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Computational Biology Journal
Volume 2013 (2013), Article ID 301369, 26 pages
http://dx.doi.org/10.1155/2013/301369
Research Article

A Composite Synergistic Systems Model for Exploring the Efficacies of Different Chemotherapeutic Strategies in Cancer

1Department of Physiology, West Bengal State University, Berunanpukuria, Malikapur, Barasat, 24 Pgs (N), Kolkata 700 126, India
2Deptartment of Electronics and Communication Engineering, Bengal College of Engineering & Technology, Sahid Sukumar Banerjee Sarani, Bidhannagar, Durgapur, Burdwan, West Bengal 713212, India

Received 17 June 2013; Revised 26 August 2013; Accepted 27 August 2013

Academic Editor: Marie-Veronique Clement

Copyright © 2013 Probir Kumar Dhar and Durjoy Majumder. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 long-term 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 long-term 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 [15]. 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 [613]. In metronomic chemotherapy (MCT), conventional CD can be applied in low, but dose-dense strategy (with frequent interval). Efficacy of MCT strategy is established in different experimental and few clinical observations [4, 1416]. 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 state-space 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 long-term toxicity development to the physiological system by the MCT schedule [1921].

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 time-varying 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.

301369.fig.001
Figure 1: Target site of different therapeutic strategies of the CSS model. In figure, , and represent the drug-sensitive, drug-resistive, and MV cells, respectively, in VG model; , , and represent the drug sensitive, drug resistive cells, and microvasculature diameter, respectively, in FD model; and HSC and BM represent the pluripotent hematopoietic stem cell and bone marrow, respectively. In figure, direct arrows, dash dotted arrows, and dotted arrows represent direct, conditional, and indirect effects of drugs.
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].

tab1
Table 1: (a) Initial parametric values for simulation. (b) Initial parametric values (assumed) of HSC for simulation. (c) () cross-section along with initial microvessel diameter in micrometer, TAF concentration in ng/L, and FNT concentration in ng/L. Positional coordinates are shown in parentheses. Presently used values are shown in bold while the other values are kept the same as those mentioned in previous works [22, 23]. (d) Change in doubling time of the tumor cells depending on the different vasculature ranges during the course of dynamics [25]. For initialization, we have considered that tumor cells have crossed a stage of dormancy but having ample microenvironment (in terms of MV/MVD) for their growth. The parametric values are the same as those depicted in the earlier work except the second column (values of MV diameters are assumed) of the table. The values in the parentheses of the third and fourth columns of the table indicate the multiplication rate corresponding to the doubling time.
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 [613], 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”.

tab2
Table 2: (a) States of drug strategy activation switches during different drug strategies. (b) States of activation switches for producing different effects during different drug strategies.

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 MTD-related killings are incorporated through (= ,   (= , and   (= . Similarly, in (10) MCT-related 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). MCT-based 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, AAG-related drug killing is incorporated through , and is the AAG drug dose on the th day. In (10), and were considered as adoptive immuno-therapy 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 sensitive-type cell) and (multiplication rate of drug resistive-type cell) and converted to another type with a rate of (conversion rate of sensitive to resistive-type cell) and (conversion rate of resistive- to sensitive-type 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.

MCT-based 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 MCT-based 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 time-varying 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 sensitive-type cells, resistive-type 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 three-cell types changes with time hence their ratio within tumor load will be a time-varying 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 2-Ib). 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).

tab3
Table 3: Simulation results in different therapeutic strategies.
301369.fig.002
Figure 2: Simulation runs in different conditions (as indicated by different rows): (I) freely growing condition, (II) MTD strategy application, (III) MTD with intermittent HSC transplantation, (IV) AAG application, (V) MTD with intermittent HSC transplantation followed by AAG application, and (VI) MCT (with Im and Ihs) application. In all the plots of 1st column (a), 2nd column (b), and 3rd column (c), -axes indicate the therapeutic profile (applied dose/unit amount) (drug profile (dose) (chemotherapeutic/AAG drug with blue line) and HSC transplantation profile (with pink line and asterisks) and the arrow in plot Va indicates the starting of AAG drug application), doubling time of sensitive (red) and resistive (black) cell types (FD model (indicated by straight line) and VG model (indicated by line with diamond/star)), and PBL TAF (ng/μL) dynamics, respectively. In all plots, -axis indicates time (in days).
fig3
Figure 3: Dynamics of MVD (in μm) at different positions of tumor milieu (a), TAF concentration (in ng/μL) at different positions of PBL from (1, 2) location of MVD (b), and tumor radius (in μm) (c) in freely growing condition.
3.2. Effect of MTD Strategy

