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

Table 1

(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.

(a)

Variable	Symbol used in model		Unit	Value	Comment
	VG	FD
Tumor cell (sensitive type) count	𝑐1	𝑆	Number of cells	^†4×10^6	Initial value at the time of diagnosis
Tumor cell (resistive type) count	𝑐2	𝑅	Number of cells	^†1×10^6	Initial value at the time of diagnosis
Microvasculature (MV) cell count	𝑣𝑣	—	Number of cells	^†1×10^4 (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)	𝐷MCT; 𝐷MTD	𝐷MCT; 𝐷MTD	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 drug; Value 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].

(b)

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

(c)

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)

(d)

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)