Lithospermum Star

What causes the branches to dry out on my Lithospermum star? It does flower but the regrowth is on the tips of the branches. I have repotted it is fresh compost and cut back some of the branches. Will this encourage growth lower down?

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Start new plants by taking cuttings from the green tips. They root easily in gritty compost. I find they are like lavender, in that if you cut back hard into brown wood, they will not resprout.