. (20) and (21) determine $g$ values in the ranges of 0.6 to $685.3\mu \mathrm{M}/\mathrm{s}$ for $P_{\mathrm{ts}}=100\mathrm{mmHg}$ and 0.4 to $1390\mu \mathrm{M}/\mathrm{s}$ for $P_{\mathrm{ts}}=50\mathrm{mmHg}$. The maximum $g$ value of $1390\mu \mathrm{M}/\mathrm{s}$ is obtained for $R_{\mathrm{c}}=10\mu \mathrm{m}$, $R_{\mathrm{t}}=18\mu \mathrm{m}$, $v_{\mathrm{z}}=200\mu \mathrm{m}/\mathrm{s}$, $P_{\mathrm{ts}}=50\mathrm{mmHg}$, $q_0=6\mu \mathrm{M}/\mathrm{s}$, and $l_{\mathrm{z}}=100\mu \mathrm{m}$. The minimum $g$ value of $0.4\mu \mathrm{M}/\mathrm{s}$ is calculated for $R_{\mathrm{c}}=2.5\mu \mathrm{m}$, $R_{\mathrm{t}}=60\mu \mathrm{m}$, $v_{\mathrm{z}}=50\mu \mathrm{m}/\mathrm{s}$, $P_{\mathrm{ts}}=50\mathrm{mmHg}$, $q_0=0.9\mu \mathrm{M}/\mathrm{s}$, and $l_{\mathrm{z}}=400\mu \mathrm{m}$. Table 5 shows the results of Eqs. (20) and (21) for the physiological parameters listed in Table 3. The $g$ values are presented with the standard deviations obtained from the actual FEM results and the calculated values from Eqs. (20) and (21). The relative errors were also measured as the percentages of the deviations divided by the FEM calculated $g$ values. The maximum error of 27% occurs for the blood vessel with $P_{\mathrm{ts}}=100\mathrm{mmHg}$, $q_0=0.9\mu \mathrm{M}/\mathrm{s}$, $v_{\mathrm{z}}=200\mu \mathrm{m}/\mathrm{s}$, $l_{\mathrm{z}}=220\mu \mathrm{m}$, $R_{\mathrm{c}}=2.5\mu \mathrm{m}$, and $R_{\mathrm{t}}=18\mu \mathrm{m}$; the minimum error of 0.03% occurs for the blood vessel with $P_{\mathrm{ts}}=100\mathrm{mmHg}$, $q_0=6\mu \mathrm{M}/\mathrm{s}$, $v_{\mathrm{z}}=100\mu \mathrm{m}/\mathrm{s}$, $l_{\mathrm{z}}=220\mu \mathrm{m}$, $R_{\mathrm{c}}=10\mu \mathrm{m}$ and $R_{\mathrm{t}}=18\mu \mathrm{m}$.

Table 5

The calculated g values using Eqs. (20) and (21) for the same microenvironment ranges examined for the actual g obtained by FEM simulation. The values are presented with the standard deviations obtained from the actual FEM results and the calculated values from Eqs. (20) and (21). The maximum relative (standard) deviation of the fits is 12.82%.