Research Article  Open Access
A Composite Synergistic Systems Model for Exploring the Efficacies of Different Chemotherapeutic Strategies in Cancer
Abstract
Different chemotherapeutic strategies like Maximum Tolerable Dosing (MTD), Metronomic Chemotherapy (MCT), and Antiangiogenic (AAG) drug are available; however, the selection of the best therapeutic strategy for an individual patient remains uncertain till now. Several analytical models are proposed for each of the chemotherapeutic strategies; however, no single analytical model is available which can make a comparative assessment regarding the longterm therapeutic efficacy among these strategies. This, in turn, may limit the clinical application of such analytical models. To address this issue here we developed a composite synergistic system (CSS) model. Through this CSS model, comparative assessment among the MTD, MCT, and AAG drug therapy can be assessed. Moreover, these chemotherapeutic strategies along with different supportive therapies like Hematopoietic Stem Cell transplantation (HSC), cellular immunotherapy as well as different combinations among these therapeutic strategies can be assessed. Fitting of initial clinical data of individual clinical cases to this analytical model followed by simulation runs may help in making such decision. Analytical assessments suggest that with the considered tumor condition MCT alone could be more effective one than any other therapeutics and/or their combinations for controlling the longterm tumor burden.
1. Introduction
Generally cancer cells survive by the growth of microvasculature (MV) around it, a process called angiogenesis. In recent time, different drugs called antiangiogenic (AAG) drugs are developed for hampering these MV cells’ growth [1–5]. Conventionally chemotherapeutic drugs (CD) are applied to the cancer patients with Maximum Tolerable Dosing (MTD) strategy. MTD application has a damaging effect on the blood vasculature (BV) around the tumor, thereby limiting the availability of the CD to the tumor. Generally, a gap period is allowed between two consecutive MTD drug applications to nullify the damaging toxic effect of MTD to the physiological system. This also helps in the restoration of the damaged BV around the tumor, so drug availability to the tumor can be expected in the consecutive MTD application. To enhance such support, autologous hematopoietic stem cell (HSC) transplantation is under clinical trials for different cancers [6–13]. In metronomic chemotherapy (MCT), conventional CD can be applied in low, but dosedense strategy (with frequent interval). Efficacy of MCT strategy is established in different experimental and few clinical observations [4, 14–16]. MTD strategy is applied to kill the tumor cells whereas MCT strategy is applied to target the growth of MV cells around the tumor mass along with the tumor cells also. Therefore, MCT strategy has an antiangiogenic effect. In an understanding of these phenomena in a quantitative manner, different analytical models are proposed.
Considering two different types of tumor cells within a tumor system having two different multiplication rates and interchangeable mutational rates between the two, a statespace dynamical model has been formulated. This model proposed that under a given tumor system MCT is equally or even more effective than MTD [17, 18]. Further, it also hypothesized that MCT possesses a low or negligible level of toxicity compared to MTD; even there is a remote chance of longterm toxicity development to the physiological system by the MCT schedule [19–21].
The hypothesis regarding the efficacy of MCT over MTD strategy is further established in a fluid dynamics (FD) based angiogenesis model. This model considered a single type of tumor cells, Tumor Associated Factors (TAF) secreted by them, influence fibronectin (FNT) at the cancer milieu. This process ultimately develops MVs around the tumor mass, which makes a link between the tumor system and nearby BV. The advantage of this FD model makes a provision of intermittent tumor system tracking through MV diameter (MVD) measurement by MRI and frequent tracking through TAF measurement from peripheral blood (PBL) as it diffuses from cancer milieu to PBL [22, 23]. However, the model has no provision for fitting the exact MV cell number, which can be obtained readily from the biopsy data for initializing the simulation runs. It is to be noted here that collection of biopsy sample is possible only with surgery and this is generally done at the initial phase of cancer diagnosis. This is also necessary for getting an idea about the tumor load. While applying FD model, cancer system also incorporates some imaginary stochastic components in terms of MVD movement, which may limit the prediction of the real system [24].
