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 [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 dose-dense 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 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 [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 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 Methods2.1. 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—c1 and c2 in VG and S and R in FD) having different multiplication rates (for sensitive and resistive type cells—f11 and f22 in VG and mS and mR in FD), different conversion rates between the two (from sensitive to resistive and vice versa are denoted by f12 and f21 in VG and gSR and gRS 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: (1) MTD, (2) MTD with intermittent autologous hematopoietic stem cell (HSC) transplantation [4, 8], (3) AAG, (4) strategy 2 (MTD + HSC) followed by strategy 3 (AAG) [26], (5) MCT without any immunoboosting (Im) and HSC mobilization inhibition (Ihs) from bone marrow, (6) MCT having Im and Ihs factor [4, 15, 16, 24], (7) strategy 5 (MCT without Im + Ihs) with intermittent strategy 3 (AAG), and (8) 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 (mst), apoptosis rate (ast), conversion rate to microvasculature (fa (to MV cell) and fb (to MVD)), and MTD chemotherapeutic drug sensitivity. The target actions of different therapeutic strategies are schematically shown in Figure 1.
Target site of different therapeutic strategies of the CSS model. In figure, c1, c2 and vv represent the drug-sensitive, drug-resistive, and MV cells, respectively, in VG model; S, R, and n 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 (c1) and drug resistive (c2) within the tumor mass and MV cells (vv) around the tumor. These cells grow with their own multiplication rates, that is, f11, f22, and fvv, respectively. Again c1 and c2 may be transformed into each other with a conversion rate of f12 andf21, respectively. Two malignant cell types (c1 and c2) respond in different ways with the change in MV cells (vv) numbers.
The dynamical relationships among these cells were represented by the following equation:
(1)[c1(k+1)c2(k+1)vv(k+1)]=[f11-f12-fim1(k)f21f1vf12f22-f21-fim2(k)f2v00fvv-ft(k)]×[c1(k)c2(k)vv(k)].
In (1), f1v = σ1×ρ1(k)×cv×[fvv-1-ft(k)] and f2v=σ2 × ρ2(k)×cv×[fvv-1-ft(k)], where σ1and σ2 are the sensitivities of two types of malignant cells to anoxia and ρ1 and ρ2 are the proportions of a specific type of tumor cells with respect to total tumor cell population and represented as ρ1(k)=c1(k)/(c1(k)+c2(k)) and ρ2(k)=c2(k)/(c1(k)+c2(k)). Again, cv represents a coefficient that is defined as the number of total tumor cells supported by each MV cell. Thus, cv is represented through the following relationship: cv=(c1(1)+c2(1))/vv(1). It was assumed that a fully grown tumor requires vv(1), amount of vasculature to support a total tumor cell count [c1(1)+c2(1)] at the time (in days) of diagnosis, that is, k=1 and is considered as constant parameter. The variables f1v and f2v, being multiplied with the amount of MV cells, translate the change in vv into a corresponding change in tumor cells number. With the application of AAG drug, MV cells are killed with a rate of ft(k) 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 fim1(k) andfim2(k), 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, f11 and f22 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) Initial parametric values for simulation. (b) Initial parametric values (assumed) of HSC for simulation. (c) (3×3) cross-section along with initial microvessel diameter in micrometer, TAF concentration in ng/μL, and FNT concentration in ng/μL. 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.
