Glypican 3-specific Chimeric Antigen Receptor Expressing T Cells for Hepatocellular Carcinoma (GLYCAR)

A maximum of 30 subjects will participate in the treatment part of this study. Up to 180ml of blood is collected from patients to grow the T cells. The T cells are grown and a retrovirus (a special virus that can carry the GPC3 CAR gene into the T cells) is used to genetically engineer them. After the CAR gene is put into the T cells, the investigators make sure that they are able to kill hepatocellular carcinoma cells in the laboratory. LYMPHODEPLETION CHEMOTHERAPY: Several studies suggest that the infused T cells need room to be able to proliferate and accomplish their functions and that this may not happen if there are too many other T cells in circulation. Because of that, participants will receive treatment with cyclophosphamide (Cytoxan) and fludarabine for 3 days before receiving the T-cell infusion. These drugs will decrease the numbers of the participants' own T cells before the investigators infuse the GLYCAR T CELLS. WHAT THE INFUSION WILL BE LIKE: After making these cells, they will be frozen. If the patient agrees to participate in this study, at the time the patient is scheduled to be treated, the cells will then be thawed and injected into the patient over 1 to 10 minutes. The participant will receive the GLYCAR T CELLS 48-72 hours after completing the chemotherapy. The participant may be pretreated with Tylenol (acetaminophen) and Benadryl (diphenhydramine). Tylenol and Benadryl are given to prevent a possible allergic reaction to the T cell administration. This is a dose escalation study, which means that the investigators do not know the highest dose of GLYCAR T cells that is safe. To find out, the investigators will give the cells to at least 3 participants at one dose level. If that is safe, they will raise the dose given to the next group of participants. The dose each patient gets will depend on how many participants get the agent before that patient and how they react. The investigator will tell each patient this information. This will help the patient think about possible harms and benefits. Since the treatment is experimental, what is likely to happen at any dose is not known. All of the Treatments will be given by the Center for Cell and Gene Therapy at Houston Methodist Hospital. Medical tests before treatment: * Physical exam and History * Blood tests to measure blood cells, kidney and liver function and blood clotting (INR). * If the patient is infected with the hepatitis B virus (HBV) we will do a test to measure the levels of the virus * Pregnancy test (if the patient is a female who can get pregnant) * Measurements of the patient's tumor by scans (within 4 weeks before chemotherapy) and the tumor marker alfa-fetoprotein (AFP), if the patient's tumor produces this protein. * Tumor biopsy Medical tests during and after treatment: * Physical exams and History * Blood tests to measure blood cells, kidney and liver function and blood clotting (INR) * If the patient is infected with the hepatitis B virus (HBV) we will do a test to measure the levels of the virus * Measurements of the tumor maker alfa-fetoprotein (AFP), if the patient's tumor produces this protein * Measurements of the patient's tumor by scans (8 weeks after the infusion +/- 1 week) * Tumor biopsy (optional) FOLLOW-UP STUDIES The investigators will follow the participant during and after the injections. To learn more about the way the T cells are working in the participant's body, up to 60 mL (12 teaspoons) of blood will be taken from the participant before the chemotherapy, before the T-cell infusion, 3 to 4 hours after the infusion, 3 to 4 days after the infusion (this time point is optional ) at 1 week, 2 weeks, 4 weeks, 6 weeks and 8 weeks after the injection, every 3 months for 1 year, every 6 months for 4 years and then every year for the next 10 years. Total time participation for this study will be 15 years. During the time points listed above, if the T cells are found in the participant's blood at a certain amount an extra 5mL of blood may need to be collected for additional testing. The investigators will use this blood to look for the frequency and activity of the cells that they have given; that is, to learn more about the way the T cells are working and how long they last in the body. They will also use this blood to see if there are any long-term side effects of putting the new gene (chimeric antigen receptor, CAR) into the cells. In addition to the blood draws, because the participants have received cells that have had a new gene put in them, they will need to have long term follow up for 15 years so the investigators can see if there are any long term side effects of the gene transfer. Once a year, participants will be asked to have their blood drawn and answer questions about their general health and medical condition. The investigators may ask participants to report any recent hospitalizations, new medications, or the development of conditions or illness that were not present when they enrolled in the study and may request that physical exams and/or laboratory tests be performed if necessary. Prior to procurement we confirmed GPC3-expression in your tumor by obtaining part of a biopsy sample that was done for clinical purposes. We will obtain an additional tumor biopsy at 2 weeks after treatment for research purposes. These samples will be used to look for GPC3-CAR T cells in the tumor and to learn more about the effects of the treatment on your disease. In the event that a tumor biopsy is performed for clinical reasons the investigators will request permission to obtain excess sample to learn more about the effects of the treatment on the participant's disease. In the event of death, the investigators will request permission to perform an autopsy to learn more about the effects of the treatment on the participant's disease and if there were any side effects from the cells with the new gene. If participants develop a second abnormal cancer growth, significant blood or nervous system disorder during the trial, a biopsy sample of the tissue will be tested (if a sample can be obtained).