One may bypass such limitation by considering MV cell growth characteristics directly as a state variable within the model [25]. Hence, two types of tumor cells having different drug/anoxia sensitivities and MV cell number have been considered as the state variables in the analytical model (Vasculature Growth (VG) model). The model assumes that the growth of each of the tumor cell types at a given instant of time will depend on the availability of MV cell numbers. Through this model, the efficacy of the AAG drug was evaluated. However, there is no provision for MTD evaluation as the model considered only that any antiangiogenic process which reduces the growth of the MV cells will decrease the availability of MV cells in the cancer milieu, which, in turn, restricts the growth of tumor cells. The presence of proportional cell number of each type of tumor cells within the transfer function matrix of the system equation makes the system a timevarying nonlinear system.
Though the present form of VG model can overcome the limitations (initialization for simulation runs) of FD model and represents the behavior of a nearly real system due to its intrinsic nonlinearity, however, it suffers from the limitations of intermittent data collection of the MV cells’ number (through biopsy) during the course of a therapeutic procedure, especially when a tumor is located within the deeper site (internal organ) of the human body. This limitation, in turn, may produce an erroneous prediction of a therapeutic outcome, as matching between the real systems dynamics and the simulation output becomes difficult [24]. Though it was suggested that MCT could be evaluated through this VG model, but it needs necessary modifications to incorporate the effect of CD on tumor cells as well.
In FD model, tumor cells (through TAF) influence the microvasculature whereas in VG model microvasculature influences the growth of tumor cells. Due to this difference in assumption of dependency between the variables, discrepancies may exist between these two models. This may enhance an additional error in the prediction of tumor growth dynamics in clinical cases. These strongly demand the necessity for the development of a more clinically aligned single analytical procedure, so that different therapeutic procedures can be assessed through the same model. The present work is targeted to this direction for the development of a composite synergistic systems (CSS) model for the assessment of different chemotherapeutic regimes, namely, MTD, AAG, and MCT that are now being widely suggested for cancer treatment.
2. Materials and Methods
2.1. Brief Descriptions of CSS Model Components
To overcome the limitations of each of the models [24], we designed a Composite Synergistic Systems (CSS) model for cancer therapy. This CSS model considers a heterogeneous tumor mass having two types of tumor cells (sensitive and resistive type cells— and in VG and and in FD) having different multiplication rates (for sensitive and resistive type cells— and in VG and and in FD), different conversion rates between the two (from sensitive to resistive and vice versa are denoted by and in VG and and in FD), and with different CD and/or anoxia sensitivities. The cancer system considers this tumor system along with MV cell (for VG model) or MVD (for FD model). CSS is formulated in a manner that two different models (VG and FD) synergies with each other. Under the same tumor system, different therapeutic strategies along with different therapeutic combinations are simulated. These strategies are as follows: MTD, MTD with intermittent autologous hematopoietic stem cell (HSC) transplantation [4, 8], AAG, strategy 2 (MTD + HSC) followed by strategy 3 [26], MCT without any immunoboosting (Im) and HSC mobilization inhibition (Ihs) from bone marrow, MCT having Im and Ihs factor [4, 15, 16, 24], strategy 5 (MCT without Im + Ihs) with intermittent strategy 3 , and strategy 6 (MCT with Im + Ihs) with intermittent strategy 3 (AAG). Immunoboosting profile by MCT application is considered same as of the earlier work [18]. HSC mobilization from bone marrow is also considered to be of the same profile. This factor is considered, as about 50% of endothelial cells in newly formed blood vessels are bone marrow derived [4, 27]. For the transplanted HSC, we have considered its multiplication rate (), apoptosis rate (), conversion rate to microvasculature ( (to MV cell) and (to MVD)), and MTD chemotherapeutic drug sensitivity. The target actions of different therapeutic strategies are schematically shown in Figure 1.
2.2. Description of the Existing Systems Models
For tumor angiogenesis, two types of analytical models are available—one, considers microvasculature (MV) cell growth (VG model) and the other considers growth of microvasculature cell diameter (MVD) based on fluid dynamic (FD model) principles [22, 23, 25]. The former model is formulated to test the effect of antiangiogenic (AAG) drug, while the latter is formulated to test the effect of maximum tolerable dosing (MTD) and metronomic dosing (MCT) of chemotherapeutic drugs.