It has been found that application of CD in MTD strategy (six cycles with 21-day interval) reduces sensitive and resistive cell count in both FD and VG models; however, after stoppage of drug application, tumor cells have exponential growth (Table 3). Similarly, their doubling time after fluctuating between higher and lower ranges finally settled to lower range (Figure 2-IIb) and system becomes unbounded soon. It is to be noted here that on the day of drug application, the malignant cells of both types (in FD and VG models) suffered sharp falling, but on the following days they were showing recovering nature and after completion of drug course the tumor system regained exponential growth which indicates the failure of MTD strategy for control of malignancy (Table 3). Hence, in long-term simulation, it is found that there is increasing of MVD and TAF concentration (FD model) and MV cell (VG model) (Table 3). TAF concentration in PBL showed the same growth dynamics as free growth indicating that MTD drug has no influence on TAF (Figure 2-IIc). The application of CD in MTD strategy is shown in (Figure 2-IIa)). To make synergism in growth dynamics between FD and VG models, the CD sensitivites for MVD and MV cell are calibrated with the factors 1 × 107 and 15, respectively. The corresponding MVD in tumor milieu, TAF recording at different locations of PBL, and total tumor growth (in terms of radius) have also been recorded (Figure 4). CD applications with high (MTD) dose do not improve the therapeutic outcome. The effects of increased dose of MTD strategy on tumor dynamics in different time instants are represented in Table 3.

fig4
Figure 4: Dynamics of MVD (in μm) at different positions of tumor milieu (a), TAF concentration (in ng/μL) at different positions of PBL from (1, 2) location of MVD (b), and tumor radius (in μm) (c) with the application of MTD chemotherapeutic drug strategy.
3.3. Effect of Conventional MTD Dose with Intermittent HSC Transplantation

The effect of MTD (drug dose kept the same as in the case of low/conventional MTD dose) with intermittent HSC transplantation is also simulated. Simulation is done with the assumption that transplanted HSCs increase the drug availability (in terms of killing) to the tumor; hence it is applied during the gap period between two consecutive CD applications (Figure 2-IIIa). Moreover, it has been assumed that the HSC transplantation favors the increased mutability rate from to and to . As expected, it has been observed that though on the day of MTD drug application, , , , and showing sharp falling due to drug effect but on following days, they were found to have a recovering/growing nature. Simulations study indicates that there is no overall beneficial effect of HSC transplantation in terms of tumor control rather it facilitates the growth of malignant cells further due to increase in microvasculature around the tumor (Table 3). Such effect is reflected in doubling time plots of tumor cells. The doubling time plots show that though during the application of therapy schedule the doubling time occasionally reached the higher stage, but finally as the therapy ends, doubling time of cells settled down to the lowest level which indicate the increased multiplication rates of the malignant cells and tumor growth (Figure 2-IIIb). PBL TAF reflects this pattern (Figure 2-IIIc). The corresponding MVD in tumor milieu, PBL TAF recording at different locations of PBL and total tumor growth (in terms of radius) has also been recorded under such condition (Figure 5).

fig5
Figure 5: Dynamics of MVD (in μm) at different positions of tumor milieu (a), TAF concentration (in ng/μL) at different positions of PBL from (1, 2) location of MVD (b), and tumor radius (in μm) (c) with the application of MTD chemotherapeutic drug strategy with intermittent HSC transplantation.
3.4. Effect of AAG Therapy

Application of AAG drug shows a control over the tumor growth characteristics after ~73 days and this is maintained for long run (Figure 2-IV); however, progression-free outcome is not achieved. It has been observed that though the multiplication rates of tumor cells were found to be lower for a longer period of time in comparison to free growth condition or MTD (conventional and high) or MTD with HSC transplantation strategies, the growth dynamics of , , , and have exponential growth. The tumor growth control that is achieved with AAG drug application is reflected in MV cell count, TAF at cancer milieu, and PBL TAF (Table 3). The corresponding MVD in tumor milieu, TAF recording at different locations of PBL, and total tumor growth (in terms of radius) have also been recorded under such condition (Figure 6).

