Solid stress and elastic energy as measures of tumour mechanopathology

Abstract

Solid stress and tissue stiffness affect tumour growth, invasion, metastasis and treatment. Unlike stiffness, which can be precisely mapped in tumours, the measurement of solid stresses is challenging. Here, we show that two-dimensional spatial mappings of solid stress and the resulting elastic energy in excised or in situ tumours with arbitrary shapes and wide size ranges can be obtained via three distinct and quantitative techniques that rely on the measurement of tissue displacement after disruption of the confining structures. Application of these methods in models of primary tumours and metastasis revealed that: (i) solid stress depends on both cancer cells and their microenvironment; (ii) solid stress increases with tumour size; and (iii) mechanical confinement by the surrounding tissue significantly contributes to intratumoural solid stress. Further study of the genesis and consequences of solid stress, facilitated by the engineering principles presented here, may lead to significant discoveries and new therapies.

Figure 1. 2-D map of solid stress in tumours

(A) The fresh tumour is embedded in 2% agarose, liquid in 40° C, and gellated in ice-water. The tumour-agarose construct is incised at the plane of interest to release solid stress. The release of solid stress results in the deformation field δ_z(x,y) with respect to the agarose reference plane; δ_z and tumour geometry are quantified by high-resolution ultrasound (resolution of 20 μm). (B) Using a finite element model, σ_zz, the mechanical stress in the z direction, and W, the elastic energy stored by the solid stress, are estimated from the deformation δ_z and the Young’s modulus of the tissue based on the Hooke’s Law. (C) The stress-induced deformation δ_z is uniform and negligible in stress-free normal tissues (e.g., kidney at the top and liver at the bottom). (D) A representative ultrasound image showing the stress-induced deformation and agarose reference line (yellow dashed line), the 2-D deformation field δ_z, the 2-D stress field σ_zz, and a representative stress profile across the tumour diameter are shown for mouse models of breast tumour (MMTV-M3C), pancreatic ductal adenocarcinoma tumour (AK4.4) and brain tumour (glioblastoma U87). (E) Enzymatic depletion of collagen in (MMTV-M3C) breast tumour results in significant reduction of solid stress and elastic energy density (control: n=5 mice; collagenase treated: n=4 mice; mean ± SEM; *: p < 0.05). The representative ultrasound image and solid stress maps are shown for control and collagenase treated tumours.

Figure 2. Solid stress and elastic energy in primary vs. metastatic tumours

(A) Solid stress is mapped in size-matched primary pancreatic ductal adenocarcinoma (PDAC, AK4.4 cells; n=4 mice), (B) liver metastasis of PDAC (AK4.4; n=3 mice) (C) primary colorectal carcinoma (SL4 cells; n=3 mice), and (D) liver metastasis of colorectal carcinoma (SL4; n=4 mice). Comparing (E) the Young’s modulus (unconfined compression), (F) average solid stress in the z-direction, σ_zz, and (G) elastic energy density in these four tumour models shows that cancer cells are not the only determinant of biomechanical abnormalities: the organ and microenvironment in which the cancer cells reside are equally important in generation of solid stress and elastic energy in tumours. The data shown are mean ± SEM; *: p < 0.05.

Figure 3. Creating tumour slices provides a sensitive measure of solid stress applicable to a wide range of specimen sizes

(A) The fresh tumour is embedded in liquid 2% agarose, then sliced with a compresstome and left in PBS at room temperature, so the stress-induced deformation occurs in the tumour slice. The deformed slice and the surrounding agarose disk are imaged via high-resolution ultrasound (resolution, 30μm) or optical frequency domain imaging (OFDI; resolution, 1μm) for small samples (e.g., lymph node). The buckling and deformation of representative slices from (B) breast tumour (MMTV-M3C), (C) pancreatic tumour (AK4.4) and (D) lymph node with macro-metastasis of breast cancer (4T1) are shown. Tissues with negligible or low levels of stress, such as (E) lymph node with micro-metastasis and (F) kidney are shown. (G) The blank space in the agarose disk represents the area of the tumour slice before the stress relaxation. (H) The 3-D reconstruction of the slices in breast and pancreatic tumours, compared to kidney. (I) The expansion ratio, an index of solid stress and elastic energy, is defined as the ratio of the surface area of the slice after stress relaxation A_tumour to the area of the blank in the agarose disk, A_blank (equivalent to the area of the slice before stress relaxation). (J) The expansion ratio in breast (n=3 slices) and pancreatic tumours (n=3 slices) are significantly higher than the expansion in the kidney slice (n=3 slices). The expansion ratio of lymph node macro-metastasis (n=4 slices) is significantly higher than the one in micro-metastatic lymph node (n=4 slices)(mean ± SEM).

Figure 4. Evolution of solid stress and stiffness as a function of tumour size

(A) Solid stress was estimated with the slicing method in breast tumours (MMTV-M3C) with mean diameters ranging from 2–7 mm. (B) The expansion ratio – an index of solid stress – increases significantly with increasing tumour size. (C) The AFM-based indentation modulus of the same tumour model does not vary significantly with tumour size (D) Representative images of collagen I staining (white) of the MMTV-M3C tumours. The collagen content (assessed by positively stained collagen I area fraction) becomes localized to the tumour periphery with increasing tumour size. The radius R_c, defined as the effective normalized radius of an area that contains 50% of the positively stained collagen, increases with tumour diameter (E) The fraction of perfused blood vessels shows a decreasing trend with increasing tumour size, consistent with the hypothesis that high solid stress (which increases with tumour size, as shown in panel B) compresses blood vessels [14]. The data shown in B, C, D are mean ± SEM. Each data point in B represents n=3 slices from each mouse; each tumour diameter associates to a different mouse. In C, each data point represents n=10 indentation sites from each mouse. In D, n=10 radial sectors have been quantified for each data point.

Figure 5. In situ measurement of solid stress – the normal tissue that surrounds the tumour significantly contributes to the state of stress inside the tumour

(A) The stress is released by punching a cylindrical void in the tumour. The stress relaxation results in deformation along the axis of the void: compressive and tensile stresses contract and enlarge the hole, respectively. The changes in the geometry of the biopsy core are also an indicator of solid stress. (B) This method is validated on breast tumour (MMTV-M3C). In liver (C) and kidney (D) – control tissues with negligible stored stress – the profile of the biopsy void and core remain constant. (E) The punching method is capable of in situ estimation of solid stress in tissues. The punch can release the solid stress in the axis of interest, without disturbing the boundary between tumour and surrounding tissue, as schematically shown for the case of brain tumour with intact normal brain and cranium. (F) The stress-induced deformation has been measured for brain tumour (glioblastoma, U87) in situ (by punching the tumour surrounded by brain, through a cranial window; n=5 mice) and ex vivo (by punching surgically excised tumour; n=4 mice). (G, H) The solid stress is estimated by an axisymmetric mathematical model that uses the stress-induced deformation and the elastic properties of the tumour and brain as inputs. The distribution of σ_rr, the stress at the surface of the punched hole in the r direction, is shown at the original state before the deformation occurs.