A multicompartment pharmacokinetic model of the interaction between paclitaxel and doxorubicin
 Kenneth JE Vos^{1},
 Angela G Martin^{1},
 Maxine G Trimboli^{1},
 Lindsay Forestell^{1},
 Khaled Barakat^{2, 3} and
 Jack A Tuszynski^{2, 4}Email author
DOI: 10.1140/epjnbp/s403660140013x
© Vos et al.; licensee Springer on behalf of EPJ. 2014
Received: 12 June 2014
Accepted: 5 September 2014
Published: 5 November 2014
Abstract
Background
In this paper the interactions between paclitaxel, doxorubicin and the metabolic enzyme CYP3A4 are studied using computational models. The obtained results are compared with those of available clinical data sets. Analysis of the drugenzyme interactions leads to a recommendation of an optimized paclitaxeldoxorubicin drug regime for chemotherapy treatment.
Methods
A saturable multicompartment pharmacokinetic model for the multidrug treatment of cancer using paclitaxel and doxorubicin in a combination is developed. The model’s kinetic equations are then solved using standard numerical methods for solving systems of nonlinear differential equations. The parameters were adjusted by fitting to available clinical data. In addition, we studied the interaction of each drug with the metabolic enzyme CYP3A4 through blind docking simulations to demonstrate that these drugs compete for the same metabolic enzyme and to show their molecular mode of binding. This provides a molecularlevel justification for the introduction of interaction terms in the kinetic model.
Results
Using docking simulations we compared the relative binding affinities for the metabolic enzyme of the two chemotherapy drugs. Since paclitaxel binds more strongly to CYP3A4 than doxorubicin, an explanation is given why doxorubicin has no apparent influence upon paclitaxel, while paclitaxel has a profound effect upon doxorubicin. Finally, we studied different time sequences of paclitaxel and doxorubicin concentrations and calculated their AUCs.
Conclusions
We have found excellent agreement between our model and available empirical clinical data for the drug combination studied here. To support the kinetic model at a molecular level, we built an atomistic threedimensional model of the ligands interacting with the metabolic enzyme and elucidated the binding modes of paclitaxel and doxorubicin within CYP3A4. Blind docking simulations provided estimates of the corresponding binding energies. The paper is concluded with clinical implications for the administration of the two drugs in combination.
Keywords
Paclitaxel Doxorubicin Metabolism Combination chemotherapy Molecular dynamics Docking binding energy Multicompartment model Pharmacokinetics PharmacodynamicsBackground
The accumulating knowledge about the molecular biology of cancer and novel delivery tools to specifically target aberrant proteins are opening up new therapeutic possibilities. One of the most common methods of increasing cure rates using chemotherapeutic agents is to administer combination chemotherapy which most often refers to the simultaneous administration of two or more medicinal compounds or modalities to treat a single disease. This approach in cancer treatment can be traced back to 1965 when James Holland, Emil Freireich, and Emil Frei hypothesized that cancer chemotherapy should follow the strategy of antibiotic therapy for tuberculosis and use combinations of drugs, each with a different mechanism of action. Cancer cells could conceivably mutate to become resistant to a single agent, but by using different drugs concurrently it would be more difficult for the tumor to develop resistance to the combination. Holland, Freireich, and Frei simultaneously administered an antifolate, a vinca alkaloid, 6MP and Prednisone  referred to as the POMP regimen  and induced longterm remissions in children with ALL. This approach was extended to the lymphomas and other types of cancer. Currently, nearly all successful cancer chemotherapy regimens use this paradigm of multiple drugs given simultaneously. Some types of cancers, previously universally fatal, are now considered to be generally curable diseases as a result of this advance.
Current clinical trials in oncology commonly focus on three key aspects: (a) extending the scope of known drugs to new types of cancer, (b) testing new compounds, and (c) optimizing treatment by using combinations of known compounds. The latter aspect is of great interest and could benefit from a mathematical modelling approach aimed at achieving an optimized set of parameters for the dose, frequency and route of administration. This is, indeed, one of our main objectives behind the present work.
Two of the most widely used chemotherapy agents are paclitaxel and doxorubicin. Paclitaxel [5] is active against many human solid tumors related to breast cancer, ovarian cancer, nonsmall cell lung cancer, head and neck cancer, and advanced forms of Kaposi’s sarcoma [6]. Because it is poorly watersoluble, the current formulation for Taxol®; incorporates a 6 mg/ml solution in a solvent consisting of 50% polyoxyethylated castor oil (Cremophor®; ELCrEL) and 50% dehydrated alcohol (USP). Paclitaxel is typically administered by IV infusion over one or three hours but has also been administered over six and twenty four hours. Because a patient may have an anaphylactic reaction to CrEL, alternative formulations of paclitaxel have been introduced, including BMS184476, oral paclitaxel in polysorbate 80, and ABI007, GenexolPM [7] and Abraxane®; (nabpaclitaxel) [8]. Paclitaxel has a long residence time within the body and can stay trapped in cancer cells for over a week [9]. Paclitaxel is also highly bound to CrEL micelles, plasma proteins, platelets, and red blood cells [10].
DNA intercalators inhibit DNA polymerases and topoisomerases, resulting in the induction of apoptosis in tumor cells. DNA intercalating agents, such as amsacrine, actinomycin, mitoxantrone, and doxorubicin, have been employed as anticancer drugs and are in routine clinical use as chemotherapeutic agents [11]. It is well accepted that the antitumor activity of doxorubicin is caused by the formation of a cleavable complex of topoisomerase II, resulting in apoptosis [12],[13]. Doxorubicin is indicated in the treatment of a broad spectrum of solid tumors (e.g. breast, bladder, endometrium, thyroid, lung, ovary, stomach, and sarcomas of the bone) and in the treatment of lymphoma, as well as acute lymphoblastic and myeloblastic leukemias [14]. One of the most important and clinically relevant side effects of doxorubicin is the induction of cardiomyopathy [15]. A number of mechanisms have been proposed to explain this effect of doxorubicin, including oxidative stress [16], the induction of mitochondrial damage [17], and changes in gene expression in cardiac myocytes and muscle cells in general [18],[19].
Pharmacokinetics is the study of the absorption, distribution, metabolism and elimination (ADME) of drugs to, in, and from the body [20]. Pharmacological data usually consist of discrete values of the concentration of a drug in the plasma as a function of time. For drugs administered by direct intravenous (IV) infusion, a plot of these values generates a concentrationversustime curve that rises during the infusion and then decreases after a maximum concentration value is reached. This decline may be relatively short or may last for several days, and it is mainly governed by the rate of elimination of the drug from the body. One of the key questions investigated is the functional dependence of the elimination curve and a single parameter that is often used to characterize the drug, namely its halflife. Another parameter of interest is the power law exponent of the elimination curve, which provides important information about the fate of the drug in the body and its efficacy. During clinical trials, the concentrationversustime curves are used to determine optimum dosing regimens, potential toxicities, and drugdrug interactions.
The implicit solution in Equation 10 must be modified for two special cases. The first special case is when a=1, the first term must be replaced with ${K}_{M}ln\left[\frac{X}{{X}_{o}}\right]$. The second special case is when a−b=1, the second term must be replaced with $ln\left[\frac{X}{{X}_{o}}\right]$.
where X_{i,j} is the concentration of drug i in compartment j, I_{i,j} is the rate of infusion divided by the volume of distribution of drug i in compartment j, compartment 0 (k=0 in the Equation 11 sum) is the environment external to the body, K_{i,j,k} is the kinetic rate coefficient for drug i moving from compartment j to compartment k, Γ_{i,j,k} is the saturation coefficient for drug i moving from compartment j to compartment k, and A(i,j,k) and B(i,j,k) are the fractal exponents for drug i moving from compartment j to compartment k.
The primary motivation for this paper is to quantitatively understand the combined action of paclitaxel and doxorubicin in view of the reported interactions when these drugs are administered simultaneously. Namely, it has been demonstrated that the combination of doxorubicin followed by infusion of paclitaxel has high antitumor activity in patients with metastatic breast cancer, but is strongly limited by the associated cardiac toxicity [24]. The question that arises is how these side effects can be reduced by optimal mode of administration of the two drugs. This requires careful analysis and model development. Both paclitaxel and doxorubicin are known to be substrates for specific metabolic enzymes. Such interactions with the metabolic enzymes raise the risk of alleviating their associated toxicity and inducing unpredictable adverse side effects due to metabolites. Given this clue, here we develop a predictive model of the pharmacokinetic and pharmacodynamic interactions between paclitaxel, doxorubicin and the metabolic enzymes in the body during multidrug chemotherapy. Our pharmacokinetic model can be used in aiding the design of optimized dosage and scheduling of the twodrug combination.
In this paper, we discuss the results obtained at both levels of computational modeling: pharmacokinetics of the two drugs under investigation and molecular dynamics of their interactions with the metabolic enzyme.
Methods
Below is a short description of the pharmacokinetic model mentioned above. The model focuses on the interactions of paclitaxel and doxorubicin with a common metabolic enzyme for the two compounds. To gain further understanding of these interactions at the molecular level, we performed a blind docking simulation protocol, which was employed to predict the binding sites, modes of binding and binding energies of the two drugs to the CYP3A4 metabolic enzyme. This was intended to explain at a molecular level why these two drugs affect each other’s pharmacokinetic profiles. Below is a description of the modeling techniques used both for pharmacokinetic analysis and molecular dynamics simulations of the two drugs, paclitaxel and doxorubicin.
Pharmacokinetic modeling
The mathematical model corresponding to Figure 4 is Equation 11, which represents a system of nonlinear differential equations and in general, the kinetic rate coefficients do not have units of inverse time. Only in the special cases when all of the A’s are equal to one and the drugs have the same units in all the compartments, will the kinetic rate coefficients have units of inverse time. We defined I as the rate of infusion divided by the volume of distribution for the entire body, which is commonly known for most drugs from empirical data. In our case the volume of distribution is an adjustable parameter used to optimize fits to the available data sets. The volume of distribution for specific organs would be desirable but is very hard to determine, especially for human patients. We would require empirical data for the drug concentrations in several compartments in order to generate reliable parameter fits.
The initial starting values for the parameters in Equation 11 were assumed to be for a linear system. i.e. the exponents were all set to one, the interaction parameters were set to zero, and the saturation parameters were set to zero. The clinical halflife for each of the studies was used to estimate the kinetic rate coefficients. The X_{i,j}(t_{ k }) in Equation 12 were determined by solving the system of first order differential equations using standard numerical methods such as the RungeKutta Method. The maximum number of differential equations solved was fourteen for this system of drugs and their metabolites. The weighted percentage variance, VAR, can then be calculated for this specific set of parameters. The kinetic rate coefficients, the saturation parameters, and the exponents are then allowed to vary with the maximum number of parameters varying not exceeding the number of clinical data values for the concentrations minus one. The appropriate values for the parameters are obtained by finding the minimum value of VAR. The VAR can be minimized by varying these parameters using the Powell’s Method in Multidimensions.
An additional aspect over and above pharmacokinetic modeling that enhances our analysis is the molecularlevel understanding of the nature of interactions between each of the two drugs and the metabolic enzyme CYP3A4. The latter effort requires an entirely different modeling methodology, namely molecular dynamics, which is briefly described below.
Molecular dynamics methodology
In order to determine if both doxorubicin and paclitaxel are substrates for the same metabolic enzyme, we used blind docking combined with molecular dynamics simulations to characterize the mode of action of the two drugs at the molecular level. We used the human microsomal cytochrome P450 3A4 crystal structure (PDB: entry 1TQN) [25]. The catalytic active site was wellcharacterized and included a HEM group. Prior to docking simulations, protonation states of the residues constituting the CYP3A4 including the HEM group were adjusted using the software PDB2PQR [25]. The protein structure was conformationally relaxed using the NAMD molecular dynamics software with constraints on the backbone atoms. The AMBER99SB force field [26] was used for protein parameterization, while the GAFF provided parameters for the HEM group.
To carry out the blind docking protocol, the entire surface of the CYP3A4 was divided into 90 focus docking regions. For each region, the center of mass of three solvent exposed neighboring atoms was used as the center of the docking box. The dimensions of the docking cube were 90×90×90 points with grid spacing of 0.03 nm. Clustering of the docking poses was performed using a 0.2 nm RMSD cutoff. To identify the most preferred binding locations, we combined all docking results and ranked them with the lowest binding energy of the largest docking cluster. As we are not only interested in the binding energies as indicators for adequate binding, we defined a hit as the docking run that includes at least 20% of the total population in its largest cluster. Finally, the solventexposed atoms that were used to construct the docking boxes have been used as markers for the binding locations.
All docking simulations were performed using AutoDock, version 4.0. Hydrogen atoms were added to all CYP3A4 and the two ligands followed by assigning their partial atomic charges using the GasteigerMarsili method. Atomic solvation parameters were assigned to the protein atoms using the AutoDock utility ADDSOL. A docking grid map with 90×90×90 points and grid point spacing of 0.03 nm has been calculated using AUTOGRID program. The grid box was centered on the active site. Rotatable bonds of each ligand were then automatically assigned using AUTOTORS utility of AutoDock. Docking was performed using the LGA method with an initial population of 400 random individuals, a maximum number of 10,000,000 energy evaluations, 100 trials, 50,000 maximum generations, a mutation rate of 0.02, a crossover rate of 0.80 and the requirement that only one individual can survive into the next generation.
Results and discussions
Pharmacokinetics of doxorubicin
The pharmacokinetics of a drug varies quite significantly between people and even in an individual person (pharmacokinetics will change with age and the health of the patient). Ideally, we would want a pharmacokinetic model with parameter values for each individual patient and the analysis would then be optimized for that specific patient; however, that is not practical. Therefore, we will calculate a set of parameters for a mean concentration versus time curve and obtain general qualitative features that will describe the people in this group as a whole.
The PK parameters for Equation 11 modeling doxorubicin
j,k  A _{D O X,j,k}  B _{D O X,j,k}  Γ _{D O X,j,k}  K _{D O X,j,k} 

