Hi Arjan Gelderblom,
The two standard ideas are Bump or Displacement. Which both work with image-based information.
Since I have no file, I can’t say anything else, regarding modeling. It seems to be only partial on the surface, speaking about the four lines.
Except, I can suggest one way, please have a look here:
Screen capture
https://www.amazon.com/clouddrive/share/Pik3rh8v4o3OOi9OgP1ArX1VnsajL9hbUrffPS8Rswg
(While the Move tool is active, it is possible to use the CMD-key and click on the spline to get more points.)
If the object is editable, points can be added with the Knife tool, and the selected can be scaled to the inside. The scaling tool should be used only flat (two axes) if you adjust more than one row of points at once.
All the best