A hybrid model of tumor-stromal interactions in breast cancer - PubMed (original) (raw)

A hybrid model of tumor-stromal interactions in breast cancer

Yangjin Kim et al. Bull Math Biol. 2013 Aug.

Abstract

Ductal carcinoma in situ (DCIS) is an early stage noninvasive breast cancer that originates in the epithelial lining of the milk ducts, but it can evolve into comedo DCIS and ultimately, into the most common type of breast cancer, invasive ductal carcinoma. Understanding the progression and how to effectively intervene in it presents a major scientific challenge. The extracellular matrix (ECM) surrounding a duct contains several types of cells and several types of growth factors that are known to individually affect tumor growth, but at present the complex biochemical and mechanical interactions of these stromal cells and growth factors with tumor cells is poorly understood. Here we develop a mathematical model that incorporates the cross-talk between stromal and tumor cells, which can predict how perturbations of the local biochemical and mechanical state influence tumor evolution. We focus on the EGF and TGF-β signaling pathways and show how up- or down-regulation of components in these pathways affects cell growth and proliferation. We then study a hybrid model for the interaction of cells with the tumor microenvironment (TME), in which epithelial cells (ECs) are modeled individually while the ECM is treated as a continuum, and show how these interactions affect the early development of tumors. Finally, we incorporate breakdown of the epithelium into the model and predict the early stages of tumor invasion into the stroma. Our results shed light on the interactions between growth factors, mechanical properties of the ECM, and feedback signaling loops between stromal and tumor cells, and suggest how epigenetic changes in transformed cells affect tumor progression.

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Figures

Fig. 1

A schematic of the steps in the transition from normal ducts to an invasive tumor in breast cancer. (From Paszek and Weaver 2004, with permission)

Fig. 2

The interaction of the EGF and TGF-β pathways in the control of proliferation in breast cancer. In normal ECs, these pathways are balanced so as to control growth, but in TECs increased secretion of TGF-β induces fibroblasts and myofibroblasts to secrete more EGF. This disrupts the proliferation-inhibition mechanism by partially blocking the TGF-β-Smad pathway and triggers proliferation

Fig. 3

Illustration of tumor invasion in vivo (after Geho et al. 2005; Kim and Friedman 2010). In response to chemical signals, possibly from tumor cells, recruited fibroblasts secrete proteinases (MMPs), which diffuse through the ECM and bind to the receptors of tumor cells. Then, a “stimulated” tumor cell secretes secondary proteinases, which degrade ECM proteins. In order for a cell to invade the stroma, the tumor cell has to sever all adhesion junctions to neighboring cells and the basal membrane as a first step, and secrete proteinases to pass through the thick layers of the duct. The building blocks of this thick shield consist of layers of epithelial cells, myoepithelial cells, and basal membrane. The basal membrane consists of various proteins, fiber bundles (collagens), and ECM components

Fig. 4

A schematic of the components in the EGF and TGF-β pathways

Fig. 5

Local dynamics of all internal variables, indicated in each panel, for high and low values of TGF-β (1, 400 pM) and EGF (0.01, 1 nM). The pSmad level (Sp) is high for the low EGF level (0.01 nM) and high TGF-β level (400 pM), while the pSmad levels are suppressed for high levels of EGF (E = 1 nM) or when both inputs are low (E = 0.01 nM, T = 1 pM). Initial conditions: _y_1(0) = 3.956, y2(0) = 0.0752, y3(0) = 4.6759, y4(0) = 12.5923, y5(0) = 17.675, y6(0) = 0.5532, yj(0) = 0.0396, y8 (0) = 1.0, _y_9(0) = 5.7833, _y_10(0) = 15.9485. In all panels, the x-axis indicates the time in hours as in (I) and the y-axis indicates the concentration of species in nM units

