The disease from which Dr. Rank suffers from is tuberculosis of the spine. Within the play's setting, this disease is seen as a result of some form of over-indulgence. In the play, Nora discusses the potential indulgences that may have brought about Dr. Rank's development of his illness, which he inherited as a result of the "sins" of his father.
This may be significant because here we have a decent and kind man suffering and dying from his backbone. A good man dying by the backbone may signify the end of the image of men, as Nora knows it. As a woman of her time, Nora believes that men take a superior and dominant role because that is what she is used to.
Dr. Rank appears to be the opposite: he is kind to her, treats her with respect, and loves her. The very ignorant Nora completely bypasses this and bestows upon her husband, Torvald, those very attributes; attributes which Torvald completely lacks. Hence, just like the last good man in Nora's life is now dying, the image of Torvald as a good, kind, man will also die, similarly, by the backbone. Nora will discover that Torvald has no backbone whatsoever.
To further prove that this is the connection between Dr. Rank's disease and Nora's situation, notice that it is through the illness that we find more information about the Helmers. While Rank admits to the gravity of his condition, he also warns Nora that he will not tell Torvald because of Torvald's dislike of "ugly" things. When you think about this, what kind of man is Torvald? One who treats his wife like a child, who devalues her opinions, and now he also has a phobia for ugly things, and prefers to ignore that a "good" friend of the house is dying! All of these things must have combined in Nora's mind when she finally makes the choice to leave him forever. Torvald is not a good man and, maybe, not even a real man. He is the one who has no "backbone". Nora will discover that quite soon.