Chemotherapeutic Dosing Implicated by Pharmacodynamic Modeling of in vitro Cytotoxic Data: a case study of paclitaxel

Abstract

Conventional maximum tolerated doses (MTD) in chemotherapy are recently challenged by an alternative dosing method with low doses and high dosing frequency (LDHF). Still, it remains unclear which chemotherapies would potentially benefit from LDHF. The pharmacokinetic (PK) differences between MTD and LDHF are drug exposure magnitude (concentration) and exposure duration (time), two fundamental PK elements that are associated with the pharmacodynamics (PD) of chemotherapies. Here we hypothesized that quantitatively analyzing the contribution of each PK element to the overall cytotoxic effects would provide insights to the selection of the preferred chemotherapeutic dosing. Based on in vitro cytotoxic data, we developed a cellular PD model, which assumed that tumor cells were generally comprised of two subpopulations that were susceptible to either concentration- or time-dependent cytotoxicity. The developed PD model exhibited high flexibility to describe diverse patterns of cell survival curves. Integrated with a PK model, the cellular PD model was further extended to predict and compare the anti-tumor effect of paclitaxel in two dosing regimens: multiple MTD bolus and continuous constant infusion (an extreme LDHF). Our analysis suggested that the ratio between drug steady-state concentrations and its cytotoxic sensitivity (C_ss /KC₅₀) was a critical factor that largely determines favored dosing regimen. LDHF would produce higher efficacy when the ratio C_ss /KC₅₀ is greater than 1. Otherwise MTD was favored. The developed PD model presented an approach simply based on in vitro cytotoxic data to predict the potentially favored chemotherapeutic dosing between MTD and LDHF.

Introduction

The magnitude and duration in drug exposure are two fundamental elements in pharmacokinetics (PK) that drive drug pharmacodynamics (PD) and anti-tumor effect in chemotherapy. However, under the philosophy of “more is better”, maximum tolerated doses (MTD) are conventionally employed in chemotherapy in order to abruptly eradicate cancerous cells and release tumor burden to the highest degree. MTD approach usually needs a long dosing interval (drug holiday) to reduce drug side effects so that normal organs would partly recover from drug cytotoxicity [1–3]. However, such an extreme dosing approach is not always satisfactory [4]. Besides severe toxicities, saturation of certain cytotoxic mechanisms, such as inhibition of DNA synthesis, might also account for certain inadequate efficacy in MTD regimen [1]. Once certain maximum cytotoxic mechanisms (plateau) are achieved, MTD would become unnecessary to further augment anti-tumor effect [1].

Duration of exposure, a critical driver of cytotoxic effect, is often neglected in design of chemotherapeutic dosing regimen. Kamen et al. suggested that certain chemotherapies (e.g., antimetabolites) largely benefited from prolonged infusion once drug concentrations were above a desired level [1]. Thereby, an alternative approach, with relatively low doses and high frequency (LDHF), has been broadly explored. LDHF is usually regarded as metronomic therapy, a promising alternative to MTD dosing approach. This regimen, because of lower system concentrations, usually has much less system toxicities, and many potential mechanisms of anti-tumor effects [5, 6]. Although MTD and LDHF regimens have been compared in many preclinical and clinical studies [4, 7–9], it is still largely unknown which dosing regimen would potentially yield higher tumor suppressive effect for a given chemotherapy.

In many in vitro cytotoxic studies, concentrations of cytotoxic agents and incubation duration are both found contributing to cell killing effects [10]. The concentration (magnitude) and time when concentration above effective levels (duration) are two fundamental PK differences among various chemotherapeutic dosing regimens. Therefore, we hypothesized that quantitatively analyzing the relative contributions of each PK element to overall cytotoxic effect for a given chemotherapy, would be potentially helpful in selection of their favored dosing regimen, i.e., finding the very fine combinations of magnitude and duration resulting in highest cytotoxic effect. Herein, with paclitaxel as an example, we developed a proof-of-concept to solely analyze in vitro cytotoxic data with a cellular PD model to implicate favored chemotherapeutic dosing regimen.