2.2.1. VG Model
VG model considered two types of malignant cells—drug sensitive () and drug resistive () within the tumor mass and MV cells () around the tumor. These cells grow with their own multiplication rates, that is, , , and , respectively. Again and may be transformed into each other with a conversion rate of and, respectively. Two malignant cell types ( and ) respond in different ways with the change in MV cells () numbers.
The dynamical relationships among these cells were represented by the following equation: In (1), = and × , where and are the sensitivities of two types of malignant cells to anoxia and and are the proportions of a specific type of tumor cells with respect to total tumor cell population and represented as and . Again, represents a coefficient that is defined as the number of total tumor cells supported by each MV cell. Thus, is represented through the following relationship: . It was assumed that a fully grown tumor requires , amount of vasculature to support a total tumor cell count at the time (in days) of diagnosis, that is, and is considered as constant parameter. The variables and , being multiplied with the amount of MV cells, translate the change in into a corresponding change in tumor cells number. With the application of AAG drug, MV cells are killed with a rate of which varies with time depending on the amount of drug present in system.
This model can be equipped with the application of chemotherapeutic drug in MCT strategy, another approach of antiangiogenic based therapy. Immunity boosting, if any, was introduced through and, respectively [15, 25]. These factors are also time varying and proportional to the amount of immunomodulatory agents present in the system at a particular instant of time. In this model, and are changed in four different ranges according to the availability of MV cells at the cancer milieu (Table 1(d)). Within each range, the selected coefficient remains constant and cell killing depends on the destruction of MV cells. Using this approach, a highly nonlinear system has been transformed into a piecewise linear model [25].
(a)  
 
Parametric value is taken from [18, 22, 23]. ^{†}Parametric value is taken from [25].  
(b)  
 
(c)  
 
(d)  

2.2.2. FD Model
Considering a single type of tumor cells, FD based model was developed [22, 23]. The growth of tumor cells is represented by the following equation:
where is the malignant cell count on th day. represents the drug sensitivity and denotes the drug dose (MTD or MCT) applied on the th day. denotes the immunity level on th day and is the vasculature diameter at location on th day and is given by following equation: where MVD is linked with concerned probabilistic cell movement at each location at the cancer milieu (here 9 locations are considered) at an instant of time. The probabilistic cell movements are influenced by the TAF (tumor associated factors secreted by the tumor cells) and FNT (fibronectin factor) concentration at the cancer milieu. Again, at each location MVD have five types of probabilistic movements (stationary probability), (right direction probability), (left direction probability), (bottom direction probability), and (up direction probability). They are represented in terms of TAF concentration , FNT concentration , and MVD [22, 23] and are represented through the following relationships:
In the above equations , , , and can be considered either as some constant factors or variables; however, they can be derived through the following equations: where , , , , and are some constant values. TAF concentration and FNT concentration at different locations of cancer milieu are represented through the following equations:
Now using (3), (2) can be modified as Considering probabilistic movement , tumor cell number , and MVD as three state variables, the system can be represented through the following equation:
2.3. Scheme for Implementing Different Drug Strategies
From the previously developed VG and FD models as represented by (1) and (9), the present work is aimed to develop a composite synergistic systems (CSS) model for the assessment of different chemotherapeutic strategies, that is, MTD, MTD with intermittent autologous hematopoietic stem cell (HSC) transplantation [6–13], AAG, strategy 2 (MTD + HSC) followed by strategy 3 (AAG) [26], MCT without any immunoboosting (Im) and HSC mobilization inhibition (Ihs) from bone marrow, strategy 5 (MCT without Im + Ihs) with intermittent strategy 3 (AAG), strategy 6 (MCT with Im + Ihs) with intermittent strategy 3 (AAG), and MCT having Im and Ihs factor [8, 15, 26]. MTD drug strategy directly affects the malignant cells and MV/MVD, whereas AAG indirectly affects malignant cells by affecting MV/MVD. MCT drug strategy also affects MV/MVD, indirectly affects malignant cells, and also has a direct damaging effect on malignant cells. Reduction in malignant cells reduces TAF production that in turn again reduces MVD gradually. Immunoboosting profile by MCT application is considered same as of the earlier work [18]. HSC mobilization from bone marrow is also considered to be of the same profile. This factor is considered, as about 50% of endothelial cells in newly formed blood vessels are bone marrow derived [15, 27]. For the transplanted HSC, we have considered its multiplication rate (), apoptosis rate (), conversion rate to vasculature ( (to MV cell) and (to MVD)), and MTD chemotherapeutic drug sensitivity (Table 1(b)). The effects of different therapeutic strategies that have been considered in the model are schematically shown in Figure 1.
