Comparison of Conventional Chemotherapy, Stealth Liposomes and Temperature-Sensitive Liposomes in a Mathematical Model

Various liposomal drug carriers have been developed to overcome short plasma half-life and toxicity related side effects of chemotherapeutic agents. We developed a mathematical model to compare different liposome formulations of doxorubicin (DOX): conventional chemotherapy (Free-DOX), Stealth liposomes (Stealth-DOX), temperature sensitive liposomes (TSL) with intra-vascular triggered release (TSL-i), and TSL with extra-vascular triggered release (TSL-e). All formulations were administered as bolus at a dose of 9 mg/kg. For TSL, we assumed locally triggered release due to hyperthermia for 30 min. Drug concentrations were determined in systemic plasma, aggregate body tissue, cardiac tissue, tumor plasma, tumor interstitial space, and tumor cells. All compartments were assumed perfectly mixed, and represented by ordinary differential equations. Contribution of liposomal extravasation was negligible in the case of TSL-i, but was the major delivery mechanism for Stealth-DOX and for TSL-e. The dominant delivery mechanism for TSL-i was release within the tumor plasma compartment with subsequent tissue- and cell uptake of released DOX. Maximum intracellular tumor drug concentrations for Free-DOX, Stealth-DOX, TSL-i, and TSL-e were 3.4, 0.4, 100.6, and 15.9 µg/g, respectively. TSL-i and TSL-e allowed for high local tumor drug concentrations with reduced systemic exposure compared to Free-DOX. While Stealth-DOX resulted in high tumor tissue concentrations compared to Free-DOX, only a small fraction was bioavailable, resulting in little cellular uptake. Consistent with clinical data, Stealth-DOX resulted in similar tumor intracellular concentrations as Free-DOX, but with reduced systemic exposure. Optimal release time constants for maximum cellular uptake for Stealth-DOX, TSL-e, and TSL-i were 45 min, 11 min, and <3 s, respectively. Optimal release time constants were shorter for MDR cells, with ∼4 min for Stealth-DOX and for TSL-e. Tissue concentrations correlated well quantitatively with a prior in-vivo study. Mathematical models may thus allow optimization of drug delivery systems to achieve a better therapeutic index.


Introduction
Current chemotherapy may be improved if sufficient levels of drug were obtained in the tumor while at the same time limiting system toxicity. Doxorubicin (DOX) is a clinically used chemotherapy agent with dose limiting toxicities [1] and a short plasma half life of five to ten minutes [2]. To overcome short plasma half-life of DOX and to reduce systemic toxicity, pegylated Stealth liposomal drug carriers for DOX (e.g., DoxilH) have been developed [3], allowing for long circulation times, up to several days [4,5]. Stealth liposomes, such as Doxil, remain an excellent example of a drug delivery system with reduced toxicity, but there have been limited benefits in terms of clinical efficacy [6,7,8]. A different liposomal approach was first proposed in the late 1970s by Yatvin and colleagues [9] called temperature sensitive liposomes (TSL). TSL rapidly release their content upon heating (within seconds to minutes) [10,11,12], while at body temperature the drug is somewhat stably encapsulated ( Figure 1). Therefore, TSL in combination with heating of the target region can selectively enhance bioavailability of the drug locally while minimizing systemic exposure.
In the last decade there has been increased interest in the TSLbased delivery, in part due to advances in image-guided hyperthermia applicators. The TSL approach requires the perfect marriage of liposomal properties, in terms of plasma pharmacokinetics and temperature dependent release, with a hyperthermia applicator that generates accurate and homogeneous spatial temperature distributions. TSL have been successfully combined in both preclinical and clinical studies with heat-based thermal therapies including radiofrequency ablation [13,14], ultrasound hyperthermia [15,16,17], and microwave hyperthermia [18]. Although many reports suggest potential of a TSL based approach, the optimal combination of TSL and hyperthermia applicator properties for a given tumor type remains unknown.
For example, TSL were formulated to provide ultrafast release within seconds [13,19] while other approaches use a longer circulating liposome with longer release times within minutes to hours [20,21,22]. Ultrafast release TSL may facilitate an intravascular triggered tumor delivery paradigm, but more stable long circulating liposomes may first accumulate in the tumor region prior to substantial temperature activated drug release.
