Inclusion Criteria:
Phase 1 part:
1. Diagnosis, allowable prior therapy, and disease measurability requirements:
1. All participants must have a pathologically documented, following advanced solid tumor known to express folate receptor alpha (FR alpha), that is resistant or refractory to standard treatment, for which no standard treatment is available, or the participant refuses standard therapy.
* Ovarian cancer
* Endometrial cancer
* Non-small cell lung cancer (NSCLC)
* Triple-negative breast cancer (TNBC)
* Cholangiocarcinoma
* Colorectal cancer (CRC)
* Gastro-esophageal adenocarcinoma Note: Participants with a solid tumor type other than the above will be eligible as long as there is prior documentation of tumor FR alpha expression.
2. All participants without prior documentation of tumor FR alpha expression by immunohistochemistry (IHC) must be willing to provide an archival tumor tissue block or slides, or undergo procedure to obtain a new biopsy using a low risk, medically routine procedure for IHC confirmation of FR alpha positivity of \>=1% of viable tumor cells with membrane staining at \>=1+ intensity for entry into Phase 1 part
3. There is no upper limit on the number of prior cytotoxic or targeted therapies the participant may have received. Participants may have received prior treatment with investigational compounds targeting folate receptor excluding MIRV.
4. Participants must have measurable or non-measurable disease (such as large abdominal masses that cannot be accurately measured) according to Response Evaluation Criteria in Solid Tumors (RECIST) 1.1.
2. Participant must have an Eastern Cooperative Oncology Group Performance Status (ECOG PS) of 0 or 1
3. Time from Prior Therapy:
* Systemic anti-neoplastic therapy: five half-lives or four weeks, whichever is shorter (6 weeks for prior nitrosoureas or mitomycin C)
* FR alpha-targeted therapy: five half-lives or four weeks, whichever is longer
* Radiotherapy: wide-field radiotherapy (e.g. affecting at least 30% of the bone marrow) completed at least four weeks, or focal radiation completed at least two weeks, prior to starting study drug
4. Participants must have stabilized or recovered (Grade 1 or baseline) from all prior therapy-related toxicities
5. Major surgery must be completed four weeks prior to first dose of TAK-853. Participants must have recovered or stabilized from the side effects prior to study treatment.
6. Participants must have adequate hematologic, liver and kidney function as defined by the following parameters:
1. Absolute neutrophil count (ANC) \>= 1.5\*10\^9/L (1,500/microliter) without granulocyte colony stimulating factor (G-CSF) in the prior 10 days or long-acting white blood cell (WBC) growth factors in the prior 20 days
2. Platelet count \>= 100.0\*10\^9/L (100,000/microliter; without platelet transfusion in the prior 10 days)
3. Hemoglobin \>= 9.0 g/dL without packed red blood cell (PRBC) transfusion in the prior 21 days
4. Serum creatinine =\< 1.5\* upper limit of normal (ULN) or estimated creatinine clearance of \>= 30 mL/minute (as calculated using the Cockcroft Gault equation),
5. Aspartate aminotransferase (AST) =\< 2.5\* ULN; alanine aminotransferase (ALT) =\< 2.5\* ULN (AST, ALT \< 5\* ULN if liver metastases), and
6. Total bilirubin =\< 1.5\* ULN (participants with documented diagnosis of Gilbert syndrome are eligible if total bilirubin \< 3.0\* ULN)
7. Participants with central nervous system (CNS) disease involvement are eligible if they have had brain metastases resected or have received radiation therapy ending at least 4 weeks prior to study Day 1 and they meet all of the following criteria:
<!-- -->
1. Residual neurological symptoms =\< Grade 1
2. No dexamethasone requirement, and
3. Follow-up MRI shows no progression of treated lesions and no new lesions appearing.
Phase 2 part:
1. Participants must have a confirmed diagnosis of high-grade serous epithelial ovarian cancer, primary peritoneal cancer, or fallopian tube cancer
2. Participants must have platinum-resistant disease:
1. Participants who have only had 1 line of platinum-based therapy must have received at least 4 cycles of platinum, must have had a response (complete response \[CR\] or partial response \[PR\]) and then progressed between \> 3 months and =\< 6 months after the last dose date of platinum
2. Participants who have received 2 or 3 lines of platinum therapy must have progressed on or within 6 months after the date of the last dose of platinum Note: Progression should be calculated from the date of the last administered dose of platinum therapy to the date of the radiographic imaging showing progression Note: Participants who are primary platinum-refractory during front-line treatment are excluded (see exclusion criteria)
3. Participants must have progressed radiographically on or after their most recent line of therapy
4. Participants must be willing to provide an archival tumor tissue block or slides, or undergo procedure to obtain a new biopsy using a low risk, medically routine procedure for IHC confirmation of FR alpha expression (reported as "positive") as defined by the Ventana FOLR1 Assay. Tumors must be confirmed FR alpha-high as defined by FR alpha positivity of \>=75% of viable tumor cells with membrane staining at \>=2+ intensity for entry into the Phase 2.
5. Participants must have at least one lesion that meets the definition of measurable disease by RECIST v1.1 criteria (radiologically measured by the Investigator).