The matrix elements of the transformation matrixes of (1) and (9) are being modified by the incorporation of necessary subtractive terms to implement the effects of different therapeutic strategies. The activation of corresponding subtractive terms of concerned therapeutic strategies are being operated in the system equation by the activation of the concerned switches: for MTD application, for MCT application, for AAG application, for MTD with intermittent HSC application, and for hematopoietic stem cell transplantation (Table 2(a)). The effects which are produced by the applications of different drug strategies are operated through the following switches: for MV/MVD damage, for immunity boosting, and for stem cell mobilization inhibition (Table 2(b)). The switches remain “1” during the application period of concerned drug application; otherwise they become “0”.
(a)  
 
(b)  

Vasculature damage occurs; that is, MV cell killing happens when the available MV cell count () (VG model) is greater than the minimum number required (a set value) to reach drug at the tumor site and a minimum amount of drug (a set value) is present within the system. In such condition, the switch is being activated (i.e., becomes “1”) depending upon the drug level and MV cell numbers () within the system. This simultaneously becomes effective MVD () damage in FD model.
MTD with intermittent HSC transplantation strategy is applied for better killing of tumor cells as drug transportation to the tumor site increases with the increase in MV/MVD around the tumor; however, drug application in MTD and/or MTD + HSC strategy causes rapid destruction of MV/MVD cells. This actually limits the chemotherapeutic efficiency in terms of killing of tumor cells, though there is high amount of drug present within the physiological system [19, 20]. This, in turn, causes unnecessary killing of other normal cells of the physiological system which, in turn, produces toxicity burden within the physiological system. So for recovering the physiological system from this toxicity burden, a gap period is necessary between two successive chemotherapeutic doses in MTD. In this intermittent period, HSC transplantation strategy is applied. Hence, in both VG and FD model, it is assumed that vasculature damage by the chemotherapeutic drug application in MTD and/or (MTD + HSC) strategy restricts the nutritional supply to the tumor cells, and this phenomenon is formulated by lowering the multiplication rates of tumor cells (, , , and ) on the day of drug application. However, if sufficient condition for microvasculature (MV/MVD) damage does not exist, the multiplication rate of tumor cells will remain unchanged (as per Table 1(d)). For synergism between two models, we considered that in the tumor milieu the (3 × 3) matrix area in FD model corresponds to 10,000 MV cells ().
2.4. Development of Composite Synergistic Systems (CSS) Model
To implement the different drug strategies into both the VG and FD models, both models are modified as follows. This ultimately leads towards the development of a composite synergistic systems (CSS) model.
2.4.1. Modification of VG Model
For the assessment of AAG drug and also MTD/MCT strategies in the VG model, (1) can be modified as follows:
In (10), the MTDrelated killings are incorporated through (= , (= , and (= . Similarly, in (10) MCTrelated killings are incorporated through (= , (= , and (= ). and are the drug doses in MTD and MCT strategies, respectively, on th day and , , and are the drug sensitivities to the two types of malignant cells and MV cells to MTD/MCT (Table 1). MCTbased immunoboosting is introduced through additional subtractive terms in and , and is the factor that represents the inhibition of stem cell mobilization from bone marrow in MCT therapy which is introduced within the term .
Similarly, AAGrelated drug killing is incorporated through , and is the AAG drug dose on the th day. In (10), and were considered as adoptive immunotherapy or other immunoboosting instead of MCT based immunoboosting (as mentioned in Section 2.2.1). For therapeutic strategies other than MTD or MTD + HSC, the terms , , and will be equal to 1, whereas in case of MTD or MTD + HSC, these terms also need necessary modification as explained later in Section 2.4.6.