Variable
Symbol used in model
Unit
Value
Comment
VG
FD
Tumor cell (sensitive type) count
c1
S
Number of cells
†4×106
Initial value at the time of diagnosis
Tumor cell (resistive type) count
c2
R
Number of cells
†1×106
Initial value at the time of diagnosis
Microvasculature (MV) cell count
vv
—
Number of cells
†1×104 (VG)
Initial value at the time of diagnosis
Microvasculature (MV) diameter
—
n
μm
10 (FD)
Initial value at the time of diagnosis
Multiplication rate of sensitive cells
f11
mS
Cells/day
Table Id
—
Multiplication rate of resistive cells
f22
mR
Cells/day
Table Id
—
Conversion rate of sensitive to resistive cells
f12
gSR
Cells/day
^{†}0
In presence of exogenous HSC, it changes to 0.001
Conversion rate of resistive to sensitive cells
f21
gRS
Cells/day
^{†}0
—
Anoxia sensitivity of sensitive-type cells
σ1
σ1
% of cells
^{†}1.0 (VG), 1.0 (FD)
Full sensitivity Value for FD is assumed
Anoxia sensitivity of resistive-type cells
σ2
σ2
% of cells
^{†}0.5 (VG), 0.5 (FD)
Half of the total cells resist Value for FD is assumed
AAG drug sensitivity of TAF
σ7
% of TAF concentration
*0.25
1/4
th amount of existing TAF will be inactivated
Duration between two successive AAG drug applications
ancycle
Stint
Day
*15
Once in two-week interval
Amount of AAG drug dose
ft
ft
Fraction of cell killing and/or TAF concentration decrease per day
^{†}0.25
On the day of drug application, fraction of MV cells destroyed and/or fraction of TAF concentration reduced
Amount of AAG drug retention on the subsequent day of drug application
anret
anret
% of drug of the preceding day
^{†}80% (VG), 80% (FD)
Effectiveness decay, reaches almost zero after 10 days Value for FD is assumed
Chemotherapeutic drug sensitivity of sensitive-type cells (MCT/MTD)
σ3
σ3
% of cells
*0.9 (MTD), *0.9 (MCT)
90% cells are sensitive to drug Value for VG is assumed
Chemotherapeutic drug sensitivity of resistive-type cells (MCT/MTD)
σ4
σ4
% of cells
*0.3 (MTD), *0.3 (MCT)
30% cells are sensitive to drug Value for VG is assumed
Chemotherapeutic drug sensitivity of MV cells (MCT/MTD)
σ5
—
% of cells
0.5 (MTD), 0.5 (MCT)
Effect of MTD on MV cells is calibrated with a multiplying factor 15 Value is assumed
Chemotherapeutic drug sensitivity of MV diameter (MCT/MTD)
—
σ5
% of cell diameter
*0.5 (MTD), *0.5 (MCT)
Effect of MTD on MV diameter is calibrated with a multiplying factor 10^{7}
Chemotherapeutic (MCT) drug sensitivity to TAF
—
σ6
% of TAF concentration
*0.25
1/4
th amount of existing TAF will be inactivated
Duration between two successive MCT/MTD chemotherapeutic drug applications
stdint
stdint
Days
*1 (MCT), *21 (MTD)
Daily application of drug in MCT and once in three weeks in MTD
Chemotherapeutic drug dose (MCT/MTD)
DMCTDMTD
DMCTDMTD
Fraction of cell killing and/or MVD reduce and/or fraction of TAF concentration decrease per day
0.0079 (MCT), 0.1667 (MTD)
Multiplying factor 1.25 (MCT) and 1.5 (MTD). Hence, effective dose ≅ 0.01 (MCT), =0.25 (MTD)Values are assumed
Amount of chemotherapeutic drug (MCT/MTD) retention on the subsequent day of drug application
chemoret
chemoret
% of drug of the preceding day
*90%
Retention of 90% drug of the previous day drug amount Value for VG is assumed
MTD drug sensitivity of sensitive-type cells in strategy 2
σ8
σ8
% of cells
*0.9
90% cells are sensitive to drugValue for VG is assumed
MTD drug sensitivity of resistive-type cells in strategy 2
σ9
σ9
% of cells
*0.3
30% cells are sensitive to drug. Value for VG is assumed
MTD drug sensitivity of MV cells in strategy 2
σ10
—
% of cells
0.5
Value is assumed Effect of MTD on MV cell is calibrated with a multiplying factor 15
MTD drug sensitivity of MV cell diameter in strategy 2
—
σ11
% of cell diameter
*0.5
Effect of MTD on MV diameter is calibrated with a multiplying factor 10^{7 }
MTD drug dose in strategy 2
DMYL
DMYL
Fraction of cell killing and/or MVD reduction per day
0.1667
Value is assumed Multiplying factor 1.5. Hence, effective dose = 0.25
Amount of MTD drug retention on the subsequent day of drug application in strategy 2
MYLret
MYLret
% of drug of the preceding day
*90%
Retention of 90% drug of the previous day drug amount
Coefficient denoting both types of tumor cells supported by one MV cell (VG)
cv
—
Ratio
^{†}500 (VG)
Derived from initial values supplied for c1, c2, and vv
*Parametric value is taken from [18, 22, 23].
^{†}Parametric value is taken from [25].