One difficulty in uncovering an optimal combination of TSL and hyperthermia applicators is that drug delivery is determined by the interplay of several transport mechanisms affected by a large number of parameters, e.g. vascular density, permeability, perfusion, and rate of cellular uptake to name a few. While it is not possible to systematically examine (or in many cases even measure) the influence of these parameters with in vivo studies, computational models offer the unique ability to efficiently perform such a multi-parameter analysis. In prior studies, mathematical models have described the pharmacokinetics of DOX resulting from different drug delivery methods [23,24,25], including intravascular and extravascular triggered release from TSL [13,17,26]. Mathematical analysis of drug delivery kinetics can thus identify the key parameters that affect drug distribution. Furthermore, these models may facilitate optimization of parameters, such as drug release rate and plasma half-life. The objective of this study was to mathematically model and compare standard chemotherapy to Stealth liposomes and TSL with different release time constants triggered intra-or extra-vascularly, and to determine plasma-and tumor concentrations of bioavailable drug. Further we examined importance of liposome extravasation and changes in cellular uptake rate. These findings are significant in that they provide a basic foundation for current activate-able drug delivery approaches and essential guidance for future drug delivery system development.

Materials and Methods
A mathematical model to simulate and compare drug delivery after administration of either liposomal encapsulated DOX or DOX alone was developed. Specifically, the following cases were modeled, where all drug types were administered as bolus injection at a dose of 9 mg/kg in mice (small animals were modeled, since for most drug formulations below only transport parameters from small animal studies data is available): The release rates were calculated as linear approximation of the data in Figure 1 at t = 0 (initial slope). The assumption of linear release time course is adequate since TSL-i only release during the few seconds of tumor transit, where curves in Figure 1 are approximately linear. The tumor was assumed to be uniformly heated to 42uC for 30 min to trigger release during this period. For the release of DOX due to liposomal instability at 37uC a biexponential fit to experimental data was used (see release of DOX at 37uC shown in Figure 1).

5)
Validation of Free-DOX, Stealth-DOX, TSL-i-DOX models: A prior in-vivo study measured tissue DOX concentration 1 h after administration of different DOX formulations, with and without hyperthermia (1 h@42 uC) [12]. Kong et al studied two TSL formulations: the slowrelease formulation from above (complete release within ,2 min, release rate 0.025/s), and one with complete release after ,1.5 h (release rate 0.00096/s). We implemented these release parameters in our model, and used same dose (5 mg/ kg). We modeled Free-DOX and Stealth-DOX without hyperthermia, and the two TSL-formulations with hyperthermia. Tissue concentration after 1 h was compared between our model and this study by Kong et al [12].
In the following text we will always refer to the fast-release TSL-i formulation, except if specified otherwise.
All simulations used the same multi compartment model, which included following compartments (see Figure 2): N Systemic plasma compartment Compartments were assumed perfectly mixed, and for each compartment the DOX concentration was calculated over a 50 h period. Additionally, the liposomal (i.e. encapsulated) DOX concentration was calculated for the plasma and tumor EES. As presented in Figure 2, liposomal release of drug, extravasation of liposomes from tumor vasculature into the tumor EES, transvascular transport of free drug and cancer cell uptake were modeled similar to prior studies [17,23,27,28].
The aggregate body tissue compartment and systemic plasma compartment were assumed at 37uC body temperature, and a uniformly distributed blood perfusion w 0 = 108 ml/100 ml/min was assumed.
All simulations were done with Matlab (R2009a) Software. The model was implemented via a system of ordinary differential equations (ODEs), similar to a prior publication [13]. The complete set of equations is listed and described in detail in the supplementary data.
Below, the assumptions underlying the model, and descriptions on how different processes were modeled, are presented: N TSL-DOX (TSL-i, TSL-e): Liposomal DOX concentration within the systemic plasma considered release of DOX at body temperature (due to liposomal instability), body clearance of both DOX and liposomes, and triggered release within the tumor (either intra-, or extravascularly for TSL-i and TSL-e, respectively). We assumed uniform transport parameters within the tumor (i.e. no spatial heterogeneity was considered). N Tumor sub-compartments: Tumor tissue was modeled in more detail than other tissues, where plasma, interstitial, and intracellular regions were each represented separately ( Figure 2). The transport of liposomes as well as unencapsulated DOX were modeled as diffusion based processes, and thus depend on the concentration difference between plasma and EES, as well as on apparent vascular permeability (P) and vascular surface area (S): Even though transport processes other than diffusion potentially contribute (e.g. convection), this approach is appropriate as long as overall transport can be represented accurately by an apparent permeability, and this method has been employed in numerous prior studies [17,23,29].