2.4.2. Modification of FD Model
Contrary to the previous model, the present model has considered a tumor system consisting of heterogeneous tumor cells— (sensitive cell type) and (resistive cell type). These cells are multiplied with (multiplication rate of drug sensitivetype cell) and (multiplication rate of drug resistivetype cell) and converted to another type with a rate of (conversion rate of sensitive to resistivetype cell) and (conversion rate of resistive to sensitivetype cell). Growth of each type of the tumor cells is influenced by the development of MVD () at different locations near the tumor milieu at different time points. In corollary with the VG model, FD model is further modified. Thus, like VG model, growth rate of each cell type has been set with four different ranges of values according to the availability of MVD at the cancer milieu (Table 1(d)).
Again, (5) is also modified as (11): Contrary to earlier works, here we have considered them as variables; however, this makes the overall systems equations nonlinear [22, 23]. This may impart extra flexibility in parametric value adjustment for aligning the model with the real system; thus, the model behavior becomes more aligned with the real system. Thus, (9) of FD system can be represented through the following equation:
The probabilistic growth becomes unrealistically large; hence, in (12), a calibration factor, (a constant fractional number), is considered to make MVD output realistic. Again, and , where and are the anoxia sensitivities of and type cells, respectively and = and = . is a subtractive term equivalent to length reduction of MVD. In MTD strategy, = = , in MTD + HSC strategy, = = , and in MCT strategy, = = .
Again in (12), for MTD strategy, = and = ; for MCT strategy, = and = . and are the drug doses in MTD and MCT strategies on th day and , , and are the drug sensitivities to the two types of malignant cells and MVD to MTD/MCT.
MCTbased immunoboosting is introduced through additional subtractive terms in and and is the factor that represent the inhibition of stem cell mobilization from bone marrow in MCT therapy is introduced within the term . The term is the drug sensitivity of MVD in MTD + HSC. The detailing of MTD + HSC incorporation is explained later in (17). In (12), and were considered as adoptive immunotherapy or other immunoboosting instead of MCTbased immunoboosting.
Thus, MVD () at different locations of cancer milieu are derived from (3) modified as (13) and TAF concentration can be obtained by modifying (6) to (14) to implement the effect of MCT and AAG:
2.4.3. Incorporation of AAG Drug in FD Model
Application of AAG drug is introduced into the system equation by introducing its effect on the TAF concentration. The effect of AAG drug on TAF has been introduced by adding a subtractive term in (14). A fraction equivalent to AAG drug, present in the system multiplied with the AAG sensitivity (), has been used to generate the effect of AAG drug on TAF. As TAF influence the probabilistic movement of MVD (as indicated by a set of equations (3) to (7)), hence, MVD related to TAF will be influenced according to (13). Therefore, sensitive and resistive cell numbers will also be modulated as they are dependent directly on MVD (as indicated by (12)).
2.4.4. Tracking of Tumor Dynamics from Peripheral Blood
Such tumor system can be tracked by measuring TAF concentration from peripheral blood as mentioned earlier [22, 23]. The TAF concentration at different positions in peripheral blood vessel is represented through the following equation: In (15), represents the change in TAF concentration per unit blood at th time followed by th time concentration at distance from location where is position at the cancer milieu, is the TAF degradation per unit time, and is TAF absorption rate per unit distance.
2.4.5. Incorporation Intermittent Hematopoietic Stem Cell (HSC) Transplantation with MTD in VG & FD Models
Model has further modified to incorporate HSC transplantation in the gap period of MTD application. This has been developed in such a manner that it incorporates the flexibility in choosing the application day of transplantation or MTD drug. It has been considered that MTD drug will influence the number of sensitive cells, resistive cells, MVD (in FD model), and MV cell number (in VG model). Transplanted HSC cell number will increase the vasculature diameter (MVD) (in FD model) and MV cell number (in VG model). It has been considered that after the day of HSC transplantation the conversion rate of sensitive to resistive cell will be increased while after application of MTD on the following days this rate will remain the same as of the initial set value, that is, at the time of diagnosis. To include the effect of MTD and HSC transplantation, the VG and FD model would be modified as follows.