PK/PD models have proven to be a powerful tool to disentangle complex pharmacological and biological problems [11]. Previously, a variety of cellular PD models have been developed for analysis of in vitro cytotoxic data [12–18]. Unfortunately, all these previous models were either highly empirical or developed primarily for the purpose of capturing diverse patterns of concentrations-tumor survival curves rather than exploring underlying cytotoxic mechanisms [14–16]. In this study, we proposed a semi-mechanism-based cellular PD model that was simply based on analysis of in vitro cytotoxic data to quantitatively assess the relative contributions of incubation time and exposure magnitude to overall cytotoxicity. The developed model was further extended into physiological contexts to simulate a wide range of clinical situations and assessed which model parameters closely determined the selection of favored dosing regimen between MTD and LDHF.

Theoretical

The susceptibility of cancerous cells to chemotherapeutic agents is dependent on both concentrations of cytotoxic drugs and incubation time, two fundamental PK elements that contribute to overall cytotoxicity. The contributions of two PK elements to overall cytotoxicity vary considerably for different cytotoxic drugs and different type of cell lines. In many cytotoxic studies, cell death was observed shortly after drug exposure, where the cytotoxic effect are closely associated with drug concentrations [12]; while some cytotoxicity is highly contributed by incubation time and the cytotoxic effect greatly increases in prolonged incubation time [10]. In most cases, both elements are involved and jointly contribute to cytotoxicity. This is why the cell survival-drug concentrations curves are often shifted left with prolonged incubation times (lower IC₅₀) in many cytotoxic studies. To simultaneously describe the contributions of both PK elements, we developed a cellular PD model, where cancerous cells were assumed to be generally comprised of two subpopulations that were relatively susceptible to either concentration-dependent or time-dependent cytotoxicity (Fig. 1). Concentration-dependent subpopulation essentially represents tumor cells that are killed at a relatively fast rate and the killing effect is largely predicted by drug concentrations, with low dependency on incubation time. For time-dependent subpopulation, on the other hand, the exposure time is a critical cytotoxic driver, usually along with a remarkable delay before the death of cancerous cells after drug exposure.

Fig. 1. Schematic diagram of the developed cellular pharmacodynamics (PD) model, extended with a one-compartment pharmacokinetic (PK) model for clinical simulations.

For the cellular PD model, the tumor cells were divided into two subpopulations that were susceptible to either time- or concentration-dependent cytotoxicity. Time-dependent cytotoxicity was described by a linear killing function followed by a time-dependent transduction model (S2 to S4). Concentration-dependent cytotoxicity was described by a typical sigmoidal K_max model. One compartment PK model was developed with a tumor biophasic compartment to simulate drug concentrations at solid tumors.

Such a “two subpopulations” assumption was mostly supported by two lines of evidence:

Cell cycle-dependent cytotoxicity.

In comparison with normal cells, tumor cells usually rapidly proliferate and are under constant cell division and growth. Most cycle-specific cytotoxic drugs, like 5-fluorouracil, interfere with DNA and RNA synthesis by mimicking the building blocks for DNA synthesis. Therefore, at the early time of drug exposure, only tumor cells at dividing phases are sensitive, with cells at other cell cycles being relatively resistant [12, 15]. For example, methotrexate, a phase-specific agent, only exhibits cytotoxicity to tumor cells at synthesis (S) phase. Therefore, for a short period of time, cells at S phase are expected to be killed with cells at other phases temporarily surviving until conversion into S phase and becoming sensitive to methotrexate. This is why many phase-specific chemotherapeutic drugs exhibit time-dependent cytotoxicity. This time delays in cytotoxicity have been broadly observed in in vitro studies. Gardner et al, developed a model that divided cells into sensitive and non-sensitive subpopulations to predict cytotoxicity of cycle-specific drugs [15].

Diverse mechanisms of cytotoxicity.