Fig. 6

(A) Activation of the A3-Luciferase reporter construct at the indicated concentration of TGF-β analyzed in EpH4 (squares) and EpRas, a Ras-transformed derivative (triangles), of mammary epithelial cells. (Modifed from Kretzschmar et al. 1999 with permission.) (B) Model predictions: inhibition of pSmad level when ECs are exposed to a higher concentration of EGF. For comparison of the model predictions with normal (EpH4) and mutant (EpRas) cell responses in (A), we assume that increased signaling due to increased EGF is equivalent to the Ras-mutant response. For fixed values of EGF levels, pSmad levels are increased when TGF-β levels are increased. The pSmad level is inhibited as EGF levels are increased (red arrow). (C) Smad2/3 localization levels of EpH4 (squares) and EpRas (triangles) cells were measured with fixed amounts of TGF-β treatments. (Modified from Kretzschmar et al. 1999 with permission.) (D) Model predictions: the time course of pSmad levels for low (E = 0.01 nM) andhigh (E = 1 nM) EGFlevels while TGF-β levels were fixed (T = 400 pM). Initial conditions for the low EGF (E = 0.01 nM) _y_1 (0) = 3.956, _y_2(0) = 0.0752, _y_3(0) = 4.6759, _y_4(0) = 12.5923, _y_5(0) = 17.6754, _y_6(0) = 0.5532, _y_7(0) = 0.0396, _y_8(0) = 1.0, _y_9(0) = 5.7833, _y_10(0) = 15.9485. Initial conditions for the high EGF (E = 1 nM) _y_1 (0) = 3.956, _y_2(0) = 0.0752, _y_3 (0) = 4.6759, _y_4 (0) = 13.0923, _y_5 (0) = 17.6754, _y_6(0) = 0.5532, _y_7 (0) = 0.0396, _y_8(0) = 0.5, _y_9(0) = 5.7833, _y_10(0) = 15.9485

Fig. 7

(A) The steady-state pSmad levels in response to an increase in EGF levels for various fixed values of TGF-β. For the fixed values of TGF-β (T = 10, 20, 400 pM), an increase in the EGF level induces a decrease in the pSmad level. (B) The steady-state pSmad levels in response to a simultaneous increase in both EGF and TGF-β levels. The high pSmad levels present at low values of both growth factors are decreased as both TGF-β and EGF levels are increased simultaneously

Fig. 8

Patterns of pSmad in EGF and TGF-β plane, with initial conditions as before. _y_1(0) = 3.956, _y_2(0) = 0.0752 _y_3(0) = 4.6759, _y_4(0) = 12.5923, _y_5(0) = 17.6754, _y_6(0) = 0.5532, _y_7(0) = 0.0396, _y_8 (0) = 1.0, _y_9(0) = 5.7833, _y_10(0) = 15.9485

Fig. 9

(A) pSmad levels in response to fluctuating TGF-β levels with a fixed EGF level (E = 0.08 nM). High (T = 400 pM; arrows) and low (T = 1 pM) levels of TGF-β were imposed in a periodic fashion. The pSmad level rises quickly immediately after a step increase of TGF-β, and decays slowly to the low state in response to small input of TGF-β. (B) pSmad levels in response to fluctuating EGF levels with the fixed TGF-β level (T = 100 pM). High (E = 4 nM; arrows) and low (0.01 nM) levels of EGF were imposed in a periodic fashion. The high level of pSmad decreases rapidly to a low level in response to a high EGF level, but the pSmad level slowly creeps up to a high level in response to low EGF. Initial conditions are as in Fig. 8

