Publisher Correction to: Three-dimensional tumor growth in time-varying chemical fields: a modeling framework and theoretical study

Correction to: BMC Bioinformatics

https://doi.org/10.1186/s12859-019-2997-9

Following publication of the original article [1], the authors noticed that the following errors were introduced by pdf/html formatting issues. The original article has been corrected. The publisher apologizes to the authors and readers for these errors.

Page 9, first column:

The paragraph

“- o b_t(A): sec) by the local vascular network within/sec) by the local vascular network within A during the previous time interval t − Δτ → t”

Should be replaced with

“- o_b_t(A) : Oxygen supply rate (pmols/sec) by the local vascular network within A during the previous time interval t − Δτ → t ”

In the subsequent paragraph, i.e. the paragraph starting starting with the phrase “ - gl_b_t(A): Glucose supply rate … ” ,

the phrase “ interval−Δτ → t ” should be replaced with “ interval t − Δτ → t ”

Page 11, second column:

The equation $\beta \ge 1-\frac{6O_{\mathit{av}}}{l_t\left(A\right){{\rm K}}_{\mathit{ATP}}\mathit{\Delta \tau }}\equiv \beta \_$ should be $\beta \ge 1-\frac{6O_{\mathit{av}}}{l_t\left(A\right){{\rm K}}_{\mathit{ATP}}\mathit{\Delta \tau }}\equiv \underset{\_}{\beta }$ .

In the subsequent sentence, i.e. “Since a_o(β) is increasing, β_ is actually … ” the “β_ ” should be “ $\underset{\_}{\beta }$ ”.

In the subsequent paragraph (first bullet point), in the sentence “ If β _  > β₂, we have that … ” the “ β_ ” should be “ $\underset{\_}{\beta }$ ”.

In the subsequent paragraph (second bullet point), i.e. “ If β _  ≤ β₂, we have that for each β ∈ [max(β_, β₁), β₂] it holds that a_o(β) ≥ 0. ” all occurences of “ β_ ” should be “ $\underset{\_}{\beta }$ ”.

In paragraph 11a, first line, “ a_o(β) ” should be “ a_gl(β) ”.

Page 12, first column:

In the sentence “ Case 2.2.1. If β > β₂ or $\overline{\beta }<{\beta }_1$ or $\mathit{min}\left(\overline{\beta },{\beta }_2\right)<\mathit{max}\left(\beta \_,{\beta }_1\right)$, the analysis above implies that … ” the inequality β > β₂ should be $\underset{\_}{\beta }>{\beta }_2$ and the inequality $\mathit{min}\left(\overline{\beta },{\beta }_2\right)<\mathit{max}\left(\beta \_,{\beta }_1\right)$ should be $\mathit{min}\left(\overline{\beta },{\beta }_2\right)<\mathit{max}\left(\underset{\_}{\beta },{\beta }_1\right)$.

In the sentence “ Case 2.2.2. If β ≤ β₂, $\overline{\beta }\ge {\beta }_1$ and $\mathit{min}\left(\overline{\beta },{\beta }_2\right)\ge \mathit{max}\left(\beta \_,{\beta }_1\right)$… ” the inequality β ≤ β₂ should be $\underset{\_}{\beta }\le {\beta }_2$ and the inequality $\mathit{min}\left(\overline{\beta },{\beta }_2\right)\ge \mathit{max}\left(\beta \_,{\beta }_1\right)$ should be $\mathit{min}\left(\overline{\beta },{\beta }_2\right)\ge \mathit{max}\left(\underset{\_}{\beta },{\beta }_1\right)$ .

In the same paragraph, the mathematical expression $\beta \in \left[\mathit{max}\left(\beta \_,{\beta }_1\right),\mathit{min}\left(\overline{\beta },{\beta }_2\right)\right]$ should be $\beta \in \left[\mathit{max}\left(\underset{\_}{\beta },{\beta }_1\right),\mathit{min}\left(\overline{\beta },{\beta }_2\right)\right]$

Page 12, second column:

In the sentence “ Again, we pick a random $\tilde{\beta }$ in $\left[\mathit{max}\left(\beta \_,{\beta }_1\right),\mathit{min}\left(\overline{\beta },{\beta }_2\right)\right]$. ” the mathematical expression $\left[\mathit{max}\left(\beta \_,{\beta }_1\right),\mathit{min}\left(\overline{\beta },{\beta }_2\right)\right]$ should be $\left[\mathit{max}\left(\underset{\_}{\beta },{\beta }_1\right),\mathit{min}\left(\overline{\beta },{\beta }_2\right)\right]$