1,2  0.90  1.99  0.0005  $8.77\left\frac{10.017{X}_{\mathit{\text{DOX}},2}^{1.78}}{1+0.189{X}_{\mathit{\text{DOX}},2}^{1.03}}\right$ 
1,3  1.08  1.71  0.0038  $3.55\left\frac{1+3.09{X}_{\mathit{\text{DOX}},3}}{1+1.53{X}_{\mathit{\text{DOX}},3}}\right$ 
1,4  1.97  0.96  0.0008  $0.354\left1+0.0002{X}_{\mathit{\text{DOX}},1}^{2.77}\right\left\frac{1+0.306{X}_{\mathit{\text{DOX}},4}^{1.51}}{1+0.0253{X}_{\mathit{\text{DOX}},4}^{0.40}}\right$ 
2,0  0.94  1.00  0.0000  0.252 
2,1  0.72  0.80  0.1466  $0.132\left\frac{1+9.35{X}_{\mathit{\text{DOX}},1}^{0.96}}{1+0.020{X}_{\mathit{\text{DOX}},1}^{0.84}}\right$ 
3,1  0.97  1.16  0.0242  $0.0325\left\frac{1+2.40{X}_{\mathit{\text{DOX}},1}^{1.34}}{1+0.165{X}_{\mathit{\text{DOX}},1}^{1.01}}\right$ 
4,1  1.91  1.00  0.0000  $0.0055\left\frac{1+6.67{X}_{\mathit{\text{DOX}},1}^{0.86}}{1+0.15{X}_{\mathit{\text{DOX}},1}^{1.27}}\right$ 
The PK parameters for Equation 11 modeling doxorubicinol
j,k  A _{D O L,j,k}  B _{D O L,j,k}  Γ _{D O L,j,k}  K _{D O L,j,k} 

