# Is There One Key Step in the Metastatic Cascade?

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

^{†}

^{‡}

## Abstract

**:**

## Simple Summary

## Abstract

## 1. Introduction

## 2. Material and Methods

#### 2.1. The Drake Equation

#### 2.2. The Metastatic Drake Equation

#### 2.3. Parameters Estimation from the Literature

^{6}or 4.05 × 10

^{6}cells per gram of tumor per day [22]. The probability $Pc$ that a CTC survives in the circulating system was calculated based on the half-life of 1 and 2.5 h, estimated by [23], and using the following equation:

#### 2.4. Numerical Simulations

#### 2.4.1. Twenty-Four-Hour Time Scale

#### 2.4.2. Six-Month Time Scale

^{b}) with an exponent b representing a very small fraction of the tumor volume (here 0.001). This scenario aimed to simulate a situation where metastases are descendants of stem-like cancer cells with stable population sizes due to asymmetric division. Finally, in the third case, we considered that the number of cells a tumor sheds scales with its surface, assuming a spherical tumor shape. An initial tumor weight of 1 g shedding 3.18 × 10

^{6}or 4.05 × 10

^{6}cells per day was assumed to have a volume of 75.7 mm

^{3}, a radius of 2.6 mm and a surface of 86.5 mm

^{2}(based on [27]). This equates to a number of 36,763 or 46,821 cells that intravasate in the vascular system per mm

^{2}per day (according to [22]).

^{3}to 456.3 mm

^{3}with a radius increasing from 2.6 mm to 4.8 mm; and, for the luminal A cancer, a tumor volume increasing from 75.7 mm

^{3}to 103.7 mm

^{3}with a radius increasing from 2.6 to 2.9 mm). For each day of the simulations, we computed the median number of cells that successfully extravasated in the target organ based on all possible parameter combinations for a given simulation.

## 3. Results

#### 3.1. 24-Hour Time Scale

#### 3.1.1. Simulating the Number of Extravasated Cells as Function of Half-Life and Time to Reach the Target Organ

#### 3.1.2. Simulating the Effect of a Treatment on the Survival and Extravasation of CTCs

#### 3.2. Six-Month Time Scale

^{3}compared to 103.7 mm

^{3}, Figure 5). This ratio range remains constant at any time between the two cancer types regardless of the half-life of CTCs or the time that CTCs take to reach the target organ. Those results suggest that reducing the half-life of CTCs early in the development of the tumor has the potential to prevent a very large number of CTCs to extravasate in the target organ, particularly for fast-growing tumors.