Page 14, second column:

The equations

$$\mathit{ob}{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)=\left(1-f_r\left(l_t\left(A\right),{\mathit{nc}}_t\left(A\right)\right)v_r+{\mathit{sw}}_t\left(A\right)f_e\left(l_t\left(A\right),{\mathit{nc}}_t\left(A\right),{\mathit{nn}}_t\left(A\right)\right)v_e\right)\bullet \mathit{ob}{\mathit{max}}_t\left(A\right)$$

$$\mathit{glb}{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)=\left(1-f_r\left(l_t\left(A\right),{\mathit{nc}}_t\left(A\right)\right)v_r+{\mathit{sw}}_t\left(A\right)f_e\left(l_t\left(A\right),{\mathit{nc}}_t\left(A\right),{\mathit{nn}}_t\left(A\right)\right)v_e\right)\bullet \mathit{glb}{\mathit{max}}_t\left(A\right)$$

should be

$$o\_b\_{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)=\left(1-f_r\left(l_t\left(A\right),{\mathit{nc}}_t\left(A\right)\right)v_r+{\mathit{sw}}_t\left(A\right)f_e\left(l_t\left(A\right),{\mathit{nc}}_t\left(A\right),{\mathit{nn}}_t\left(A\right)\right)v_e\right)\bullet o\_b\_{\mathit{max}}_t\left(A\right)$$

$$\mathit{gl}\_b\_{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)=\left(1-f_r\left(l_t\left(A\right),{\mathit{nc}}_t\left(A\right)\right)v_r+{\mathit{sw}}_t\left(A\right)f_e\left(l_t\left(A\right),{\mathit{nc}}_t\left(A\right),{\mathit{nn}}_t\left(A\right)\right)v_e\right)\bullet \mathit{gl}\_b\_{\mathit{max}}_t\left(A\right).$$

The equations

$$\mathit{ob}{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)=\left(1-f_r\left(l_t\left(A\right),{\mathit{nc}}_t\left(A\right)\right)r_3{v}_r+{\mathit{sw}}_t\left(A\right)f_e\left(l_t\left(A\right),{\mathit{nc}}_t\left(A\right),{\mathit{nn}}_t\left(A\right)\right)r_4{v}_e\right)\bullet \mathit{ob}{\mathit{max}}_t\left(A\right)$$

$$\mathit{glb}{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)=\left(1-f_r\left(l_t\left(A\right),{\mathit{nc}}_t\left(A\right)\right)r_3{v}_r+{\mathit{sw}}_t\left(A\right)f_e\left(l_t\left(A\right),{\mathit{nc}}_t\left(A\right),{\mathit{nn}}_t\left(A\right)\right)r_4{v}_e\right)\bullet \mathit{glb}{\mathit{max}}_t\left(A\right)$$

should be

$$o\_b\_{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)=\left(1-f_r\left(l_t\left(A\right),{\mathit{nc}}_t\left(A\right)\right)r_3{v}_r+{\mathit{sw}}_t\left(A\right)f_e\left(l_t\left(A\right),{\mathit{nc}}_t\left(A\right),{\mathit{nn}}_t\left(A\right)\right)r_4{v}_e\right)\bullet o\_b\_{\mathit{max}}_t\left(A\right)$$

$$\mathit{gl}\_b\_{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)=\left(1-f_r\left(l_t\left(A\right),{\mathit{nc}}_t\left(A\right)\right)r_3{v}_r+{\mathit{sw}}_t\left(A\right)f_e\left(l_t\left(A\right),{\mathit{nc}}_t\left(A\right),{\mathit{nn}}_t\left(A\right)\right)r_4{v}_e\right)\bullet \mathit{gl}\_b\_{\mathit{max}}_t\left(A\right)$$

Page 15, first column:

