If you don't need the rectangle you can easyly find the line for Intensity Profile as you mention.
img=import["https://i.sstatic.net/r6cDj.png"]; mf1 = MeanFilter[img, 5]; ht1 = HistogramTransform[mf1]; i = SelectComponents[ SelectComponents[ColorNegate@Binarize[ht1], Small], Large] lineByCenter[center_, semi_, angle_] := Rotate[Line[{#1 - #2, #1 + #2}], angle, #1] &[center, {0, semi}] sa = 1 /. ComponentMeasurements[i, "SemiAxes"] {113.783, 44.4184}
angle = 1 /. ComponentMeasurements[i, "Orientation"] -3.14122
bb = Last /@ ComponentMeasurements[i, "BoundingBox"] // First {{0., 320.}, {235., 410.}}
bbc = Mean /@ bb {117.5, 365.}
Show[i, Graphics[{Yellow, Opacity [0.5], Rectangle[bb[[1]], bb[[2]]], Thick, Red, Rotate[Circle[bbc, sa], angle, bbc], Blue, lineByCenter[bbc, sa[[1]], angle + Pi/2], Green, lineByCenter[bbc, sa[[2]], angle]}]] 