Tumor cells are highly heterogeneous and they are essentially a constantly evolving population. The same cell line, even on the same plate, may display dramatically different sensitivity to a given cytotoxic agent [19–21]. After exposure to a cytotoxic drug, tumor cells often go through distinct paths towards death, including apoptosis, necrosis, mitotic catastrophe, autophagy, and premature senescence following DNA damage [22]. For necrosis, there are usually few intracellular signaling events prior to death. The lag time before necrosis after exposure of a cytotoxic agent is usually brief, which we often regard as “immediate cytotoxicity”. Whereas, for other forms of death, such as apoptosis, multiple cascades of receptor-signaling pathways are involved and a relatively longer time towards death is expected than necrosis, which we often regard as “delayed cytotoxicity” [23]. This is why cisplatin, even not a cycle-specific agent, also showed highly time-dependent cytotoxic effect [24, 25].

Therefore, in our model, concentration-dependent subpopulation generally represents cells at either cytotoxic sensitive phase and/or cells that are subject to “immediate cytotoxicity”; time-dependent subpopulation signifies tumor cells that are either at cytotoxic insensitive phases or being subject to “delayed cytotoxicity”. However, it is worth to note that two subpopulations in our model assumption are not completely distinct. Time-dependent subpopulation is also subject to drug concentrations to death, even though may not as dramatic as concentration-dependent subpopulation. Likewise, concentration-dependent subpopulation also need certain duration of incubation time before death upon drug exposure.

The fraction of two subpopulations is denoted as f_u (0 < f_u < 1), where f_u is the fraction of time-dependent subpopulation and the remainder 1- f_u indicates the fraction of concentration-dependent subpopulation. Of note, f_u is a parameter not only associated with chemotherapeutic drugs, but also related to types of cell lines. In other words, f_u is a parameter that needs to be optimized for a specific cytotoxic drug to a given cell line.

Both linear and nonlinear models were tested to describe the drug killing effect on two subpopulations. A linear model was finally chosen for time-dependent cytotoxicity considering exposure time is more predictive than concentrations for this subgroup of cells. A typical sigmoidal K_max model with a hill factor γ was selected for concentration-dependent cytotoxicity. To fully capture the delays in time-dependent cytotoxicity, a time-dependent transduction model (with 3 transits) was applied to simulate cells that are either at a cascade of signal transductions towards death or slowly converting into drug sensitive phases through cell cycle [15, 23]. The differential equations are:

$$\frac{d{S}_1}{dt}=G(S_1)-k_l\cdot C\cdot S_1{S}_1\left(0\right)=f_u$$ (1)

$$\frac{d{S}_2}{dt}=k_l\cdot C\cdot S_1-k_t\cdot S_2{S}_2\left(0\right)=0$$ (2)

for S₃ and S₄,

$$\frac{d{S}_n}{dt}=k_t\cdot \left(S_{n-1}-S_n\right)S_n\left(0\right)=0,n=3or4$$ (3)

$$\frac{d{S}_5}{dt}=G\left(S_5\right)-\frac{k_{max}\cdot C^{\gamma }}{\left(K{C}_{50}^{\gamma }+C^{\gamma }\right)}\cdot S_5{S}_5\left(0\right)=1-f_u$$ (4)

$$G\left(S_n\right)=k_g\cdot S_n\cdot \left(1-\frac{S_n}{S_{max}}\right)$$ (5)

where S represents the survive fraction of tumor cells or the relative tumor volume in in vivo situations. S₁ is the survival fraction of time-dependent subpopulation and S₅ is the survival fractions of concentration-dependent subpopulation. S_2–4 denotes cells in transit processes towards to death, k_t is the transit rate. The total survival fraction of two subpopulations was denoted by W(S), the sum of S₁ to S₅. G(S_n) is the proliferative fractions of tumor cells.

At in vivo situations, logistic model was adopted to describe the capacity-limited tumor growth (eq. 5). S_max is maximum survival (50-fold larger than original volume). The same tumor growth rate (k_g) and maximum survival rate (S_max) were applied for both time- and concentration-dependent subpopulations. To be consistent with literature data, the survival rate (fraction) of tumor cells was normalized by control data (without drugs), herein G (S_n) was omitted. K_l is rate constant of killing to time-dependent subpopulation and k_max is maximal killing rate to concentration-dependent subpopulation. KC₅₀ is drug concentrations yielding 50% of k_max, γ is hill factor. C is extracellular drug concentration. The concentration-cell survival curves in in vitro cytotoxic assays were quantitatively analyzed by the developed cellular PD model.