Variable
Symbol used in model
Unit
Value
VG
FD
Count
THSC
THSC
Number of cells
1
Multiplication rate
mst
mst
% of cells
2%
Apoptosis rate
ast
ast
% cells
2.1%
Conversion rate to vasculature
fa
fb
% change to vv cell number (VG), % change to MVD (FD)
10% (VG), 1% (FD)
MTD drug sensitivity in strategy 2
σ12
σ12
% of cells
0.5
Microvessel diameter (μm)
0.2 (1, 1)
0.23 (1, 2)
0.25 (1, 3)
0.45 (2, 1)
10 (2, 2) (previous value: 0.44)
0.49 (2, 3)
0.57 (3, 1)
0.59 (3, 2)
0.62 (3, 3)
TAF concentration (ng/μL)
0.1 (1, 1)
0.1 (1, 2)
0.1 (1, 3)
0.1 (2, 1)
0.1 (2, 2)
0.1 (2, 3)
0.15 (3, 1) (previous value: 0.1)
0.15 (3, 2) (previous value: 0.1)
0.15 (3, 3) (previous value: 0.1)
FNT concentration (ng/μL)
0.005 (1, 1)
0.005 (1, 2)
0.005 (1, 3)
0.005 (2, 1)
0.005 (2, 2)
0.005 (2, 3)
0.005 (3, 1)
0.005 (3, 2)
0.005 (3, 3)
MV cell number range (tumor type)
MV diameter range (μm) (tumor type)
Doubling time (days)
f11 (VG)/mS (FD)
f22 (VG)/mR (FD)
0–500 (dormant)
0<n≤1(dormant)
25 (1.02811)
20 (1.03526)
501–5000 (regressive)
1<n≤5(regressive)
15 (1.04729)
12 (1.05942)
5001–20000 (progressive)
5<n≤10(progressive)
10 (1.0718)
6 (1.1225)
20001–above(vigorous)
10<n(vigorous)
5 (1.1487)
3 (1.2599)
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:
(2)M(k+1)=(1-H×K(k)-N(k))×M(k)+(n(i,j)(k+1)-n(i,j)(k))×103,
where M(k) is the malignant cell count on kth day. H represents the drug sensitivity and K(k) denotes the drug dose (MTD or MCT) applied on the kth day. N(k) denotes the immunity level on kth day and n(i,j)(k) is the vasculature diameter at (i,j) location on kth day and is given by following equation:
(3)n(i,j)(k+1)=[n(i,j)(k)+(P0+P1+P2+P3+P4)],
where MVD (n) is linked with concerned probabilistic cell movement (P) at each (i,j) 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 P0 (stationary probability), P1 (right direction probability), P2 (left direction probability), P3 (bottom direction probability), and P4 (up direction probability). They are represented in terms of TAF concentration (c), FNT concentration (f), and MVD (n) [22, 23] and are represented through the following relationships:
(4)P0=(1-(4×l)+x-y-z),P1=(l-((14×h2)×((0.38×c(i,j)(k)×(c(i+1,j)(k)-c(i-1,j)(k)))+(0.34×(f(i+1,j)(k)-f(i-1,j)(k))))(14×h2))),P2=(+(0.34×f(i-1,j)(k))))l+((14×h2)×((0.38×c(i,j)(k)×c(i-1,j)(k))+(0.34×f(i-1,j)(k)))(14×h2))),P3=(l-((14×h2)×((0.38×c(i,j)(k)×(c(i,j+1)(k)-c(i,j-1)(k)))+(0.34×(f(i,j+1)(k)-f(i,j-1)(k))))(14×h2))),P4=(l-((14×h2)×((0.38×c(i,j)(k)×(c(i,j+1)(k)-c(i,j-1)(k)))+(0.34×(f(i,j+1)(k)-f(i,j-1)(k))))(14×h2))).
In the above equations l, x, y, and z can be considered either as some constant factors or variables; however, they can be derived through the following equations:
(5)l=(μ×D)h2,x=(μ×α×χ)×[{c(i+1,j)(k)-c(i-1,j)(k)}2+{c(i,j+1)(k)-c(i,j-1)(k)}2],y=(μ×β×χ)×[{c(i+1,j)(k)+c(i-1,j)(k)}-4c(i,j)(k)+c(i,j+1)(k)+c(i,j-1)(k)],z=(μ×ρ)h2×[f(i+1,j)(k)+f(i-1,j)(k)-4f(i,j)(k)+f(i,j+1)(k)+f(i,j-1)(k)],
where μ, h, α, β, and χ are some constant values. TAF concentration (c) and FNT concentration (f) at different locations (i,j) of cancer milieu are represented through the following equations:
(6)c(i,j)(k+1)=0.018×c(i,j)(k),(7)f(i,j)(k+1)=f(i,j)(k)1/h2.