While uptake of liposomes by macrophages and subsequent release of DOX is another possible mechanism enhancing tissue concentration [30,31], this mechanism is not considered here. For TSL-i, macrophage uptake is not expected to have significant impact due to the comparably slow rate, though it may contribute in the case of Stealth-DOX and TSL-e. N Although it is known that DOX binds extensively (,70%) to plasma proteins [2], experimental data [32] suggest that plasma protein binding of DOX is weak and does not affect transvascular transport significantly. Thus we did not include any protein binding of DOX in this current modeling study. N Note that vascular permeability for liposomes is by three orders of magnitude lower than permeability for unencapsulated DOX (Table 1). Further, rate of tumor perfusion -which delivers liposomes and unencapsulated DOX to tumor as well as removes those -affects liposomal and drug transport and is considered.
N DOX cell uptake was modeled according to a mathematical cell uptake model suggested in a prior publication [27], which is based on in-vitro studies in human non-small cell lung tumor cells [33]. For modeling MDR resistant cells, DOX cellular efflux was increased by a factor of ten compared to nonresistant cells [24].
N Maximum tumor intracellular concentration was considered as measure of efficacy of a particular liposomal formulation, since intracellular concentration has been shown to directly correlate with cell survival independent on cell type, contrary to EES concentration [34]. N Body and heart tissue compartments: The systemic plasma compartment concentration considered inflow and outflow from tumor, as well as clearance, and uptake by body tissue. All body tissues except tumor and heart were lumped together in one compartment (Figure 2), and transport to and from tissue were described each by a rate constant. No specific tissue types, and no separation into EES and cellular compartments were considered. Heart tissue was modeled in the same manner, but with different transport rate constants than the lump tissue. Heart tissue concentration served as marker of cardiac toxicity for our purposes, which limits DOX lifetime exposure.
N For all liposomal carriers, we assumed a bolus injection at time = 0, and perfect mixing within systemic plasma after administration. Similarly, for administration of Free-DOX, a bolus injection was assumed, with initial volume of distribution defining initial DOX concentration. Table 1 shows the primary model parameters, including mouse physiological variables. A complete list of parameters is presented in the supplementary data. Figure 3 shows the concentration of liposomal DOX in body plasma resulting from administration of Stealth-DOX and TSL-i-DOX, showing comparably shorter plasma half-life of TSL-i due to lower stability of this fast-release TSL formulation (see Figure 1). The highest peak tumor concentration of total DOX (i.e. liposomal and released DOX) was reached with TSL-i-DOX, followed by Stealth-DOX and TSL-e-DOX ( Figure 4 and Table 2). For Stealth-DOX, only a small fraction of total DOX in tumor was bioavailable ( Figure 5B); bioavailable tumor DOX concentration was highest for TSL-i with a peak value of 77.34 mg/g ( Figure 5A and Table 2). Figure 6 compares time course of bioavailable DOX concentration in plasma, EES, and inside tumor cells, each for Free-DOX, TSL-i, and TSL-e. Plasma and EES concentrations of bioavailable DOX equilibrate within ,2 min due to comparably high vascular permeability of unencapsulated DOX. Free-DOX results in short drug exposure of cells, and low cellular uptake ( Figure 6A). TSL-i allow for extended duration of high plasma concentration within the tumor, with considerably higher cell uptake. For TSL-i, rapid release formulations (within seconds) are ideal ( Figure 6C), allowing for higher EES concentration and cell uptake than those with slower release ( Figure 6B). TSL-e did not achieve as high cellular uptake as TSL-i, as most drug released within the interstitium by extravasated TSL-e diffused back into the vasculature, and was carried away by blood flow. Figure 7 compares time course of intracellular concentration of all formulations. The maximum intracellular tumor drug concentration was highest for TSL-i (100.55 mg/g) compared to TSL-e (15.9 mg/g), Free-DOX (3.39 mg/g) and Stealth-DOX (0.44 mg/g) ( Figure 7 and Table 2). Stealth liposomes released their content slowly; therefore the maximum bioavailable DOX concentration was reached much later than with other formulations ( Table 2).

