The answer to your question really depends on a lot of factors. The first of which is where exactly are you measuring the glutathione reductase (GR)? For example, serum levels may increase while tissue specific levels decrease... so you have to be more specific.
A quick PubMed search shows some variability though when it comes to the actual HCC cells. One study demonstrated no change in GR in liver tissue from patients with HCC, but another study using cell lines found differences in expression depending upon the differentiation of the cancer (PMIDs: 16052474, 16404476). This is very much a theme with an array of antioxidant enzymes; sometimes they go up, sometimes they go down, and sometimes they stay the same. It is really dependent upon A LOT of factors, and very rarely can you generalize through an entire disease or population.
Also, to make it more complicated, remember there are plenty more players in glutathione metabolism than just GR. The amount of peroxidase activity, total glutathione, ratio of oxidized/reduced glutathione, as well as levels of H2O2 and superoxide all play a role in its activity.
Last, you ask why? Well, not knowing how it actually changes, I am unsure if I can answer that. One in vitro study offers the hypothesis that HCC increases GR as a protective measure from apoptosis, but they demonstrate this effect by artificial over-expression of GR (PMID: 20302905). So the answer to "why" is still elusive from my standpoint, so you have an open door to a new project :).
Actually i'm trying to measure its activity in serum but i didn't understand how its level increases in cirrhostic and HCC patients although it is considered as an antioxidant biomarker and all GSH enzyme family actually decrease which means that GSH/GSSG ratio should increase!!!!!!!
IN (PMID: 20302905) said that GR is actually may play protective role against oxidative stress and at the same time it's called anti-apoptotic agent although I should enhance apoptosis in able to fight malignant cells!!!!!
Now that I know you want to measure it in the serum, maybe I can help a little bit more.
Many studies have examined GR in the serum of patients with disease, and in fact some have proposed that it is a more relevant biomarker than LDH when it comes to malignant disease. The question you ask is why would it increase in patients with cancer, if in fact it is an antioxidant enzyme and should protect? Remember, GR is an intracellular enzyme, and as such should not reside in the serum. The reason it increases during times of disease (cancer, cirrhosis, hemolytic anemia, etc.) is because cells are dying and releasing their components into the serum (this is true for many serum biomarkers of stress/disease like LDH, AST, ALT, etc). So even though GR is increased in the serum, this tells us nothing about the levels in the cancer (where it is truly important). Here is an old paper that started to measure GR in serum of patients, and you may find some of the data interesting. PMID: 2624.
Another VERY important thing to remember about antioxidant enzymes is that protein levels do not always correlate with activity. You state that the amount of GR goes up, but other GSH enzymes go down which should mean a preservation of the GSH/GSSG ratio. First, remember you are looking in the serum, so this tells us nothing about the cancer. Second, because protein levels mean nothing about activity we cannot infer anything about the GSH/GSSG ratio based on this (and in fact it has been shown repeatedly that GSSG increases in cancer). Last, the amount of total glutathione and available NADPH also play a role in the activity of GR, which are also factors to consider (and are highly altered in cancer as well).
It seems that the publication regarding over-expression of GR was confusing to you, but this is probably because of the "cancer tissue versus serum" argument. That paper was just trying to show that if the CANCER increases GR, it is protective from apoptosis... not increases in serum. This makes complete sense when you reference the other paper about GR in various differentiated HCC cell lines (PMID: 16052474). They found that the more differentiated (aka less malignant, less invasive) had more GR. Highly differentiated cancers are usually more resistant to death and metastasis, which would mean less GR in the serum (even though higher levels inside the actual cells). As the cells become more malignant, they down-regulate GR (and also can down/up-regulate other antioxidant enzymes like catalase, SOD's, etc), metastasize throughout the body, but also have an increased rate of death increasing the GR in the serum.
Remember, cancer cells have altered metabolism and are highly oxidatively stressed compared to normal tissue. Because of this, some researchers have attempted to exploit this by eliminating the little antioxidant capacity they have left, and thus pushing them over the edge to die (while sparing the normal cells). I recommend you read the literature by Dr. Douglas Spitz at the University of Iowa. He has been attempting to alter glutathione metabolism by various methodologies to increase cancer oxidative stress, and has shown some promising results.
I hope this was of some help. Keep reading, and I hope it will start to make some more sense.