Ch other. Results in Fig. 1H and Fig. 1I show the

Ch other. Results in Fig. 1H and Fig. 1I show the detected leading edges at t 72 hours for a high threshold (S 0:0565) and a low threshold (S 0:0135), respectively. Both detected edges 10781694 at t 72 hours appear to be reasonable approximations to the location of the leading edge of the spreading population, however they are very different to each other which indicates that the results are sensitive to S. To qualitatively compare the two leading edges detected at t 0 hours (Fig. 1B and Fig. 1C) we superimpose the two detected leading edges in Fig. 1D and show a magnified portion of these edges in Fig. 1E. The superimposed edges confirm that the choice of S has relatively little influence at t 0 hours. We now compare equivalent results at t 72 hours from Fig. 1H and Fig. 1I. Superimposing the two leading edges for high and low S thresholds in Fig. 1J indicates that there is a distinct difference between them. A magnified portion of the detected leading edges is shown in Fig. 1K which also supports our initial observation that it is difficult to visually delineate the leading edge of the spreading population when the leading edge is diffuse. Our edge detection results at t 0 hours and t 72 hours, in Fig. 1A and Fig. 1G , qualitatively indicate that the threshold value is important in detecting the edge at a later time. To quantitatively compare our edge detection results, we calculate the area enclosed by the detected leading edge and convert this area pffiffiffiffiffiffiffiffiffi into an equivalent circle with radius A=p. Results in Fig. 1F show the equivalent circular areas for low and high thresholds at t 0 hours. The area of the low and high thresholds are 32:2 mm2 and 31:1 mm2 , respectively, giving a relatively small difference of 1:1 mm2 . These two circles are almost visually indistinguishable at the scale shown in Fig. 1F, confirming there is very relatively little difference regardless of the threshold. Equivalent circular areas in Fig. 1L show the low and high threshold areas at t 72 hours superimposed on the initial area. The area of the two outer circles in Fig. 1L is 52:9 mm2 and 60:8 mm2 , giving a relatively large difference of 7:9 mm2 . If we take the initial area to be A(0) 31:1 mm2 then equation (1) gives us M(72) 70:1 for the high threshold leading edge in Fig. 1H and M(72) 95:5 for the low threshold leading edge in Fig. 1I. These results indicate that the0.3 Mathematical Modeling ToolsTo provide a physical interpretation of our image analysis results, we use a mathematical model to relate the edge detection results to the spatial distribution of the cell density. We model the spreading population of cells using a linear diffusion equation [3?5], with previously determined values of the cell diffusivity [17]. The effects of cell proliferation are neglected in our mathematical model, and this is consistent with our experimental protocol where cells were pretreated with Mitomycin-C to suppress cell proliferation [32]. To relate our edge detection results to the cell density, we consider the solution of the two-dimensional axisymmetric diffusion equation. ! Lc L2 c 1 Lc D , z Lt Lr2 r Lr??where r is radial position, t is time, c(r,t) is the non-dimensional cell MedChemExpress Imazamox MedChemExpress JW-74 density and D is the cell diffusivity, which is a measure of random, undirected, cell motility [17,37]. The non-dimensional cell density is obtained by scaling the dimensional cell density, (r,t), by the carrying capacity density K. This gives c c(r,t) (r,t)=K, with c.Ch other. Results in Fig. 1H and Fig. 1I show the detected leading edges at t 72 hours for a high threshold (S 0:0565) and a low threshold (S 0:0135), respectively. Both detected edges 10781694 at t 72 hours appear to be reasonable approximations to the location of the leading edge of the spreading population, however they are very different to each other which indicates that the results are sensitive to S. To qualitatively compare the two leading edges detected at t 0 hours (Fig. 1B and Fig. 1C) we superimpose the two detected leading edges in Fig. 1D and show a magnified portion of these edges in Fig. 1E. The superimposed edges confirm that the choice of S has relatively little influence at t 0 hours. We now compare equivalent results at t 72 hours from Fig. 1H and Fig. 1I. Superimposing the two leading edges for high and low S thresholds in Fig. 1J indicates that there is a distinct difference between them. A magnified portion of the detected leading edges is shown in Fig. 1K which also supports our initial observation that it is difficult to visually delineate the leading edge of the spreading population when the leading edge is diffuse. Our edge detection results at t 0 hours and t 72 hours, in Fig. 1A and Fig. 1G , qualitatively indicate that the threshold value is important in detecting the edge at a later time. To quantitatively compare our edge detection results, we calculate the area enclosed by the detected leading edge and convert this area pffiffiffiffiffiffiffiffiffi into an equivalent circle with radius A=p. Results in Fig. 1F show the equivalent circular areas for low and high thresholds at t 0 hours. The area of the low and high thresholds are 32:2 mm2 and 31:1 mm2 , respectively, giving a relatively small difference of 1:1 mm2 . These two circles are almost visually indistinguishable at the scale shown in Fig. 1F, confirming there is very relatively little difference regardless of the threshold. Equivalent circular areas in Fig. 1L show the low and high threshold areas at t 72 hours superimposed on the initial area. The area of the two outer circles in Fig. 1L is 52:9 mm2 and 60:8 mm2 , giving a relatively large difference of 7:9 mm2 . If we take the initial area to be A(0) 31:1 mm2 then equation (1) gives us M(72) 70:1 for the high threshold leading edge in Fig. 1H and M(72) 95:5 for the low threshold leading edge in Fig. 1I. These results indicate that the0.3 Mathematical Modeling ToolsTo provide a physical interpretation of our image analysis results, we use a mathematical model to relate the edge detection results to the spatial distribution of the cell density. We model the spreading population of cells using a linear diffusion equation [3?5], with previously determined values of the cell diffusivity [17]. The effects of cell proliferation are neglected in our mathematical model, and this is consistent with our experimental protocol where cells were pretreated with Mitomycin-C to suppress cell proliferation [32]. To relate our edge detection results to the cell density, we consider the solution of the two-dimensional axisymmetric diffusion equation. ! Lc L2 c 1 Lc D , z Lt Lr2 r Lr??where r is radial position, t is time, c(r,t) is the non-dimensional cell density and D is the cell diffusivity, which is a measure of random, undirected, cell motility [17,37]. The non-dimensional cell density is obtained by scaling the dimensional cell density, (r,t), by the carrying capacity density K. This gives c c(r,t) (r,t)=K, with c.