Results
The highest cardiac tissue concentration resulted from the administration of Free-DOX. Peak cardiac tissue concentration of DOX in the case of TSL-i (1.84 mg/g) was only about a third of the peak concentration resulting from administration of Free-DOX (6.62 mg/g) ( Table 2), and was mainly due to leakage at body temperature ( Figure 1). The best ratio between peak bioavailable tumor tissue drug concentration and cardiac tissue concentration was reached with TSL-e ( Table 2).
The optimization of Stealth liposomes in order to maximize the intracellular tumor drug concentration resulted in an optimal release time constant of t = 45 min compared to clinically used Stealth-DOX (Doxil) which has a release time constant of t = 454.5 h [4] (Figure 8, Figure 9). The optimized Stealth liposomes reached a peak intracellular drug concentration of 7.54 mg/g, compared to 0.44 mg/g with Doxil. However, the maximum plasma concentration and AUC were also higher in the case of optimized Stealth liposomes compared to Doxil (see Table 2) suggesting higher toxicity. The ratio between maximum DOX concentration in tumor tissue to cardiac tissue was 0.71 and 2.35 for optimized Stealth and Doxil, respectively (Table 2). Figure 8 and Figure 9 show tumor intracellular DOX concentration for different Stealth-DOX release time constants (t = 1-500 h), with optimum at t = 45 min. The release time constant was also optimized for TSL-e (Figure 10), and was optimal at t = 10.9 min. Note however that this optimal time constant is tied to the 30-min hyperthermia duration. For extended hyperthermia durations longer than 4 h, the optimum release time constant approaches ,30 min.
All simulations were repeated for the case of MDR tumor cells, which resulted, as expected, in lower intracellular drug concentrations in all cases, but showed similar levels of EES and plasma DOX concentrations (results shown in Figure 8 and Table 2) as for DOX-sensitive cells. In the case of MDR cells, the optimal release time constant for Stealth liposomes and TSL-e was considerably shorter (t = 4 min in both cases, shown in Figure 8) in order to maximize intracellular drug concentration.
When we modified the model in that we didn't allow extravasation of liposomal drug into the tumor EES demonstrated negligible contribution of TSL extravasation in the case of TSL-i. In contrast, for Stealth-DOX, tumor intracellular concentration was reduced 3.7 fold if extravasation was not allowed. Table 2 summarizes concentrations for all formulations.

Discussion
Despite dose limiting toxicities, DOX is one of the most commonly used anticancer drugs. Several recent studies show that liposomal encapsulated DOX can overcome some of these negative side effects of unencapsulated DOX, resulting in reduced toxicity but not necessarily higher treatment efficacy [43]. Stealth liposomes release their content slowly over time and have prolonged circulation times, up to several days. In contrast, TSL are designed to release most of their contents within seconds to minutes upon heating, whereas at normal body temperature only a small amount of drug is released (Figure 1). Thus the toxicity compared to Free-DOX can be reduced while increasing drug accumulation at the target site through locally triggered release.
In this study we used mathematical models to compare administration of conventional chemotherapy (unencapsulated DOX), DOX loaded TSL and Stealth liposomes in mice. Drug concentrations in plasma, normal tissue and tumor tissue, as well as various transport mechanisms were investigated. For all liposomal formulations, different release time constants were examined in order to maximize intracellular peak concentration of DOX. Intracellular DOX concentration was used as measure of efficacy, since it directly correlates with cell kill, contrary to EES concentration [33]. We examined the ratio of drug exposure between tumor tissue and cardiac tissue for each DOX formulation as a measure of selectivity.

