Various mathematical approaches have been devised to relate the cytotoxic effect of drugs in cell culture to the drug concentration added to the cell culture medium. Such approaches can satisfactorily account for drug response when the drugs are free in solution, but the approach becomes problematic when the drug is delivered in a drug delivery system, such as a liposome. To address this problem, we have developed a simple model that assumes that the cytotoxic potency of a drug is a function of the intracellular drug level in a critical compartment. Upon exposure to drug, cell death commences after a lag time, and the cell kill rate is dependent on the amount of drug in the critical intracellular compartment. The computed number of cells in culture, at any time after exposure to the drug, takes into account the cell proliferation rate, the cell kill rate, the average intracellular drug concentration, and a lag time for cell killing. We have applied this model to compare the cytotoxic effect of doxorubicin (DOX), or DOX encapsulated in a liposome that is targeted to CD44 on B16F10 melanoma cells in culture. CD44 is the surface receptor that binds to hyaluronan and is overexpressed on various cancer cells, including B16F10. We have shown previously that the drug encapsulated in hyaluronan-targeted liposomes was more potent than was the free drug. The model required the determination of the cell-associated DOX after the cells were incubated with various concentrations of the free or the encapsulated drug for 3 h, and the quantification of cell number at various times after exposure to the drug. The uptake of encapsulated drug was greater than that of the free drug, and the ratio of cell association of encapsulated:free drug was 1.3 at 0.5 μg/ml and increased to 3.3 at 20 μg/ml DOX. The results demonstrate that the enhanced potency of the encapsulated drug could stem from its enhanced uptake. However, in certain cases, where larger amounts of the free drug were added, such that the intracellular amounts of drug exceeded those obtained from the encapsulated drug, the numbers of viable cells were still significantly smaller for the encapsulated drug. This finding demonstrates that for given amounts of intracellular DOX, the encapsulated form was more efficient in killing B16F10 cells than the free drug. The outcome was expressed in the kinetic model as a 5–6-fold larger rate constant of cell killing potency for the encapsulated drug versus the free drug. The model provides a quantitative framework for comparing the cytotoxic effect in cultured cells when applying the drug in the free form or in a delivery system.

An unresolved challenge in experimental therapeutics is how to quantitatively compare alternative drug administration protocols or drug delivery strategies for cytotoxic agents. Generally applicable pharmacokinetic (PK)/pharmacodynamic (PD) approaches for cytotoxic drugs have proven difficult to develop because the toxic effect often occurs considerably after the exposure of the cell to the drug. The drug may initiate a cascade of events that combine to kill the cell; simple mathematical models do not easily describe the resulting complexity. Although the PK properties of the drug may be well defined, the best way to couple the PK to the PD effect remains elusive.

One way to connect the in vivo drug concentration to an in vivo cytotoxic effect is to determine the cytotoxic effects of the drug in cell culture and assume the PDs are similar in vivo. A widely accepted assumption is that the cell kill rate is a function of the product of the concentration of the drug in the cell culture medium × the exposure time (CxT; Refs. 1, 2). This approach provides one way to compare the potency of cytotoxic agents in cell culture that goes beyond the simple measurement of an ED50 during continuous drug exposure (2). A modification to the classical approach uses a Hill sigmoidal function (3, 4). More recently, transit compartment models have been applied to drugs of which the effects occur after a time delay (5). In the case of cytotoxic drugs, Lobo and Balthasar (6) deduced that a transit compartment model gave the best simulation for the cytotoxic effect of methotrexate in vitro.

In such PD models, the assumption that the drug concentration in the medium is related to the effect works well for drugs that are added in the free form to the cell culture medium but is inadequate when the drug is added in a delivery system. This is because it becomes difficult to know if the alteration in effect is due to a change in the amount of drug that enters the cell or to a change in the apparent potency of the drug due to the delivery method.

We have experienced this difficulty in evaluating the cause of the increased potency when doxorubicin (DOX) is delivered in a liposome that is targeted to the CD44 receptors on melanoma cells (7). DOX is a widely used anticancer drug of which the cytotoxicity has been well studied in cell culture using the CxT protocol (2, 4, 8), which satisfactorily described the cytotoxic effects of DOX. In the earliest of these studies, the cell-associated levels of DOX at various exposure concentrations and times were measured (8). The DOX concentration in the medium gave a better fit to the CxT model for cell killing than did DOX concentration in the cells (8).

There have been few attempts to apply PD analyses to liposome-encapsulated drugs (9, 10, 11). In most cases, the liposome-encapsulated drug was much less potent in cell culture than the free drug. This is because only a small fraction of the liposomes were internalized into cells, and the encapsulated drug in the medium did not have an opportunity to interact with the cells. In effect, encapsulation was reported to decrease the in vitro cytotoxic effect (9). The therapeutic benefit of nontargeted liposome drug delivery only becomes apparent in animals (9). Therefore, PKPD modeling of the antitumor effect of sterically stabilized liposomal DOX has been most advantageously applied in mice (11, 12, 13). In these studies a cell kill kinetic model was used to determine the antitumor effect. The authors assumed that the cytotoxic effect was due to DOX that had been released from the liposome, because sterically stabilized liposomes are not avidly internalized into tumor cells (11, 12, 13). This early work of Harashima and coworkers (11, 12, 13) is an important step in the development of a PKPD model for targeted liposomes. This is a field that is growing in importance because of an increased availability of targeting ligands for cell surface receptors (10). Such ligands can be attached to drug carriers such as liposomes to improve the selective toxicity of anticancer therapeutics (9, 10).

Several investigators have studied the intracellular fate, and PKs of free drug and drug administered in ligand-targeted systems (14, 15, 16, 17, 18). However, there is no framework to compare the results or to estimate how such systems will behave in vivo(14, 15, 16, 17, 18), particularly because the cellular membrane permeability of anthracyclines (DOX among them) does not correlate with their delivery in a tissue-isolated tumor (19).

In this article we present a modified starting point in the analysis of cell killing by DOX delivered in a targeted liposome. We assume that the rate of cell killing is dependent on the concentration (or amount) of drug within the cell, which can be determined experimentally either for liposomal or free drug (8). The emphasis in the model is on testing the ability of a semiempirical equation, which uses a minimal number of parameters to yield simulations and predictions for the kinetics of cell viability as a function of intracellular drug. This approach also enables one to compare the efficiency of cell killing by the mode of entry of the drug, i.e., free versus loaded in liposomes, irrespective of the rate of its entry into the cells.

Chemicals.

