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Stress generation, relaxation and size control in confined tumor growth

['Huaming Yan', 'Department Of Mathematics', 'University Of California', 'Irvine', 'California', 'United States Of America', 'Daniel Ramirez-Guerrero', 'Center For Multiscale Cell Fate Studies', 'John Lowengrub', 'Department Biomedical Engineering']

Date: 2022-01 

When the tumor reaches its equilibrium size R(t) = R ∞ , one can derive y ,r (R ∞ ) = β/(β + λ(R ∞ ) − λ A (R ∞ )). This shows that (i) β = 0 (no relaxation) leads to a singularity in the elastic strain and (ii) increasing β restores the proportionality between y(r, t) and r and thus decreases the elastic energy (Fig C in S1 Text ) and stress anisotropy (Fig D in S1 Text ), which are both 0 when y(r, t) = r. Similar considerations hold at the tumor center. The cell density ρ c (r, t) is highest in regions near the tumor boundary ( Fig 1E ). This is due to the synergistic effect among the rate of cell division , the rate of net local volume gain λ net (r, t) = λ(r, t)c(r, t) − λ A (r, t), and the local flux of cells. One can see from Eq (6) that cell division tends to increase the cell density, while the local volume gain tends to decrease the cell density. When this two rates match, the cell density is uniform. However, when the compression near the tumor boundary slows down the local volume gain via mechanotransduction and water efflux, the local division rate becomes faster than that of the volume gain, which leads to an increase in cell number density. Notice the advection and local random neighbor exchange of cells can further adjust the cell density distribution. In the central region with inward radial velocity v(r, t) < 0, local cell density increases due to the advection of the cells. At the tumor center with v(0, t) ≡ 0, the random neighbor exchange of cells is necessary to describe local cell accumulation. This higher-order derivative term allows us the implement the no-flux boundary condition at the tumor center, which facilitates changes in cell density at the tumor center according to the cell density in its neighborhood. Interestingly, we have found the value of D does not visibly affect the result in the cell density distribution (see Fig E in S1 Text ). Nevertheless, the effect from the cell flux is secondary to the competition between volume gain and cell proliferation rates (in this case), due to the high proliferative activity near the free tumor boundary.

The spatial distribution of the circumferential stress at any given time t seems contradictory to the experimental and theoretical results in [ 34 , 36 ], which found compressive stresses in the tumor interior and tensile stresses at the tumor boundary. In the experiments, these differences may be due to the specific tissue properties, external confinement of the implanted tumors and the growth stage of the implanted tumor. Note that in Fig 1 , the tumor is freely growing and there is no external confinement. Differences between the theoretical results in [ 34 , 36 ] and ours may be due to the different model assumptions of the stress-dependent volumetric growth and loss. Our results are consistent with stress patterns observed in implanted brain tumors in [ 14 ] except that we do not see tensile stresses near the tumor-host interface since our tumor is not confined. As we will see below, when the tumor grows in a confined region in our model, tensile stresses emerge at the tumor-host interface in the confining tissue and stresses in the tumor interior can be compressive or tensile depending on the tissue properties.

We first simulate numerically (see Section 1.3 in S1 Text for the algorithm and parameters), the unconfined (free) growth of an initially unstressed tumor spheroid (F ext = 0). As seen in Fig 1A the tumor radius increases over time and approaches a steady-state, which is independent of the initial size or initial stress state (Fig B in S1 Text ). Because of diffusion-limited nutrient transport, the net volume rate of change λ net (r, t) = λ(r, t)c(r, t) − λ A (r, t) is spatially varying ( Fig 1B ). At early times when the tumor is small, the volume increases all throughout the tumor spheroid as nutrients are readily available. At later times, volume gain (due to cell growth) dominates at the spheroid boundary and volume loss (due to cell apoptosis and water efflux) dominates at the spheroid center ( Fig 1B ) where nutrient levels are low. Correspondingly, cells move outward at early times but at late times, as in previous models (e.g., [ 12 , 32 , 35 ]), cells divide at the boundary and move inward to compensate for the loss of volume at the center ( Fig 1C ), which may explain the presence of long-lasting apoptotic markers in the core of the tumor spheroids. The total circumferential stresses are compressive throughout the tumor spheroid at early times while at later times the stresses are compressive near the tumor edge and tensile in the center ( Fig 1D ). Further, the stresses equilibrate as soon as the tumor radius reaches equilibrium.