Fig. 10

Sensitivity Analysis: General Latin Hypercube Sampling (LHS) scheme and Partial Rank Correlation Coefficient (PRCC) performed on the model. The reference outputs are the variables R1T¯,R2E¯,S,SR1T¯,A,AR2E¯,A*,Sp,A*S¯,A*Sp¯ at the final time (t = 150). The y-value in subplots (A-B) indicates PRCC values of a first subset (SR1T¯,S,Sp,A*S¯,A*Sp¯) and the second subset (R1T¯,R2E¯,A,AR2E¯,A*) for model parameters (kT+,kT−,kT0,kE+,kE−,kE0,kB+,kB−,kA,kS), respectively. The analysis was carried out using the method of Marino et al. (2008) with sample size 1,000, and Matlab files available from

http://malthus.micro.med.umich.edu/lab/usadata/

Fig. 11

(Left) The model domain: fibroblasts, myofibroblasts, and TECs interact via EGF and TGF-β. (Right) Changes in the length of the a-axis of a cell (the ellipsoid) under a given stress (fa; arrow) consist of the passive change in the first component, and the change due to the growth (uag) (from Kim et al. 2011, with permission)

Fig. 12

The growth rate function

Fig. 13

Reported patterns of DCIS. (A) Normal (Wells and El-Ayat 2007), (B) Micropapillary (Chivukula et al. 2009), (C) Apocrine (Wells and El-Ayat 2007), (D) Cribriform (Burrai et al. 2010), (E) Micropapillary (Burrai et al. 2010), (F) Solid (Burrai et al. 2010)

Fig. 14

Tumor growth patterns under fixed levels of growth factors. TECs begin to grow from one, two, three or all peripheral ECs, in (A), (B), (C), and (D), resp. Red arrowheads in (A–C) indicate the initial location of TECs. Green circles are non-proliferating ECs, and red circles are the initial TECs that generate the lineage of proliferating TECs (gray circles). L = lumen in the duct structure. S = stromal tissue. The scales of x-axis and y-axis in subplots (B-D) are same as those in the subplot (A)

Fig. 15

(A) The time course of the distance between the present position on the trajectories in (B) and the center (50, 50) of the lumen (L). (B) The four trajectories (T 1, T 2, T 3, T 4) of four TECs (red cells in Fig. 14(D)) were marked in black, blue, red, and green dots, respectively. Black arrows indicate the forces acting on the wall due to expansion of the growing tumor. (C) The time courses of the distance from the center (A Rad; B Rad) of red cells in the upper right corner in Fig. 14(A–B), and the total distance traveled between the current position and the initial position

Fig. 16

(A–C) The growth pattern of TECs in the presence of paracrine signaling. The upper panels show the evolution of TECs (gray circles) and the stromal mesh at 92 h (A), 240 h (B), 420 h (C). Initially, ECs produce more TGF-β, which stimulates EGF production in the fibroblasts, and this in turn leads to activated TECs that begin to grow when pSmad values drop below the threshold Spth=5.8nM (~90 h). The locations of fibroblasts and myofibroblasts are indicated by blue and red squares, respectively. Myofibroblasts are activated when the TGF-β level exceeds th_T_ = 0.7567 nM (~76 h). (D-I) The concentrations (in nM) of EGF (D-F) and TGF-β (G-I), on the dimensionless domain [0, 1]2 at time t = 4 h (D, G), 82 h (E, H), 480 h (F, I). (J) The temporal evolution of the pSmad concentration at one TEC site. (K) An enlarged portion of (J) corresponding to the red dotted circle in (J). The pSmad level drops significantly when myofibroblasts are activated in the neighborhood (blue arrowhead). A transition from an EC to a TEC (red arrowhead) occurs when the pSmad level drops below the threshold (Spth=5.8nM; red dotted line)

