Bubble pushing of nand gates
starting with:

(a AND1 b) OR1 (c AND2 d) OR1 (e AND3 f)    (#'s refer to gates in your drawing)


substituting NOT (x NAND y) for (x AND y):

(NOT (a NAND b)) OR (NOT (c NAND d)) OR (NOT (e NAND f))


By De Morgan, (NOT x) OR (NOT y) OR (NOT z) is same as NOT (x AND y AND z) 

NOT ((a NAND b) AND (c NAND d) AND (e NAND f))


substituting (x NAND y NAND z) for NOT (x AND y AND z):

(a NAND1 b) NAND4 (c NAND2 d) NAND4 (e NAND3 f)    (#'s refer to gates in your drawing)