The equations

$$\mathit{ob}{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)=\left(1-\frac{l_t\left(A\right)+{\mathit{nc}}_t\left(A\right)}{M}{r}_3{v}_r+{\mathit{sw}}_t\left(A\right)\frac{M-l_t\left(A\right)-{\mathit{nc}}_t\left(A\right)-{\mathit{nn}}_t\left(A\right)}{M}{r}_4{v}_e\right)\bullet \mathit{ob}{\mathit{max}}_t\left(A\right)$$

$$\mathit{glb}{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)=\left(1-\frac{l_t\left(A\right)+{\mathit{nc}}_t\left(A\right)}{M}{r}_3{v}_r+{\mathit{sw}}_t\left(A\right)\frac{M-l_t\left(A\right)-{\mathit{nc}}_t\left(A\right)-{\mathit{nn}}_t\left(A\right)}{M}{r}_4{v}_e\right)\bullet \mathit{glb}{\mathit{max}}_t\left(A\right)$$

should be

$$o\_b\_{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)=\left(1-\frac{l_t\left(A\right)+{\mathit{nc}}_t\left(A\right)}{M}{r}_3{v}_r+{\mathit{sw}}_t\left(A\right)\frac{M-l_t\left(A\right)-{\mathit{nc}}_t\left(A\right)-{\mathit{nn}}_t\left(A\right)}{M}{r}_4{v}_e\right)\bullet o\_b\_{\mathit{max}}_t\left(A\right)$$

$$\mathit{gl}\_b\_{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)=\left(1-\frac{l_t\left(A\right)+{\mathit{nc}}_t\left(A\right)}{M}{r}_3{v}_r+{\mathit{sw}}_t\left(A\right)\frac{M-l_t\left(A\right)-{\mathit{nc}}_t\left(A\right)-{\mathit{nn}}_t\left(A\right)}{M}{r}_4{v}_e\right)\bullet \mathit{gl}\_b\_{\mathit{max}}_t\left(A\right)$$

Page 15, second column:

The equations

$$\mathit{ob}{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)=\left(1-\frac{l_t\left(A\right)+{\mathit{nc}}_t\left(A\right)}{M}{r}_3{v}_r+{\mathit{sw}}_t\left(A\right)\frac{M-l_t\left(A\right)-{\mathit{nc}}_t\left(A\right)-{\mathit{nn}}_t\left(A\right)}{M}{r}_4{v}_e\right)\bullet \mathit{ob}{\mathit{max}}_t\left(A\right)$$

$$\mathit{glb}{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)=\left(1-\frac{l_t\left(A\right)+{\mathit{nc}}_t\left(A\right)}{M}{r}_3{v}_r+{\mathit{sw}}_t\left(A\right)\frac{M-l_t\left(A\right)-{\mathit{nc}}_t\left(A\right)-{\mathit{nn}}_t\left(A\right)}{M}{r}_4{v}_e\right)\bullet \mathit{glb}{\mathit{max}}_t\left(A\right)$$

Should be

$$o\_b\_{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)=\left(1-\frac{l_t\left(A\right)+{\mathit{nc}}_t\left(A\right)}{M}{r}_3{v}_r+{\mathit{sw}}_t\left(A\right)\frac{M-l_t\left(A\right)-{\mathit{nc}}_t\left(A\right)-{\mathit{nn}}_t\left(A\right)}{M}{r}_4{v}_e\right)\bullet o\_b\_{\mathit{max}}_t\left(A\right)$$

$$\mathit{gl}\_b\_{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)=\left(1-\frac{l_t\left(A\right)+{\mathit{nc}}_t\left(A\right)}{M}{r}_3{v}_r+{\mathit{sw}}_t\left(A\right)\frac{M-l_t\left(A\right)-{\mathit{nc}}_t\left(A\right)-{\mathit{nn}}_t\left(A\right)}{M}{r}_4{v}_e\right)\bullet \mathit{gl}\_b\_{\mathit{max}}_t\left(A\right)$$

The second equation appearing in this column, i.e.

$$\mathit{gl}\_b_{t+\mathit{\Delta \tau }}\left(A\right)=B_{o\_b\_{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)}\left(\mathit{gl}\_b_t\left(A\right)+r_2\left(\left(\overline{{\mathit{gl}}_0}-{\mathit{gl}}_t\left(A\right)\right)/\mathit{\Delta \tau }\right)\right)$$

Should be

$$\mathit{gl}\_b_{t+\mathit{\Delta \tau }}\left(A\right)=B_{\mathit{gl}\_b\_{\mathit{max}}_{t+\mathit{\Delta \tau }}\left(A\right)}\left(\mathit{gl}\_b_t\left(A\right)+r_2\left(\left(\overline{{\mathit{gl}}_0}-{\mathit{gl}}_t\left(A\right)\right)/\mathit{\Delta \tau }\right)\right)$$