Now using (3), (2) can be modified as
(8)M(k+1)=(1-H×K(k)-N(k))×M(k)+((n(i,j)(k)+(P0+P1+P2+P3+P4))-n(i,j)(k))×103=(1-H×K(k)-N(k))×M(k)+(P0+P1+P2+P3+P4)×103=(1-H×K(k)-N(k))×M(k)+P(i,j)(k)×103.
Considering probabilistic movement (P), tumor cell number (M), and MVD (n) as three state variables, the system can be represented through the following equation:
(9)[P(i,j)(k+1)M(i,j)(k+1)n(i,j)(k+1)]=[000103(1-H(k)×K(k)-N(k))0101]×[P(i,j)(k)M(i,j)(k)n(i,j)(k)].
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, (1) MTD, (2) MTD with intermittent autologous hematopoietic stem cell (HSC) transplantation [6–13], (3) AAG, (4) strategy 2 (MTD + HSC) followed by strategy 3 (AAG) [26], (5) MCT without any immunoboosting (Im) and HSC mobilization inhibition (Ihs) from bone marrow, (6) strategy 5 (MCT without Im + Ihs) with intermittent strategy 3 (AAG), (7) strategy 6 (MCT with Im + Ihs) with intermittent strategy 3 (AAG), and (8) 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 (mst), apoptosis rate (ast), conversion rate to vasculature (fa (to MV cell) and fb (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: swD for MTD application, swM for MCT application, swA for AAG application, swMYL for MTD with intermittent HSC application, and swT 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: swvd for MV/MVD damage, swim for immunity boosting, and swihs 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) States of drug strategy activation switches during different drug strategies. (b) States of activation switches for producing different effects during different drug strategies.
Switches
Free growth
MTD
MCT
AAG
MTD + HSC
swD
0
1
0
0
0
swM
0
0
1
0
0
swA
0
0
0
1
0
swMYL
0
0
0
0
1
swT
0
0
0
0
1
Switches
Free growth
MTD
MCT
AAG
MTD + HSC
Condition
swim
0
0
1
0
0
If immunity is being developed
swihs
0
0
1
0
0
If stem cell mobilization inhibition is developed
swvd
0
1
0
0
1
Depending on the vv cell number and drug present (in MTD or MTD + HSC condition) within the system
Vasculature damage occurs; that is, MV cell killing happens when the available MV cell count (vv) (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 swvd is being activated (i.e., becomes “1”) depending upon the drug level and MV cell numbers (vv) within the system. This simultaneously becomes effective MVD (n) 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 (c1, c2, S, and R) 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 (vv).
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:(10)[c1(k+1)c2(k+1)vv(k+1)]=[{G1(k)×f11-f12-fim1(k)-swD×chc1D(k)-swM×chc1M(k)}f21f1vf12{G2(k)×f22-f21-fim2(k)-swD×chc2D(k)-swM×chc2M(k)}f2v00{G3(k)×fvv-swA×ft(k)-swD×swvd×chvvD(k)-swM×chvvM(k)}][c1(k+1)c2(k+1)vv(k+1)]=×[c1(k)c2(k)vv(k)].
In (10), the MTD-related killings are incorporated through chc1D(k) (= σ3×DMTD(k)), chc2D(k) (= σ4×DMTD(k)), and chvvD(k) (= σ5×DMTD(k)). Similarly, in (10) MCT-related killings are incorporated through chc1M(k) (= σ3×DMCT(k)+swim×Im(k)), chc2M(k) (= σ4×DMCT(k)+swim×Im(k)), and chvvM(k) (= σ5×DMCT(k)+swihs×Ihs(k)). DMTD(k) and DMCT(k) are the drug doses in MTD and MCT strategies, respectively, on kth day and σ3, σ4, and σ5 are the drug sensitivities to the two types of malignant cells and MV cells to MTD/MCT (Table 1). MCT-based immunoboosting [Im(k)] is introduced through additional subtractive terms in chc1M(k) and chc2M(k), and Ihs(k) is the factor that represents the inhibition of stem cell mobilization from bone marrow in MCT therapy which is introduced within the term chvvM(k).