Methods

In vitro cytotoxic data

There are many studies in the literature reporting various patterns of cytotoxic survival curves. Among these studies, Au et al study was very representative because it clearly indicated both concentration- and time-dependent cytotoxicity. In Au et al study, the in vitro cytotoxicity of paclitaxel was assessed in cell lines of RT4, MCF-7, SKOV-3, FaDu, DU145 and PC3 [10]. Both immediate and delayed cytotoxic effects were noticed, suggesting both time-dependent and concentration-dependent cytotoxicity of paclitaxel to these cell lines. These data was extracted using Digitizer [26] and re-analyzed with the proposed cellular PD model.

Model sensitivity analysis

Parameter sensitivity in shaping concentration-cell survival profiles was analyzed. The central values of parameters assessed were: k_max = 0.2, KC₅₀ = 5, γ = 1, k_l = 0.01, k_t = 0.04, and f_u = 0.5. Concentration-cell survival profiles were simulated by adjusting these parameters (except f_u) up and down by 1/5 ~ 5-fold around central values. As f_u is between 0 and 1, the dynamic range for f_u was set as 0.1 or 0.9 with two extreme conditions at 0 or 1.

Clinical simulations of Paclitaxel

In the developed cellular PD model, the concentrations in in vitro system are equivalent to drug concentrations in tumor interstitium in in vivo situations. Therefore, in clinical simulations, we developed a tumor compartment, to simulate drug concentrations at tumor interstitium. One compartment PK model was used. Paclitaxel PK parameters (CL and V) were obtained in the literature [27]. The PK model was integrated with the developed cellular PD model to simulate tumor suppressive effect in physiological contexts (Fig. 1).

Paclitaxel efficacy in treatment of breast cancers was evaluated with the integrated PK-PD model. Given the fact that tumor growth rate in vivo was relatively lower than that in vitro system, three growth parameters (k_max, k_l, and k_t) in clinics were set to literature values (about 1/5 of in vitro values) [28–29]. The tumor growth rate was set as 0.1 day⁻¹, a value close to clinical observations [28].

Tumor suppressive effects at two clinically tested dosing schedules were then simulated and compared: 175 mg/m² every 3 weeks for 4 doses vs 80 mg/m² weekly for 12 doses [29]. All PK and PD parameters used in this simulation are listed in Table S1 (Supplementary materials). Equations for plasma and tumor kinetics are:

$$\frac{d{C}_P}{dt}=(K_0-C_P\cdot CL)/V{C}_{P(0)}=0$$ (6)

$$\frac{d{C}_T}{dt}=K_{eo}\cdot \left(C_P-C_T\right)C_{T(0)}=0$$ (7)

where C_P and C_T are the drug concentrations in plasma and solid tumors.

Selection between MTD and LDHF

At the same total dosage level (weekly 80 mg/m²), two dosing regimens were compared: MTD and LDHF. The duration of “tumor-free” (tumor volume < 1% of original size, DTF) was calculated as a PD marker to indicate tumor suppressive effect.

Two dosing regimens at four scenarios were systematically compared: Class I: High (C_ss /KC₅₀)^γ, Low f_u; Class II: High (C_ss /KC₅₀)^γ, High f_u; Class III: Low (C_ss /KC₅₀)^γ, High f_u; Class IV: Low (C_ss /KC₅₀)^γ, Low f_u. Parameters used in this analysis were summarized in Table S2 (Supplementary materials).

