A mathematical model of tumour growth with Beddington–DeAngelis functional response: a case of cancer without disease

Figures & data

Figure 1. The effect of the search time on the steepness coefficient, s, and tumour cell lysis, $D_1$, when $w={10}^{-8}$.

Table 1. Parameter values in Equations (9(9) $\begin{array}{rl}\dot{T}& =T(1-T)-cNT-DT-K_T(1-e^{-M})T,\end{array}$(9) )–(14(14) $\begin{array}{rl}D& =\frac{\mathrm{d}{L}^l}{w+s{T}^l+L^l}.\end{array}$(14) ) for patients 9 and 10 [18].

	c	d	f	l	g	q
Patient 9	$1.25\times {10}^{-7}$	5.429	0.0956	2.09	0.029	3230
Patient 10	$8.32\times {10}^{-8}$	4.362	0.0956	1.81	0.029	3617
	p	s	h	j	k
Patient 9	7779	0.0839	$2.1\times {10}^{-11}$	0.0578	$2.05\times {10}^{-10}$
Patient 10	8166	0.512	$2.1\times {10}^{-11}$	0.0578	$3.17\times {10}^{-10}$
	$r_1$	$r_2$	$K_T$	$K_L$	$K_N$	$K_C$
Patient 9	$2.14\times {10}^{-4}$	0.2624	2.088	1.392	1.392	1.392
Patient 10	$1.43\times {10}^{-4}$	0.175	2.088	1.392	1.392	1.392
	m	β	γ	α	u	w
Patient 9	0.4733	0.0278	2.088	$7.5\times {10}^8$	573.1	${10}^{-8}$
Patient 10	21.16	0.0186	2.088	$\phantom{0.}5\times {10}^8$	382.1	${10}^{-8}$

Figure 2. Bifurcation diagrams using Equations (15(15) $\begin{array}{rl}\dot{T}& =T(1-T)-cNT-DT,\end{array}$(15) )–(18(18) $\begin{array}{rl}D& =\frac{\mathrm{d}{L}^l}{w+s{T}^l+L^l},\end{array}$(18) ) and parameter values of patient 10 with (a) d, m, and β as the bifurcation parameters, (b) d=4.36 and m and β as the bifurcation parameters, and (c) $\beta =0.0186$ and m as the bifurcation parameter.

Figure 3. Bifurcation diagrams for treatments using chemotherapy alone, Equations (9(9) $\begin{array}{rl}\dot{T}& =T(1-T)-cNT-DT-K_T(1-e^{-M})T,\end{array}$(9) )–(14(14) $\begin{array}{rl}D& =\frac{\mathrm{d}{L}^l}{w+s{T}^l+L^l}.\end{array}$(14) ) with $v_L=0$, and parameter values of patient 10 with (a) m and $d_M$ as the bifurcation parameters, and (b) m=3 and $d_M$ as the bifurcation parameter.

Figure 4. A bifurcation diagram using parameter values of patient 10 with m=3 and $d_L$ and $d_M$ as the bifurcation parameters. (b) The magnification of the portion $0\le d_L\le 1.5\times {10}^{-3}$ in Figure 4(a). (c) A bifurcation diagram using $d_L=2$ and $d_M$ as the bifurcation parameter.

L.G. de Pillis, W. Gu, and A.E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretations, J. Theoret. Biol. 238 (2006), pp. 841–862. doi: 10.1016/j.jtbi.2005.06.037

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