Inhibition of growth through external confinement

In [10], it was shown that the growth capacity of tumor spheroids (human colon adenocarcinoma, LS174T) in agarose gels decreases as the concentration of agarose is increased (Fig 2A and A in S2 Text); the stiffness of the gels is positively correlated with the agarose concentration. However, tumors suspended in gels with lower growth rates regain their free-growth capacity once the gels are removed (Fig 2E, symbols). We use our model to fit the experimental data from tumors grown in free suspension (0% gel) and 0.7% and 1.0% agarose gels using the same set of the tumor-associated parameters (which characterize the base rates and chemomechanical responses of LS174T) but different shear moduli of the gel (e.g., c H = 0 for 0% and c H > 0 for the 0.7% and 1.0% gels). The experimental data consists of both tumor radius (shown as symbols in Fig 2A), and average cell densities of compressed tumors as a ratio to that of the free tumor (shown as vertical line segments denoting the mean and standard deviations of experimental measurements in Fig 2D). Since [10] suggests that cells does not adjust their rate of proliferation in response to the spatial confinement, we assume compressive stresses increase local water efflux (γ A , Δ A > 0 and γ λ = 0). Then, using the same tumor-associated parameters, we fit the other gel concentrations (Fig A in S2 Text) by changing only c H . See S2 Text for details on the fitting process.

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TIFF original image Download: Fig 2. Fitting the model to data from [10] of tumor spheroids grown in free suspension (blue) and 0.7% (red) and 1.0% (brown) agarose gels with (solid) and without (dashed) feedback from the elastic stress (A) and when stresses are released by removing the gels in a narrow interval around the times reported in [10] (E). Bands show the results within 10% of the best fit with feedback in all panels. (B) and (C): The distributions of stresses and net volume growth rates from the model. (D) and (F): Average cell density of tumors in (A) and (E) respectively. The bars in (D) shows the experimental average density for 0.7% and 1.0% tumors, relative to the free tumor. See S2 Text for details. https://doi.org/10.1371/journal.pcbi.1009701.g002

We find that the nondimensional relaxation rate β ∼ 1, which suggests that a fully nonlinear elastic model is needed to describe tumor biomechanics, rather than a fluid model or a linear elastic model. Such reduced models arise as limits of our model where β >> |λnet|. See S1 Text for limiting cases in different parameter regimes. The results are presented in Fig 2 (and Fig A in S2 Text). There is good quantitative agreement between the model (curves) and experiments (symbols) for the dynamics of the tumor spheroid radii (Fig 2A) where the bands show the results using parameters for which the results are within 10% of the best fit, which corresponds to β = 0.4 and and (see S2 Text for all the fitted parameters). Fits to other gel concentrations are shown in Fig A in S2 Text. For the case of the 1.0% gel at equilibrium, we predict the circumferential stress ( ) in the tumor at equilibrium is compressive and quasi-uniform (Fig 2C). We find that the average elastic energy in the tumor spheroid decreases even though c H increases (Fig B in S2 Text), which shows that stress and stiffness are not always positively correlated. This result is consistent with the findings in [14]. Our results suggest that as c H increases, the stresses in the tumor become mainly hydrostatic (see Fig B in S5 Text). When c H >> 1, it can be seen analytically that the leading-order stress distribution is uniform and is given by the compression at the boundary. As the agarose concentration of the gel is decreased, c H decreases and the stresses are less compressive and less uniform (Fig 2B, solid curves). In the freely growing case, the stress becomes tensile in the tumor interior and is compressive only at the spheroid boundary similar to that observed in Fig 1D. In contrast, the stresses in the gel (dotted curves) are tensile, as the growth of the tumor stretches the surrounding gels circumferentially, with the maximum stress occurring at the spheroid boundary. This case is more like the situation in [36], where there is a periphery layer with slower growth than the intratumor region. Increasing the gel concentration, reduces the magnitude of the circumferential stresses outside the tumor because even though c H increases, the smaller tumors displace the gel less. At equilibrium, there is a net volume loss in the tumor center, which is balanced by volume gain at the boundary (Fig 2C, solid curves). As seen in Figs 2 and A-D in S5 Text, the stress patterns in the tumor and surrounding gel depend on the tissue stiffnesses, the tumor growth rates and stress relaxation. Consistent with in vivo experimental results of orthotopically-implanted tumors in mice [14], the stresses in the tumor center can be compressive or tensile (depending on tissue properties) while tensile circumferential stresses occur at the slower-growing (e.g., gel) side of the tumor-host interface. Because of the discontinuity in growth and stress relaxation between the tumor and surrounding gel, the circumferential stresses are discontinuous at the tumor-host interface. If continuity were imposed, as might be expected in vivo, there should be a smooth transition of circumferential stress from compression to tension at the tumor-host interface.