1,2  2.11  1.41  0.5836  $12.1\left\frac{1+10.5{X}_{\mathit{\text{DOL}},2}^{0.89}}{1+4.65{X}_{\mathit{\text{DOL}},2}^{1.67}}\right$ 
1,3  1.19  1.00  0.0000  $3.41\left1+0.0702{X}_{\mathit{\text{DOL}},1}^{0.90}\right\left\frac{1+0.967{X}_{\mathit{\text{DOL}},3}^{1.92}}{1+0.0466{X}_{\mathit{\text{DOL}},3}^{1.42}}\right$ 
2,0  0.79  1.36  1.8532  $8.86\left10.0228{X}_{\mathit{\text{DOL}},2}^{1.20}\right$ 
2,1  1.03  1.08  8.5081  $3.30\left1+0.0368{X}_{\mathit{\text{DOL}},2}^{2.09}\right\left\frac{1+0.0943{X}_{\mathit{\text{DOL}},1}^{1.43}}{1+3.86{X}_{\mathit{\text{DOL}},1}^{0.76}}\right$ 
3,1  0.64  1.00  0.0000  $0.0601\left1+0.0054{X}_{\mathit{\text{DOL}},3}^{1.29}\right\left\frac{15.87{X}_{\mathit{\text{DOL}},1}^{1.68}}{1+2.38{X}_{\mathit{\text{DOL}},1}^{1.20}}\right$ 
Note that some of the parameters for doxorubicin in Table 1 do not play a significant roll in fitting Equation 11 to the clinical data. One of A parameters in Table 1 is close to 1, and can be set to one. Two of the saturation parameters, Γ, are very small and two are zero. In addition, the 0.0002 in row 1,4 can be set to zero. These parameters can be ignored without losing much information. This would reduce the number of doxorubicin parameters to 42. In the case of doxorubicinol, all the parameters in Table 2 have an effect with one exception. The exponent A=1.03 could be set to 1 without a significant change to the results. Also, one of the three exponents in Equation 14, 0.98, is close to one and could be set to one as well. This reduces the number of doxorubicinol parameters to 43.
j,k  A _{D O X,j,k}  B _{D O X,j,k}  Γ _{D O X,j,k}  K _{D O X,j,k} 