Similarly, AAG-related drug killing is incorporated through ft(k), and ft(k) is the AAG drug dose on the kth day. In (10), fim1(k) andfim2(k) 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 G1, G2, and G3 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—S (sensitive cell type) and R (resistive cell type). These cells are multiplied with mS (multiplication rate of drug sensitive-type cell) and mR (multiplication rate of drug resistive-type cell) and converted to another type with a rate of gSR (conversion rate of sensitive to resistive-type cell) and gRS (conversion rate of resistive- to sensitive-type cell). Growth of each type of the tumor cells is influenced by the development of MVD (n) at different locations (i,j) 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):
(11)l=0.00035h,x=(0.6×c(i,j)(k)4×h×(1+c(i,j)(k)))×[{c(i+1,j)(k)-c(i-1,j)(k)}2+{c(i,j+1)(k)-c(i,j-1)(k)}2],y=(0.38×c(i,j)(k)h)×[{c(i+1,j)(k)+c(i-1,j)(k)}-4c(i,j)(k)+c(i,j+1)(k)+c(i,j-1)(k)],z=(0.34h)×[f(i+1,j)(k)+f(i-1,j)(k)-4f(i,j)(k)+f(i,j+1)(k)+f(i,j-1)(k)].
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:(12)[P(i,j)(k+1)S(i,j)(k+1)R(i,j)(k+1)n(i,j)(k+1)]=[00000{G1(k)×mR-gSR-swD×chSD(k)-swM×chSM(k)-fimS(k)}gRSvS(k)0gSR{G2(k)×mS-gRS-swD×chRD(k)-swM×chRM(k)-fimR(k)}vR(k)Cf00{1-swD×swvd×chnD(k)-swM×chnM(k)}]×[P(i,j)(k)S(i,j)(k)R(i,j)(k)n(i,j)(k)].
The probabilistic growth P(i,j)(k) becomes unrealistically large; hence, in (12), a calibration factor, Cf (a constant fractional number), is considered to make MVD output realistic. Again, vS(k)=σ1×ρS(1)×[P(i,j)(k)-D(k)] and vR(k)=σ2×ρR(1)×[P(i,j)(k)-D(k)], where σ1 and σ2 are the anoxia sensitivities of S and R type cells, respectively and ρS(1) = S(1)/(S(1)+R(1)) and ρR(1) = R(1)/(S(1)+R(1)). D(k) is a subtractive term equivalent to length reduction of MVD. In MTD strategy, D(k) = swD×swvd×chnD(k) = swD×swvd×(σ5×DMTD(k)), in MTD + HSC strategy, D(k) = swMYL×swvd×chnMYL(k) = swMYL×swvd×(σ11×DMYL(k)), and in MCT strategy, D(k) = swM×chnM(k) = swM×(σ5×DMCT(k)+swihs×Ihs(k)).
Again in (12), for MTD strategy, chSD(k) = σ3×DMTD(k) and chRD(k) = σ4×DMTD(k); for MCT strategy, chSM(k) = σ3×DMCT(k)+swim×Im(k) and chRM(k) = σ4×DMCT(k)+swim×Im(k). DMTD(k) and DMCT(k) are the drug doses in MTD and MCT strategies on kth day and σ3, σ4, and σ5 are the drug sensitivities to the two types of malignant cells and MVD to MTD/MCT.
MCT-based immunoboosting [Im(k)] is introduced through additional subtractive terms in chSM(k) and chRM(k) and Ihs(k) is the factor that represent the inhibition of stem cell mobilization from bone marrow in MCT therapy is introduced within the term chnM(k). The term σ11 is the drug sensitivity of MVD in MTD + HSC. The detailing of MTD + HSC incorporation is explained later in (17). In (12), fimS(k) andfimR(k) were considered as adoptive immunotherapy or other immunoboosting instead of MCT-based immunoboosting.
Thus, MVD (n) at different locations (i,j) of cancer milieu are derived from (3) modified as (13) and TAF concentration (c) can be obtained by modifying (6) to (14) to implement the effect of MCT and AAG:
(13)n(i,j)(k+1)=[n(i,j)(k)+Cf×(P0+P1+P2+P3+P4)],(14)c(i,j)(k+1)=1.018×c(i,j)(k)×[1-swA×{σ7×ft(k)}-swM×{σ6×DMCT(k)}].