Fig. 17

The effect of varying the diffusion coefficient of EGF when a signaling fibroblast is located at the midline of the upper boundary. (A–D) Profiles of TECs within the duct for the base diffusion coefficient (D E) of EGF at time t = 2, 8, 10, 80 h. Here all ECs on the wall are activated within 10 h, and TECs grow inward without a directional bias. (F–L) Profiles of TECs within the duct for 10- (in (F–H)) and 100-fold (in (J–L)) smaller diffusion coefficients of EGF (D E / 10, D E/100) at time t = 8, 12, 80 h and t = 20, 34, 80 h, respectively. Lower diffusion coefficients lead to larger cell aggregates in the upper section of the duct where ECs are activated earlier. (E) The number of “activated” TECs on the periphery as a function of time showing that the slower diffusion of EGF leads to slower activation of TECs. (I) The time course of the tumor population for the three cases. Tumors with lower EGF diffusion coefficient (_DE/_100) grow more slowly due to slow activation of TECs. (M–N) The time course of pSmad levels for three cases at the north (11th cell; marked in ‘_N_’ in (A)) and west (22nd cell; marked in ‘ _W_’ in (A)) locations, respectively. (O) The ratio of the cell populations in the north (P N) and south (P S) section of the duct for those three cases in percentage (%): Pn * 100/Ps. S = stroma, L = Lumen

Fig. 18

(A–C) Profiles of EGF concentrations at t = 8,20,34 h for the slow diffusion case (De/100). The peak point in panels (A–C) indicate the location of fibroblasts/myofibroblasts. (D–F) Profiles of TGF-β concentrations at t = 8, 20, 34 h. (G–H) A time course of the EGF level at the 11th (G) and 22nd (H) cell site (N) for various diffusion coefficients of EGF (D E, D E/10, D E/100). (I–K) pSmad levels at t = 8, 20, 34 h for three cases at the north (11th cell; marked in ‘_N_’ in (A)) and west (22nd cell; marked in ‘ _W_’ in (A)) locations, respectively. Black arrows in (J) and (K) indicate the cells at the interface between ECs and activated TECs in Figs. 17(J) and (K), respectively

Fig. 19

The effect of changing the diffusion coefficients of EGF (D E) and TGF-β (D T). (A) Activation time of TECs and myofibroblasts as a function of diffusion coefficients (D

, D T) of regulating molecules (EGF and TGF-β). The diffusion coefficients of both EGF and TGF-β were reduced by 5-, 10-, and 100-fold compared to the control case. There are delays in activation of TECs and myofibroblasts when the diffusion coefficients are reduced. (B) The TEC population at day 10 compared to the control (arrow). (C) The time course of pSmad concentration at one EC near (0.4, 0.5) for the four cases of decreased diffusion coefficients considered. Myofibroblasts are activated (arrow) around 194, 86, 80, 78 h, respectively, when TGF-β reaches the threshold, and the pSmad level decreases slowly for smaller values of the diffusion coefficients, leading to a delay of the TEC activation time. For instance TEC is activated around 106 h in the case of 10-fold reduction relative to 90 h in the control case. (D) Initially slow increase of EGF concentration at (0.42, 0.5) near breast duct membrane begins to accelerate (arrow) when myofibroblasts are activated at t = 78, 80, 86 h for relatively smaller reductions of the diffusion coefficients (D, D/5 ,D/10), respectively. However, myofibroblasts are activated at a much later time (194 h) for the significantly stiffened tissue (very small diffusion, _D/_100)

Fig. 20

Growth patterns and the effect of stiffer stromal tissue on tumor growth in the longitudinal direction. (A–F) The time course of tumor growth that originates from two cells (one TEC in the upper stroma and another one in the lower stroma; marked in blue arrowheads). Green circles represent ECs, and TECs are represented by gray circles. TECs initially grow in the radial direction until they feel reciprocal resistance (black arrowheads in (F)) from the upper and lower stroma. Then these cells reorganize to grow in the longitudinal direction. (G) The upper stromal tissue has a larger stiffness (S u) than the lower stromal tissue (S_l_), and proliferating TECs feel a larger mechanical resistance from the upper stroma (red arrowheads) compared to the reciprocal stromal resistance from the lower stroma (black arrowheads). In general, TECs grow in the longitudinal direction but generate a bulb shape in the less stiff (lower) stromal direction. (H) The time course of the tumor population for the cases of uniform stiffness (solid black line; (F)) and different stiffness (dotted blue line; (G)). (I) Closer look of population evolution from (H). Overall growth patterns in both cases look similar to each other. Frames in panels (B–G) are same as in (A). D = breast duct, S = stromal tissue. (J) Tumor population in upper and lower bulge areas, respectively (y value of center of cells > 54 and < 46, respectively). (K) population ratio (= (population in nonuniform case)/(population in uniform case) in percentage (%)) of tumor population in upper and lower sections in (J), respectively

