Shell Script: How can I recreate the directory tree from source on destination of SCP command?

I am writing a script to read a relative path from a text file, check to see if the file exists in the source directory, and then copy the file from source to desination. So far, I can read, check, and copy files just fine but when the files get to the destination directory, only the files are there and the directory structure is lost. Here is my script so far:

SOURCE='/c/source'
DEST='/c/dest'
YOURFILE="filestomove.txt"
while read -r path
do
if [ -f "$SOURCE$path" ]
then
scp -r "$SOURCE$path" $DEST/new/
else
echo "$SOURCE$path" >> notfound.txt
fi
done < $YOURFILE

Here are the contents of filestomove.txt:
/sub/test1.txt
/sub/test2.txt
/sub/sub 1/test3.txt
/sub/test4.txt
/sub/sub 2/test5.txt
/sub/test6.txt
EOF

Right now I end up with a destination directory containing:
test1.txt
test2.txt
test3.txt
test4.txt
test5.txt
test6.txt

I can't just do a recursive scp because I do not want to move all contents of any directory. I only want to copy specific files and they will reside in various directories.

Thanks,
Adam