The sensitivity of C_ss /KC₅₀ ratio, hill factor γ, and f_u in change of DTF between LDHF and MTD regimens were assessed. Considering tumor cells at compartment S₁ to S₃ would eventually die based on our model assumptions, the total cytotoxic effects could be simply reflected by the cytotoxic function in eq. 8. The assessed range for C_ss /KC₅₀ was from 0.065 to 65, γ was from 0.1 to 10, and f_u was from 0 to 1. Total driving force to cytotoxicity is:

$$\text{Cytotoxicity}=K_l\cdot C\cdot S_1+\frac{K_{max}\cdot C^{\gamma }}{(K{C}_{50}^{\gamma }+C^{\gamma })}\cdot S_5$$ (8)

The interactions between hill factor γ and C_ss /KC₅₀ was also evaluated at C_ss /KC₅₀ from 0.065 to 65 and γ from 0.5 to 8. Other parameters were set as: f_u = 0.1, k_l = 0.006, and k_max = 0.6.

Data analysis

Model fittings and optimizations were conducted using ADAPT 5 (http://bmsr.usc.edu/Software/ADAPT/) with maximum likelihood optimization algorithm. The root mean square (RMS) deviation between model predicted and actual data was used to indicate model performance and make comparisons between the developed cellular PD model and previous PD models. Berkeley Madonna (version 8.3.23; http://www.berkeleymadonna.com/) was used for all simulations.

Results

Cellular PD model

The developed cellular PD model adequately described various patterns of concentration-cell survival curves observed by Au et al (Fig. 2) [10], with precisely estimated model parameters (Table 1). Killing plateau on cell survival curves that was often observed in the literature was well characterized by the developed PD model. The fraction of time-dependent subpopulation (f_u) was estimated in the range of 0.0618 ~ 0.668, both time- and concentration-dependent cytotoxicity were suggested in paclitaxel killing effects on these breast cancer cells.

Fig. 2. Model fittings using the developed cellular pharmacodynamic (PD) model of in vitro cytotoxicity data of paclitaxel in RT4, MCF-7, SKOV-3, FaDu, DU145, and PC3 cell lines.

RMS: root mean square.

Table 1.

Parameter estimates (CV %) of the developed cellular pharmacodynamics (PD) model by analyzing in vitro cytotoxic data of paclitaxel in six cancer cell lines.

Cell lines	kmax (h−1)	KC50 (nM)	γ (−)	kl ((h⋅nM)−1)	kt (h−1)	fu (−)
Bladder RT4	0.017	9.407	2.054	0.001	0.118	0.194
	(6.93)	(12.9)	(31.8)	(57.4)	(21.7)	(7.21)
Breast MCF-7	0.059	2.604	3.217	0.009	0.039	0.529
	(16.5)	(6.65)	(12.2)	(19.7)	(4.79)	(9.00)
Ovarian SKOV-3	0.027	5.294	1.625	3.790	0.170	0.062
	(7.42)	(113)	(163)	(203)	(61.2)	(3.70)
Pharynx FaDu	0.048	10.56	1.052	0.029	0.051	0.668
	(41.2)	(51.8)	(33.7)	(37.6)	(4.09)	(24.4)
Prostate DU145	0.024	5.400	2.421	0.003	0.218	0.391
	(11.4)	(16.9)	(20.1)	(25.3)	(13.6)	(11.8)
Prostate PC3	0.034	4.083	1.633	0.007	0.182	0.531
	(13.1)	(27.5)	(15.1)	(22.2)	(9.61)	(12.2)

Moreover, the developed PD model exhibited high flexibilities in prediction of many other patterns of survival curves. As shown in Fig. 3, the model could describe survival curves with time-associated killing plateau, double Hill “roller coaster” observed by Levasseur et al [12], and many other patterns. Parameters, f_u, k_t and k_max, were closely associated with the survival fractions at killing plateau. The killing plateau was particularly predicted by f_u and when f_u decreased from 1 to 0, the killing plateau gradually disappeared until the model eventually collapsed to a K_max model. k_max largely influenced tumor killing rate at the early phase, particularly at low drug concentrations. KC₅₀ shaped the survival curves in a similar way as k_max, the lower KC₅₀, the faster killing rate at low concentrations. K_l showed a modest effect in shaping killing curves, mainly because k_t and f_u dictated time-dependent cytotoxicity.