All of the phospholipids were purchased from Avanti Polar Lipid (Birmingham, AL) or synthesized in our laboratory. Cholesterol, sulforhodamine B (SRB), trichloroacetic acid, Bee venom, human umbilical cord hyaluronic acid, and sodium cyanoborohydride, were purchased from Sigma Chemical Co. (St. Louis, MO). Cholesterol was recrystallized from methanol. Dowex 50WX4 resin was purchased from Aldrich (Milwaukee, WI). Doxorubicin-HCl (DOX) was purchased from Bedford Laboratories (Bedford, OH). Culture medium (MEM Eagle’s with Earle’s balanced salt solution) was obtained from University of California San Francisco Cell Culture Facility. All of the other reagents were of analytical grade.

Cell Culture Conditions.

B16F10 murine melanoma cell line was obtained from University of California San Francisco Cell Culture Facility. B16F10 cells were maintained in MEM Eagle’s with Earle’s balanced salt solution medium containing 10% fetal bovine serum, 1:100 MEM nonessential amino acids, 1:100 sodium pyruvate (11 mg/ml), and 1:100 penicillin-streptomycin (0.1 μm sterile filtered). Cells were cultured with complete medium at 37°C in a humidified atmosphere of 5% CO2 in air. For all of the experiments, cells were harvested from subconfluent cultures using trypsin and were resuspended in fresh complete medium before plating. Cells with >90% viability, as determined by trypan blue exclusion, were used.

Ligand Preparation.

The preparation of hyaluronan oligosaccharides and their covalent attachment to phosphatidylethanolamine has been described previously (7).

Liposome Preparation.

Lipid films were prepared by drying 10 μmol of lipid including POPE-hyaluronan from solvent (butanol saturated with distilled water or chloroform:methanol 7:3, v/v, respectively) under vacuum using a rotary evaporator at room temperature. Liposomes (composed of POPC:Cholesterol:HAn-POPE 60:40:3) were prepared as described previously (7). Liposomes were stored at 4°C under argon and used within 1 day of preparation. The hydrodynamic diameter of the liposomes was determined by dynamic light scattering (Malvern Instruments, Southborough, MA). The net surface potential was determined with a Malvern Zetasizer IV (Malvern Instruments). The ζ potential of liposomes containing 3 mol% HAn-PE was -9.9 mV.

Liposome Uptake Assay.

Cells (2 × 105) were placed in each well in a 24-well plate and grown overnight at 37°C and 5% CO2 in medium. The cell monolayer was rinsed with FCS-free medium, and medium-containing liposomes was added. Liposomes containing trace amounts, ∼0.01 mol% of 125I-p-hydroxy-benzamidine dihexadecylphosphatidylethanolamine (20), were diluted in serum-free, antibiotic-free medium and incubated with cells for 3 h at 4°C or 37°C. At the end of the incubation, the medium was removed, and the cells were washed with three successive aliquots of 0.5 ml ice-cold PBS. The medium and washes were pooled and assayed for radioactivity. The cells were lysed and removed from the well with 1 ml of 0.5 n NaOH. The well was then washed two additional times with 1-ml PBS aliquots, and the cell lysate and washes were pooled. Radioactivity associated with the cell lysate and washes was determined in a Beckman γ scintillation spectrometer. Each time course experiment was repeated two independent times with triplicate wells of cells (n = 6).

Preparation of DOX-Loaded Liposomes.

Liposomes were prepared by thin lipid film hydration followed by sonication and extrusion as described (7). DOX was encapsulated using the procedure of Bolotin et al.(21), and encapsulation efficiency was usually >90%, with drug:phospholipid ratio of ∼100 μg/μmol. Mean vesicle diameter as measured by dynamic light scattering using the multimodal program was 110–140 nm (SD <35% of the mean) with a monodisperse particle size distribution.

Chemosensitivity Assay.

The cytotoxic effect of free DOX or liposome-encapsulated DOX on the cells was assayed colorimetrically by the SRB staining method (22), as described previously (7). Samples containing 16,000 B16F10 cells (from an exponentially growing culture) in 100-μl aliquots were plated onto 96-well flat-bottomed microtiter plates. The culture plates were incubated for 24 h at 37°C and 5% CO2, and then the medium in each well was replaced with 100 μl serum-free and antibiotic-free medium containing various concentrations of free or liposome-encapsulated DOX. For each 10-fold increase in drug concentration, four drug concentration levels were tested. Each test was performed in triplicate wells and was repeated in an independent experiment at least once. The cells were incubated for 3 h (transient protocol) at 37°C and 5% CO2. The drug was removed at 3 h, complete medium lacking drug was added, and the incubation was continued for 24 h at 37°C and 5% CO2. At the end of the incubation period, the cells were washed once with complete (growth) medium, and 100 μl of complete drug-free medium was added to each well. The cultures were fixed by gently layering 25 μl of ice-cold 50% trichloroacetic acid (4°C) on top of the growth medium in each well to produce a final trichloroacetic acid concentration of 10%. The cultures were incubated at 4°C for 1 h, and then washed and analyzed for SRB staining of the monolayers as described (22). The measurement of the absorbance of the SRB at 564 nm in the monolayers was determined using an Optimax microplate reader (Molecular Devices, Sunnyvale, CA). Each experiment was repeated twice in triplicate (n = 6). We assume cell number is proportional to the SRB staining level (3, 22) 

Immediate and Delayed Overall Effects.

To determine whether the higher activity observed for the longer treatments (continuous protocol; Ref. 7) is due to a delayed exhibition of drug effects and/or a reflection of cumulative effects that require a continuous drug exposure, cells were treated with free DOX or DOX encapsulated in hyaluronan-targeted liposomes (HAL; HAL-DOX) for 3–96 h and then either: (a) immediately processed for drug effect measurement (immediate effect); or (b) washed, incubated in drug-free medium, and processed for drug effect measurements at 96 h (delayed effect). After seeding, the cells were incubated with complete growth medium for 24 h. At this time, the medium was removed, and the cells were incubated with 100 μl serum-free and antibiotic-free medium containing 0.005 to 100 μg/ml of free or liposome-encapsulated DOX, for seven treatment durations ranging from 3 to 96 h as described by Au et al.(3) for free paclitaxel. There were three replicates for each concentration per plate. For the delayed effect, DOX-containing medium or HAL-DOX-containing medium was removed at the end of treatment, and the culture plates were rinsed three times with 200 μl of serum-free growth medium. Afterward, cells were incubated with 100 μl of drug-free medium until 96 h. The cell number was measured using the SRB assay as described above. The immediate effect was determined immediately after drug treatment. The delayed effect was determined at 96 h, irrespective of treatment duration. The SRB absorbance is proportional to the number of cells attached to the culture plates (3, 22). Therefore, these results represent the overall drug effect, i.e., the combination of cytostatic and apoptotic effects.

DOX Uptake Assay.