fig6
Figure 6: Dynamics of MVD (in μm) at different positions of tumor milieu (a), TAF concentration (in ng/μL) at different positions of PBL from (1, 2) location of MVD (b), and tumor radius (in μm) (c) with the application of AAG drug strategy.
3.5. Effect of MTD with Intermittent HSC Transplantation Followed by AAG

Under the same tumor characteristics, simulation is also carried out with an assumption that AAG drug is applied after the conventional chemotherapy (conventional dose of MTD) with intermittent HSC transplantation (Figure 2-Va). Under the same tumor condition, this therapeutic strategy can improve the tumor growth control after day ~580 compared to free growth, MTD, MTD with HSC transplantation strategies; this may be due to the application of AAG drug (as 1st AAG is applied on day 241) (Figures 2-Vb and 2-Vc) (Table 3). This indicates that the previous application of MTD with intermittent HSC transplantation has no beneficial effect on the control of tumor growth. The corresponding MVD in tumor milieu, TAF recording at different locations of PBL, and total tumor growth (in terms of radius) have also been recorded under such condition (Figure 7).

fig7
Figure 7: Dynamics of MVD (in μm) at different positions of tumor milieu (a), TAF concentration (in ng/μL) at different positions of PBL from (1, 2) location of MVD (b), and tumor radius (in μm) (c) with the application of MTD chemotherapeutic drug strategy with intermittent HSC transplantation followed by AAG drug application.
3.6. Effect of MCT Strategy (without Im and Ihs)

Effect of CD application on MCT strategy considering that it has no effect on Im and Ihs is also simulated with the same tumor characteristics. Simulation runs show that and reduced to zero on ~410th and ~543rd days while and growth rate reduced but showing exponential growth in MCT strategy (without Im and Ihs). MVD, TAF, and MV cell count are showing decaying nature (Table 3). Tumor growth though was found to be reduced compared to previous strategies, yet it has an exponential growing nature. Long-term simulation run shows that doubling times of , , , and are finally settled to a higher range.

3.7. Effect of MCT (without Im and Ihs) with Intermittent AAG

The effect of application of MCT with intermittent AAG drug has been observed through the model. In the simulation, application of drug in MCT strategy is applied for 14 days continually and on every consecutive 15th day, AAG drug is applied. Here, it is also assumed that during the application of MCT there is no Im and Ihs. Simulations suggest that this drug strategy can make a control over the tumor growth (in terms of doubling time) after ~560 days; however, it failed to remove malignancy. Tumor characteristics at different time points are tabulated in Table 3.

3.8. MCT (with Im and Ihs) with Intermittent AAG

Application of chemotherapy in MCT strategy with Im and Ihs is also simulated for the same tumor condition. Immunoboosting profile (Im) by MCT application is considered same as of the earlier work [10]. Inhibition of HSC mobilization from bone marrow (Ihs) is also considered to be of the same profile. It is assumed that initial immunity and stem cell mobilization inhibition levels are zero for the first ~12 days, and with the continuous application of chemotherapeutic drug in MCT strategy it gradually boosts Im and Ihs factors from day 13 and finally reaches to 40% of the normal population level within a period of ~2 months, and then those levels are maintained. Simulation results suggest that with this drug strategy there is a significant reduction of tumor growth and may remove malignant cells totally (Table 3).