To examine the effect of water efflux, we also fit the data without considering feedback from the elastic stress (γ A = 0). In this case, the predicted radii (dashed curves in Fig 2A) also provide a good fit of the data, but the stress distributions and net volume growth rates are more heterogeneous when considering growth regulated by nutrient level alone (Fig 2B and 2C, dashed). Further, using the corrected Akaike information criterion (AICc) [37] suggests the model with feedback provides a better fit to the experiment (see S2 Text). For the water-efflux feedback function (Eq (9)), we have considered the effect of the circumferential stress , reasoning that it dominates the mechanical state of the cells because it represents stresses from two principal directions orthogonal to the radial direction. One could change to the stress invariant and obtain similar results. See S2 Text and Fig C in S2 Text.

We calculate the average cell density in the tumor by and plot this as a function of time. The average density increases with time. For more constrained tumors with increased gel stiffness, the average cell density increases, consistent with experimental data (vertical line segments in Fig 2D). Additionally, our simulation reveals that the region with largest cell density shifts towards the center of tumor (Fig 3A), indicating that tumor cells are most packed inside due to the inward cell flow. In this case, the effect of cell flux on the local cell density becomes primary, because volume growth and proliferative activities slow down due to globally elevated compression from the spatial confinement.

To model the gel-removal experiments from [10] (Fig 2E and 2F), we set c H = 0 in all the gel cases and use the common set of fitted tumor-associated parameters (see S2 Text for details). Again, there is good quantitative agreement between the numerics and experiments, which both tend to recover the growth of the unconstrained spheroid. The average cell density also recovers to the same level as in the free-boundary case. (Fig 2E and 2F) In addition, we also show that the spatial distribution of the cell density reverse to the distribution in the free-boundary case upon gel-removal, where the cell density near the tumor boundary becomes higher again (compare Figs 3B with 1E).

We also use the data from [12, 13], where colon carcinoma tumor spheroids containing mouse CT26 cell lines where grown under isotropic compression from an osmotically-induced external pressure. As suggested by [12, 13], the compressive stresses reduce proliferation rates without increasing apoptosis. Therefore, we fit our model with feedback on the proliferation rate to data in Figure 1 in [13]. In this case, the tumor radius is determined by the sensitivity to feedback from mechanical stresses (γ λ ) and pressure boundary condition ( ), in addition to the volume loss rate λ A and other parameters. We fit the experimental data in tumor radius (shown as symbols in Fig 4A) and data of relative cell density, where the density in the compressed tumor at tumor center is approximately 20% larger than that in the free tumor (see the triangle (mean) with error bars in Fig A in S4 Text). Our fitting yields good quantitative agreement between the model and experiments for the dynamics of the tumor spheroid radii (Fig 4A), as well as the cell density (Fig A in S4 Text). Similar to Fig 2B, the circumferential stress is also compressive at the tumor boundary, and becomes tensile towards the tumor center. Both the stress and the net volume growth are more uniform when feedback is considered. This can be seen by comparing the solid curve (with feedback) and dashed curve (without feedback) in Fig 4B and 4C.

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TIFF original image Download: Fig 4. Fitting the model to data from [12, 13] of tumor spheroids grown free or with external pressure, with and without (dashed) feedback from the elastic stress (A). External pressure is released at the time reported in [13] (E). Bands show the results within 10% of the best fit with feedback in all panels. (B) and (C): The distributions of stresses and net volume growth rates from the model. (D) and (F): Average cell density of tumors in (A) and (E) respectively. See S4 Text for the list of parameters. https://doi.org/10.1371/journal.pcbi.1009701.g004

Consistent to the trend in Fig 2D, the average cell density is higher in the more compressed tumor (Fig 4D). Again, external extra compression shifts the region with higher cell density towards tumor center, as can be seen in Fig A in S4 Text. The tumor radius, the average cell density, as well as the cell density distribution are all reversible upon the pressure removal (Fig 4E and 4F and Fig A in S4 Text).

[END]

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