What relationship do you see between the prologue's reference of "the flame" and Prufrock's monologue?

The prologue of the famous poem "The Love Song of J. Alfred Prufrock" by T.S. Eliot is a quotation in Italian from The Divine Comedy by Dante Alighieri. It's from Canto XXVII of The Inferno, which describes Dante's visit to hell. The precise wording varies from translator to translator, but in essence, it says this:

If I believed that my reply were made
To one who to the world would e're return,
This flame without more flickering would stand still;
But inasmuch as never from this depth
Did any one return, if I hear true,
Without the fear of infamy I answer . . .

By this time, Dante and his guide Virgil have reached the eighth circle of hell, where sinners who have committed fraud or deceit are confined. The damned souls are wrapped in flames, and when they speak, the flames move, but when they are silent, the flames are still. The sinner who speaks to Dante is telling him that if he thought that Dante would go out and tell others what he is about to say, he would not speak but be silent. However, because he has heard that no one has ever escaped from this depth of hell, he feels free to tell Dante his story because Dante will never escape to inform others of the nature of his sin.
In using this prologue before Prufrock's monologue, Eliot is telling the readers of his poem that the narrator, Prufrock, is in his own private hell, and if he thought that anyone would ever be able to hear what he is saying, he would not utter these words.
The poem then goes on to describe the nature of Prufrock's hell. For instance, instead of saying that the evening is pleasant, he describes it as "a patient etherized upon a table." The streets he walks through are ominous with yellow fog that seems almost alive and the dark soot from chimneys. Prufrock is insecure and feels that he has to prepare a face to meet others instead of behaving like himself. He wants to be bold and well-thought-of but instead he finds himself growing old. His life has been measured in trivialities—he has never done anything significant. He is drawn to women but his self-consciousness makes him unable to approach them. He is so intensely lonely that he imagines that he should have been a crab scuttling along at the bottom of the ocean. In despair he longs for his fantasies to become reality, but he can only wish for his dreams from afar. That's the hell that in which Prufrock is caught up.
http://www.gutenberg.org/files/1001/1001-h/1001-h.htm