@FleurDeLis: There was a discussion about the gradual disappearance of the weak declension on LEO some weeks ago. Quite interesting, actually: http://dict.leo.org/forum/viewGeneraldiscussion.php?idThread=1158027&idForum=4&lp=ende&lang=de

So nouns are only inflected to show gender and number, but not cases, is that what you mean?

1. Most of the time, the nouns themselves do not change in the different cases. Only the articles change. (There are different forms of the noun for singular and plural, though, but that's a different matter).
2. However, masculine and neutral nouns change in the genitive singular, where an –s or –es is added to the noun. Ex. das Buch des Vaters (=the father’s book)
3. Moreover, nouns add the ending –n in the dative plural (if they don't end with an "n" or "s"). Ex. Ich helfe den Männern. (= I help the men)
4. So far, I've explained how things normally work. The word "Herz", however, belongs to a special group of nouns called "weak nouns". Weak nouns change much more frequently than normal nouns - they add the ending "(e)n" in most cases. See: http://www.cynthiahodges.com/german/pages/WeakNouns.htm
5. So, to answer your original question: "Arzt" is a normal noun and thus remains unchanged after the verb "brauchen", i.e. in the accusative singular. "Herz", however, belongs to a special group of nouns called weak nouns and thus adds an "-en" after "mit", i.e. in the dative singular.

@MJPerttula: the female form of 'der Arzt' is 'die 'Ärztin' (not 'die Arzten'). 'Arzt' is never used as a neuter noun in German. 'Ein paar Ärzte' (not 'Arzten') is masculine plural. For mixed groups of people the male form is used.

This really reminds me of how complicated German really is! If it's any help, I've found that often, German speakers themselves aren't sure whether to inflect a given noun or not, and the overall trend seems to go towards lack of inflection.