Solid stress and elastic energy as measures of tumour mechanopathology

Transcription

1 VOLUME: 1 ARTICLE NUMBER: 0004 In the format provided by the authors and unedited. Solid stress and elastic energy as measures of tumour mechanopathology Hadi T. Nia 1, Hao Liu 1,2, Giorgio Seano 1, Meenal Datta 1,3, Dennis Jones 1, Nuh Rahbari 1, Joao Incio 1,4, Vikash P. Chauhan 1, Keehoon Jung 1, John D. Martin 1, Vasileios Askoxylakis 1, Timothy P. Padera 1, Dai Fukumura 1, Yves Boucher 1, Francis J. Hornicek 5, Alan J. Grodzinsky 6, James W. Baish 7, Lance L. Munn 1, and Rakesh K. Jain 1,* 1. Edwin L. Steele Laboratories, Department of Radiation Oncology, Massachusetts General Hospital, Harvard Medical School, Boston, MA 02114, USA 2. Leder Human Biology and Translational Medicine, Biology and Biomedical Sciences, Harvard Medical School, Boston MA 02115, USA 3. Department of Chemical and Biological Engineering, Tufts University, Medford, MA 02155, USA 4. Department of Internal Medicine, Hospital S. Joao; I3S, Institute for Innovation and Research in Health; and Faculty of Medicine, Porto University; Porto, Portugal 5. Orthopedic Oncology Service, Center for Sarcoma and Connective Tissue Oncology, Massachusetts General Hospital, Harvard Medical School, Boston, MA 02114, USA 6. Center for Biomedical Engineering, Departments of Mechanical, Electrical and Biological Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 7. Department of Biomedical Engineering, Bucknell University, Lewisburg, PA 17837, USA NATURE BIOMEDICAL ENGINEERING DOI: /s

2 Supplementary Notes: Effect of fluid pressure: The total Cauchy stress acting on an element of the material in the tissue is the sum of the elastic stress on the solid network, and the fluid pore pressure p [46]: σ ij = 2Gε ij + δ ij λ ε kk δ ij p (S1) where ε ij is the strain tensor and p accounts for the effects of deformation on pressurization of the fluid phase under non-equilibrium conditions. In equilibrium, the total fluid pressure within the tumor tissue is the same as that in the adjacent environment. Since we focus here on the solid (elastic) stresses within the tumor tissue, the Eq. (S1) simplifies to Eq. (1) for i, j = z. In non-equilibrium conditions, the dynamic relaxation of the stress after the cut as shown in Fig. S9, the fluid-solid interactions needs to be considered via poroelasticity theory, in which the following equations need to be solved together with the stress-strain constitutive law in Eq. S1: Darcy s Law (constitutive law for fluid-solid interactions): (S2) Conservation of Mass: (S3) Conservation of Momentum: (S4) where U is the fluid velocity, v s is the local velocity of solid element, and k is the tissue hydraulic permeability. Effect of nonlinearity: The stress-strain data for the studied tumors follow a linear relationship for strains less than 0.2 (Fig. S8). Since most of the stress-induced deformations in this study are within 20% of the original associated dimension, the use of a linear constitutive law is justified. For larger deformation, a nonlinear constitutive law for stress-strain relationship and elastic energy needs to be utilized in the finite element model [53]. NATURE BIOMEDICAL ENGINEERING DOI: /s

3 Supplementary Figures: Figure S1 Demonstration of the cut method on human chondrosarcoma. (A, B) Ultrasound images and (C,D) 2-D stress-induced deformation maps of freshly excised human chondrosarcoma tumors from two patients. Human samples have been collected under the protocol approved by the Institutional Review Board of Dana-Farber/Harvard Cancer Center. The subjects gave written informed consent to participate in the study and to have their resected tissue used for research. NATURE BIOMEDICAL ENGINEERING DOI: /s

