How is the effective dose calculated when only part of the body is irradiated?
Not sure that I understood your question correctly.

The concept of "effective dose" was specially introduced to provide a mechanism for assessing the radiation detriment from partial body irradiations in terms of data derived from whole body irradiations. The effective dose is the mean absorbed dose from a uniform whole-body irradiation that results in the same total radiation detriment as from the nonuniform, partial-body irradiation in question.

What is probably misleading you is the sum of the weighting factors to be 1. This is because we try to liken the partial body irradiation and the whole-body irradiation and the weighting factors represent the relative tissue sensitivity and susceptibility to the radiation. Your radiation should not go over all possible tissues and organs, you just assume the irradiation to be zero at them that cancels their summands so that they can be ignored in the final calculation. In other words, it is absolutely fine (and it is even so "by design") to have only some of the WT*HT factors in your formula.