1,2  0.79  2.52  0.0006  $8.14\left\frac{10.0254{X}_{\mathit{\text{DOX}},2}^{2.59}}{1+0.319{X}_{\mathit{\text{DOX}},2}^{1.59}}\right$ 
1,3  0.83  2.43  0.0039  $3.08\left\frac{1+3.12{X}_{\mathit{\text{DOX}},3}^{0.97}}{1+1.44{X}_{\mathit{\text{DOX}},3}^{0.97}}\right$ 
1,4  1.59  1.62  0.0012  $0.359\left1+0.0001{X}_{\mathit{\text{DOX}},1}^{1.55}\right\left\frac{1+0.277{X}_{\mathit{\text{DOX}},4}^{1.63}}{1+0.0282{X}_{\mathit{\text{DOX}},4}^{0.25}}\right$ 
2,0  1.33  1.00  0.0000  0.264 
2,1  1.01  0.53  0.0994  $0.135\left\frac{1+13.9{X}_{\mathit{\text{DOX}},1}^{0.87}}{1+0.1672{X}_{\mathit{\text{DOX}},1}^{0.66}}\right$ 
3,1  1.04  1.12  0.0287  $0.0325\left\frac{1+2.02{X}_{\mathit{\text{DOX}},1}^{1.74}}{1+0.143{X}_{\mathit{\text{DOX}},1}^{0.61}}\right$ 
4,1  1.79  1.00  0.0000  $0.0046\left\frac{1+7.92{X}_{\mathit{\text{DOX}},1}^{1.04}}{1+0.243{X}_{\mathit{\text{DOX}},1}^{1.95}}\right$ 
j,k  A _{D O L,j,k}  B _{D O L,j,k}  Γ _{D O L,j,k}  K _{D O L,j,k} 

1,2  1.76  2.43  0.3371  $14.0\left\frac{1+10.6{X}_{\mathit{\text{DOL}},2}^{0.72}}{1+4.54{X}_{\mathit{\text{DOL}},2}^{1.92}}\right$ 
1,3  1.07  1.00  0.0000  $3.92\left1+0.1150{X}_{\mathit{\text{DOL}},1}^{0.55}\right\left\frac{1+1.02{X}_{\mathit{\text{DOL}},3}^{1.52}}{1+0.0482{X}_{\mathit{\text{DOL}},3}^{2.43}}\right$ 
2,0  0.72  1.83  1.5722  $8.58\left10.0167{X}_{\mathit{\text{DOL}},2}^{1.55}\right$ 
2,1  0.91  1.34  6.8599  $3.90\left1+0.0646{X}_{\mathit{\text{DOL}},2}^{1.18}\right\left\frac{1+0.1282{X}_{\mathit{\text{DOL}},1}^{0.95}}{1+3.97{X}_{\mathit{\text{DOL}},1}^{1.07}}\right$ 
3,1  0.79  1.00  0.0000  $0.0594\left1+0.0073{X}_{\mathit{\text{DOL}},3}^{2.26}\right\left\frac{16.94{X}_{\mathit{\text{DOL}},1}^{1.17}}{1+3.19{X}_{\mathit{\text{DOL}},1}^{0.96}}\right$ 
j,k  A _{D O X,j,k}  B _{D O X,j,k}  Γ _{D O X,j,k}  K _{D O X,j,k} 

1,2  0.86  2.14  0.0005  $8.86\left\frac{10.0231{X}_{\mathit{\text{DOX}},2}^{3.12}}{1+0.157{X}_{\mathit{\text{DOX}},2}^{0.89}}\right$ 
1,3  1.06  1.87  0.0036  $3.69\left\frac{1+3.09{X}_{\mathit{\text{DOX}},3}}{1+1.61{X}_{\mathit{\text{DOX}},3}}\right$ 
1,4  2.07  0.86  0.0007  $0.375\left1+0.0003{X}_{\mathit{\text{DOX}},1}^{2.99}\right\left\frac{1+0.375{X}_{\mathit{\text{DOX}},4}^{1.59}}{1+0.0222{X}_{\mathit{\text{DOX}},4}^{0.31}}\right$ 
2,0  1.38  1.00  0.0000  0.264 
2,1  0.91  0.51  0.2295  $0.102\left\frac{1+11.2{X}_{\mathit{\text{DOX}},1}^{0.90}}{1+0.181{X}_{\mathit{\text{DOX}},1}^{0.91}}\right$ 
3,1  1.02  0.90  0.0264  $0.0369\left\frac{1+1.74{X}_{\mathit{\text{DOX}},1}^{0.96}}{1+0.298{X}_{\mathit{\text{DOX}},1}^{1.48}}\right$ 
4,1  1.82  1.00  0.0000  $0.0052\left\frac{1+6.42{X}_{\mathit{\text{DOX}},1}^{1.02}}{1+0.154{X}_{\mathit{\text{DOX}},1}^{2.17}}\right$ 
j,k  A _{D O L,j,k}  B _{D O L,j,k}  Γ _{ DOL,j,k }  K _{D O L,j,k} 

1,2  1.57  2.51  0.3261  $15.9\left\frac{1+14.0{X}_{\mathit{\text{DOL}},2}^{0.50}}{1+8.04{X}_{\mathit{\text{DOL}},2}^{1.24}}\right$ 
1,3  1.20  1.00  0.0000  $4.82\left1+0.0749{X}_{\mathit{\text{DOL}},1}^{0.95}\right\left\frac{1+0.711{X}_{\mathit{\text{DOL}},3}^{1.22}}{1+0.0433{X}_{\mathit{\text{DOL}},3}^{1.26}}\right$ 
2,0  0.66  2.36  1.3731  $13.8\left10.0162{X}_{\mathit{\text{DOL}},2}^{2.07}\right$ 
2,1  0.61  2.02  4.5141  $5.03\left1+0.0690{X}_{\mathit{\text{DOL}},2}^{1.11}\right\left\frac{1+0.135{X}_{\mathit{\text{DOL}},1}^{0.91}}{1+2.61{X}_{\mathit{\text{DOL}},1}^{1.37}}\right$ 
3,1  0.60  1.00  0.0000  $0.0479\left1+0.0047{X}_{\mathit{\text{DOL}},3}^{1.10}\right\left\frac{17.87{X}_{\mathit{\text{DOL}},1}^{1.45}}{1+2.33{X}_{\mathit{\text{DOL}},1}^{1.22}}\right$ 
Pharmacokinetics of paclitaxel
The PK parameters for Equation 11 modeling paclitaxel
j,k  A _{P T X,j,k}  B _{P T X,j,k}  Γ _{P T X,j,k}  K _{P T X,j,k} 