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, ft(k) present in the system multiplied with the AAG sensitivity (σ7), 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:
(15)cϕ(k+1)=[c(i,j)(k+1)-(ϕ×γ)]+(cϕ(k)-τ).
In (15), cϕ(k+1) represents the change in TAF concentration per unit blood at (k+1)th time followed by kth time concentration cϕ(k) at distance ϕ from (i,j) location where (i,j) is (1,2) position at the cancer milieu, τ is the TAF degradation per unit time, and γ 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:(16)[c1(k+1)c2(k+1)vv(k+1)]=[{G1(k)×f11-f12-fim1(k)-swD×chc1D(k)-swM×chc1M(k)-swMYL×chc1MYL(k)}f21f1vf12{G2(k)×f22-f21-fim2(k)-swD×chc2D(k)-swM×chc2M(k)-swMYL×chc2MYL(k)}f2v00{G3(k)×fvv-swA×ft(k)-swD×swvd×chvvD(k)-swM×chvvM(k)-swMYL×chvvMYL(k)+swT×fa×THSC(k)}]×[c1(k)c2(k)vv(k)].
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),(18)chc1MYL(k)=σ8×DMYL(k),chc2MYL(k)=σ9×DMYL(k),chvvMYL(k)=σ10×swvd×DMYL(k),chSMYL(k)=σ8×DMYL(k),chRMYL(k)=σ9×DMYL(k),chnMYL(k)=σ11×swvd×DMYL(k),
where σ8, σ9, and σ10/σ11 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 fa and fb are fractional numbers. Again THSC(k) is the number of HSC cells on kth day and is given by(19)THSC(k)={ST+(mst-ast-σ12×DMYL(k-1))×THSC(k-1)whenkisthedayoftransplantation
(mst-ast-σ12×DMYL(k-1))×THSC(k-1)whenkisthedayotherthanthetransplantationday,
where ST is the initial number of transplanted HSC cells. mst and ast are the multiplication and apoptosis rates of transplanted HSC cells. In MTD + HSC strategy, σ12 is the MTD drug sensitivity and DMYL(k) is the MTD drug present in the system on kth 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, G1(k), G2(k), and G3(k), that modulate, the multiplication rates of these three cell types (c1/S, c2/R, and vv). 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:(20)[c1(k+1)c2(k+1)vv(k+1)]=[{G1(k)×f11-f12-fim1(k)-H1(k)}f21f1vf12{G2(k)×f22-f21-fim2(k)-H2(k)}f2v00{G3(k)×fvv-H3(k)-swA×ft(k)+swT×fa×THSC(k)}]×[c1(k)c2(k)vv(k)].
Similarly, (17) can be simplified as(21)[P(i,j)(k+1)S(i,j)(k+1)R(i,j)(k+1)n(i,j)(k+1)]=[00000{G1(k)×mR-gSR-H1(k)-fimS(k)}gRSvS(k)0gSR{G2(k)×mS-gRS-H2(k)-fimR(k)}vR(k)Cf00{1-H3(k)+swT×fb×THSC(k)}]×[P(i,j)(k)S(i,j)(k)R(i,j)(k)n(i,j)(k)].
In (20) and (21), H1(k), H2(k), and H3(k) 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. H1(k), H2(k), and H3(k) 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 (S), resistive cells (R), and microvasculature cells (vv) have been considered. Each cell type and tumor load have been assumed to be spherical in shape. Let Rrad, Srad, vvrad and trad be the radii of resistive cell, sensitive cell, MV cell, and total tumor volume, respectively. Again trad is proportional to the total cell population of the above three cell types. That is,
(22)trad(k)=q×[S(k)+R(k)+vv(k)],
where q 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 q is calibrated as q′ 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 Rrat(k):Srat(k):vvrat(k). 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 R(k)/T(k), S(k)/T(k), vv(k)/T(k) or Rrat(k), Srat(k), vvrat(k). Moreover, the total cell population in tumor is given by
(23)T(k)=S(k)+R(k)+vv(k).
Now, the volumes of single resistive cell, sensitive cell, MV cell, and total tumor are given by Rvol (= (4/3)πRrad3), Svol(=(4/3)πSrad3), vvvol(=(4/3)πvvrad3), and tvol(k) (= (4/3)πtrad3(k)), respectively.
