I already watched this drama few years ago. And it was on of good japanese drama based on based on Aya Kito (木藤亜也 Kitō Aya?, July 19, 1962 – May 23, 1988)'s diary. published shortly before her death. The diary, a true story based on her own life, was originally written in first person.
It is about a girl coping with her teenage life along with a degenerative disease. She keeps a diary of not only what she does but how she feels and the hardships she must endure. Initially, the diary's purpose was for Kitō to chronicle impressions she had about how the disease was affecting her daily life. As the disease progressed, however, the diary became Kitō's outlet for describing the intense personal struggles she underwent in coping, adapting, and ultimately trying to survive her disease. As she notes in one entry, "I write because writing is evidence that I am still alive." [Source : WIKI]

OST Japanese Drama :  1 Litre of Tears' (With Japanese Lyrics and English Translation)

Only Human by K:

Kanashimi no mukou kishi ni
(On the opposite coast of sadness)

Hohoemi ga aru toiu yo
(is something called a smile)
Kanashimi no mukou kishi ni
(On the opposite coast of sadness)

Hohoemi ga aru to iu yo
(is something called a smile)

Tadori tsuku sono saki ni wa
(But before we can go there,)

Nani ga bokura wo matteru?
(is there something we’re waiting for?)

Nigeru tame ja naku yume ou tame ni
(In order to chase our dreams, we can’t have a reason to run away)

Tabi ni deta hazusa tooi natsu no ano hi
(We’ve got to go, to that far away summer’s day)

Ashita sae mieta nara tame iki mo nai kedo
(If we find it tomorrow, we can’t sigh)

Nagare ni sakarau fune no you ni
(Because like a boat that opposes the stream)

Ima wa mae he susume
(we have to walk straight on)

Kurushimi no tsukita basho ni
(In a place worn down by sadness)

Shiawase ga matsu toiu yo
(something called a miracle, is waiting)

Boku wa mada sagashite iru
(Yet we are still searching)

Kisetsu hazure no himawari
(for the sunflower that grows at the end of spring)

Kobushi nigirishime asahi wo mateba
(The warrior who awaits the morning light)

Akai tsume ato ni namida kirari ochiru
(before he can clasp it with red nails, his tears glitter and fall)

Kodoku ni mo nareta nara
(Even if we’ve grown used to loneliness)

Tsuki akari tayori ni
(only relying on the light of the moon)

Hane naki tsubasa de tobi tatou
(We have to fly away with featherless wing)

Motto mae he susume
(just go foward, just a little further)

Amagumo ga kireta nara
(As the rainclouds break)

Nureta michi kagayaku
(the wet streets sparkling)

Yami dake ga oshiete kureru
(Although it brings only darkness)

Tsuyoi tsuyoi hikari
(A powerful, powerful light)

Tsuyoku mae he susume
(helps push us to walk on)