After the policeman had ran for several miles, he finally caught up with the man who had stolen the purse.

The choice "had ran" is incorrect because it does not use the correct past participle form of the verb "run." The correct phrase should be "had run," as "run" is the past participle, while "ran" is the simple past form of the verb. In English, the perfect tense requires the use of the past participle. Therefore, the correct structure is "after the policeman had run for several miles."

Using the correct past participle ensures that the tenses are properly aligned and that the sentence maintains grammatical consistency. This aspect is crucial for conveying the sequence of events accurately, indicating that the action of running occurred before the action of catching up with the man.