1,2  1.06  1.79  0.0001  $13.8\left\frac{1+0.0117{X}_{\mathit{\text{PTX}},2}^{1.05}}{1+3.37{X}_{\mathit{\text{PTX}},2}^{0.96}}\right$ 
1,3  1.40  1.62  0.0000  $29.1\left10.0002{X}_{\mathit{\text{PTX}},1}^{1.68}\right\left\frac{1+0.0004{X}_{\mathit{\text{PTX}},3}^{1.24}}{1+0.551{X}_{\mathit{\text{PTX}},3}^{1.22}}\right$ 
1,4  1.23  1.10  0.0000  $2.85\left\frac{1+0.0082{X}_{\mathit{\text{PTX}},4}^{1.01}}{1+0.117{X}_{\mathit{\text{PTX}},4}^{1.22}}\right$ 
2,0  1.03  1.99  0.0876  $2.98\left10.0023{X}_{\mathit{\text{PTX}},2}^{2.09}\right$ 
2,1  0.94  1.09  0.0028  $0.0523\left\frac{10.0023{X}_{\mathit{\text{PTX}},1}^{0.97}}{1+0.0015{X}_{\mathit{\text{PTX}},1}^{1.32}}\right$ 
3,1  1.66  1.93  0.00003  $0.0230\left\frac{1+0.0423{X}_{\mathit{\text{PTX}},1}^{1.75}}{1+0.0199{X}_{\mathit{\text{PTX}},1}^{1.60}}\right$ 
4,1  1.78  1.83  0.0009  $0.0061\left\frac{10.0054{X}_{\mathit{\text{PTX}},1}^{0.89}}{1+6.36{X}_{\mathit{\text{PTX}},1}^{0.74}}\right$ 
The PK parameters for Equation 11 modeling 6 α hydroxypaclitaxel
j,k  A _{O H P,j,k}  B _{O H P,j,k}  Γ _{O H P,j,k}  K _{O H P,j,k} 

1,2  1  1  21.3666  $12.2\left\frac{10.523{X}_{\mathit{\text{OHP}},2}}{1+0.204{X}_{\mathit{\text{OHP}},2}}\right$ 
1,3  1  1  0.0808  1.671+0.166X_{O H P,3} 
2,0  1  1  0.0000  23.9 
2,1  1  1  0.0000  0.1031+0.0182X_{O H P,2}1+2.54X_{O H P,1} 
3,1  1  1  0.0000  $0.0468\left1+0.0975{X}_{\mathit{\text{OHP}},3}\right\left\frac{115.2{X}_{\mathit{\text{OHP}},1}}{1+28.9{X}_{\mathit{\text{OHP}},1}}\right$ 
The coefficient 37.5 in Equation 17 represents the fraction of eliminated paclitaxel being converted into 6αhydroxypaclitaxel times the volume of distribution for paclitaxel divided by the volume of distribution for 6αhydroxypaclitaxel. The weighted percentage variance for 6αhydroxypaclitaxel was calculated to be V A R=0.043 and the weighted variance was determined to be S=0.043 micromoles squared per liter squared without the paclitaxel. The combined weighted percentage variance for paclitaxel and its metabolite is V A R=0.036. The combined weighted variance is S=0.16 micromoles squared per liter squared for paclitaxel and its metabolite.
These models describe the pharmacokinetic effects of the drugs when there is no interaction between the drugs. The clinical data shows that they do influence each other’s metabolism. We intend to find out what the nature of their molecular interactions with the key metabolic enzyme are. In particular, we have hypothesized that both of the drugs compete for the same liver enzyme and the metabolic action of this enzyme, CYP3A4, can be saturated by one type of the drug molecules leaving the other unmetabolized. The drug with a higher binding free energy would have a greater probability of being metabolized. This is discussed in some detail below based on atomic level MD simulations.
The pharmacodynamic interaction of Doxorubicin and Paclitaxel with CYP3A4
Based on the above simulations, it appears that both paclitaxel and doxorubicin strongly bind to the same metabolic enzyme, CYP3A4. Their binding locations are distinct, but conformational changes associated with the binding process may result in their competitive inhibition of each other. Furthermore, paclitaxel is predicted to bind significantly more strongly to CYP3A4 than doxorubicin giving a greater probability of being metabolized when administered simultaneously with doxorubicin. It is, therefore, reasonable to recommend the administration of doxorubicin after paclitaxel. This is further analyzed and quantified below.
The pharmacokinetic interaction between Doxorubicin and Paclitaxel
In contrast, we expect the presence of paclitaxel to have a strong influence on the metabolism of doxorubicin. As predicted in the previous subsection, paclitaxel will bind to CYP3A4 more strongly than doxorubicin and if the two drugs are simultaneously present, we expect paclitaxel to replace doxorubicin that is bound to CYP3A4. The doxorubicin and doxorubicinol parameters given in Tables 3 and 4 are for the best fit to the data from Gianni et al. [27], and the doxorubicin and doxorubicinol parameters in Tables 5 and 6 are for the best fit to the data from Moreira et al. [28]. The influence of paclitaxel on doxorubicin can be included in the model given in Equation 11 by modifying the rate coefficients, K_{i,j,k} given in these tables. The theoretical curve was fitted to the data by minimizing VAR in Equation 12. The number of data points in the Gianni et al. [27] study during the interaction of the drugs is 46 for doxorubicin (N_{ p }=568) and 48 for doxorubicinol (N_{ p }=494). The weighted percentage variance is V A R=0.069 and the weighted variance is S=1.6 micromoles squared per liter squared for doxorubicin. The weighted percentage variance is V A R=0.063 and the weighted variance is S=0.00010 micromoles squared per liter squared for doxorubicinol. The combined weighted percentage variance for all the drugs (145 data values and N_{ p }=1670) is V A R=0.068 and the combined weighted variance is S=1.1 micromoles squared per liter squared.
j,k  K _{D O X,j,k} 