3.9. Effect of MCT Strategy (with Im and Ihs)

Simulation runs show that counts of , , , and reduced to zero with MCT strategy only. Doubling time plots of tumor cells show that the tumor system is under control within a short period of time, and this has been continued for longer period of time (Figures 2-VIa and 2-VIb) (Table 3). Here, it is assumed that the action of MCT has the capacity of immunoboosting (ImB) along with the inhibition of stem cells mobilization (Ihs) from bone marrow as mentioned previously. MVD (and MV cell count) are showing a decaying nature with time as reflected by PBL TAF (Figure 2-VIc). Simulation reveals that with this therapeutic strategy malignancy may be controlled most successfully than any other type of therapy provided that MCT could boost immunity against the malignant cells and inhibition of stem cells mobilizations from bone marrow. The corresponding MVD in tumor milieu, TAF recording at different locations of PBL, and tumor growth (in terms of radius) have also been recorded under such condition (Figure 8).

fig8
Figure 8: Dynamics of MVD (in μm) at different positions of tumor milieu (a), TAF concentration (in ng/μL) at different positions of PBL from (1, 2) location of MVD (b), and tumor radius (in μm) (c) with the application of chemotherapeutic drug in MCT strategy.

4. Analytical Studies for Different Therapeutic Schemes

In reality, biological systems exhibit nonlinear behavior. In the present work, the FD model was standardized in such a way that it will follow dynamical behavior of the VG modeling output. In doing so, the FD model becomes time-varying nonlinear system as the factors , , , and have been considered as variables (as mentioned previously); therefore, the probabilistic movements of MV cells’ diameter grow in a nonlinear fashion. This makes the dynamical behavior of other factors like FNT, TAF, and tumor cells’ growth nonlinear.

For a nonlinear system, assessment of controllability criteria is difficult. However assessment of controllability criteria is important, as oncologists may be interested to know whether a specific therapeutic scheme would bring the tumor system under control or not. The assessment of controllability criteria is possible if a nonlinear system can be represented through an approximated linear system.

The nonlinear VG model has already been linearized through an approximation by using fixed cellular ratio, that is, and = [25]. These fixed ratios have been utilized in the iteration process of (1) and thus it has been shown that the dynamics of approximated system will precede much ahead of time with respect to the actual system.

The FD model has also followed the same scheme of fixed cellular ratios. Now to include all the variables within the state transition matrix of the systems equation, the following approximations can be done. The position of the (3 × 3) cross-sectional area at the cancer milieu has been considered only. This , position is mathematically linked with the tumor system and the rest of the other points are terminal points; hence, those points may have error in the probabilistic movement of MVD [22, 23]. At the point   , , , , and are approximated as a function of TAF concentration of position only instead of , , , and . Moreover, TAF influences the FNT production and continuous tracking of FNT is difficult in individual patient cases. Therefore, FNT can be omitted in approximated systems equation and in place of FNT an equivalent amount of TAF concentration has been considered in the equation. The probabilistic growth equation at point becomes In (27) is proportionality constant and hence, probabilistic movement in each direction is equal to . The approximated FD model is represented by the following equation:

Whether or not the approximated system will proceed ahead of time with respect to the actual system being dependent on , increase in the value of will shift the dynamical behavior of the approximated system much ahead of time with respect to the actual system. With the existing initial parametric settings (Table 1) under free growth condition, probabilistic movement of MVD (n) of the approximated system leading the actual system for first ~258 days then it lags from the actual system when = 1 × 105; however, sensitive () cell type, resistive () cell type count of the approximated system remains in the leading position with respect to the actual system for longer period of time (Figure 9).

fig9
Figure 9: Dynamics of linearized approximated system (blue line with star) and actual system (red line). In all the plots, -axis indicates time (in days) and -axis indicates microvasculature diameter (in μm) (in (a)), sensitive cell count (in (b)), resistive cell count (in (c)), and total (sensitive + resistive) tumor cell count (in (d)).

Eigenvalue analysis of approximated FD model has been done using the transfer function matrix of (28) for different therapeutic strategies. Analysis shows that in case of MCT drug strategy if immunity boosting and stem cell mobilization inhibition were activated the Eigenvalue becomes <1 which signifies that the system becomes controllable (strategy 8). Similar observation is also observed for strategy 7, except the day of (intermittent) AAG application. For all other therapeutic strategies, tumor system is not controllable as the Eigenvalues remain above one. Eigenvalue analysis of approximated VG model, that is with fixed cellular ratio as mentioned in the earlier work [25], also reflects similar observations regarding the Eigenvalue analysis of the approximated FD model except strategy 7 (where it has been observed that tumor system is controllable).