Hence, the expected ratio of each cell type with respect to their volume in unit volume of collected biopsy sample is given by Rrat(k)×Rvol:Srat(k)×Svol:vvrat(k)×vvvol or Rrat(k)×(4/3)πRrad3:Srat(k)×(4/3)πSrad3:vvrat(k)×(4/3)πvvrad3 or Rrat(k)×Rrad3:Srat(k)×Srad3:vvrat(k)×vvrad3. Therefore, the expected number of resistive cells, sensitive cells, and MV cells in tumor is given by (24)–(26), respectively:
(24)NR(k)=tvol(k)×R(k)·Rrad3R(k)·Rrad3+S(k)·Srad3+vv(k)·vvrad3×1Rvol,(25)NS(k)=tvol(k)×S(k)·Srad3R(k)·Rrad3+S(k)·Srad3+vv(k)·vvrad3×1Svol,(26)Nvv(k)=tvol(k)×vv(k)·vvrad3R(k)·Rrad3+S(k)·Srad3+vv(k)·vvrad3×1vvvol.
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 (3×3) 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).
Simulation results in different therapeutic strategies.
Different strategies
VG model
FD model
Condition
Days
c1
c2
vv
S
R
n(2,2)
c(2,2)
(0) Free growth
50
1.5×109
5.6×109
9.0×104
8.2×108
5×109
10.47
0.248
Freely growing condition
150
1.6×1015
6.1×1019
7.4×106
8.64×1014
5.40×1019
11.7864
1.525
300
1.7×1024
6.8×1034
5.43×109
9.28×1023
6.07×1034
266.23
23.26
500
1.9×1036
7.9×1054
3.6×1013
1.02×1036
7.08×1054
3.65×105
879.54
(1a) MTD strategy (conventional dose)
50
3.57×106
1.85×107
4.74×103
1.15×107
4.06×108
0.039
0.248
Drug application as per Table 1(a), drug cycle = 6
150
5.65×104
6.3×109
9.12×103
2.76×108
3.25×1016
0.5160
1.525
300
1.45×1013
8.36×1023
6.72×106
2.49×1017
3.47×1031
204.07
23.26
500
1.6×1025
9.76×1043
4.47×1010
2.74×1029
4.05×1051
2.92×105
879.54
(1b) MTD strategy (high dose)
50
3.22×106
1.7×107
4.34×103
1.07×107
3.99×108
0.039
0.248
Drug dose = 0.255; other condition is the same as strategy 1a
150
2.67×104
1.9×109
7.35×103
2.1×108
3.02×1016
0.516
1.525
300
4.81×1012
1.41×1023
5.41×106
1.88×1017
3.22×1031
204.07
23.257
500
5.29×1024
1.64×1043
3.61×1010
2.07×1029
3.75×1051
3.35×105
879.54
(2) MTD with HSC transplantation
50
1.37×108
4.42×108
3.9×105
2.27×107
2.99×108
0.14
0.248
MTD drug (as per strategy 1a) is applied and HSC transplantation was applied alternatively (total 5 times) after a gap period of 20 days
150
7.26×1014
1.71×1016
6.22×107
9.45×109
5.22×1015
0
1.525
300
1.40×1023
2.65×1028
2.16×1010
1.09×1015
4.44×1027
192.53
23.26
500
1.54×1035
3.1×1048
2.27×1016
1.2×1027
5.18×1047
3.59×105
879.54
(3) AAG strategy
50
1.8×107
1.22×107
890.54
3.61×107
1.74×107
11.1948
0.0834
Applied continuously with a strategy as per Table 1(a)
150
3.99×108
5.88×108
29.02
6.68×108
6.31×108
14.1133
0.0709
300
2.55×1010
1.06×1011
0.08
4.33×1010
1.15×1011
19.99
0.0502
500
6.52×1012
1.09×1014
0
1.11×1013
1.17×1014
31.84
0.0295
(4) MTD with HSC transplantation followed by AAG
50
1.37×108
4.42×108
3.9×105
2.72×107
2.99×108
0.14
0.25
MTD with HSC as per strategy 2 and AAG as per strategy 3, AAG started after 8 days of the completion of strategy 2
150
7.26×1013
1.71×1016
6.22×107
9.45×109
5.22×1015
0
1.525
300
3.39×1023
3.46×1028
2.83×107
3.83×1012
5.41×1022
40.18
6.38
500
1.07×1035
5.06×1047
7.36×103
9.82×1014
5.53×1025
113.03
3.75
(5) MCT (without Im and Ihs)
50
7.71×106
5.83×107
1.29×104
2.88×106
1.28×107
1.99
0.09
Continuous (daily) application of drug as per Table 1(a) but without Im and Ihs
150
1.4×106
4.25×1011
8.18×103
7.19×105
4.13×107
1.3045
0.05
300
3.35×104
1.97×1016
4.09×103
1.00×103
2.47×108
0.0045
0.016
500
6.29
6.82×1018
1.63×103
0
7.38×108
0
0.004
(6) MCT (without Im and Ihs) with intermittent AAG (15-day interval)
50
7.22×106
5.50×107
1.40×104
3.67×106
1.41×107
2.25
0.10
MCT drug is applied daily as per strategy 5 with AAG application in every 15-day interval as per strategy 3
150
2.34×106
4.76×1011
1.13×104
8.26×105
8.82×107
1.29
0.057
300
4.10×104
3.75×1017
8.32×103
1.42×104
1.43×109
0.03
0.026
500
4.09×104
2.73×1025
5.61×103
0
6.38×109
0
0.009
(7) MCT (with Im and Ihs) with intermittent AAG
50
5.04×106
3.95×107
1.04×104
2.61×106