1,2  $8.14\left\frac{10.0254{X}_{\mathit{\text{DOX}},2}^{2.59}}{1+0.319{X}_{\mathit{\text{DOX}},2}^{1.59}}\right\left1+\frac{1.70{X}_{\mathit{\text{PTX}},1}}{1+0.000323{X}_{\mathit{\text{PTX}},1}}+\frac{0.233{X}_{\mathit{\text{PTX}},2}}{1+0.00193{X}_{\mathit{\text{PTX}},2}}\right$ 
1,3  $3.08\left\frac{1+3.12{X}_{\mathit{\text{DOX}},3}^{0.97}}{1+1.45{X}_{\mathit{\text{DOX}},3}^{0.97}}\right$ 
1,4  $0.359\left1+0.0001{X}_{\mathit{\text{DOX}},1}^{1.55}\right\left\frac{1+0.277{X}_{\mathit{\text{DOX}},4}^{1.63}}{1+0.0282{X}_{\mathit{\text{DOX}},4}^{0.25}}\right$ 
2,0  $0.264\left1\frac{0.314{X}_{\mathit{\text{PTX}},2}}{1+0.0119{X}_{\mathit{\text{PTX}},2}}\right$ 
2,1  $0.135\left\frac{1+13.9{X}_{\mathit{\text{DOX}},1}^{0.87}}{1+0.167{X}_{\mathit{\text{DOX}},1}^{0.66}}\right\left1+\frac{3.08{X}_{\mathit{\text{PTX}},1}}{1+0.0441{X}_{\mathit{\text{PTX}},1}}\frac{0.0003{X}_{\mathit{\text{PTX}},2}}{1+0.0078{X}_{\mathit{\text{PTX}},2}}\right$ 
3,1  $0.0325\left\frac{1+2.02{X}_{\mathit{\text{DOX}},1}^{1.74}}{1+0.143{X}_{\mathit{\text{DOX}},1}^{0.61}}\right$ 
4,1  $0.0046\left\frac{1+7.92{X}_{\mathit{\text{DOX}},1}^{1.04}}{1+0.243{X}_{\mathit{\text{DOX}},1}^{1.95}}\right$ 
j,k  K _{D O L,j,k} 

1,2  $14.0\left\frac{1+10.6{X}_{\mathit{\text{DOL}},2}^{0.72}}{1+4.54{X}_{\mathit{\text{DOL}},2}^{1.67}}\right\left1\frac{0.00008{X}_{\mathit{\text{PTX}},1}^{0.39}}{1+0.00007{X}_{\mathit{\text{PTX}},1}^{3.64}}\frac{0.0368{X}_{\mathit{\text{PTX}},2}^{3.10}}{1+0.00001{X}_{\mathit{\text{PTX}},2}^{0.60}}\right$ 
1,3  $3.92\left1+0.115{X}_{\mathit{\text{DOL}},1}^{0.55}\right\left\frac{1+1.02{X}_{\mathit{\text{DOL}},3}^{1.52}}{1+0.0482{X}_{\mathit{\text{DOL}},3}^{2.43}}\right\left1\frac{0.00095{X}_{\mathit{\text{PTX}},1}^{3.26}}{1+0.00006{X}_{\mathit{\text{PTX}},1}^{5.09}}\frac{0.0003{X}_{\mathit{\text{PTX}},3}^{2.64}}{1+0.0001{X}_{\mathit{\text{PTX}},3}^{2.58}}\right$ 
2,0  $8.58\left10.0167{X}_{\mathit{\text{DOL}},2}^{1.55}\right\left1\frac{0.0426{X}_{\mathit{\text{PTX}},2}^{0.457}}{1+0.00004{X}_{\mathit{\text{PTX}},2}^{2.28}}\right$ 
2,1  $3.90\left1+0.0646{X}_{\mathit{\text{DOL}},2}^{1.18}\right\left\frac{1+0.128{X}_{\mathit{\text{DOL}},1}^{0.95}}{1+3.97{X}_{\mathit{\text{DOL}},1}^{1.07}}\right\left1+\frac{0.0223{X}_{\mathit{\text{PTX}},1}^{2.22}}{1+0.00004{X}_{\mathit{\text{PTX}},1}^{0.21}}+\frac{1.11{X}_{\mathit{\text{PTX}},2}^{0.52}}{1+0.00001{X}_{\mathit{\text{PTX}},2}^{1.12}}\right$ 
3,1  $0.0594\left1+0.0073{X}_{\mathit{\text{DOL}},3}^{2.26}\right\left\frac{16.94{X}_{\mathit{\text{DOL}},1}^{1.17}}{1+3.19{X}_{\mathit{\text{DOL}},1}^{0.96}}\right\left1\frac{0.00003{X}_{\mathit{\text{PTX}},1}^{2.50}}{1+0.00002{X}_{\mathit{\text{PTX}},1}^{0.60}}+\frac{0.0010{X}_{\mathit{\text{PTX}},3}^{3.31}}{1+0.00004{X}_{\mathit{\text{PTX}},3}^{5.16}}\right$ 
The doxorubicin and doxorubicinol parameters given in Tables 5 and 6 are for the best fit to the data from Moreira et al. [28]. The influence of paclitaxel on doxorubicin and doxorubicinol can be included in the model given in Equation 11 by modifying the rate coefficients, K_{i,j,k} given in these tables. The theoretical curve was fitted to the data by minimizing VAR in Equation 12. The number of data points in the Moreira et al. [28] study during the interaction of the drugs is 14 for doxorubicin (N_{ p }=392) and 16 for doxorubicinol (N_{ p }=448). The weighted percentage variance is V A R=0.0028 and the weighted variance is S=1.2×10^{−5} micromoles squared per liter squared for doxorubicin. The weighted percentage variance is V A R=0.028 and the weighted variance is S=0.00019 micromoles squared per liter squared for doxorubicinol. The combined weighted percentage variance for all the drugs (69 data values and N_{ p }=1932) is V A R=0.042 and the combined weighted variance is S=0.082 micromoles squared per liter squared.
j,k  K _{D O X,j,k} 