Free-DOX (Conventional Chemotherapy)
Unencapsulated DOX has a short initial plasma half life (,1 min in mice, see Figure 6A) [44]. The administration of unencapsulated DOX results in a high initial peak plasma concentration followed by a fast decrease. Most of the drug is cleared out of the body before the tumor cellular compartment has sufficient time to take up substantial amounts of drug. Hence, peak EES concentration in the tumor compartment is 2.33 mg/g after 1.4 min, followed by a fast decrease according to the decrease of the plasma concentration ( Figure 6A). Drug uptake by EES results from transvascular transport of drug, and is therefore limited by the short period of high plasma concentration. In fact, the time constant for DOX transvascular transport (i.e. equilibration between EES and plasma concentration) is about two minutes ( Figure 6), which is significantly longer than the plasma half-life. That is for example why slow infusion of DOX is more effective than rapid bolus [23]. As tumor intracellular uptake of DOX occurs at a slower rate than the rate of transvascular transport, peak intracellular concentration of 3.39 mg/g occurs delayed after 2.8 h (Figure 7). Intracellular tumor drug concentration is low and heart tissue concentration high compared to liposomal formulations, as shown in Table 2.

Stealth-DOX
Increasing the period for which EES DOX concentration is elevated would be expected to result in higher intracellular DOX concentrations. This is more likely to be obtained with liposomal DOX than with the administration of unencapsulated DOX. Liposomes extravasate from tumor vasculature and accumulate in tumor tissue due to their prolonged plasma circulation time and the higher permeability of tumor vasculature compared to most noncancerous tissues [45]. Our results show that for Stealth liposomes, extravasation is a dominant transport mechanism as indicated by a 3.7-fold reduction in intracellular DOX when liposome extravasation was not included in the model. Due to the slow release (t = 454.4 h) of DOX from Stealth liposomes, the peak concentrations of DOX in the plasma and EES are low compared to concentrations after administration of Free-DOX. Several prior animal studies have shown increased tumor drug    accumulation with Stealth liposomes compared to Free-DOX administration [46,47], similar to our results ( Figure 4). While more drug is accumulated in tumor tissue when administered via Stealth liposomes, importantly, most of the accumulated drug is present as encapsulated liposomal drug in the EES with only a small fraction released and being bioavailable ( Figure 5B). This may explain why in general Stealth liposomes result in reduced toxicity, but not higher efficacy compared to free drug administration [43]. The reduced toxicity of Stealth liposomes compared to free drug was also observed here in terms of reduced cardiac tissue concentration (Table 2). El-Kareh and Secomb [23] compared different infusion times of DOX, as well as different release time constants of DOX from Stealth liposomes for 4 different doses. We performed a similar parametrical analysis for different release times t of Stealth liposomes (Figure 9). For very   small release time constants t (,1 min) the results closely match those after the administration of Free-DOX. Similar to the prior study by El-Kareh and Secomb [23], our results show that the optimal release time constant of Stealth liposomes in order to maximize intracellular tumor concentration is 45 min, which is much lower than the release time constant of current clinically used Stealth liposomes (e.g. Doxil). However, these optimized Stealth liposomes also had higher plasma drug concentrations (peak concentration as well as AUC) compared conventional Stealth-DOX, which might implicate higher toxicity. Therefore not only intracellular tumor concentration, but also systemic toxicity should be considered in choice of the optimal release properties to achieve the optimal therapeutic index. Table 2 and Figure 11(B) show that the ratio between peak concentrations of tumor tissue and cardiac tissue is better for Doxil, than for the optimized Stealth liposomes. The optimal liposomal release rate also depends on the intracellular DOX uptake mechanism of cells, as this affects uptake rate. The cellular uptake mechanisms are likely to be different for different cell types (e.g. MDR tumor cells as discussed below), and therefore the presented results are only valid for the cell type we assumed in this study (human non-small cell lung tumor cells).