Fig. 21

(Left) A schematic of collective cell migration in a tissue (from Friedl and Alexander 2011 with permission). (Right) A schematic of the invasion model used in the computations. In response to diffusing FSP, a TEC becomes activated (large red circle) and begins to form a microtrack for the invading front by proteolysis of the stromal tissue (ΩS). The follower cells (gray circles) create the macrotrack by proteolyt-ically degrading the ECM. Those two cell types show collective migration (blue arrows) in the invasion region (ΩI) and have to penetrate the initial barrier Ωεinv (basal membrane + layer of myoepithelial cells;Ωεinv⊂Ωs ) and invade the stromal tissue (Ωs\Ωεinv). TECs inside the duct preferentially grow in the longitudinal direction due to low resistive forces (black arrows). Mechanical forces acting on the intact wall are marked as red dashed arrows. Myoepithelial cells are not modeled specifically but rather are included in the continuum region Ωεinv

Fig. 22

(A) A schematic of active force generation of cell i. An invasive cell i is able to generate the traction force in the migration direction (black arrow; di) when enough space is available (case 1; Niinv=0 ). Where there are other neighboring cells (Niinv=0 ) in the migration direction, the active force Tia is turned off (case 2)

Fig. 23

Tumor invasion in response to signals from fibroblasts in stromal tissue: t = 1 h (A), 45 h (B), 90 h (C), 135 h (D), 180 h (E), 210 h (F). Fibroblasts, which secrete the invasion signal, are marked by the red ellipse on the top of the domain in each panel. The black arrowhead denotes the Invasion front in panels (C–F)

Fig. 24

Profiles of ECM, TAP, FSP at t = 1 h (A, D, G), 135 h (B, E, H), 210 h (C, F, I). The spatial scales of the x-axis and y-axis in each row are the same as in the first panel in the row

Fig. 25

The effect of the FSP secretion rate λ5. (A) The activation time of tumor invasion in response to various FSP secretion rates. (B) The area of ECM degradation, estimated by calculating the area of the region where the ECM level is below a threshold (ρ<ρth=0.8). (C) TAP activity, which in turn was estimated by the area of the region where the TAP level is above a threshold Pt>Ptth=2.0. (D) The average FSP level in the invasion region [48, 52] × [56, 58] as a function of time for different secretion rates

Fig. 26

The effect of FSP diffusion. (A) The activation time of tumor invasion for three values of FSP diffusion coefficient (D s). (B) The ECM degradation area: the area was estimated by calculating the area of the region where the ECM level is below a threshold (ρ<ρth=0.8). (**C**) TAP activity: the activity was estimated by the area of region where the TAP level is above the threshold Pt>Ptth=2.0 (. (D) The FSP level: the average value of FSP levels was estimated in the localized invasion region [48,52] × [56,58]

Fig. 27

Simulated therapy—the effect of blocking TGF-β secretion from ECs. (A) The time evolution of pSmad concentration at one epithelial cell near (0.4, 0.5) for four cases of decreased TGF-p secretion rate (k T n): 46, 60 and 80 %, compared to the normal rate (100 %). Myofibroblasts are activated around 48, 58, 77, 105 h (black arrowheads) when TGF-β reaches the threshold, while the pSmad level decreases slowly for a smaller degree of the TGF-β blocking, leading to a delay of the TEC activation time (green arrowheads). (B) The initially slow increase of EGF levels at (0.42, 0.5) near a breast duct membrane begins to accelerate when myofibroblasts are activated (arrowheads) at the corresponding times for all four cases

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A hybrid model of tumor-stromal interactions in breast cancer - PubMed (original) (raw)