1,2  $8.86\left\frac{10.0231{X}_{\mathit{\text{DOX}},2}^{3.12}}{1+0.157{X}_{\mathit{\text{DOX}},2}^{0.89}}\right\left1+\frac{0.0010{X}_{\mathit{\text{PTX}},1}}{1+0.00002{X}_{\mathit{\text{PTX}},1}}\frac{0.0003{X}_{\mathit{\text{PTX}},2}}{1+0.106{X}_{\mathit{\text{PTX}},2}}\right$ 
1,3  $3.69\left\frac{1+3.09{X}_{\mathit{\text{DOX}},3}^{1.02}}{1+1.61{X}_{\mathit{\text{DOX}},3}^{0.95}}\right$ 
1,4  $0.375\left1+0.0003{X}_{\mathit{\text{DOX}},1}^{2.99}\right\left\frac{1+0.375{X}_{\mathit{\text{DOX}},4}^{1.59}}{1+0.0222{X}_{\mathit{\text{DOX}},4}^{0.31}}\right$ 
2,0  $0.264\left1+0.00002{X}_{\mathit{\text{DOX}},2}^{1.90}\right\left1\frac{0.498{X}_{\mathit{\text{PTX}},2}}{1+0.101{X}_{\mathit{\text{PTX}},2}}\right$ 
2,1  $0.102\left\left(1+0.00001{X}_{\mathit{\text{DOX}},2}^{1.71}\right)\frac{1+11.2{X}_{\mathit{\text{DOX}},1}^{0.90}}{1+0.181{X}_{\mathit{\text{DOX}},1}^{0.91}}\right\left1\frac{0.0010{X}_{\mathit{\text{PTX}},1}}{1+0.00002{X}_{\mathit{\text{PTX}},1}}+\frac{0.240{X}_{\mathit{\text{PTX}},2}}{1+0.350{X}_{\mathit{\text{PTX}},2}}\right$ 
3,1  $0.0369\left\frac{1+1.74{X}_{\mathit{\text{DOX}},1}^{0.96}}{1+0.298{X}_{\mathit{\text{DOX}},1}^{1.48}}\right$ 
4,1  $0.0052\left\left(10.00001{X}_{\mathit{\text{DOX}},4}^{1.14}\right)\frac{1+6.42{X}_{\mathit{\text{DOX}},1}^{1.02}}{1+0.154{X}_{\mathit{\text{DOX}},1}^{2.17}}\right$ 
j,k  K _{D O L,j,k} 