5. Discussion

Previously developed FD systems model has some limitations. Fitting of initial parametric values to the FD model is difficult as from the 2D cross-sectional (microscopic) view of biopsy, it is very difficult to evaluate the overall microvessel diameter in the tumor milieu. Moreover, the considered probabilistic movements of microvessels at the cancer milieu are imaginary, as this sort of statistical measurement may not be equivocally implemented in reality due to the fact that microvasculature movements cannot be assessed in individual clinical cases. This may augment noncongruency between the model and the real system. Previously, the model was developed with the consideration of that the probabilistic movement of microvessel cell diameter (MVD) depends on some constant factors [22, 23]. Contrary to the previous works, here we have included those factors into the systems equations as time-varying variables. Therefore, the probabilistic cell movement of MVD at different grid points depends on the TAF and FNT concentrations at the concerned and surrounding grid points in a time-varying condition at the tumor milieu. Once again this abolishes the linearized relationship between the variables. Inclusions of such nonlinear and stochastic feature augment the unpredictability regarding the tumor dynamics.

VG model does not have such above-mentioned components and is advantageous in a sense that cell counts of different cell types (vasculature cell, sensitive cell, and resistive cell) can directly be fitted into this model. It is needless to point out here that clinical diagnosis is readily made with the biopsy sample and from biopsy material, immuno-histochemistry, or single cell suspension followed by flow cytometric analysis or that analysis of gene expression study for multidrug resistance proteins can predict the number of individual cell types in a tumor sample. However, through the VG model, it is difficult to track the tumor dynamics during a course of a therapeutic regime, as intermittent biopsy cannot be possible if the tumor is located within the deeper site (internal organ) of the human body. Intermittent tracking is possible through different noninvasive techniques like MRI and/or CT scan. Such procedures can give a quantitative measure of tumor in terms of vessel diameter and tumor radius. To fit such data into a model, FD model is advantageous regarding the prediction of a therapeutic outcome. However, the mentioned noninvasive procedure may not reveal the exact tumor load in terms of cell number and two successive MRI at frequent intervals of time may not be conducted. If conducted, it may not make any significance in difference between two closely spaced data acquisition in terms of tumor load and/or microvasculature. Hence, to overcome the limitations of each of the analytical model, a CSS model for tumor dynamics is a necessity.

Our simulation studies reveal that under the same tumor condition(i)MTD can kill the tumor cells and reduce the tumor load but is unable to eradicate it completely, as drug application in MTD strategy cannot be continued for longer time period due to high level of toxicity development [19].(ii)Application of conventional MTD regime followed by hematopoietic stem cell (HSC) transplantation does not provide any extra benefit on tumor growth control. The plausible reason may be that in the present parametric setting the time gap between transplantation and MTD drug application allows continuation of tumor growth due to multiplication of tumor cells in absence of any CD and introduces mutability between sensitive and resistive cells and finally, at the time of transplantation the presence of drug content within the system is almost zero (due to drug clearance system). These factors enhance the vigorosity of malignancy. Hence, the benefit of HSC transplantation (i.e., increase in microvasculature cell number or MVD) has no effect to decrease malignant cell killing; rather it increases the malignant cell count.(iii)Application of AAG drug can maintain a reduced level of tumor growth by reducing microvasculature but cannot remove tumor load completely from the system and a residual number of tumor cells exist. This observation corroborates the less promising outcomes of different clinical results with different AAG drugs [2831].(iv)As MTD drug application with intermittent HSC transplantation cannot produce any extra benefit as mentioned in case (ii); hence, delay in the applications of AAG drug may fail to control the vigorousity (doubling time) and growth of tumor cells.(v)MCT without immunoboosting effect and inactivation of stem cell mobilization inhibition effect has almost the same effect as that of AAG application (case (iii)). But this may provide a better quality of life compared to other therapeutic strategies, as it may not impose any toxicity burden to the system [19].(vi)MCT with the activation of immunoboosting effect and stem cell mobilization inhibition effect can be able to remove (microscopic) tumor completely.