## 4. Discussion

_{C}= aM

^{b}where −1 < b < 0 scales this release by total tumor mass. Under such a formulation, the number of tumor cells released per day per gram of primary tumor, $Nc$, declines with tumor mass, M, but the overall number of released cells, $M\xb7Nc$, increases with tumor mass.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Dillekås, H.; Rogers, M.S.; Straume, O. Are 90% of deaths from cancer caused by metastases? Cancer Med.
**2019**, 8, 5574–5576. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bohl, C.R.; Harihar, S.; Denning, W.L.; Sharma, R.; Welch, D.R. Metastasis suppressors in breast cancers: Mechanistic insights and clinical potential. J. Mol. Med.
**2014**, 92, 13–30. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Khan, I.; Steeg, P.S. Metastasis suppressors: Functional pathways. Lab. Investig.
**2018**, 98, 198–210. [Google Scholar] [CrossRef] [Green Version] - Stoletov, K.; Beatty, P.H.; Lewis, J.D. Novel therapeutic targets for cancer metastasis. Expert Rev. Anticancer Ther.
**2020**, 20, 97–109. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lambert, A.W.; Pattabiraman, D.R.; Weinberg, R.A. Emerging biological principles of metastasis. Cell
**2017**, 168, 670–691. [Google Scholar] [CrossRef] [Green Version] - Eslami, S.Z.; Majidzadeh, A.K.; Halvaei, S.; Babapirali, F.; Esmaeili, R. Microbiome and breast cancer: New role for an ancient population. Front. Oncol.
**2020**, 10, 120. [Google Scholar] [CrossRef] [Green Version] - Franssen, L.C.; Lorenzi, T.; Burgess, A.E.F.; Chaplain, M.A.J. A mathematical framework for modelling the metastatic spread of cancer. Bull. Math. Biol.
**2019**, 81, 1965–2010. [Google Scholar] [CrossRef] [Green Version] - Ganesh, K.; Massagué, J. Targeting metastatic cancer. Nat. Med.
**2021**, 27, 34–44. [Google Scholar] [CrossRef] - Chambers, A.F.; Groom, A.C.; MacDonald, I.C. Dissemination and growth of cancer cells in metastatic sites. Nat. Rev. Cancer
**2002**, 2, 563–572. [Google Scholar] [CrossRef] - Mehlen, P.; Puisieux, A. Metastasis: A question of life or death. Nat. Rev. Cancer
**2006**, 6, 449–458. [Google Scholar] [CrossRef] - Arnal, A.; Ujvari, B.; Crespi, B.; Gatenby, R.A.; Tissot, T.; Vittecoq, M.; Ewald, P.W.; Casali, A.; Ducasse, H.; Jacqueline, C.; et al. Evolutionary perspective of cancer: Myth, metaphors, and reality. Evol. Appl.
**2015**, 8, 541–544. [Google Scholar] [CrossRef] [PubMed] - Valastyan, S.; Weinberg, R.A. Tumor metastasis: Molecular insights and evolving paradigms. Cell
**2011**, 147, 275–292. [Google Scholar] [CrossRef] [Green Version] - Dujon, A.M.; Aktipis, A.; Alix-Panabières, C.; Amend, S.R.; Boddy, A.M.; Brown, J.S.; Capp, J.; DeGregori, J.; Ewald, P.; Gatenby, R.; et al. Identifying key questions in the ecology and evolution of cancer. Evol. Appl.
**2021**, 14, 877–892. [Google Scholar] [CrossRef] [PubMed] - Luzzi, K.J.; MacDonald, I.C.; Schmidt, E.E.; Kerkvliet, N.; Morris, V.L.; Chambers, A.F.; Groom, A.C. Multistep nature of metastatic inefficiency: Dormancy of solitary cells after successful extravasation and limited survival of early micrometastases. Am. J. Pathol.
**1998**, 153, 865–873. [Google Scholar] [CrossRef] - Warner, H.V.; Sivakumar, N.; Peirce, S.M.; Lazzara, M.J. Multiscale computational models of cancer. Curr. Opin. Biomed. Eng.
**2019**, 11, 137–144. [Google Scholar] [CrossRef] - Drake, F.D. Discussion at space science board, national academy of sciences. In Proceedings of the Conference on Extraterrestrial Intelligent Life, Green Bank, WV, USA, 1–2 November 1961. [Google Scholar]
- Sandberg, A.; Drexler, E.; Ord, T. Dissolving the Fermi Paradox. arXiv