Fig 3. Parameter sensitivity in the developed cellular pharmacodynamic (PD) model in shaping cell survival curves.

Simulations were performed around k_max = 0.2 h⁻¹, KC₅₀ = 5 nM, γ = 1, k_l = 0.01 h⁻¹, k_t =0.04 ${(h\cdot nM)}^{-1}$, and f_u = 0.5, unless otherwise specified for each panel.

Clinical simulations of Paclitaxel

At a similar accumulative dosage, two clinically tested dosing regimens (Q1W vs Q3W) yielded significantly different tumor suppressive effect (DTF). DTF was 167 days for patients at weekly treatment of paclitaxel, significantly longer than 16 days in another group of patients who received paclitaxel Q3W (Fig. 4). The tumor suppressive effect on ovarian and prostate cancers were also simulated based on the in vitro data. For ovarian and prostate cancers, the minimal tumor volumes after dosing were compared to evaluate the tumor suppressive effect between Q1W and Q3W regimens for these cancers. Similar with breast cancers, Q1W was also the superior dosing regimen in ovarian cancer (74% VS 11%) and prostate cancer (53% VS 16%) when the minimal tumor volume was used as the indicator to evaluate the tumor suppressive effect. Model parameters used in these simulations were summarized in online Supplement Table S1. These results suggest that short dosing interval (or high dosing frequency) would be promising for better tumor control in paclitaxel chemotherapy. Similar conclusion was made in clinical studies to indicate that Q1W dosing regimen produced higher efficacy than Q3W regimen for paclitaxel in treatment of breast, ovarian and prostate cancers, even though the difference in efficacy between two dosing regimens in clinical trials was not as high as our model prediction due to many undefined confounding factors in clinical studies [30–32].

Fig. 4. Simulated tumor suppressive effects of paclitaxel in breast cancer at two different dosing regimens (Q1W and Q3W).

Simulations were done based on the developed cellular pharmacodynamics (PD) model extended with a one-compartment pharmacokinetic (PK) model. Two clinically tested dosing regimens (1 h infusion of 80 mg/m² weekly and 3 h infusion of 175 mg/m²/3 weeks) were compared. Simulations indicated that Q1W produced much higher tumor suppressive effects than Q3W. Blue arrows and red arrows denotes the dosing time of Q1W or Q3W.

Selection between MTD and LDHF

At the same cumulative dose and therapeutic duration, we compared tumor suppressive effect between LDHF and MTD at four conditions. As shown in Fig. 5, when C_ss /KC₅₀ was 1.5, above the predefined cutoff 1.0, Class I and II both showed higher tumor suppression (higher DTF) in LDHF regimen than MTD regimen. On the other hand, Class III and IV, when C_ss /KC₅₀ equal 0.5, selected MTD over LDHF regimen. The difference in DTF between two dosing regimens was greatly predicted by f_u, a parameter reflecting the fraction of time-dependent subpopulation. The higher f_u, i.e., the higher fraction of time-dependent subpopulation, predicted the less difference in DTF between two dosing regimens. When f_u was changed from 0.1 to 0.9, their difference in DTF declined from 80 to 38 days in groups of Class I and II, and declined from 64 to 37 days in Class IV and III.

Fig. 5. Tumor suppressive effects of paclitaxel with two dosing regimens (MTD and LDHF) at four scenarios.

Class I (high C_ss /KC₅₀, low f_u), Class II (high C_ss /KC₅₀, high f_u), Class III (low C_ss /KC₅₀, high f_u), and Class IV (low C_ss /KC₅₀, low f_u). Paclitaxel was dosed at 80 mg/m²/week in both regimens. DTF is duration of tumor free, where tumor free was defined as tumors shrinkage to less than 1% of the original tumor size. ΔDTF, the difference in DTF between MTD and LDHF.

For better comparisons, killing capacity k_max (= 1.5) was set to 2.5-fold higher in Class III and IV than that (= 0.6) in Class I and II, considering Class I and II have relatively higher cytotoxic sensitivity (lower KC₅₀). As shown in Fig. 6, the selection between MTD and LDHF regimens was largely predicted by C_ss /KC₅₀. When C_ss /KC₅₀ > 1, LDHF would probably produce higher anti-tumor effect than MTD regimen, otherwise, MTD regimen worked potently.

