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

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The original article was published in BMC Bioinformatics 2019 20:442

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 bt(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_bt(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_bt(A): Glucose supply rate … ” ,

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

Page 11, second column:

The equation \( \beta \ge 1-\frac{6{O}_{av}}{l_t(A){K}_{ATP}\varDelta \tau}\equiv \beta \_ \) should be \( \beta \ge 1-\frac{6{O}_{av}}{l_t(A){K}_{ATP}\varDelta \tau}\equiv \underset{\_}{\beta } \) .

In the subsequent sentence, i.e. “Since ao(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 ao(β) ≥ 0. ” all occurences of “ β_ ” should be “ \( \underset{\_}{\beta } \) ”.

In paragraph 11a, first line, “ ao(β) ” should be “ agl(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 \( \overset{\sim }{\beta } \) 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

$$ ob\ {\mathit{\max}}_{t+\varDelta \tau}(A)=\left(1-{f}_r\left({l}_t(A),{nc}_t(A)\right){v}_r+{sw}_t(A)\ {f}_e\left({l}_t(A),{nc}_t(A),{nn}_t(A)\right){v}_e\right)\bullet ob\ {\mathit{\max}}_t(A) $$
$$ glb\ {\mathit{\max}}_{t+\varDelta \tau}(A)=\left(1-{f}_r\left({l}_t(A),{nc}_t(A)\right){v}_r+{sw}_t(A){f}_e\left({l}_t(A),{nc}_t(A),{nn}_t(A)\right){v}_e\right)\bullet glb\ {\mathit{\max}}_t(A) $$

should be

$$ o\_b\_{\mathit{\max}}_{t+\varDelta \tau}(A)=\left(1-{f}_r\left({l}_t(A),{nc}_t(A)\right){v}_r+{sw}_t(A)\ {f}_e\left({l}_t(A),{nc}_t(A),{nn}_t(A)\right){v}_e\right)\bullet o\_b\_{\mathit{\max}}_t(A) $$
$$ gl\_b\_{\mathit{\max}}_{t+\varDelta \tau}(A)=\left(1-{f}_r\left({l}_t(A),{nc}_t(A)\right){v}_r+{sw}_t(A){f}_e\left({l}_t(A),{nc}_t(A),{nn}_t(A)\right){v}_e\right)\bullet gl\_b\_{\mathit{\max}}_t(A). $$

The equations

$$ ob\ {\mathit{\max}}_{t+\varDelta \tau}(A)=\left(1-{f}_r\left({l}_t(A),{nc}_t(A)\right){r}_3{v}_r+{sw}_t(A)\kern0.5em {f}_e\left({l}_t(A),{nc}_t(A),{nn}_t(A)\right){r}_4{v}_e\right)\bullet ob\ {\mathit{\max}}_t(A) $$
$$ glb\ {\mathit{\max}}_{t+\varDelta \tau}(A)=\left(1-{f}_r\left({l}_t(A),{nc}_t(A)\right){r}_3{v}_r+{sw}_t(A){f}_e\left({l}_t(A),{nc}_t(A),{nn}_t(A)\right){r}_4{v}_e\right)\bullet glb\ {\mathit{\max}}_t(A) $$

should be

$$ o\_b\_{\mathit{\max}}_{t+\varDelta \tau}(A)=\left(1-{f}_r\left({l}_t(A),{nc}_t(A)\right){r}_3{v}_r+{sw}_t(A)\kern0.5em {f}_e\left({l}_t(A),{nc}_t(A),{nn}_t(A)\right){r}_4{v}_e\right)\bullet o\_b\_{\mathit{\max}}_t(A) $$
$$ gl\_b\_{\mathit{\max}}_{t+\varDelta \tau}(A)=\left(1-{f}_r\left({l}_t(A),{nc}_t(A)\right){r}_3{v}_r+{sw}_t(A){f}_e\left({l}_t(A),{nc}_t(A),{nn}_t(A)\right){r}_4{v}_e\right)\bullet gl\_b\_{\mathit{\max}}_t(A) $$

Page 15, first column:

