Phase II Study of Atezolizumab and Tiragolumab With ctDNA for HPV-positive SCC

Inclusion Criteria: * Histologically or cytologically confirmed stage I-III HPV+ squamous cell carcinoma of the anus (anal canal, rectum, or anal margin), cervix, head \& neck, penis, vagina or vulva. An HPV/p16 status will be reviewed from outside testing if available or obtained on available tumor tissue as confirmed in a CLIA-certified environment within 28 days of enrollment. * At least 12 weeks following completion of standard-of-care treatment with radiation and/or chemotherapy and/or surgical resection with curative intent, according to accepted standard practices for the disease site * No clinical evidence of disease at the assessment of the treating investigator * Detectable HPV ctDNA on a commercially available ctDNA assay or an MD Anderson-created HPV ctDNA assay at least 12 weeks following completion of curative-intent treatment * Results of positive ctDNA test to confirm study eligibility may be performed within 42 days prior to enrollment * Age ≥18 years * Ability to understand and the willingness to sign a written informed consent document. A medical interpreter may be used when obtaining written informed consent * Ability to comply with the study protocol * ECOG performance status ≤2 (Karnofsky ≥60%). * Adequate hematologic and end organ function, defined by the following laboratory test results, obtained within 14 days prior to initiation of study treatment: * ANC \>/= 1.0 x 109/L (1500/µL) without granulocyte colony-stimulating factor support. * Lymphocyte count \> 0.5 x 109/L (500/µL). * Platelet count \>/= 75 x 109/L (100,000/µL) without transfusion. * Hemoglobin \>/= 90 g/L (9 g/dL). (Patients may be transfused to meet this criterion.) * AST, ALT, and ALP \</= 2.5 x upper limit of normal (ULN). * Total bilirubin \</= 1.5 x ULN with the following exception: * Patients with known Gilbert disease: total bilirubin \</= 3 x ULN. * Creatinine clearance \>/= 45 mL/min (calculated using the Cockcroft-Gault formula). * Serum albumin \>/= 25 g/L (2.5 g/dL). * For patients not receiving therapeutic anticoagulation: INR and aPTT \< 1.5 x ULN. * For patients receiving therapeutic anticoagulation: stable anticoagulant regimen * Negative HIV antibody at screening (within 28 days of enrollment) with the following exception: individuals with a positive HIV antibody test at screening are eligible provided they are stable on anti-retroviral therapy, have a CD4+ T cell count ≥200/µL, and have an undetectable viral load * Negative hepatitis B surface antigen (HBsAg) test at screening (within 28 days of enrollment) * Negative hepatitis B surface antibody (HBsAb) test at screening (within 28 days of enrollment, or if the HBsAb is positive at screening, The positive HBsAb test must be accompanied by either of the following within 28 days of enrollment: * Negative total hepatitis B core antibody (HBcAb). * Positive total HBcAb test followed by quantitative hepatitis B virus (HBV) DNA \< 500 IU/mL. * Negative hepatitis C virus (HCV) antibody test at screening (within 28 days of enrollment), or positive HCV antibody test followed by a negative HCV RNA test at screening (within 28 days of enrollment) o The HCV RNA test will be performed within 28 days of enrollment only for patients who have a positive HCV antibody test. * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial. * The effects of atezolizumab and tiragolumab on the developing human fetus are unknown. For this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. (Refer to Pregnancy Assessment Policy MD Anderson Institutional Policy # CLN1114). This includes all female patients, between the onset of menses (as early as 8 years of age) and 55 years unless the patient presents with an applicable exclusionary factor which may be one of the following: * Postmenopausal (no menses in greater than or equal to 12 consecutive months). * History of hysterectomy or bilateral salpingo-oophorectomy. * Ovarian failure (Follicle Stimulating Hormone and Estradiol in menopausal range, who have received Whole Pelvic Radiation Therapy). * History of bilateral tubal ligation or another surgical sterilization procedure. o For women: agreement to refrain from donating eggs, remain abstinent or use contraceptive methods with a failure rate of \<1% per year during the treatment period and for 90 days after the final dose of tiragolumab, 5 months after the final dose of atezolizumab. The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not adequate methods of contraception. If required per local guidelines or regulations, locally recognized adequate methods of contraception and information about the reliability of abstinence will be described in the local Informed Consent Form. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. * Examples of contraceptive methods with a failure rate of \<1% per year include bilateral tubal ligation, male sterilization, hormonal contraceptives that inhibit ovulation, hormone-releasing intrauterine devices, and copper intrauterine devices. * For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use a condom, and agreement to refrain from donating sperm, as defined below: * With a female partner of childbearing potential, men who are not surgically sterile must remain abstinent or use a condom plus an additional contraceptive method that together result in a failure rate of \<1% per year during the treatment period, for 90 days after the final dose of tiragolumab, and 5 months after the final dose of atezolizumab. Men must refrain from donating sperm during this same period. * With a pregnant female partner, men must remain abstinent or use a condom during the treatment period for 90 days after the final dose of tiragolumab or cisplatin and 5 months after the final dose of atezolizumab, to avoid exposing the embryo. * The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not adequate methods of contraception. If required per local guidelines or regulations, locally recognized adequate methods of contraception and information about the reliability of abstinence will be described in the local Informed Consent Form. * Men who would like to father a child after study treatment initiation should be advised regarding the conservation of sperm prior to treatment because of the possibility of irreversible infertility resulting from chemotherapies used in this study Exclusion Criteria: * Patients with a non-squamous cell carcinoma histology at the primary tumor type. Examples include, but are not limited to, adenocarcinoma, mixed adenosquamous carcinoma, melanoma. * Patients with clinical, radiographic or pathologic evidence of squamous cell carcinoma at the time of study enrollment at the discretion of the treating investigator. An exception can be made for patients with radiographically measurable disease noted at the 12 week time point after completion of (chemo)radiation that is demonstrating an improvement/reduction in size at the discretion of the treating investigator consistent with treatment response. * Uncontrolled or symptomatic hypercalcemia within 14 days of study registration (ionized calcium 1.5 mmol/L, calcium 12 mg/dL or corrected serum calcium ULN) * Prior surgery as salvage treatment for radiation-refractory SCC. * Patients who are receiving any other investigational agents or who received treatment with investigational therapy within 12 weeks prior to initiation of study treatment * Patients who are receiving active therapy for cancer, including immunotherapy, chemotherapy, or radiotherapy. Participants should not have had received prior standard treatments within 12 weeks of study registration. * Patients with psychiatric illness/social situations that would limit compliance with study requirements. * Illness or condition that may interfere with a patient's capacity to understand, follow, and/or comply with study procedures o Active or history of autoimmune disease or immune deficiency, including, but not limited to, myasthenia gravis, myositis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, antiphospholipid antibody syndrome, Wegener granulomatosis, Sjögren syndrome, Guillain-Barré syndrome, or multiple sclerosis (see Appendix 8 for a more comprehensive list of autoimmune diseases and immune deficiencies), with the exceptions listed below: * Patients with a history of autoimmune-related hypothyroidism who are on thyroid replacement hormone are eligible for the study. * Patients with controlled Type 1 diabetes mellitus who are on an insulin regimen are eligible for the study. * Patients with eczema, psoriasis, lichen simplex chronicus, or vitiligo with dermatologic manifestations only (e.g., patients with psoriatic arthritis are excluded) are eligible for the study provided all of following conditions are met: i. Rash must cover \<10% of body surface area. ii. Disease is well controlled at baseline and requires only low-potency topical corticosteroids. iii. There has been no occurrence of acute exacerbations of the underlying condition requiring psoralen plus ultraviolet A radiation, methotrexate, retinoids, biologic agents, oral calcineurin inhibitors, or high potency or oral corticosteroids