We determined the uptake of the free DOX into the cells as follows: 2.5 × 105 cells/well were seeded in six-well plate. The culture plates were incubated for 24 h at 37°C and 5% CO2, and then the medium in each well was replaced with 2 ml of phenol red-free, serum-free, and antibiotic-free medium containing various concentrations of free DOX. The cells were incubated for 3 h, then the medium was removed and assayed for free drug, by adding the 2–6 ml acidified isopropanol 90% (0.075 m HCl). The cells were washed three times with 2 ml of cold PBS. Then the cells were lysed by adding 0.5 ml of double distilled water and rotated on orbital shaker for 5 min at room temperature. Then the lysed cells were added to 1.5 ml of acidified isopropanol, and incubated at 4°C in the dark for 3 h. Total DOX content was then measured fluorometrically using a Perkin-Elmer LS-50-B spectrofluorometer (excitation: 480 nm; emission: 590 nm; Ref. 23). Fluorescence intensity was translated to DOX concentration, using a standard curve prepared from DOX solutions in cell lysates that were not exposed to the drug previously. Mass balance calculations based on the amount of drug in the cells and the amount of drug recovered in the medium confirmed the accuracy of the method. Results are the mean ± SD of at least three replicates for each experiment.

Efflux of DOX from treated cells was determined by measuring DOX released into the medium at 12 h after incubation of treated cells with drug-free medium. For efflux calculations, the contribution of DOX released from dead cells was subtracted from the DOX released into the medium. In all of the cases, a mass balance analysis of DOX added to the medium, released into the medium during the efflux period, and extracted from the cells was performed and confirmed that 95% ± 5% of the drug could be accounted for. Because DOX and several metabolic products of DOX are also fluorescent the values we measure are designated DOX equivalent concentrations.

Analysis of Cell Killing by DOX.

The main goal was to find a simple functional form, which could be ultimately applied in the PK analysis of in vivo results. The general form proposed for model 1 is:

dN/dt = {ks − k·f [Cin (t)]}·N and we used the simplified version

\[(\mathrm{dN/N})/\mathrm{dt}\ {=}\ {-}\mathrm{k{\cdot}Cin}\ (\mathrm{t})^{\mathrm{n}}\ {+}\ \mathrm{k}_{\mathrm{s}}\]

In Eq. A, N is the number of cells, ks is the rate constant of cell proliferation, k is the rate constant of cell kill, f is a function of the average concentration of drug in the cell, and Cin (t) is the average concentration of drug inside the cell at time t. Thus, f is assumed to be a function of Cin (t). The value n is an adjustable parameter related to the intensity of the cytotoxic effect of the drug. The value of ks was determined in the absence of added drug. A starting point in our approach has been the explicit consideration of drug concentration (or amount) inside the cell. This quantity was obtained by measuring the uptake of free drug or the uptake of liposomes (7) into cells at several times up to 3 h.

We considered variations in n, but a sufficiently adequate description of the kinetics of cell viability could be achieved by setting n = 1, i.e., a linear effect. Thus, Eq. A uses a single parameter, the rate constant of cell killing. However, because the experimental results demonstrated a lag time in cell killing, we included in the analysis an option for another parameter, tl, time lag. The lag time was assumed to be independent of drug concentration. Accordingly, the program defines a time, T, T = t − tl, and cell killing starts from T > 0.

For t < tl Eq. A becomes

\[(\mathrm{dN/N})\ \mathrm{dt}\ {=}\ \mathrm{k}_{\mathrm{s}}\]

which gives an exponential growth.

We used in these calculations Cin (t) as an amount of intracellular drug rather than concentration, because the average amount of drug uptake per cell was known, whereas a transformation to concentration required knowledge of cellular volume. Accordingly, the unit of k is (mol*sec)−1, (for n = 1), whereas if concentration is used, the unit is (M* Sec)−1.

The numerical solution of Eq. A was as follows. In the current program the time t is divided into units of minutes. After 3 h the free or liposomal drug is removed from the medium. Hence, Cin (t), or Cin (T) becomes independent of time. We ignore efflux of the drug from living cells, in accord with our direct measurements of efflux. During the first 3 h the dependence of Cin (t) on time is assumed to be known at certain time points and is interpolated between two consecutive time points. For T > 0 no time dependence of Cin exists in our experimental procedure, but even if it did exist, the right side of Eq. A could be assumed constant within a narrow time interval, e.g., 1 min. The numerical solution of Eq. A is:

\[\mathrm{Ln}\ {[}\mathrm{N}\ (\mathrm{T}\ {+}\ \mathrm{h})/\mathrm{N}\ (\mathrm{T}){]}\ {=}\ {-}\mathrm{k{\cdot}Cin}\ (\mathrm{T})^{\mathrm{n}}\mathrm{{\cdot}h\ {+}\ h{\cdot}k}_{\mathrm{s}},\]

in which h is a time increment (e.g., 1-min). In a matrix notation, let V = Ln [N(T)],

\[\mathrm{V}\ (\mathrm{I}\ {+}\ 1)\ {=}\ \mathrm{V}\ (\mathrm{I})\ {-}\ \mathrm{k}{\cdot}\mathrm{Cin}\ (\mathrm{I})^{\mathrm{n}}\mathrm{{\cdot}h}{+}\mathrm{h}{\cdot}\mathrm{k}_{\mathrm{s}}\]

At the beginning of the experiment V = Ln (100). The vector VCAL (I) = exp [V (I)] gives the percentage of cell viability at time I. In the program V is represented by a tridimensional matrix, in which one index is the time, a second index corresponds to a given final Cin (and corresponding interim values), and the third index allows for a distribution of cells containing different amounts (rather than average values) of the uptaken drug. An example of such a distribution was demonstrated in Nir et al.(24) for the uptake of liposomes by cells. In the current simulation of cell viability we avoided an explicit consideration of cell distributions. The program (in Fortran IV; can be obtained from S. N.) was run on both VAX UNIX (University of California, San Francisco, CA) and VMS (Hebrew University, Rehovot, Israel).

The determination of lag time is explained in “Results.” The determination of the parameter k is as follows. The program generates calculations corresponding to input k values. An overestimation of k yields underestimates in the percentages of viable cells and vice versa. A gross fitting is obtained after a few steps. Hence, finding an initial guess is not important. However, it is clear from Eq. A, e.g., for n = 1, that a reduction in the number of cells (relative to the initial number) for a certain drug uptake requires that k> ks/Cint, and vice versa, if no such reduction occurs. The user can choose whether to select the k value that gives the best fit in terms of Root Mean Square Error (RMSE) and R2, or to emphasize the good fit to certain points.