The equations

$$ ob\ {\mathit{\max}}_{t+\varDelta \tau}(A)=\left(1-\frac{l_t(A)+{nc}_t(A)}{M}{r}_3{v}_r+{sw}_t(A)\kern0.5em \frac{M-{l}_t(A)-{nc}_t(A)-{nn}_t(A)}{M}{r}_4{v}_e\right)\bullet ob\ {\mathit{\max}}_t(A) $$
$$ glb\ {\mathit{\max}}_{t+\varDelta \tau}(A)=\left(1-\frac{l_t(A)+{nc}_t(A)}{M}{r}_3{v}_r+{sw}_t(A)\frac{M-{l}_t(A)-{nc}_t(A)-{nn}_t(A)}{M}{r}_4{v}_e\right)\bullet glb\ {\mathit{\max}}_t(A) $$

should be

$$ o\_b\_{\mathit{\max}}_{t+\varDelta \tau}(A)=\left(1-\frac{l_t(A)+{nc}_t(A)}{M}{r}_3{v}_r+{sw}_t(A)\kern0.5em \frac{M-{l}_t(A)-{nc}_t(A)-{nn}_t(A)}{M}{r}_4{v}_e\right)\bullet o\_b\_{\mathit{\max}}_t(A) $$
$$ gl\_b\_{\mathit{\max}}_{t+\varDelta \tau}(A)=\left(1-\frac{l_t(A)+{nc}_t(A)}{M}{r}_3{v}_r+{sw}_t(A)\frac{M-{l}_t(A)-{nc}_t(A)-{nn}_t(A)}{M}{r}_4{v}_e\right)\bullet gl\_b\_{\mathit{\max}}_t(A) $$

Page 15, second column:

The equations

$$ ob\ {\mathit{\max}}_{t+\varDelta \tau}(A)=\left(1-\frac{l_t(A)+{nc}_t(A)}{M}{r}_3{v}_r+{sw}_t(A)\kern0.5em \frac{M-{l}_t(A)-{nc}_t(A)-{nn}_t(A)}{M}{r}_4{v}_e\right)\bullet ob\ {\mathit{\max}}_t(A) $$
$$ glb\ {\mathit{\max}}_{t+\varDelta \tau}(A)=\left(1-\frac{l_t(A)+{nc}_t(A)}{M}{r}_3{v}_r+{sw}_t(A)\frac{M-{l}_t(A)-{nc}_t(A)-{nn}_t(A)}{M}{r}_4{v}_e\right)\bullet glb\ {\mathit{\max}}_t(A) $$

Should be

$$ o\_b\_{\mathit{\max}}_{t+\varDelta \tau}(A)=\left(1-\frac{l_t(A)+{nc}_t(A)}{M}{r}_3{v}_r+{sw}_t(A)\kern0.5em \frac{M-{l}_t(A)-{nc}_t(A)-{nn}_t(A)}{M}{r}_4{v}_e\right)\bullet o\_b\_{\mathit{\max}}_t(A) $$
$$ gl\_b\_{\mathit{\max}}_{t+\varDelta \tau}(A)=\left(1-\frac{l_t(A)+{nc}_t(A)}{M}{r}_3{v}_r+{sw}_t(A)\frac{M-{l}_t(A)-{nc}_t(A)-{nn}_t(A)}{M}{r}_4{v}_e\right)\bullet gl\_b\_{\mathit{\max}}_t(A) $$

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

$$ gl\_{b}_{t+\varDelta \tau}(A)={B}_{o\_b\_{\mathit{\max}}_{t+\varDelta \tau}(A)}\left(\kern0.5em gl\_{b}_t(A)+{r}_2\left(\left(\ \overline{gl_0}-{gl}_t(A)\ \right)/\varDelta \tau \right)\kern0.5em \right) $$

Should be

$$ gl\_{b}_{t+\varDelta \tau}(A)={B}_{gl\_b\_{\mathit{\max}}_{t+\varDelta \tau}(A)}\left(\kern0.5em gl\_{b}_t(A)+{r}_2\left(\left(\ \overline{gl_0}-{gl}_t(A)\ \right)/\varDelta \tau \right)\kern0.5em \right) $$

Reference

  1. 1.

    Antonopoulos M, Dionysiou D, Stamatakos G, Uzunoglu N. Three-dimensional tumor growth in time-varying chemical fields: a modeling framework and theoretical study. BMC Bioinformatics. 2019;20:Article number: 442.

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Correspondence to Markos Antonopoulos.

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Antonopoulos, M., Dionysiou, D., Stamatakos, G. et al. Publisher Correction to: Three-dimensional tumor growth in time-varying chemical fields: a modeling framework and theoretical study. BMC Bioinformatics 20, 500 (2019) doi:10.1186/s12859-019-3102-0

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