1.02×107
1.66
0.096
MCT drug (having Im and Ihs) is applied, except the day of AAG drug application; AAG is applied with a 15-day interval as per strategy 3
150
0
0
0
8.83×103
5.13×103
0.13
0.057
300
0
0
0
0
0
0
0.026
500
0
0
0
0
0
0
0.009
(8) MCT strategy (with Im and Ihs)
50
5.40×106
4.19×107
9.66×103
2.05×106
8.94×106
1.47
0.09
Drug is applied as per strategy 5 and with Im and Ihs
150
0
0
0
7.73
16.79
0
0.04
300
0
0
0
0
0
0
0.016
500
0
0
0
0
0
0
0.004
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), Y-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, x-axis indicates time (in days).
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 × 10^{7} 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.
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 S to R and c1 to c2. As expected, it has been observed that though on the day of MTD drug application, S, R, c1, and c2 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).
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 S, R, c1, and c2 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).
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).
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 S and c1 reduced to zero on ~410th and ~543rd days while R and c2 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 S, R, c1, and c2 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 S, R, c1, and c2 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).
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 l, x, y, and z 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, ρ1(1)=c1(1)/(c1(1)+c2(1)) and ρ2(1) = c2(1)/(c1(1)+c2(1)) [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 (2,2) position of the (3 × 3) cross-sectional area at the cancer milieu has been considered only. This (2,2), 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 (2,2)P0, P1, P2, P3, and P4 are approximated as a function of TAF concentration of (2,2) position only instead of l, x, y, and z. 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 (2,2) point becomes
(27)p(i,j)(k+1)=Papx×c(i,j)(k).
In (27) Papx is proportionality constant and hence, probabilistic movement in each direction is equal to p(i,j)(k+1)/5. The approximated FD model is represented by the following equation:(28)[c(i,j)(k+1)P(i,j)(k+1)S(i,j)(k+1)R(i,j)(k+1)n(i,j)(k+1)]=[{1.018-swA×σn×ft(k)-swM×(σ6×DMCT(k)+ImCF×Im(k))}0000Papx000000{1+G1(k)×mR-gSR-swD×chSD(k)-swM×chSM(k)-swMYL×chSMYL(k)-fimS(k)}gRSvS(k)00gSR{1+G2(k)×mS-gRS-swD×chRD(k)-swM×chRM(k)-swMYL×chRMYL(k)-fimR(k)}vR(k)0Cf00{1-swD×swvd×chnD(k)-swM×chnM(k)-swMYL×chnMYL(k)+fb×T(k)}]×[c(i,j)(k)P(i,j)(k)S(i,j)(k)R(i,j)(k)n(i,j)(k)].
Whether or not the approximated system will proceed ahead of time with respect to the actual system being dependent on Papx, increase in the value of Papx 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 Papx = 1 × 10^{5}; however, sensitive (S) cell type, resistive (R) 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).
Dynamics of linearized approximated system (blue line with star) and actual system (red line). In all the plots, x-axis indicates time (in days) and y-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
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].
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.
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 [28–31].
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.
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].
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 [6–13]. 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 [20–25]. 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 [33–36]. 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, 28–30]. 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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