1,2  $15.9\left\frac{1+14.0{X}_{\mathit{\text{DOL}},2}^{0.50}}{1+8.04{X}_{\mathit{\text{DOL}},2}^{1.24}}\right\left1+\frac{7.23{X}_{\mathit{\text{PTX}},1}^{0.66}}{1+0.0070{X}_{\mathit{\text{PTX}},1}^{1.46}}\frac{0.701{X}_{\mathit{\text{PTX}},2}^{0.28}}{1+0.0005{X}_{\mathit{\text{PTX}},2}^{6.85}}\right$ 
1,3  $4.82\left1+0.0749{X}_{\mathit{\text{DOL}},1}^{0.95}\right\left\frac{1+0.711{X}_{\mathit{\text{DOL}},3}^{1.22}}{1+0.0433{X}_{\mathit{\text{DOL}},3}^{1.26}}\right$ 
2,0  $13.8\left10.0162{X}_{\mathit{\text{DOL}},2}^{2.07}\right\left1\frac{0.131{X}_{\mathit{\text{PTX}},2}^{1.37}}{1+0.0004{X}_{\mathit{\text{PTX}},2}^{6.68}}\right$ 
2,1  $5.03\left1+0.0690{X}_{\mathit{\text{DOL}},2}^{1.11}\right\left\frac{1+0.135{X}_{\mathit{\text{DOL}},1}^{0.91}}{1+2.61{X}_{\mathit{\text{DOL}},1}^{1.37}}\right\left1\frac{0.0434{X}_{\mathit{\text{PTX}},1}^{1.10}}{1+0.0103{X}_{\mathit{\text{PTX}},1}^{2.41}}+\frac{10.9{X}_{\mathit{\text{PTX}},2}^{1.12}}{1+0.00007{X}_{\mathit{\text{PTX}},2}^{6.94}}\right$ 
3,1  $0.0479\left1+0.0047{X}_{\mathit{\text{DOL}},3}^{1.10}\right\left\frac{17.87{X}_{\mathit{\text{DOL}},1}^{1.45}}{1+2.33{X}_{\mathit{\text{DOL}},1}^{1.22}}\right$ 
The presence of paclitaxel within a compartment causes an increase in the rate at which doxorubicin flows from the compartment to the other compartment(s). In addition, the presence of paclitaxel slows the rate that doxorubicin is being eliminated from the body. The paclitaxel also causes the flow rate of the doxorubicinol to increase. The expulsion of the doxorubicin and doxorubicinol into the blood/plasma compartment is in agreement with what is observed experimentally with the spikes in the concentrationversustime curves.
The amount of metabolite doxorubicinol being produced is also strongly affected by the presence of paclitaxel. We see from the expression for I_{D O L,2} that the fraction being metabolized into doxorubicinol is increased by the presence of paclitaxel; but the amount of doxorubicin in compartment 2 available to be converted into doxorubicinol is reduced as well as the amount being eliminated. Doxorubicinol is a highly cardiotoxic metabolite and the reduction of its production will reduce the chances of serious side effects [35].
The parameters in Tables 9 and 10 compared to Tables 11 and 12 are similar with a few differences. These differences do not change the overall qualitative behavior and produces similar results. For example, if doxorubicin is present and then paclitaxel is introduced to the body then both predict that the flow rate of doxorubicin into the elimination organs compartment will increase. Once the paclitaxel concentration in the elimination organ has increased to a sufficient level then the flow rate out into the blood plasma has increased more than the flow rate into the organ, which causes a spike in the concentration levels seen in the experimental data.
To investigate the time sequence we use the dosage introduced by Gianni et al. [27] and Moreira et al. [28]. In other words, 60 m g/m^{2} of doxorubicin is administered in a fiveminute IV bolus and 150 m g/m^{2} or 200 m g/m^{2} of paclitaxel in a threehour IV. The AUC is integrated over the period of infusion plus a oneweek interval after the infusion has stopped instead of calculating the integral over all time. The portion of the curve neglected after a week has a very small contribution to the AUC, which does not change the results significantly.
The AUC for doxorubicin and doxorubicinol was calculated for the blood plasma compartment (1). In the case when doxorubicin is given first and paclitaxel is administered to the patient afterwards, the AUC of the doxorubicin is larger the closer in time the two are administered at. Unfortunately, the AUC of doxorubicinol is also very large the closer the two drugs are administered to each other reaching a maximum when both started at the same time. Therefore, we expect the toxicity of doxorubicin and doxorubicinol to be enhanced with this drug sequence. In the case when the order is switched and paclitaxel is given first there are two different situations. The first situation is when the doxorubicin is given during the first three hours while the paclitaxel is being administered. The AUC of the doxorubicin is a maximum during this interval. If the doxorubicin is given after the third hour the AUC becomes smaller the larger the interval between when the paclitaxel finishes and the doxorubicin is given. The AUC of the doxorubicinol is reduced by having the doxorubicin given after the paclitaxel. This reduces the toxicity of doxorubicinol while maximizing the effects of doxorubicin. Therefore, we maximize the effects of the combination if paclitaxel is given over three hours followed by doxorubicin given between 1 to 4 hours after the paclitaxel is started.
We conclude that doxorubicin should be administered optimally within four hours after the paclitaxel infusion is complete. These results suggest that the efficacy of doxorubicin can be increased by giving the patient paclitaxel prior to infusion of doxorubicin.
Conclusions
One of the major challenges in dose optimization is nonlinear behavior in one or more drug processes. In the current study, we investigated new ways to assess and quantify nonlinear pharmacokinetic behavior, with emphasis on their origins and therapeutic applications of the drug combination involving paclitaxel and doxorubicin. We have justified the development of a saturable compartment model with competing interactions between the two drugs by demonstrating through molecular dynamics simulations that they compete for the same metabolic enzyme with different binding affinities.
Molecular pathways are complex and the correlation of these pathways with malignancy is still an open issue, a complication that has led to systems biology approaches to cancer research [36]. While the system biology approach is promising, we focused on a wellunderstood and timetested approach involving pharmacokinetic equations for a multicompartment model leaving a more complex issue of multiple molecular pathways for future development.
Since no single drug is sufficiently efficacious to become a “silver bullet” cure for large patient populations, combinations of drugs addressing different molecular pathways appear to be capable of increasing patient survival. Optimization of combinations in terms of their sequences, schedules and dosages is a daunting task and needs a formal mathematical development. In the present paper we focused on a specific combination of two drugs and made the following two assumptions. First, saturation kinetics is in principle allowed in all compartments. Second, drug interactions are included to the extent of affecting kinetic rates of other drugs; specifically some drugs may either enhance or inhibit the transition kinetics of others depending on the builtin parameter values.
We have developed a relatively simple but robust fourcompartment saturable kinetics model that has been shown to reproduce a number of empirical data sets very well. This is of practical importance in view of the toxic effects these drugs have on patients. In particular, it has been reported that doxorubicin is strongly cardiotoxic and doxorubicinol is ten times more cardiotoxic than doxorubicin [37]. Therefore, by keeping doxorubicin from being metabolized by liver enzymes, the drug will be more effective at killing the tumor as well as being less toxic to the body. This suggests that the optimal sequence is to give paclitaxel first and then almost immediately afterwards, give a dose of doxorubicin to the patient.
Appendix A: Doxorubicin experimental data
Equation 11 is the basis of our pharmacokinetic model and was fitted to several data sets that were digitized from concentrationversustime curves. Some of the data represent mean values obtained from multiple patient sets, some of the data come from a representative patient out of a group, and some of the data describe a single patient. Due to this diversity of origin comparing different data sets is fraught with problems. However, our intention was to demonstrate the robustness of our model in fitting the data. In total, we have digitized 463 data points for doxorubicin. Multiplying the data points that represent mean values by the number of patients gave 1341 patientdata points. Similarly, for the metabolite doxorubicinol, we obtained 126 data points, which become 869 patientdata points when multiplied by the number of patients. The figures below show the individual data sets and our best fit.
Appendix B: Paclitaxel experimental data
Equation 11 is the basis of our pharmacokinetic model and was fitted to several sets of paclitaxel data that were digitized from figures representing experimental concentrationversustime curves. Some of the data represents mean values obtained from multiple patients, some of the data came from a representative patient out of a group, and some of the data describe a single patient. Due to this diversity of origin, we weighted each data set based on the number of patients it represented. We did not attempt to fit to a single set of data, but used all the data; our intention was to demonstrate the robustness of our model in fitting the data. In total, we have digitized 352 data points for paclitaxel. If we multiply the data points that represent mean values by the number of patients then we have 1104 patientdata points. Similarly, for the metabolite 6 αhydroxylpaclitaxel, we obtained 20 data points, which is still 20 patientdata points when multiplied by the number of patients. The lack of data for the metabolite means we can use only 19 parameters and the fit is not as robust as it could have been with more experimental data. The figures below show the individual data sets and our best fits using Equation 11 with the parameter values given in Tables 7 and 8.
Abbreviations
 ADME:

Absorption, distribution, metabolism and elimination
 ALA:

Alanine
 ALL:

Acute lymphoblastic leukaemia
 AMBER:

Assisted model building with energy Refinement
 ARG:

Arginine
 AUC:

Area under the curve
 CREL:

Cremphor EL
 CYP:

Cytochrome P450
 DNA:

Deoxyribonucleic acid
 DOL:

Doxorubicinol
 DOX:

Doxorubicin
 GAFF:

General amber force field
 HEM:

Hemoglobin
 ILE:

Isoleucine
 IV:

Intravenous
 KEGG:

Kyoto Encyclopedia of Genes and Genomes
 LEU:

Leucine
 LGA:

Lamarckian genetic algorithm
 MD:

Molecular dynamics NAMD: Not (just) Another molecular dynamics program
 OHP:

6ahydroxypaclitaxel
 PDB:

Protein data bank
 PHE:

Phenylalanine
 PK:

Pharmacokinetics
 PTX:

Paclitaxel
 RMSD:

Root mean square deviation
Declarations
Acknowledgements
J.A.T. acknowledges funding support for this project from NSERC, the Allard Foundation, the Canadian Breast Cancer Foundation and the Alberta Advanced Education and Technology. K.J.E.V. would like to thank J.A.T., the Cross Cancer Institute and the University of Alberta for their hospitality.
Authors’ Affiliations
References
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