**2018**, arXiv:1806.02404. [Google Scholar] - Hunter, K.W.; Crawford, N.P.; Alsarraj, J. Mechanisms of metastasis. Breast Cancer Res.
**2008**, 10, S2. [Google Scholar] [CrossRef] [Green Version] - Dujon, A.M.; Bramwell, G.; Roche, B.; Thomas, F.; Ujvari, B. Transmissible cancers in mammals and bivalves: How many examples are there? BioEssays
**2020**, 43, 2000222. [Google Scholar] [CrossRef] - Naxerova, K.; Jain, R.K. Using tumour phylogenetics to identify the roots of metastasis in humans. Nat. Rev. Clin. Oncol.
**2015**, 12, 258–272. [Google Scholar] [CrossRef] - Bray, F.; Ferlay, J.; Soerjomataram, I.; Siegel, R.L.; Torre, L.A.; Jemal, A. Global cancer statistics 2018: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries. CA Cancer J. Clin.
**2018**, 68, 394–424. [Google Scholar] [CrossRef] [Green Version] - Butler, T.P.; Gullino, P.M. Quantitation of cell shedding into efferent blood of mammary adenocarcinoma. Cancer Res.
**1975**, 35, 512–516. [Google Scholar] [PubMed] - Meng, S.; Tripathy, D.; Frenkel, E.P.; Shete, S.; Naftalis, E.Z.; Huth, J.F.; Beitsch, P.D.; Leitch, M.; Hoover, S.; Euhus, D.; et al. Circulating tumor cells in patients with breast cancer dormancy. Clin. Cancer Res.
**2004**, 10, 8152–8162. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chen, M.C.; Whisler, J.A.; Jeon, J.S.; Kamm, R.D. Mechanisms of tumor cell extravasation in an in vitro microvascular network platform. Integr. Biol.
**2013**, 5, 1262–1271. [Google Scholar] [CrossRef] [Green Version] - Jeon, J.S.; Bersini, S.; Gilardi, M.; Dubini, G.; Charest, J.L.; Moretti, M.; Kamm, R.D. Human 3D vascularized organotypic microfluidic assays to study breast cancer cell extravasation. Proc. Natl. Acad. Sci. USA
**2015**, 112, E818. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Martin, M.D.; Kremers, G.-J.; Short, K.W.; Rocheleau, J.V.; Xu, L.; Piston, D.W.; Matrisian, L.M.; Gorden, D.L. Rapid extravasation and establishment of breast cancer micrometastases in the liver microenvironment. Mol. Cancer Res.
**2010**, 8, 1319–1327. [Google Scholar] [CrossRef] [Green Version] - Esteva-Font, C.; Jin, B.J.; Verkman, A.S. Aquaporin-1 gene deletion reduces breast tumor growth and lung metastasis in tumor-producing MMTV-PyVT mice. FASEB J.
**2014**, 28, 1446–1453. [Google Scholar] [CrossRef] [Green Version] - Lee, S.H.; Kim, Y.S.; Han, W.; Ryu, H.S.; Chang, J.M.; Cho, N.; Moon, W.K. Tumor growth rate of invasive breast cancers during wait times for surgery assessed by ultrasonography. Medicine
**2016**, 95, e4874. [Google Scholar] [CrossRef] - Lloyd, M.C.; Gatenby, R.A.; Brown, J.S. Ecology of the Metastatic Process. In Ecology and Evolution of Cancer; Academic Press: Cambridge, MA, USA, 2017; pp. 153–165. ISBN 9780128043806. [Google Scholar]
- Nguyen, D.X.; Bos, P.D.; Massagué, J. Metastasis: From dissemination to organ-specific colonization. Nat. Rev. Cancer
**2009**, 9, 274–284. [Google Scholar] [CrossRef] - Yu, M.; Stott, S.; Toner, M.; Maheswaran, S.; Haber, D.A. Circulating tumor cells: Approaches to isolation and characterization. J. Cell Biol.
**2011**, 192, 373–382. [Google Scholar] [CrossRef] - Alix-Panabières, C.; Pantel, K. Liquid biopsy: From discovery to clinical application. Cancer Discov.
**2021**, 11, 858–873. [Google Scholar] [CrossRef] - Rack, B.; Schindlbeck, C.; Jückstock, J.; Andergassen, U.; Hepp, P.; Zwingers, T.; Friedl, T.W.P.; Lorenz, R.; Tesch, H.; Fasching, P.A.; et al. Circulating tumor cells predict survival in early average-to-high risk breast cancer patients. JNCI J. Natl. Cancer Inst.
**2014**, 106. [Google Scholar] [CrossRef] [PubMed] - Pantel, K.; Alix-Panabières, C. Liquid biopsy and minimal residual disease—Latest advances and implications for cure. Nat. Rev. Clin. Oncol.
**2019**, 16, 409–424. [Google Scholar] [CrossRef] - Pantel, K.; Speicher, M.R. The biology of circulating tumor cells. Oncogene
**2016**, 35, 1216–1224. [Google Scholar] [CrossRef] [PubMed] - Pantel, K.; Alix-Panabières, C. Tumour microenvironment: Informing on minimal residual disease in solid tumours. Nat. Rev. Clin. Oncol.
**2017**, 14, 325–326. [Google Scholar] [CrossRef] - Hanin, L.; Bunimovich-Mendrazitsky, S. Reconstruction of the natural history of metastatic cancer and assessment of the effects of surgery: Gompertzian growth of the primary tumor. Math. Biosci.