We also considered calculations according to an “Emax phase nonspecific model” (6),

\[(\mathrm{dN/N})\ \mathrm{dt}\ {=}\ \mathrm{k}_{\mathrm{s}}\ {-}\ \mathrm{K}\]

in which

\[\mathrm{K}\ {=}\ \mathrm{kmax\ C}/(\mathrm{EC}_{50}\ {+}\ \mathrm{C})\]

Here, kmax has a unit of min−1; C and the Michaelis constant EC50 have a unit of concentration. In the application of Eqs. E and F we used the same numerical procedure as described for Eqs.(A–D). We considered C in Eq. F as the drug concentration in the medium (6), but we also added calculations in which C = Cin. The calculations were performed with or without explicit lag time.

The results in Table 1 give the amount of DOX equivalent accumulated per single cell after 3 h of incubation with free drug or with targeted liposomes loaded with the drug. In the case of free drug, the results were obtained by a direct measurement and satisfied the mass balance for total DOX. In the case of cell-associated liposomal drug, the results were computed from the amount of cell-associated liposomes (7) and the amount of DOX per liposome. Because leakage of DOX was <10% during the 3 h incubation period (7), for the computation we assumed that there was no leakage of DOX from the liposomes. Our results also neglect DOX leakage from noninternalized liposomes during the incubation period. The results in Table 1 demonstrate that the uptake of the free drug is approximately proportional to its added values for the lower drug concentrations, 0.5 to 5 μg/ml, but not for the larger concentrations, where cell killing is more pronounced.

During the 3-h incubation, the uptake of encapsulated DOX exceeded that of the free drug, the excess being more extensive for 20 μg/ml DOX, (a factor of 3.3), whereas at 0.5 μg/ml DOX the ratio was reduced (a factor of 1.35). Furthermore, when added to the cells in the free form, once DOX had become cell-associated, there was no efflux of DOX (within experimental error of 10%) from the cells over the subsequent 12 h period.

We considered two types of concentration-effect relationships to describe the effect of DOX on cells. The first analysis considered the immediate effect, and the second analysis considered the delayed effects. For the remainder of this report, immediate effect refers to the drug effect that was measured immediately after termination of treatment. Delayed effect refers to the drug effect that was measured after an additional growth period after drug removal from the medium. The delayed effects were measured as follows. The cells were exposed to drug for 3 h, the drug-containing medium was removed, and the cells were permitted to grow for the additional periods of 6, 12, 24, 48, 72, and 96 h. This measurement reveals the kinetics of the delayed drug effect. Note that the immediate effect and the delayed effect at 96 h are the same for the 96-h treatment, because the effect was measured only once at 96 h.

The left panels of Fig. 1 show the immediate overall effect of free DOX (top panel) and HAL-DOX (bottom panel) treatments of 3–96 h. For both cases, the treatment produced sigmoidal concentration-effect relationship; drug effect increased with increasing drug concentration and increasing treatment time for free DOX, as well as for HAL-DOX. Prolonging the treatment duration significantly increased the maximum effect (cytotoxicity) and decreased the IC50.

The right panels of Fig. 1 show the delayed overall effects measured at 96 h. Similar to the immediate effects, increasing the treatment duration enhanced the delayed effect. The delayed drug effects were significantly greater than the immediate effects, as indicated by the steeper concentration-effect relationships, higher maximum effect, and lower IC50 values associated with the delayed effect.

The main observation from Fig. 1 is that HAL-DOX is much more potent than the free drug in cell killing, in both immediate and delayed effects. A comparison of the immediate effects for the 3-, 6-, 12-, 24-, 48-, and 72-h treatments, and their delayed effects measured at 96 h shows significantly higher effect for treatment ranging from 3 h to 48 h for free DOX, and only from 3 h to 12 h for HAL-DOX. A comparison of the delayed effects resulting from different treatment durations shows the IC50s for treatments of ≥24 h were indistinguishable for free DOX from the 96 h treatment, whereas for HAL-DOX they were indistinguishable already for treatments ≥12 h. This suggests that the DOX added in targeted liposomes reached a critical toxic site in the cell more rapidly than did the free drug regardless of the drug concentration added to the cell.

The applicability of the program based on Eq. A for simulating cell viability results when DOX was added to the medium as a free drug can be viewed in Table 2 and Fig. 2. The calculations used values for the uptake of free drug listed in Table 1. The percentages of viable cells are given relative to initial cell number.

The introduction of lag time in the equation was necessitated by the experimental results presented here (Tables 2 and 4; Fig. 1), and in Eliaz and Szoka (7), which indicated no cell killing by DOX at times of 3 h even for very large drug concentrations (20 and 100 μg/ml), and very little cytotoxic effect at 6 h. An absence of drug effect after 3 h was noted before for several cases (3). In our case, the lag time in cell killing by DOX was not due to the low accumulation of the drug in the cells. In the liposomal case ignoring the lag time in the calculations resulted in a very large percentage of cell killing at 3 h, in contrast with the typical experimental values of cell numbers, which were ∼108% of the initial number. Furthermore, the number of viable cells at 3 h was the same when free drug (Table 2) or liposomal drug (Table 4) was administered, despite 3.3-fold more uptake in the liposomal case for 20 μg/ml DOX (Table 1). Consequently, the employment of a lag time in the calculations reflects a straightforward and important part of the simulation procedure. To reduce the number of adjustable parameters we fixed the lag time at 3 h, which was the minimal lag time required for obtaining agreement at 3 h between calculated and experimental values for the 20 μg/ml case, but it cannot be ruled out that the lag time might be somewhat larger, albeit smaller than 6 h. The value of lag time may be more important when focusing on results at 6, 9, and 12 h, but the effect of lag time is minimal at 27 h and at 96 h. The important fact is that a constant value of 3 h is applicable for the lag time for both free and encapsulated drug.

The results in Table 2 show that even at the highest concentration of free drug (20 μg/ml) the percentage of viable cells at 27 h after application of the drug is still rather large, 69.9% of the initial, whereas for 1 μg/ml the number of cells is 147.9% of the initial. This value above 100% is due to natural rate of proliferation of the cells, which was determined as 4.62*10−4 min −1.

Table 3 presents the results of cell viability at 96 h from the moment of application of the drug as a percentage of control. The percentages of viable cells in this representation are still rather high, and because the cells were from an exponentially growing culture, there was in fact little reduction in total number of cells treated by the free drug until a drug concentration of 0.25 μg/ml. In Table 3, we also illustrate the predictions (or extrapolation) of the model for viable cell numbers relative to the control at 96 h. The function f (Cin) used in Eq. A was simply linear, and the fit is quite good with R2 = 0.95 and RMSE = 8.6.