In previously developed VG or FD model, comparison between the efficacies of MCT and AAG drug application strategies was not tested. Nowadays, in conventional clinical practice, to enhance the efficacy of MTD therapy intermittent autologous HSC transplantations are being suggested along with the MTD scheduling [613]. So far no analytical model is available to assess its efficacy particularly in comparison with the other therapeutic strategies like AAG or MCT. This analytical model has a provision to test these different sorts of therapies, combinations of different therapies, and/or different sequential therapeutic strategies. Therefore, the present model has that flexibility to fix up the therapy choice and therapy day for designing a therapy schedule according to patients’ requirement/clinician choice. Any parametric value of any variable can be changed according to the need of the clinical scenario. Hence, overall this CSS model is flexible in nature.

The rationale of acquisition of parametric values of different variables is mentioned previously [2025]. The model can be initialized with the biopsy (cell number) and 3D data (volume) that can be readily obtained by routine clinical investigations along with the suggested/applied therapeutic data followed by simulation can predict the therapeutic outcome as well. Even this can be done at any time point during the course of a therapy. Through this analytical model, tumor load can be determined from micrometer range to centimeter range. In clinical practice, different investigation procedures are carried out to measure different parametric values in different scales and performed at different time points. This CSS model is aligned with such sorts of practices and therefore, one can get a meaningful prediction through this analytical model by fitting those clinical data.

In clinical practice, several therapeutic regimes are available and the majority of them are established with population-based analysis. However, which therapeutic regime and when it would be appropriate to an individual is still uncertain [32]. Therefore, in silico modeling for cancer therapy took the focal point in recent time [3336]. The developed CSS model has been targeted to address these issues. The advantage of this model is the flexibility with respect to the setting of clinical parametric values for the individual cases. Simulation runs will be carried out with the initial parametric values that are obtained at the time of diagnosis for the individual cases followed by matching of the simulation outputs with the intermittent clinical investigations. If any deviation is found from the predicted output value, then proper parametric adjustment can be made and further simulation runs has to be carried out for further prediction regarding the cancer treatment dynamics (subject to predict-observe-correct cycle). Such optimization method may be used for the measurements and estimations of the parametric values of the nonstate variables. Thus, this CSS model can be implemented in clinical situation.

Drug resistance is due to the presence of cancer stem cells; our model considered two types (sensitive and resistance) of cells with an interconversion rates between the two having the advantage of making a rationalization of a therapeutic scheme for drug resistant tumor [20]. The another advantage of the developed CSS model is that through this analytical model one can predict the tumor load in terms of tumor cell number, microvessel cell number, tumor radius, tumor volume, microvessel diameter.

In reality, the biological systems have a time-varying nonlinear nature. However, from oncological point of view, clinicians may be interested to know whether a specific therapeutic scheme would bring tumor system under control or not. But, for a nonlinear system, assessment of controllability criteria is difficult. Hence, for the assessment of the controllability criteria, the nonlinear system can be represented through an approximated linear system. This has already been utilized in VG model with a notion that the dynamics of the approximated system will precede much ahead of time with respect to the actual system [25]. In a similar fashion, FD model has also been approximated to a linear system and after approximation, synergism between the two models has also been made. Under the same tumor condition the Eigenvalue analysis showed that only MCT scheme (with Im and Ihs) (strategy 6) has values less than one. Though Eigenvalue analysis with strategy 8 (MCT along with AAG combination) for VG model showed similar observation of strategy 6, for FD model Eigenvalues exceeds one on the day of (intermittent) AAG application. This sort of discrepancy is due to consideration of five state variables in the approximated FD model instead of three state variables in the approximated VG model.

Several AAG drugs are in clinical trial and in recent time application of AAG after conventional MTD is also in the Phase II clinical trial; however, reports indicate the possible limitations of AAG application [26, 2830]. Our simulation studies with the therapeutic strategies of MTD with intermittent HSC transplantation (strategy 2) and AAG application (strategy 3) corroborate such hypotheses [8, 28, 29, 31]. Simulation study also suggests that under the present tumor condition MCT alone (strategy 6) is more effective in controlling long-term tumor burden than any other chemotherapeutic strategies and/or their combinations. In recent time, it is of the opinion of some oncologists that tumor controllability is more desirable than curing it [37]. To target this, MCT could be an interesting alternative for primary systemic therapy or maintenance therapy [15, 16, 30]. To make a careful investigation among different available therapeutic options for cancer, the developed CSS model may provide an analytical tool to test the effectiveness of a chosen therapy in individual cancer cases.

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