To add to Ryan's answer,

$2$--knots usually don't have aspherical complements, see [Dyer & Vasquez, The sphericity of higher dimensional knots, Canad. J. Math. 25(1973), 1132-1136][1]. Which suggests a complicated answer for slice disks in general.

On the other hand, if the disk is ribbon, then the complement is aspherical, see [Asano, Marumoto, and Yanagawa,  T, Ribbon knots and ribbon disks, Osaka J. Math.
18 (1981), 161-174][2]


  [1]: http://math.ca/10.4153/CJM-1973-121-5
  [2]: http://ir.library.osaka-u.ac.jp/metadb/up/LIBOJMK01/ojm18_01_12.pdf