TSL-i-DOX (Intravascular Triggered TSL)
The dominant mechanism of delivery by fast-releasing TSL-i is dissimilar from Stealth liposomes, in that DOX is released inside the tumor plasma compartment at the location of heating, and subsequently transported into the tissue's EES as free drug. The extravasation of TSL-i into the EES is not relevant due to the short plasma half-life of TSL-i and slow extravasation rate of nanoparticles. The release of DOX from the fast-release TSL within systemic plasma at body temperature results in a higher systemic plasma drug concentration compared to Stealth liposomes, though still considerably lower than for administration of Free-DOX ( Figure 6). Liposomal body plasma concentration of TSL-i decreases much faster compared to Stealth liposomes due to the lower stability of the fast-release TSL-i formulation (Figure 3). The mechanism for the shorter plasma half-life is unknown but it is most likely a combination of DOX leakage from the TSL-i and clearance of the TSL-i containing DOX. Tumor plasma concentration of bioavailable DOX is highest for fast-release TSL-i due to quick release of drug from heated TSL while the liposomes are transiting the heated tumor vasculature. Rapid intravascular release of DOX upon heating provides locally maintained elevated tumor intravascular concentration during heating ( Figure 6C); this is conceptually similar to local infusion of unencapsulated drug into the tumor's vascular supply. Thus, the systemic plasma serves as large reservoir of non-bioavailable drug that becomes bioavailable once entering the heated tumor.
The amount of released drug within the heated region not taken up by the tumor, and carried by perfusion into the systemic plasma, is negligibly small compared to drug leaked from TSL-i systemically at body temperature; this is due to the much larger volume of the systemic plasma compared to tumor plasma compartment [48]. However, this intravascular release approach may be improved by using drugs that are completely extracted by the heated tissue. In addition to fast-release TSL-i, a slow-release TSL-i formulation [22] with release rate of R = 0.025 s 21  (complete release within a few minutes) was modeled, which resulted in a 2.7 fold lower heart tissue DOX concentration (due to lower leakage at body temperature) and 5.1 fold lower tumor tissue DOX concentration compared to the fast release TSL-i formulation. This is in agreement with a prior study, where very slowrelease TSL-i (release within ,1 h) showed no benefit over Stealth-DOX, compared to TSL-i that release within 1 min [12] (Figure 12). An ideal TSL-i would therefore have no release at body temperature, and very quick complete release (within , a second) when heated.

TSL-e-DOX (Extravascular Triggered TSL)
This TSL formulation combines the stability and long circulation time of Stealth liposomes with heat-activated release after liposomal tumor accumulation. Liposomes were allowed to extravasate for 24 h until they reach maximum EES concentration. At this time point we initiatde release of drug from the liposomes within the tumor EES and plasma by local heating ( Figure 6D). Following release, some bioavailable DOX is taken up by cells, but most DOX diffuses back into plasma due to fast transvascular transport. Cellular DOX uptake is therefore lower than for TSL-i. The optimal release time constant for TSL-e was 11 min (Figure 10). While Figure 10 shows highest tumor drug uptake at very short release time constants, this is due to contribution of intravascular release (i.e. would correspond to TSL-i described below). The maximum indicated in Figure 10 is at the release rate where contribution of extravasated TSL-e is optimal, according to the intended delivery mechanism for TSL-e. Note that this optimal time constant is tied to the 30-min hyperthermia duration. For extended hyperthermia duration longer than 4 h, the optimum release time constant approaches ,30 min (graph not shown).
Validation. A prior in-vivo study in a small animal tumor model measured tissue concentration after administration of different DOX formulations (Free-DOX, Stealth-DOX, two TSL-DOX formulations with varying release rates), both with and without hyperthermia [12]. After adjusting parameters in our model to fit this prior study (TSL release rates, dose), tissue concentration correlates well quantitatively with this prior in-vivo study. Importantly, the prediction for the two TSL-i-DOX formulations with different release rates match well. Note that for the fast TSL-i-DOX formulation studied above, our model predicts 2.7 times higher tissue concentration than for the slow TSL formulation shown in Figure 12.
