Quantitative analysis of tumour spheroid structure
- School of Mathematical Sciences, Queensland University of Technology, Australia
- ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Australia
- The University of Queensland Diamantina Institute, The University of Queensland, Australia
- Department of Computer Science, University of Oxford, United Kingdom
Figures
Figure 1
Experimental data and mathematical model.
(a–f) Growth of WM983b and WM793b spheroids over three weeks, initiated using approximately 2500, 5000 and 10,000 cells. The solid curve represents average outer radius and the coloured region …
Figure 2
Late-time progression of WM983b spheroids, randomly sampled from the 10 spheroids imaged from each condition (additional images in Supplementary file 2).
Overlaid are the three boundaries identified by the image processing algorithm: the entire spheroid, the inhibited region and the necrotic region. Each image shows a 800 × 800 µm field of view. …
Figure 3
We calculate approximate confidence intervals (CI) using profile likelihood and confidence regions (CR) using contours of the normalised likelihood function.
Results demonstrate estimates of and using the structural model, Equation 6, and data from WM983b spheroids at day 14 initiated using 5000 cells. Point estimates are calculated using the …
Figure 4
Estimates of parameters using the structural model with data from various time points.
In (a–c), parameters are the mean of each observation: . In (d–e), parameters are those in the structural model: . In (f), estimates of are obtained by calibrating observations to the …
Figure 5
Comparison of WM983b spheroids between each initial seeding density at day 18 (spheroids seeded with 5000 or 10000 cells) and day 21 (2500).
(a–c) Profile likelihoods for each parameter, which are used to compute approximate confidence intervals (Table 1). (d) 95% confidence region for the full parameter space. 95% confidence regions for …
Figure 6
Data from days 3 to 21 (WM983b) and days 4 to 24 (WM793b) for all initial conditions.
Solid curves in (a) show the solution to the mathematical model (Equation 6) using the maximum likelihood estimate calculated using the steady-state data (Table 1). Solid curves in (c) show the …
Appendix 1—figure 1
Number of solutions of Equation 20.
(a) Number of solutions to Equation 20 subject to the constraint 0 ≤ ρ ≤ 1. Dashed line indicates the region of interest, where γ > 0 and 0 < Q < 1. (b) Comparison between a long-term solution to …
Appendix 2—figure 1
Fitting experimental data to linear model.
(a) Comparison between typical least-squares error (blue dashed), and total-least-squares error (blue solid). (b) Square error observed in the data and fitted gamma distribution.
Appendix 3—figure 1
Estimates of parameters using the structural model with data from various time points.
In (a–c), parameters are the mean of each observation: . In (d–e), parameters are those in the structural model: . In (f), estimates of are obtained by calibrating observations to the …
Tables
Table 1
Parameter estimates and approximate confidence intervals for each initial conditions.
Also shown are p-values for likelihood-ratio-based hypothesis tests for parameter equivalence between seeding densities.
| Parameter | θ2500 | θ5000 | θ10000 | p 2500,5000 | p 5000,10000 |
|---|---|---|---|---|---|
| 340.0 (331.0, 349.0) | 353.0 (344.0, 361.0) | 356.0 (347.0, 365.0) | 0.0420 | 0.617 | |
| 0.899 (0.889, 0.908) | 0.895 (0.886, 0.905) | 0.901 (0.891, 0.911) | 0.617 | 0.406 | |
| 0.719 (0.674, 0.764) | 0.716 (0.671, 0.761) | 0.742 (0.696, 0.788) | 0.940 | 0.438 | |
| μ | 0.202 | 0.687 | |||
| 0.75 (0.696, 0.811) | 0.758 (0.704, 0.818) | 0.771 (0.711, 0.838) | 0.854 | 0.767 | |
| 149.0 (127.0, 171.0) | 156.0 (133.0, 178.0) | 145.0 (121.0, 168.0) | 0.672 | 0.503 | |
| 0.737 (0.598, 0.916) | 0.768 (0.624, 0.953) | 0.657 (0.532, 0.816) | 0.792 | 0.308 | |
| 0.202 | 0.687 | ||||
Additional files
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Transparent reporting form
- https://cdn.elifesciences.org/articles/73020/elife-73020-transrepform1-v2.docx
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Supplementary file 1
Spheroid count per experimental condition (harvest day, seeding density and cell line).
- https://cdn.elifesciences.org/articles/73020/elife-73020-supp1-v2.docx
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Supplementary file 2
Additional cross-sectional confocal images of spheroids; 10 per experimental condition (harvest day, seeding density and cell line).
- https://cdn.elifesciences.org/articles/73020/elife-73020-supp2-v2.pdf
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Supplementary file 3
Reproduction of Figure 5 using data from day 21 for all initial seeding densities.
- https://cdn.elifesciences.org/articles/73020/elife-73020-supp3-v2.pdf