Fig. 6. Parameter sensitivity to cytotoxic effect at four clinical scenarios.

The influence of C_ss /KC₅₀, hill factor γ, and f_u on tumor suppressive effects between two dosing regimens was simulated. Simulations were performed with parameters listed in Table S2, unless other specified. The values of KC₅₀ were set as 0.6 day⁻¹ for the first Class I and II and 1.5 day⁻¹ for Class III and IV.

In addition to C_ss /KC₅₀, hill factor γ and f_u are two other critical predictors to favored dosing. The influence of f_u on DTF difference was found monotonous, the higher f_u, the smaller DTF difference. When f_u = 1, i.e. time-dependent subpopulation is the only cell population, there would be no DTF difference between two dosing regimens. For Class I and II, the higher hill factor γ predicted the larger DTF difference; For Class III and IV, the influence of γ on DTF was bidirectional, when γ was small, continuous infusion was slightly preferred; otherwise, multiple MTD bolus seemed to be a better option.

As shown in Fig. 7, when γ was low (< 1.0), DTF difference between two dosing regimens was negligible, when C_ss /KC₅₀ slightly affected. However, with the increase of γ, DTF difference became larger with a stronger dependency on C_ss /KC₅₀. This result also suggests that cytotoxic drugs with steeper survival curves (high γ) would likely have schedule-dependent cytotoxicity, for which C_ss /KC₅₀ could adequately predict the preferable dosing schedule in antitumor efficacy.

Fig. 7. The impact of hill factor γ on cytotoxicity predicted by C_ss /KC₅₀ at two dosing regimens.

Simulation were performed with k_max = 0.6 day⁻¹, k_l = 0.006 ${(day\cdot nM)}^{-1}$, k_t = 0.05 day⁻¹ and f_u = 0.1, unless otherwise specified.

Discussion

Cytotoxic anti-cancer effect is associated with both drug concentrations and treatment duration (corresponding to incubation time in in vitro system). Two components are actually intertwined and it is hardly to assess their independent contributions to overall cytotoxic effects. However, generally defining the whole tumor cells population into relatively higher time- or concentration-dependent subpopulation would help us to approximately evaluate their relative contribution and select the favored combinations of two elements for high cytotoxicity. We believe that one chemotherapy, which is a result of various combinations of exposure magnitude and duration, could be potentially tailored by adjusting dose level and dosing frequency to achieve maximal cytotoxic effect and tumor suppression. In contrast to conventional MTD-based methods [33], we sought for a fine combinations of exposure magnitude and duration in this study. We found that quantitatively analyzing the relative contributions of both factors to overall cytotoxicity could help to implicate the favored dosing schedules. And the developed a cellular PD model provided a proof-of-concept to select favored dosing regimen between MTD and LDHF.

The developed PD model was applied to simulate a virtual clinical trial based on the PD parameters obtained by analyzing the in vitro cytotoxic data of paclitaxel on breast cancer cell line MCF-7 and several others. The simulations suggested that weekly dosing of paclitaxel at a similar accumulative dose level was favored over triweekly regimen in treatment of breast cancer and several others. This is consistent with clinical observations, even though the clinical trial only observed modest survival benefit (81.5% vs 76.9% at 5 years) from weekly dosing [30] in comparison with triweekly dosing. Given that our model just predicted benefit in tumor shrinkage rather than survival, poor translation from tumor sizes to patients survive may partly explain the difference between our prediction and clinical observations. In addition, it is important to note that we simulated the trajectory of tumor shrinkage in order to provide a proof-of-concept to implicate favored dosing regimen rather to make accurate predictions of tumor growth trajectories. Therefore, cautions should be possessed to make straightforward comparison between our simulations with clinical observed tumor trajectories.