The results in Table 4 again demonstrate that DOX encapsulated in liposomes was significantly more potent in killing B16F10 murine cancer cells. Here the observed percentages of viable cells relative to the initial number are 17.5-, 14.2-, and 7-fold smaller than the corresponding values for the free drug on addition of 20, 5, and 1 μg/ml DOX, respectively (Table 2). The results from the simulation for the case where targeted liposomes were used are presented in Table 4. In this case, the best fit was obtained for a quadratic relation in model 1 Eq. A, n = 2, which gave R2 = 0.98 and RMSE = 7.6. The fit is also acceptable (R2 = 0.94; RMSE = 14) for the linear dependence; the rate constant of cell killing for DOX delivered in targeted liposomes was ∼6-fold larger than in the case of free DOX. The results in Fig. 2,B illustrate a fit to the data (Table 2) with a CxT protocol, but with a lag time of 3 h. This calculation gave R2 = 0.744 and RMSE = 22, which is significantly inferior to the fit shown by model 1 in Table 2. Ignoring the time lag worsened the fit, giving R2 = 0.713 and RMSE = 23.3.The predictions of the CxT analysis for larger incubation times are unacceptable; even for 6 h this model predicts complete cell death after an additional 21 h for most of the concentrations used, in contrast with the results in Fig. 1. A graphical presentation of the fits between calculated and experimental values of viable cells is given in Fig. 2, which also demonstrates the sensitivity of calculated values to variations in k-values for free drug (Fig. 2,A) and liposomal drug (Fig. 2,C). The sensitivity of the various models is also presented in Table 5.

The results of simulations and predictions obtained by using the Emax phase nonspecific model (Eqs. E and F) are presented in Tables 2,3,4 and Fig. 2, B and D. We only show in these tables results of calculations, which do not use a lag time and use the media concentrations as seen elsewhere (6). A summary of the performance of the various computations is given in Table 6. This Emax phase nonspecific model gives a better fit to the results of cell killing by the free drug at 27 h (Table 2) than the model based on Eq. A, but the predictions of the model 1 (Eq. A) for 96 h are better (Table 3). The phase nonspecific model based on Eqs. E and F is significantly inferior to that based on Eq. A for the liposomal drug (Tables 4,5,6).

It may be of interest to convert the rate constants of cell killing to units of (M*Min)−1. If we estimate the volume of a single B16F10 cell by 104 (μm)3, then the conversion to mol/liter introduces a factor of 10−11. Hence, the values of k are 1 and 6 (M*Min)−1, for the free and liposomal drug, respectively, whereas according to the phase nonspecific model the corresponding values of kmax are 0.001 and 0.005 min−1.

In summary, these results indicate a significantly higher efficiency of DOX in killing cancer cells when applied via targeted liposomes for the same amount of intracellular drug. Thus the improved cytotoxic effect from the targeted liposomes compared with the free DOX after a 3-h incubation is because more DOX is internalized by the B16F10 cells when added via targeted liposomes and the targeted DOX is more effective.

We observed a pronounced improvement in the cytotoxic effect of DOX when delivered in liposomes targeted to CD44-overexpressed cells compared with the free drug (7). A classical CxT analysis (1, 2, 4, 8) was incapable of satisfactorily describing the results, so we devised a simple alternative approach to analyze the data that takes into account the amount of drug in the cell. This approach is conceptually akin to the modification that Ozawa et al.(25) used wherein they corrected for the degradation of the drug in the medium before applying the CxT analysis. In that case, the cells were actually exposed to less drug; in our case there is, in effect, more drug available in the cell.

In the analysis presented here, Eq. A is similar in form to that used by Harashima and coworkers (11, 12, 13). However, they consider the concentration of the free drug outside the cancer cells and ignore the process of uptake of liposomes carrying the drug by cancer cells. By evaluating cell killing as a function of the concentration of intracellular drug, we can focus on the effect of targeting, rather than introduce from the onset the added complexity of kinetics of uptake of the drug by the cells, which is a related, but a separate undertaking. We were able to compute the amount of cell-associated DOX because the kinetics of liposome uptake has been well described by a model, which considers binding (including dissociation) followed by endocytosis (24, 26). This model has also been able to simulate the uptake of HAL liposomes by B16F10 cells(27), where most of the uptake is temperature sensitive and appears to be via endocytosis (7). During this uptake we assume there is little loss of DOX from the liposome. It should be noted that the analysis of liposome uptake was limited to 3 h. The uptake of free drug (Table 1) as has been noted previously (8, 28) appears to obey a solubility-diffusion mechanism for the smaller added concentrations, but the situation is certainly more complex for the larger concentrations. Furthermore, at longer incubation times (>6 h) the uptake of free DOX was reduced when added at 10 μg/ml, due to the effect of the drug on the cells.

The model based on Eq. A might be considered an oversimplification. However, we have shown its capability to yield simulations and predictions for cell viability as a function of time and drug concentration (Tables 2,3,4; Fig. 2). We illustrate this outcome in a different type of presentation by comparing cell numbers (% of initial) in Tables 2 and 4. After a 3-h treatment and an additional 24-h incubation, at 5 μg/ml DOX the numbers of viable cells (% of initial) were 89.8% and 6.3% for free and liposomal drug, respectively. The targeted liposomes deliver more DOX into the cell than does the free form. Thus, the targeted liposomes have an advantage due to a more efficient uptake of DOX than that of the free drug. However, the results also point out an additional dramatic advantage of the targeted liposomes. Table 1 shows that after 3 h, the amount of DOX per cell is >3-fold larger when added as a free drug at 5 μg/ml than as liposomal drug at 1 μg/ml. Yet, after 27 h the corresponding percentages of viable cells (relative to initial) were 89.8% (free drug, Table 2) and 21.1% (liposomal drug; Table 4). Hence, the drug loaded via targeted liposomes is significantly more efficient in killing the cells per amount of intracellular drug. Model 1 (Eq. A) expresses this efficiency in terms of a 6-fold larger rate constant of cell killing via liposomal DOX, whereas a factor of 5 is obtained for kmax in the application of Eqs. E and F.

The introduction of a lag time into the model was necessary to account for the delay in the cytotoxicity at 3 h either with free DOX or with targeted liposomes, even for the largest loads. Hence, the results dictate that the explicit consideration of lag time in Eq. A is essential. The origin of this lag period has not yet been elucidated and could reflect the duration of a certain crucial process that culminates in cell killing. The assumption that the lag time is independent of drug concentration is the simplest one that could be used, but its test requires a more complete set of experimental results at times ranging from 3 to 12 h, because the results at 27 and 96 h are not sufficiently sensitive to the values of lag times in the range of 3–6 h. More complicated PD schemes that incorporate a transit compartment model to account for delays have been suggested for those cases where the measured initiation of drug effect is substantially displaced from drug exposure, such as in the case of methotrexate cytotoxicity (6). Such an approach is perhaps more refined, but it adds complexity and does not appear to us to provide more information to assess targeted DOX activity studied here.