**2014**, 247, 47–58. [Google Scholar] [CrossRef] - Hanin, L.; Korosteleva, O. Does extirpation of the primary breast tumor give boost to growth of metastases? Evidence revealed by mathematical modeling. Math. Biosci.
**2010**, 223, 133–141. [Google Scholar] [CrossRef] - Kumar, S.; Weaver, V.M. Mechanics, malignancy, and metastasis: The force journey of a tumor cell. Cancer Metastasis Rev.
**2009**, 28, 113–127. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Aceto, N.; Bardia, A.; Miyamoto, D.T.; Donaldson, M.C.; Wittner, B.S.; Spencer, J.A.; Yu, M.; Pely, A.; Engstrom, A.; Zhu, H.; et al. Circulating tumor cell clusters are oligoclonal precursors of breast cancer metastasis. Cell
**2014**, 158, 1110–1122. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Castro-Giner, F.; Aceto, N. Tracking cancer progression: From circulating tumor cells to metastasis. Genome Med.
**2020**, 12, 31. [Google Scholar] [CrossRef] [Green Version] - Celià-Terrassa, T.; Kang, Y. Metastatic niche functions and therapeutic opportunities. Nat. Cell Biol.
**2018**, 20, 868–877. [Google Scholar] [CrossRef] [Green Version] - Pienta, K.J.; Hammarlund, E.U.; Brown, J.S.; Amend, S.R.; Axelrod, R.M. Cancer recurrence and lethality are enabled by enhanced survival and reversible cell cycle arrest of polyaneuploid cells. Proc. Natl. Acad. Sci. USA
**2021**, 118, e2020838118. [Google Scholar] [CrossRef] - Pienta, K.J.; Hammarlund, E.U.; Axelrod, R.; Brown, J.S.; Amend, S.R. Poly-aneuploid cancer cells promote evolvability, generating lethal cancer. Evol. Appl.
**2020**, 13, 1626–1634. [Google Scholar] [CrossRef] - Mallin, M.M.; Pienta, K.J.; Amend, S.R. Cancer cell foraging to explain bone-specific metastatic progression. Bone
**2020**, 115788. [Google Scholar] [CrossRef] - Ibrahim-Hashim, A.; Robertson-Tessi, M.; Enriquez-Navas, P.M.; Damaghi, M.; Balagurunathan, Y.; Wojtkowiak, J.W.; Russell, S.; Yoonseok, K.; Lloyd, M.C.; Bui, M.M.; et al. Defining cancer subpopulations by adaptive strategies rather than molecular properties provides novel insights into intratumoral evolution. Cancer Res.
**2017**, 77, 2242–2254. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Robey, I.F.; Baggett, B.K.; Kirkpatrick, N.D.; Roe, D.J.; Dosescu, J.; Sloane, B.F.; Hashim, A.I.; Morse, D.L.; Raghunand, N.; Gatenby, R.A.; et al. Bicarbonate increases tumor pH and inhibits spontaneous metastases. Cancer Res.
**2009**, 69, 2260–2268. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chiang, S.P.H.; Cabrera, R.M.; Segall, J.E. Tumor cell intravasation. Am. J. Physiol. Cell Physiol.
**2016**, 311, C1–C14. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Abdul Pari, A.A.; Singhal, M.; Augustin, H.G. Emerging paradigms in metastasis research. J. Exp. Med.
**2021**, 218. [Google Scholar] [CrossRef] - Yang, H.; Kuo, Y.; Smith, Z.I.; Spangler, J. Targeting cancer metastasis with antibody therapeutics. WIREs Nanomed. Nanobiotechnol.
**2021**, 13, e1698. [Google Scholar] [CrossRef] - Jacot, W.; Mazel, M.; Mollevi, C.; Pouderoux, S.; D’Hondt, V.; Cayrefourcq, L.; Bourgier, C.; Boissiere-Michot, F.; Berrabah, F.; Lopez-Crapez, E.; et al. Clinical correlations of programmed cell death ligand 1 status in liquid and standard biopsies in breast cancer. Clin. Chem.
**2020**, 66, 1093–1101. [Google Scholar] [CrossRef] - Artzy-Randrup, Y.; Epstein, T.; Brown, J.S.; Costa, R.L.; Czerniecki, B.J.; Gatenby, R.A. Novel evolutionary dynamics of small populations in breast cancer adjuvant and neoadjuvant therapy. NPJ Breast Cancer
**2021**, 7, 26. [Google Scholar] [CrossRef] - Cunningham, J.J.; Brown, J.S.; Vincent, T.L.; Gatenby, R.A. Divergent and convergent evolution in metastases suggest treatment strategies based on specific metastatic sites. Evol. Med. Public Health
**2015**, 2015, 76–87. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Illustration of the different steps of the metastatic cascade. Each step is represented by a red arrow. The blue text represents the associated parameters of the metastatic Drake equation.