4 Figure S2 The relaxation of in-plane stresses in the slice method results in expansion in area of the slice. The expansion ratio, defined as the ratio of the area of the slice after the relaxation to the area of the slice before the relaxation (obtained from the agarose blank), is an indicator of area strain. The average of the in-plane stresses can be obtained from the product of the average Young s modulus and the area strain. The inhomogeneity in the in-plane stresses (as shown representatively in Fig. S3), results in out-of-plane bending and buckling of the slice. NATURE BIOMEDICAL ENGINEERING DOI: /s

5 Figure S3 The solid stress components σ xx (A), and σ yy (B) in a MMTV-M3C breast tumor, partially released after the planar cut method. After creating the planar cut, σ zz, shown in Fig. 1D for the same tumor, is completely released since all the resistance in the z-direction is removed. However, σ xx and σ yy are only partially released, since the major resistances in x- and y-directions are still present. The above heat maps show this partial release of the in-plane stresses after the planar cut. In order to fully release σ xx and σ yy, the slice method is used as shown in Fig. 3. NATURE BIOMEDICAL ENGINEERING DOI: /s

6 Figure S4 Histology of breast cancer lymph node micro- and macrometastases. NATURE BIOMEDICAL ENGINEERING DOI: /s

7 Figure S5 Representative images of collagen I staining (white) of the MMTV-M3C tumors. The collagen content (assessed by positively stained collagen I area fraction) becomes localized to the tumor periphery with increasing tumor size. NATURE BIOMEDICAL ENGINEERING DOI: /s

8 Figure S6 Perfusion analysis. Representative processed histology image of vascular perfusion in orthotopic MMTV-M3C breast tumor. Following lectin injection and animal killing, perfusion was quantified as the fraction of vessels that are both lectin- and CD31-positive out of all CD31-positive vessels. NATURE BIOMEDICAL ENGINEERING DOI: /s

9 Figure S7 Pushing versus infiltrative growth pattern. (A) Example of Haematoxylin & Eosin staining of pancreatic ductal adenocarcinoma liver metastasis with clear boundary between metastasis (Ca) and liver parenchyma (LP), a phenotypic marker of pushing growth pattern. (B) Example of colorectal liver metastasis with infiltrative phenotype. NATURE BIOMEDICAL ENGINEERING DOI: /s

10 Figure S8 Unconfined compression test for macroscale estimation of the Young s modulus. The stress relaxation is measured after application deformation in four consecutive steps, each with strain of 0.05 (total strain = 0.2). The equilibrium stress is then plotted as a function of strain, and the Young s modulus is estimated as the slope of the linear fit to the stress-strain data. NATURE BIOMEDICAL ENGINEERING DOI: /s

11 Figure S9 Dynamics of stress relaxation. Relaxation time constant of the stress-induced deformation, estimated as ~50 s by fitting an exponential curve, is consistent with relaxation time of the poroelastic relaxation in unconfined compression, estimated as ~40 s. Since the poroelastic length scale and material properties of the tumor (MMTV-M3C breast tumor) such as hydraulic permeability and equilibrium modulus are the similar in both cases of the stress-relaxations, we expect that linear poroelasticity governs the dynamics of solid stress relaxation in solid tumors. NATURE BIOMEDICAL ENGINEERING DOI: /s

12 Figure S10 AFM-based indentation test for microscale estimation of the Young s modulus. A representative force-displacement curve (black) using AFM-based indentation is shown. The Young s modulus is estimated by the best fit Hertzian model (blue). NATURE BIOMEDICAL ENGINEERING DOI: /s

13 Table S1 Capabilities and limitations of the three developed methods to quantify solid stress. Capabilities Limitations Planar Cut Direct estimation of solid stress and Ex vivo measurement Method elastic energy High-resolution 2-D distribution map of solid stress Distinguish tension from compression Less sensitive than slice method Slice Method Needle Biopsy Method Sensitive to measure small values of solid stress Applicable to small specimens In situ measurement 1-D profile of solid stress Potential use in clinic Ex vivo measurement Indirect estimation of solid stress as area expansion ratio Bulk estimation of the average stress in the slice Less sensitive than slice method NATURE BIOMEDICAL ENGINEERING DOI: /s