The enhanced potency mediated by the HAL-DOX at longer incubation times is unusual for targeted DOX preparations. For instance, DOX targeted on an antibody was less effective than the free drug (29) as was DOX targeted in various immunoliposomes (14, 30, 31). Moreover, DOX targeted by attachment to an 11.8 kDa hyaluronan was less active than free DOX (32). Other methods of targeting liposomal DOX, such as the use of folate, have generated variable results; one group reported an increased potency for DOX delivered in a folate liposome (33, 34), whereas another group did not observe an increase in comparison with free DOX (35).

Early work suggested that endocytosis does not augment the cytotoxicity of DOX when delivered in immunoliposomes (17). Indeed, DOX delivered to cells in antibody-targeted liposomes has been observed to have a different distribution between cellular compartments than does the free drug (14). This group also found a significant correlation between the rate of nuclear accumulation of DOX and its in vitro cytotoxicity. The free drug entered the nucleus more rapidly than did the encapsulated drug (15). Thus, it would be of interest to relate the efficiency of HAL-DOX to the cellular site where the drug is deposited. One possible reason for the enhanced potency of HAL-DOX is that the DOX reaches a critical compartment more efficiently when it is delivered as the HAL-DOX. Targeting DOX via a hyaluronan ligand may result in uptake of the delivery system via a nonclathrin-coated endosome as has been reported to occur in the case of hyaluronan catabolism (36). The identification of this critical compartment will need additional investigation but there are three possibilities: (a) HAL-DOX traffics faster to the nucleus (15); (b) HAL-DOX traffics faster to the mitochondrial membrane and induces collapse of the mitochondrial transmembrane potential leading to apoptotic signals (37); and (c) the cytotoxicity of HAL-DOX is due not only to its nuclear toxicity but to a cytocidal effect directed against cell membranes, as suggested for polymeric DOX systems (18, 38). To differentiate among these possibilities requires a thorough comparison of the time course of the subcellular distribution of DOX delivered by the two routes (liposomal and free) as has been undertaken for the immunoliposomes (14, 15).

Grant support: Cancer Research Coordinating Committee Award No. 2-519850 (R. E.), and NIH GM61851 (F. S.).

Notes: Present address for Rom E. Eliaz, Alza Corp., 1501 California Avenue, Palo Alto, CA 94304; Present address for Shlomo Nir, Seagram Center for Soil and Water Sciences, Faculty of Agricultural, Food and Environment Quality Sciences, Hebrew University, Rehovot 76100, Israel; Contact Shlomo Nir with questions concerning the model; Present address for Cornelia Marty Paul Scherrer Institute, Molecular Cell Biology Institute, 5232 Villigen-PSI, Switzerland.

The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Requests for reprints: Francis Szoka, Departments of Biopharmaceutical Sciences and Pharmaceutical Chemistry, University of California, San Francisco, CA 94143-0446. Phone: (415) 476-3895; Fax: (415) 476-0688; E-mail: [email protected]

Fig. 1.

Immediate and delayed cytotoxic effects of doxorubicin (DOX) exposure as the free drug or encapsulated in HAL liposomes. Cells were incubated with free DOX or DOX encapsulated in hyaluronan-targeted liposomes (HAL-DOX) for 3–96 h. The immediate effect was determined immediately after drug treatment. The delayed effect was determined at 96 h, irrespective of treatment durations. For example, the delayed effect of the 3-h treatment was measured with a 93-h delay. ▪, 3 h; ○, 6 h; ▴, 12 h; ⋄, 24 h; •, 48 h; ▵, 72 h; ♦, 96 h. Results are means; bars, ±SD.

Fig. 1.

Immediate and delayed cytotoxic effects of doxorubicin (DOX) exposure as the free drug or encapsulated in HAL liposomes. Cells were incubated with free DOX or DOX encapsulated in hyaluronan-targeted liposomes (HAL-DOX) for 3–96 h. The immediate effect was determined immediately after drug treatment. The delayed effect was determined at 96 h, irrespective of treatment durations. For example, the delayed effect of the 3-h treatment was measured with a 93-h delay. ▪, 3 h; ○, 6 h; ▴, 12 h; ⋄, 24 h; •, 48 h; ▵, 72 h; ♦, 96 h. Results are means; bars, ±SD.

Close modal
Fig. 2.

Experimental and calculated effect on cell viability expressed as percentage of initial cell number at 27 h for a 3-h exposure. Cells were treated with free doxorubicin (DOX; A and B) or with encapsulated DOX (C and D) for 3 h and then processed for drug effect measurements at 27 h (delayed effect; 3 h treatment + 24 h incubation). A, ▪, E (experimental); ⋄, (R2 = 0.94, RMSE = 11) model 1 from Table 2. The sensitivity of the model to changes in k-values is illustrated by Δ (R2 = 0.6, RMSE = 25) and ○ (R2 = 0.68, RMSE = 25), corresponding, respectively, to calculations using k-values which are 2- and 0.5-fold of the values in model 1 from Table 2. B, ▪, E (experimental); the calculations ∗ (R2 = 0.744, RMSE = 22) are the best fits of a model which used the free drug rather than Cint values. □ (R2 = 0.98, RMSE = 5.4) calculations according to the phase nonspecific model Eqs. E and F using drug concentrations inside the cell and the time lag; parameter values: kmax = 0.0014 min−1, EC50 = 7.5 femtomole DOX/cell. C, ▪, E (experimental); ▵ (R2 = 0.94, RMSE = 14.2) model 1, n = 1, from Table 4. ⋄ (R2 = 0.72, RMSE = 30.6) and □ (R2 = 0.72, RMSE = 30.5) correspond, respectively, to calculations using k-values which are 2- and 0.5-fold of the values in model 1, n = 1, Table 4. D, ▪, E (experimental); ∗ (R2 = 0.98, RMSE = 7.6) model 1, n = 2, from Table 4; ○ (R2 = 0.83, RMSE = 23.7) phase nonspecific model from Table 4.

Fig. 2.

Experimental and calculated effect on cell viability expressed as percentage of initial cell number at 27 h for a 3-h exposure. Cells were treated with free doxorubicin (DOX; A and B) or with encapsulated DOX (C and D) for 3 h and then processed for drug effect measurements at 27 h (delayed effect; 3 h treatment + 24 h incubation). A, ▪, E (experimental); ⋄, (R2 = 0.94, RMSE = 11) model 1 from Table 2. The sensitivity of the model to changes in k-values is illustrated by Δ (R2 = 0.6, RMSE = 25) and ○ (R2 = 0.68, RMSE = 25), corresponding, respectively, to calculations using k-values which are 2- and 0.5-fold of the values in model 1 from Table 2. B, ▪, E (experimental); the calculations ∗ (R2 = 0.744, RMSE = 22) are the best fits of a model which used the free drug rather than Cint values. □ (R2 = 0.98, RMSE = 5.4) calculations according to the phase nonspecific model Eqs. E and F using drug concentrations inside the cell and the time lag; parameter values: kmax = 0.0014 min−1, EC50 = 7.5 femtomole DOX/cell. C, ▪, E (experimental); ▵ (R2 = 0.94, RMSE = 14.2) model 1, n = 1, from Table 4. ⋄ (R2 = 0.72, RMSE = 30.6) and □ (R2 = 0.72, RMSE = 30.5) correspond, respectively, to calculations using k-values which are 2- and 0.5-fold of the values in model 1, n = 1, Table 4. D, ▪, E (experimental); ∗ (R2 = 0.98, RMSE = 7.6) model 1, n = 2, from Table 4; ○ (R2 = 0.83, RMSE = 23.7) phase nonspecific model from Table 4.