**Figure 2.**The number of extravasated cells predicted by calculating all avalaible parameter combinations obtained from the literature as function of the time CTCs take to reach the target organ and their half-life (ranging from 0.25 to 6 h). Each solid line corresponds to one combination of parameters, and the bold red line to the median number of extravasated cells over time for each half-life value. Values for Pe (the probability of a CTC extravasating in the target organ) were set to 0.236, 0.384, 0.56 or 0.22. A tumor mass M of 1 g was assumed, and the number of cells shed from a tumor this size was assumed as 3.18 × 10

^{6}or 4.05 × 10

^{6}cells per gram of tumor per day.

**Figure 3.**Proportion of CTCs dying in the vascular system as a function of their half-life (ranging from λ = 0.25 to 6 h) and the time required to reach the target organ (0 to 200 min). The values for M, Pe and the number of shed cells are the same as in Figure 2.

**Figure 4.**The effect of a hypothetical treatment on the proportion of CTCs dying in the vascular system as function of their half-life (initially ranging from 0.25 to 6 h) and the time required to reach the target organ. The red lines correspond to mortality calculated from simulations in which no treatment was applied, the blue lines from simulation in which the half-life was reduced by 15 min and the green lines from simulation in which the half-life was reduced by 30 min. The values for M, Pe and the number of shed cells are the same as in Figure 2. Note that in the simulation with a half-life of 0.25 h, the green line (half-life reduced by 30 min) overlaps the blue line (half-life reduced by 15 min).

**Figure 5.**The number of extravasated cells per day for two different breast cancer types: triple-negative cancer which is fast growing and luminal A which is slow-growing. CTCs with half-lives of 1 and 2.4 h and a time for those cells to reach the target organ of 1, 60 or 120 min were considered in the simulation. The solid lines represent the median number of extravasated cells per day for a given simulation. Luminal A tumors grew by 0.175% per day while triple-negative cancer tumors grew by 1.003% per day. Note that the two lines for the simulations scaling to a small fraction of the tumor mass M (here M

^{b}with b = 0.001) (in green and dark green) are overlapping closely on the plots. The values for M, Pe and the number of shed cells are the same as in Figure 2.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dujon, A.M.; Capp, J.-P.; Brown, J.S.; Pujol, P.; Gatenby, R.A.; Ujvari, B.; Alix-Panabières, C.; Thomas, F.
Is There *One* Key Step in the Metastatic Cascade? *Cancers* **2021**, *13*, 3693.
https://doi.org/10.3390/cancers13153693

**AMA Style**

Dujon AM, Capp J-P, Brown JS, Pujol P, Gatenby RA, Ujvari B, Alix-Panabières C, Thomas F.
Is There *One* Key Step in the Metastatic Cascade? *Cancers*. 2021; 13(15):3693.
https://doi.org/10.3390/cancers13153693

**Chicago/Turabian Style**

Dujon, Antoine M., Jean-Pascal Capp, Joel S. Brown, Pascal Pujol, Robert A. Gatenby, Beata Ujvari, Catherine Alix-Panabières, and Frédéric Thomas.
2021. "Is There *One* Key Step in the Metastatic Cascade?" *Cancers* 13, no. 15: 3693.
https://doi.org/10.3390/cancers13153693