For VG model, (10) has been modified to (16) to reflect the influence of HSC transplantation during the gap period of MTD drug:
Similarly for FD model, (12) has been modified to (17) to reflect the influence of HSC transplantation during the gap period of MTD drug application:
In (16) and (17), where , , and are the MTD drug sensitivities in MTD + HSC therapeutic strategies to the two types of malignant cells and MV cells/MV diameter, respectively (Table 1) and and are fractional numbers. Again is the number of HSC cells on th day and is given by
where is the initial number of transplanted HSC cells. and are the multiplication and apoptosis rates of transplanted HSC cells. In MTD + HSC strategy, is the MTD drug sensitivity and is the MTD drug present in the system on th day.
2.4.6. Drug Resistance Feature in MTD and/or (MTD + HSC) Therapy in VG and FD Models
In both VG and FD models, it is assumed that during the course of MTD or MTD + HSC therapy, there is a chance of development of drug resistance, so that this condition may affect the multiplication rate of these three cell types. This feature has been introduced into the system model by incorporating three timevarying multiplication factors, , , and , that modulate, the multiplication rates of these three cell types (, , and ). In the systems model, it is assumed that on the day of the application of MTD or MTD + HSC, the multiplication rates will be reduced by 1% of the previous day; however, in the successive days, it will go on increasing by 0.005% of the previous day.
2.5. Simplified Form of VG and FD Model
Equation (16) can be simplified as follows:
Similarly, (17) can be simplified as
In (20) and (21), , , and represent the subtractive terms for sensitivetype cells, resistivetype cells, and MV cells or MVD will be activated depending on the application of chosen therapeutic scheme for MTD/(MTD + HSC)/MCT. , , and will be updated depending on the concerned therapeutic scheme. Each therapeutic scheme will be activated through different switches as mentioned earlier (Table 2).
2.6. Tumor Load Analysis from Two Models
For tumor volume calculation, three different types of cells—sensitive cells (), resistive cells (), and microvasculature cells () have been considered. Each cell type and tumor load have been assumed to be spherical in shape. Let , , and be the radii of resistive cell, sensitive cell, MV cell, and total tumor volume, respectively. Again is proportional to the total cell population of the above three cell types. That is, where is a calibration factor and small fractional number. In chemotherapy with long gap, that is, MTD or MTD + HSC therapy when applied, the tumor radius shows an unrealistic numerical value, so is calibrated as for getting a realistic value. Again, the ratio of these three cell types in terms of their population per unit volume (as in biopsy sample analysis) can be . Again as the number of threecell types changes with time hence their ratio within tumor load will be a timevarying quantity. The proportion may be represented as , , or , , . Moreover, the total cell population in tumor is given by Now, the volumes of single resistive cell, sensitive cell, MV cell, and total tumor are given by (= ()), , ), and (= ()), respectively.
Hence, the expected ratio of each cell type with respect to their volume in unit volume of collected biopsy sample is given by or or . Therefore, the expected number of resistive cells, sensitive cells, and MV cells in tumor is given by (24)–(26), respectively: Thus, CSS model could be helpful to indicate the quantitative assessment of individual cell types within a tumor mass in a dynamical manner and this assessment thus encompasses both invasive (biopsy) and noninvasive (MRI) data. Thus, better assessment can be made [24].
3. Results
With the developed CSS model, as described in Section 2, rigorous simulation exercises are carried out using MATLAB 6.5. The initial parametric values used for simulations are mentioned in Table 1. Different therapeutic strategies are implemented in the model using different activation switches as mentioned in Table 2. For synergism in simulation, we have considered that in the tumor bed (milieu), the () matrix of FD model corresponds to 10,000 MV cell (VG model).
3.1. Free Growth of Tumor
Malignant cells if left untreated grow exponentially. This is reflected in the enhanced multiplication rate or decrease in the doubling time of the malignant cells (Figure 2Ib). This is due to the exponential growth of microvasculature (MV cells/MVD) by the incremental effect of TAF concentration at the cancer milieu. The changes of tumor characteristics (tumor cells, MV cell numbers, MVD, and TAF concentration) in different time points as observed through simulations are indicated in Table 3. In the model there is provision of recording of TAF dynamics at different positions of the cancer milieu and at different locations of the peripheral blood. In this condition, the corresponding MVD growth and the total tumor growth (in terms of radius) are recorded to be in the growing stage; however, the TAF concentration in PBL becomes saturated after a period of time (Figure 3).