Close modal
Table 1

Uptake of doxorubicin (DOX) into B16F10 cells after a 3-h exposure

Cells were treated with hyaluronan-targeted liposomes (HAL)-DOX or free DOX for 3 h and then the drug level in the cells was determined. Values presented are the mean of three experiments, six replicates per data point for each experiment. SD were <10% (n = 3).

DOX conc. (μg/ml) [μm]Femtomole DOX/cell
HAL-DOXFree DOX
20 [34.48] 32 9.6 
10 [17.24] 16 8.7 
5 [8.62] 5.7 
1 [1.72] 1.6 1.2 
0.5 [0.86] 0.8 0.59 
DOX conc. (μg/ml) [μm]Femtomole DOX/cell
HAL-DOXFree DOX
20 [34.48] 32 9.6 
10 [17.24] 16 8.7 
5 [8.62] 5.7 
1 [1.72] 1.6 1.2 
0.5 [0.86] 0.8 0.59 
Table 2

Experimental and predicted effect on cell viability of a 3-h exposure to free doxorubicin (DOX) at 3 and 27 h

Cells were treated with free DOX for 3 h and then either: (a) immediately processed for drug effect measurement (immediate effect); or (b) washed, incubated in drug-free medium, and processed for drug effect measurements at 27 h (delayed effect; 3-h treatment + 24-h incubation). Values presented are the mean ± SD of six replicates. Rate constant of cell proliferation was 4.62 E-4 (min−1); Rate constant of cell killing was 1.04 × 1011 (mol × min)−1 for model 1 (Eq. A), which employed a time lag of 3 h. The value of R2 was 0.94, and the value of RMSE of the fit of calculated to experimental values of % of initial cell numbers was 11. The values for the phase nonspecific model were computed according to Eqs. E and F. The parameters were kmax = 0.001 min−1 and EC50 = 8.6 μm, R2 was 0.96 and RMSE = 8.6.

DOX conc. (μg/ml) [μm]Cell number: % of initial
3 h27 (3 + 24) h
Exp.Model 1Phase nonspecific modelExp.Model 1Phase nonspecific model
20 [34.48] 107.7 ± 2.5 108.6 95.2 69.9 ± 9.3 50.3 58.5 
5 [8.62] 108.7 ± 0.3 108.6 100.5 89.8 ± 9.9 90.1 95.1 
1 [1.72] 109.1 ± 0.5 108.6 105.9 147.9 ± 2.8 176.5 162. 
0.5 [0.86] 108.7 ± 1.6 108.6 107.1 190.8 ± 2.3 193.4 182.8 
0.25 [0.43] 109.0 ± 0.3 108.6 107.8 202.1 ± 9.5 202.1 195.9 
0.125 [0.22] 109.8 ± 3.0 108.6 108.2 204.3 ± 1.3 206.6 203.3 
DOX conc. (μg/ml) [μm]Cell number: % of initial
3 h27 (3 + 24) h
Exp.Model 1Phase nonspecific modelExp.Model 1Phase nonspecific model
20 [34.48] 107.7 ± 2.5 108.6 95.2 69.9 ± 9.3 50.3 58.5 
5 [8.62] 108.7 ± 0.3 108.6 100.5 89.8 ± 9.9 90.1 95.1 
1 [1.72] 109.1 ± 0.5 108.6 105.9 147.9 ± 2.8 176.5 162. 
0.5 [0.86] 108.7 ± 1.6 108.6 107.1 190.8 ± 2.3 193.4 182.8 
0.25 [0.43] 109.0 ± 0.3 108.6 107.8 202.1 ± 9.5 202.1 195.9 
0.125 [0.22] 109.8 ± 3.0 108.6 108.2 204.3 ± 1.3 206.6 203.3 
Table 3

Experimental and predicted effect on cell viability of a 3-h exposure to free doxorubicin (DOX) at 96 h

Cells were incubated with free DOX at various concentrations for 3 h, and then washed, incubated in drug-free medium, and processed for drug effect measurements at 96 h (delayed effect). Values presented are the mean ± SD of three replicates. The calculations used the same parameters as in Table 2. The statistical measures were R2 = 0.95 and RMSE = 8.6 for Model 1, and R2 = 0.92 and RMSE = 10.7 for the phase nonspecific model.

DOX conc. (μg/mL) [μM]Cell number % of control
Exp.Model 1Phase nonspecific model
10 [17.24] 29.1 ± 0.5 1.0 2.2 
1 [1.72] 57.0 ± 0.5 48.4 38.4 
0.5 [0.86] 70.3 ± 2.5 70.6 59.4 
0.25 [0.43] 87.0 ± 2.6 84.2 76.1 
0.125 [0.22] 93.1 ± 1.6 91.7 87.1 
0.05 [0.086] 99.8 ± 0.4 96.2 94.5 
0.005 [0.0086] 99.9 ± 0.3 99.5 99.5 
DOX conc. (μg/mL) [μM]Cell number % of control
Exp.Model 1Phase nonspecific model
10 [17.24] 29.1 ± 0.5 1.0 2.2 
1 [1.72] 57.0 ± 0.5 48.4 38.4 
0.5 [0.86] 70.3 ± 2.5 70.6 59.4 
0.25 [0.43] 87.0 ± 2.6 84.2 76.1 
0.125 [0.22] 93.1 ± 1.6 91.7 87.1 
0.05 [0.086] 99.8 ± 0.4 96.2 94.5 
0.005 [0.0086] 99.9 ± 0.3 99.5 99.5 
Table 4

Experimental and predicted effect on cell viability at 3 h and 27 h of a 3-h exposure to doxorubicin (DOX) administered in hyaluronan-targeted liposomes

Cells were treated with encapsulated DOX for 3 h and then either: (a) immediately processed for drug effect measurement (immediate effect); or (b) washed, incubated in drug-free medium, and processed for drug effect measurements at 27 h (delayed effect; 3-h treatment + 24-h incubation). Values presented are the mean ± SD of six replicates. Rate constant of cell proliferation was as in Table 2. Calculations for model 1 assumed either n = 2 and n = 1 in Eq. A. Calculation for the phase nonspecific model was according to Eqs. E and F. Time lag: 3 h in calculations for model 1 and 0 for the phase nonspecific model. Rate constants of cell killing were: Model 1, n = 2, k = 5.5 × 1026 × mol−2 × min−1, R2 = 0.98, RMSE = 7.6; Model 1, n = 1, k = 6 × 1011 × mol−1 × min−1, R2 = 0.94, RMSE = 14.2. Phase nonspecific model, kmax = 0.005 min−1 and EC50 = 8.6 μm, R2 was 0.83 and RMSE = 23.7.