Toxicities. The administration of all liposomal DOX formulations resulted in significantly reduced peak plasma concentrations and plasma AUC's of bioavailable drug (Table 2), reduced heart tissue concentration (Table 2), and presumably reduced toxicity, compared to administration of free drug. Cardiac toxicity is of relevance, since it limits lifetime dose of DOX that can be administered. The suggested reduced toxicity indicated by peak cardiac concentration is in accordance with published experimental data [3,14]. Highest concentration of DOX in cardiac tissue resulted from administration of Free-DOX while lowest levels were reached with Stealth liposomes. While the fact that high cardiac concentrations are obtained with free DOX may be contradictory in light of the low tumor concentrations, this can be explained by the very rapid DOX uptake by cardiac cells compared to cancer cells. The ratio between DOX concentration in tumor tissue and heart tissue was best for TSL-e, followed by fast-release TSL-i, slow-release TSL-i, and worst for Free-DOX ( Figure 11, Table 2). The differences in cardiac tissue uptake are primarily resulting from differences in plasma stability of the various liposomal formulations. Other dose limiting toxicities of Figure 10. Optimization of release time constant for TSL-e. The optimum (optimum indicated by line at ,11 min) is where contribution of release from extravasated TSL-e towards drug delivery is maximal (intended delivery mechanism). While higher intracellular concentrations are observed at very short release time constants, this is due to contribution from intravascular release, i.e. the TSL-e become equivalent to TSL-i. doi:10.1371/journal.pone.0047453.g010 DOX, such as myelosuppression, mucositis and PPE, correlate with plasma AUC. Plasma AUC was lowest for Doxil, followed by TSL-e, slow TSL-i, fast TSL-i, optimized Stealth, and highest for Free-DOX (Table 2).
MDR. Many cancers develop resistance to chemotherapy drugs such as DOX. Literature suggests that MDR is due to presence of membrane proteins (p-glycoprotein and the so-called multidrug resistance-associated protein (MRP)), which facilitate DOX transport out of the tumor cells. We simulated this active efflux mechanism by increasing the transport of DOX from the intracellular tumor compartment to the tumor EES ten times [24].
As expected, the model demonstrated for all DOX formulations a decreased tumor intracellular concentration ( Table 2). For Stealth liposomes, optimal release time constant for MDR cells was t = 4 min compared to 45 min for DOX-sensitive cells. In fact, even Free-DOX was more effective in MDR cells than the slower Stealth-DOX optimized for DOX sensitive cells (t = 45 min) ( Figure 8B). Note that the optimal release time constant matches approximately the time constant of cellular uptake (i.e. time it takes for intracellular concentration to saturate), which is in the minute range for MDR cells in our model (uptake time constant data not shown) compared to hours for DOX-sensitive cells [23].
A study that fitted data from several prior in-vitro studies to a cell uptake model found cellular uptake time constants varying between a few minutes to many hours, depending largely on cell type [27]. Combined with our results, this suggests (1) the ideal release time constant may depend on cell type, (2) the release time constant may be tailored to more optimally target a specific cell sub-population, and (3) for optimal delivery to a heterogeneous cell population, a cocktail of carriers with multiple release time constants may be beneficial.

Conclusion
Slow release Stealth liposomes reduced systemic toxicity, but did not increase antitumor efficacy as measured by maximum intracellular concentration, which is in accordance with clinical observations [43]. Optimized Stealth liposomes resulted in higher tumor intracellular peak concentration, but also in a higher peak plasma concentration and higher plasma AUC, suggesting higher toxicity. The optimal release time constant for Stealth liposomes as well as of TSL-e for maximizing intracellular tumor drug concentration depended on cell uptake kinetics, which varies between different types of tumor cells (e.g. MDR status of tumor cells). In addition, optimal release time constant highly depends on drug delivery paradigm (passive accumulation, intra-, or extravascular triggered release). Between the compared formulations, TSL-i resulted in highest tumor tissue drug concentration by locally keeping plasma concentration of bioavailable DOX high. Note that many of the presented results are directly relevant to other intra-or extra-vascular triggered drug carriers, in addition to TSL. Optimization of liposomal drug delivery systems (in terms of release time constant, plasma half-life, etc.) clearly may improve therapeutic index providing new motivation to the drug delivery community. However, many of these liposomal properties are interdependent, presenting a formidable development challenge. Mathematical models as the one presented here may thus be a valuable tool to aid design and optimization of drug formulations to achieve a better therapeutic index.

Supporting Information
Equations S1 Detailed description and equations of mathematical model. (DOC) Author Contributions Figure 12. Comparison of mathematical model to a prior invivo study [12] (note that the fast TSL-i formulation discussed above was not employed in this prior study). Free-DOX and Stealth-DOX were compared without hyperthermia; for the two TSL-i-DOX formulations, 1 h hyperthermia was assumed. Administered dose was 5 mg/kg (compared to 9 mg/kg in all other models). doi:10.1371/journal.pone.0047453.g012