It seems not to be known whether X4 + Y4 = 4481 has a rational solution( From [1].Cohen. P.409).

If Z becomes a square number in X4 + Y4 = 4481Z2,then X4 + Y4 = 4481 will have rational solutions. 

So,I searched the solutions for X4 + Y4 = 4481Z2.

First,we transform the equation (1) to the elliptic curve (3).
Next,we look for the generators of (3) by using Cremona's mwrank.
And,we can get the several small solutions by generators.




X4+Y4=4481Z2...........................................................(1)

Equation (1) is finally transformed to the next equation (2).

V2 = 4481U4+4481.......................................................(2)

If we obtain a rational solution of equation (2),we can get a solution of (1)
with the next process.

U= (-1714750x-306118977280-1140136y)/(-255455y+152450048x+68587978400)
V= (-46550241544085509440y+234598870023187200x2-6280793868516872212505600+87125914693889x3+6997730774375206899716x)/((-255455y+152450048x+68587978400)2)
X/Y=U
4481*Z/Y2=V

Next, we must get the rational solutions U of (2).

First,we transform (2) to Minimal Weierstrass form (3),and get the rational solutions of (3).

After that,by using group law of generators, we can get infinitely many rational solutions of (3).

Minimal Weierstrass form: y2 = x3-80317444x.............................(3)

We can get the generators of (3) by Cremona's mwrank.


-------------------------------------------------------------------------------
 Rank = 2
 After descent, rank of points found is 2
 
 Generator 1 is [-127971424400834 : -7680455997963840 : 17434421857]; height 24.0
 777949026435
 Generator 2 is [162599634272318343598960 : 22794533266636311016786881 : 70311004
 66750976000]; height 30.728667870629
 
 The rank has been determined unconditionally.
 The basis given is for a subgroup of full rank of the Mordell-Weil group
  (modulo torsion), possibly of index greater than 1.
 Regulator (of this subgroup) = 506.387417938697
-------------------------------------------------------------------------------


We got two generators p1,p2 of (3) by Cremona's mwrank.

I found several solutions under the condition of Z<10^100.


p1= [-49352651138/6723649, -7680455997963840/17434421857]
p2= [84874741237064321/3670136377600, 22794533266636311016786881/7031100466750976000]


p=n1*p1+n2*p2

                                           X4 + Y4 = 4481Z2




[n1,n2] [X ,Y ,Z]
[-2, 0] [4307945468596333240619530023616774856, 5673107283875478000643188428496996895, 554994619686552715747108822976186495684013075971136937206088884548368129]
[-2, 1] [80341616496488168, 17880918178460095, 96544111685930771175910444442561]
[-2, 2] [436181415933776791298824, 299194976486407934618785, 3141037838612151815393067700303535911612687489]
[-1, -1][3862562334375596758423669592973048398411105, 1117294564420039832533280175798741935935976, 223654948382014255812145312270027589420969293494934461238945912989436396276860261889]
[-1, 0] [1126909825, 1646826952, 44735995667375681]
[-1, 1] [95, 424, 2689]
[-1, 2] [41917972072957820641855, 31758706583636465243528, 30266107940867973549898232313663936081822401]
[0, -1] [31758706583636465243528, 41917972072957820641855, 30266107940867973549898232313663936081822401]
[0, 1]  [1646826952, 1126909825, 44735995667375681]
[0, 2]  [1117294564420039832533280175798741935935976, 3862562334375596758423669592973048398411105, 223654948382014255812145312270027589420969293494934461238945912989436396276860261889]
[1, -1] [299194976486407934618785, 436181415933776791298824, 3141037838612151815393067700303535911612687489]
[1, 0]  [17880918178460095, 80341616496488168, 96544111685930771175910444442561]
[1, 1]  [5673107283875478000643188428496996895, 4307945468596333240619530023616774856, 554994619686552715747108822976186495684013075971136937206088884548368129]
[2, -1] [4032679945230549380418279063282653174011563112, 909579681417369784714122177816034549316783615, 243254745111871438691942785873648325884286809032322745901905312142099169406903410609503041]
 


Z did not become a square number in this search range. 
I think that perhaps, there is not a rational solution in X4 + Y4 = 4481.

[1].Henri Cohen:Number Theory Volume 1:Tools and Diophantine Equations.