The favored dosing method in chemotherapy was found to be highly dependent on the ratio of effective steady-state drug concentrations (C_ss) and the cytotoxic sensitivity (KC₅₀). For chemotherapies with high MTD (relatively low system toxicity), when C_ss is higher than KC₅₀, prolonged exposure time (high dosing frequency) is suggested potentially favored with higher tumor suppression. Drug concentrations (C_ss) at solid tumors are largely predicted by therapeutic doses and drug PK properties, and KC₅₀ is a parameter that is specific to a given drug in a certain cell line. Therefore, the favored dosing regimen for a given chemotherapy is determined not only by the types of chemotherapeutic drugs and their therapeutic doses, but also by the tumor types and their cytotoxic sensitivity to chemotherapy. KC₅₀ is a parameter readily obtained by fitting in vitro cytotoxic data using the developed cellular PD model. Information on drug PK and tumor bio-distribution is mostly available in the literature. In addition, with the developed PD model, other parameters in relation to drug sensitivity and the fractions of subpopulation are also attainable. Thus, we think such a proof-of-concept presented in this study is readily translatable and could be easily evaluated in many other types of chemotherapies.

Surprisingly, we found that tumor suppressive differences between two dosing regimens (MTD and LDHF) significantly decreased with the increase of time-dependent subpopulation (higher f_u in Class II and III, lower f_u in Class I and IV). This finding is contradictory to our previous thought that high time-dependency in cytotoxicity expected strong schedule-dependency [24], wherein prolonged exposure duration (LDHF) is usually therapeutically preferred. In the literature, the ratio of IC₅₀ at two incubation times (e.g., 1 vs 24 hr) was ever taken to indicate the degree of time-dependency in cytotoxicity, which was further used to suggest the optimal dosing approach [24]. However, this traditional concept is not well supported by our simulations. This is mainly because the time delays in in vitro studies is usually on a short temporal scale (hours to days) than clinical dosing (weeks to months). Such shorter delays observed in in vitro studies would not significantly affect clinical situations and clinical dosing schedules. In contrast, our model suggested that the ratio of concentrations to sensitivity (C_ss /KC₅₀) was a key factor in selection of two dosing regimens. Theoretically, when cytotoxicity becomes completely time-dependent (f_u = 1), MTD and LDHF regimens would exert almost the same efficacy (Fig. 6). Even though more investigations are warranted, there have been certain studies that have suggested that prolonged duration did not always resulted in improved antitumor effect for time-dependent chemotherapeutic agents [10]. As suggested by the study of Au et al., similar antitumor effects were observed when paclitaxel treatment longer than 12 h (12 h - 96 h) [10].

Nevertheless, we should be aware that this is just a proof-of-concept study and several caveats of this approach should be clarified. The model is not applicable to cases with intrinsic resistance, as in this model assumptions, tumor cells would die sooner or later upon the exposure of drugs and there is no fraction of resistant cells surviving over therapeutic treatment. In addition, even though we strongly believe that in vitro cytotoxicity data would implicate dosing regimens as it reveals the relative contributions of two fundamental PD drivers, many other factors in relation to tumor complex physiopathology, drug toxicity, drug mechanism of actions, and dynamic tumor microenvironments would also influence the chemotherapeutic dosing regimen [34]. Of note, this study was to provide an approach to get a quick speculation on dosing regimen simply based on in vitro cytotoxicity data. Another caveat of our approach is associated with the model structure and assumptions. After many rounds of model competitions, we finally selected linear killing model to describe time-sensitive subpopulation, which partly explained why C_ss / KC₅₀ was found as a critical determinant for dosing selection. But such a model assumption of linear killing effect on time-sensitive subpopulation would make model parameters fully identifiable from in vitro cytotoxic data.

Conclusions

The present study developed a cellular PD model to select chemotherapeutic dosing between MTD and LDHF. Our results suggested that the favored dosing approach was largely predicted by the ratio of drug concentrations at tumors (C_ss) to the drug cytotoxic sensitivity (KC₅₀) on tumor cells. This is a proof-of-concept study providing a simple approach to speculate the favored chemotherapeutic regimens solely based on analysis of in vitro cytotoxic data.