unit iso-quant curve and unit iso-value line

This question is about international trade theory.

For the labor L and capital K, I know the theory of iso-quant and iso-cost. The price will be determined where iso-cost line touches iso-quant curve. When price changes, the iso-quant curve itself doesn’t change.

However, I am confused unit iso-quant and unit iso-value theory. Iso-value line is different from iso-cost line? Unit iso-quant curve is different from iso-quant curve? When price change, the unit iso-quant curve changes?