DOX conc. (μg/ml) [μm]Cell number: % of initial
3 h27 (3 + 24) h
Exp.Model 1 n = 2Model 1 n = 1Phase nonspecific modelExp.Model 1 n = 2Model 1 n = 1Phase nonspecific model
20 [34.48] 106.5 ± 1.4 108.6 108.6 62.1 4.0 ± 0.4 0.0 0.0 0.4 
5 [8.62] 108.7 ± 0.3 108.6 108.6 80.3 6.3 ± 0.7 0.0 0.2 3.8 
1 [1.72] 108.6 ± 0.4 108.6 108.6 99 21.1 ± 3.0 28.1 44.7 47.8 
0.5 [0.86] 108.8 ± 0.7 108.6 108.6 103.3 84.5 ± 5.8 96.4 97.2 93.0 
0.25 [0.43] 109.1 ± 3.5 108.6 108.6 105.9 177.1 ± 13.1 159.7 149.7 143.1 
0.125 [0.22] 108.7 ± 1.7 108.6 108.6 107.3 200.0 ± 6.3 195.3 177.8 173.1 
DOX conc. (μg/ml) [μm]Cell number: % of initial
3 h27 (3 + 24) h
Exp.Model 1 n = 2Model 1 n = 1Phase nonspecific modelExp.Model 1 n = 2Model 1 n = 1Phase nonspecific model
20 [34.48] 106.5 ± 1.4 108.6 108.6 62.1 4.0 ± 0.4 0.0 0.0 0.4 
5 [8.62] 108.7 ± 0.3 108.6 108.6 80.3 6.3 ± 0.7 0.0 0.2 3.8 
1 [1.72] 108.6 ± 0.4 108.6 108.6 99 21.1 ± 3.0 28.1 44.7 47.8 
0.5 [0.86] 108.8 ± 0.7 108.6 108.6 103.3 84.5 ± 5.8 96.4 97.2 93.0 
0.25 [0.43] 109.1 ± 3.5 108.6 108.6 105.9 177.1 ± 13.1 159.7 149.7 143.1 
0.125 [0.22] 108.7 ± 1.7 108.6 108.6 107.3 200.0 ± 6.3 195.3 177.8 173.1 
Table 5

Sensitivity of models

Cell number: % of initial for doxorubicin (DOX) concentration of 0.5 μg/ml. In the case of the phase nonspecific model based on Eqs. E and F, the parameter k is kmax. In all the calculations a time lag of 3 h was used.

ModelA. Free drug at 27 h and 96 h
27 h96 h
Exp.kk/22kExp.kk/22k
Model 1, n = 1 190.8 193.4 202.5 177.2 70.3 70.6 84.0 49.9 
(CT) protocol 190.8 195.5 202.4 177.8 70.3 71.6 84.7 51.2 
Emax phase nonspecific model, cell-associated drug 190.8 182.5 196.3 157.6 70.3 56.2 74.9 31.5 
Emax phase nonspecific model, drug in medium 190.8 183.0 190.6 139.8 70.3 57.2 75.7 32.8 
ModelA. Free drug at 27 h and 96 h
27 h96 h
Exp.kk/22kExp.kk/22k
Model 1, n = 1 190.8 193.4 202.5 177.2 70.3 70.6 84.0 49.9 
(CT) protocol 190.8 195.5 202.4 177.8 70.3 71.6 84.7 51.2 
Emax phase nonspecific model, cell-associated drug 190.8 182.5 196.3 157.6 70.3 56.2 74.9 31.5 
Emax phase nonspecific model, drug in medium 190.8 183.0 190.6 139.8 70.3 57.2 75.7 32.8 
ModelB. Liposomal drug at 27 h
Exp.kk/22k
Model 1, n = 1 84.5 97.2 143.3 44.7 
Model 1, n = 2 84.5 96.4 153.4 58.7 
Phase nonspecific model 84.5 97.8 122.1 23.6 
ModelB. Liposomal drug at 27 h
Exp.kk/22k
Model 1, n = 1 84.5 97.2 143.3 44.7 
Model 1, n = 2 84.5 96.4 153.4 58.7 
Phase nonspecific model 84.5 97.8 122.1 23.6 
Table 6

A comparison of the performance of models

ModelFree drugLiposomal drug
(3 h + 27 h)Prediction: 96 h(3 h + 27 h)
R2RMSER2RMSER2RMSE
Model 1; n = 1 0.94 11 0.95 8.6 0.94 14.2 
Model 1; n = 2     0.98 7.6 
(CT) protocol; no lag time 0.71 23.3 Poora  Poor  
With lag time 0.74 22 Poor  Poor  
Emax phase nonspecific, 0.96 8.6 0.92 10.7 0.83 23.7 
With lag timeb 0.97 7.1 0.91 11.5 0.9 17.3 
Emax phase nonspecific, 0.98 6.3 0.91 11.5   
Consideration of cell-associated drug/no lag time       
With lag time 0.98 5.4 0.9 12.2   
ModelFree drugLiposomal drug
(3 h + 27 h)Prediction: 96 h(3 h + 27 h)
R2RMSER2RMSER2RMSE
Model 1; n = 1 0.94 11 0.95 8.6 0.94 14.2 
Model 1; n = 2     0.98 7.6 
(CT) protocol; no lag time 0.71 23.3 Poora  Poor  
With lag time 0.74 22 Poor  Poor  
Emax phase nonspecific, 0.96 8.6 0.92 10.7 0.83 23.7 
With lag timeb 0.97 7.1 0.91 11.5 0.9 17.3 
Emax phase nonspecific, 0.98 6.3 0.91 11.5   
Consideration of cell-associated drug/no lag time       
With lag time 0.98 5.4 0.9 12.2   
a

The lag time when applicable was 3 h. The notation poor indicates values of R2 below 0.7 and RMSE above 25.

b,c

The value of kmax in this case was 0.0013. In all other cases the values of parameters are given in Tables 2,3,4 or legend to Fig. 2.

c

The value of kmax in this case was 0.0014 min−1. In all other cases the values of parameters are given in Tables 2,3,4 or legend to Fig. 2.

We thank Dr. Carl T. Redemann for synthesis of the HAn-PE ligand, and Lucie Gagne for setting up the SRB assay and technical assistance.

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