within the previous 12 months. iv. Active idiopathic pulmonary fibrosis or organizing pneumonia (e.g., bronchiolitis obliterans) on screening chest CT scan at the discretion of the treating investigator * Significant cardiovascular disease (such as New York Heart Association Class II or greater cardiac disease, myocardial infarction, or cerebrovascular accident) within 3 months prior to initiation of study treatment, unstable arrhythmia, or unstable angina. Patients must be NYHA class 2B or better at the time of study registration. * Major surgical procedure, within 4 weeks prior to initiation of study treatment * History of malignancy other than HPV+ squamous cell carcinoma within 3 years prior to screening, with the exception of malignancies with a negligible risk of metastasis or death (e.g., 5 year OS rate \>90%), including but not limited to, non-melanoma skin carcinoma, localized prostate cancer, ductal carcinoma in situ, or Stage I uterine cancer * Severe infection within 4 weeks prior to initiation of study treatment, including, but not limited to, hospitalization for complications of infection, bacteremia, or severe pneumonia, or any active infection that could impact patient safety. Treatment with therapeutic oral or IV antibiotics within 2 weeks prior to initiation of study treatment. Patients receiving prophylactic antibiotics (e.g., to prevent a urinary tract infection or chronic obstructive pulmonary disease exacerbation) are eligible for the study. * Prior allogeneic stem cell or solid organ transplantation requiring immunosuppressive therapy * Any other disease, metabolic dysfunction, physical examination finding, or clinical laboratory finding that contraindicates the use of an investigational drug, may affect the interpretation of the results, or may render the patient at high risk from treatment complications * Treatment with a live, attenuated vaccine within 4 weeks prior to initiation of study treatment, or anticipation of need for such a vaccine during study treatment, within 90 days after the final dose of tiragolumab, or within 5 months after the final dose of atezolizumab * Patients being treated with chemotherapy (i.e., carboplatin, cisplatin, pemetrexed, paclitaxel, or gemcitabine) should not receive live vaccines. * Positive Epstein-Barr virus (EBV) viral capsid antigen immunoglobulin M (IgM) test at screening * An EBV polymerase chain reaction (PCR) test should be performed as clinically indicated to screen for acute infection or suspected chronic active infection. Patients with a positive EBV PCR test are excluded. * Prior treatment with CD137 agonists or investigational immune checkpoint blockade therapies, including, but not limited to, anti-PD(L)1, anti-CTLA-4, anti-TIGIT, anti-LAG3 antibodies\] * Any prior treatment administered for treatment of minimal residual disease according to an HPV ctDNA-detected status * Treatment with systemic immunostimulatory agents (including, but not limited to, interferon and IL-2) within 4 weeks or 5 drug-elimination half-lives (whichever is longer) prior to initiation of study treatment * Treatment with systemic immunosuppressive medication (including, but not limited to, corticosteroids, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor fnecrosis factor - á \[TNF-á\] agents) within 2 weeks prior to initiation of study treatment, or anticipation of need for systemic immunosuppressive medication during study treatment, with the following exceptions: * Patients who received acute, low-dose systemic immunosuppressant medication or a one-time pulse dose of systemic immunosuppressant medication (e.g., 48 hours of corticosteroids for a contrast allergy) are eligible for the study. The corticosteroid dose cannot exceed 10 mg prednisone daily or equivalent. * Patients who received mineralocorticoids (e.g., fludrocortisone) or corticosteroids for chronic obstructive pulmonary disease (COPD) or asthma, or low-dose corticosteroids for orthostatic hypotension or adrenal insufficiency are eligible for the study. * History of severe allergic anaphylactic reactions to chimeric or humanized antibodies or fusion proteins * Known hypersensitivity to Chinese hamster ovary cell products or to any component of the atezolizumab or tiragolumab formulation * Pregnancy or breastfeeding, or intention of becoming pregnant during study treatment, within 90 days after the final dose of tiragolumab and/or 5 months after the final dose of atezolizumab. * Women of childbearing potential must have a negative serum pregnancy test result within 14 days prior to initiation of study treatment.