6. Participants must have received at least 1 but no more than 3 prior systemic lines of anticancer therapy, and for whom single-agent therapy is appropriate as the next line of treatment:
a. Neoadjuvant +- adjuvant considered one line of therapy b. Maintenance therapy (e.g., bevacizumab, poly-ADP ribose polymerase \[PARP\] inhibitors) will be considered as part of the preceding line of therapy (i.e., not counted independently) c. Therapy changed due to toxicity in the absence of progression will be considered as part of the same line (i.e., not counted independently) d. Hormonal therapy will be counted as a separate line of therapy unless it was given as maintenance
7. Participant must have an ECOG PS of 0 or 1
8. Time from prior therapy:
1. Systemic antineoplastic therapy (5 half-lives or 4 weeks, whichever is shorter)
2. Focal radiation completed at least 2 weeks prior to first dose of study drug
9. Participants must have stabilized or recovered (Grade 1 or baseline) from all prior therapy-related toxicities
10. Major surgery must be completed at least 4 weeks prior to first dose and the participant must have recovered or stabilized from the side effects of prior surgery
11. Participants must have adequate hematologic, liver, and kidney functions defined as:
1. ANC \>= 1.5\* 10\^9/L (1,500/microliter) without G-CSF in the prior 10 days or long-acting WBC growth factors in the prior 20 days
2. Platelet count \>= 100\* 10\^9/L (100,000/microliter) without platelet transfusion in the prior 10 days
3. Hemoglobin \>= 9.0 g/dL without PRBC transfusion in the prior 21 days
4. Serum creatinine =\< 1.5\* ULN or estimated creatinine clearance of \>= 30 mL/minute (as calculated using the Cockcroft Gault equation).
5. AST and ALT =\< 3.0\* ULN
6. Total bilirubin =\< 1.5\* ULN (participants with documented diagnosis of Gilbert syndrome are eligible if total bilirubin \< 3.0\* ULN)
7. Serum albumin \>= 2 g/dL
Exclusion Criteria:
Phase 1 part:
1. Participant with \> Grade 1 peripheral neuropathy per National Cancer Institute Common Terminology Criteria for Adverse Events (NCI CTCAE) v5.0
2. Participants with active or chronic corneal disorders, history of corneal transplantation, or active ocular conditions requiring ongoing treatment/monitoring such as uncontrolled glaucoma, wet age-related macular degeneration requiring intravitreal injections, active diabetic retinopathy with macular edema, macular degeneration, presence of papilledema, and/or monocular vision.
3. Serious concurrent illness, including, but not limited to the following:
1. Clinically relevant active infection including - Active hepatitis B or C infection (whether or not on active antiviral therapy)
\- Human Immunodeficiency Virus (HIV) infection
\- Active cytomegalovirus infection
\- Active COVID-19/SARS-CoV-2 infection. Although SARS-CoV-2 testing is not mandatory for study entry, testing should follow local clinical practice guidelines and standards
\- Any other known concurrent infectious disease requiring IV antibiotics within 2 weeks before starting study drug Note: Testing at screening for hepatitis is required, while not required for the remaining infections above unless clinically indicated. Participants with known hepatitis B surface antigen seropositivity and/or detectable hepatitis C virus RNA will be excluded. Participants who have positive hepatitis B core antibody and/or hepatitis B surface antibody can be enrolled but must have an undetectable serum hepatitis B virus DNA. Participants who have positive hepatitis C virus antibody must have an undetectable hepatitis C virus RNA serum level. Participants will be monitored and managed according to Guideline for the prevention of immunosuppressive therapy or chemotherapy-induced reactivation of hepatitis B virus infection (The Japan Society of Hepatology 2022).
2. Participants with clinically significant cardiac disease including, but not limited to, any one of the following:
\- Myocardial infarction =\< 6 months prior to first dose of study medication
\- Unstable angina pectoris
\- Uncontrolled congestive heart failure (New York Heart Association \> class II)
\- Uncontrolled \>= Grade 3 hypertension (per NCI CTCAE v5.0)
\- Uncontrolled cardiac arrhythmias
\- Severe aortic stenosis
* \>= Grade 3 cardiac toxicity following prior chemotherapy
3. History of multiple sclerosis or other demyelinating disease, Lambert-Eaton syndrome (paraneoplastic syndrome), history of hemorrhagic or ischemic stroke within the last six months, or alcoholic liver disease.
4. Previous clinical diagnosis of interstitial lung disease (ILD), including pneumonitis.
4. Any other concomitant anti-cancer treatment such as immunotherapy, biotherapy, radiotherapy, chemotherapy, investigative therapy, or high-dose steroids; however, low-dose steroids and Luteinizing Hormone Releasing Hormone (LHRH) at doses that have been stable for \>= 14 days are permitted for participants with prostate cancer
5. Known hypersensitivity to previous monoclonal antibody therapy or maytansinoids, or study drugs and/or any of their excipients
6. Prior history of solid tumor malignancy within the last 3 years except for adequately treated basal cell or squamous cell skin cancer, in situ cervical cancer, in situ breast cancer, in situ prostate cancer (participants must have shown no evidence of active disease for 2 years prior to enrollment)
7. Participants with required use of folate-containing supplements (e.g., folate deficiency)
8. Participants who have received prior allogeneic or autologous bone marrow transplants
Phase 2 part:
1. Participants with endometrioid, clear cell, mucinous, or sarcomatous histology, mixed tumors containing any of the above histologies, or low-grade or borderline ovarian tumor
2. Participants with primary platinum-refractory disease, defined as disease that did not respond to (CR or PR) or has progressed within 3 months of the last dose of first line platinum-containing chemotherapy
3. Participants with prior wide-field radiotherapy affecting at least 20% of the bone marrow
4. Participants with \> Grade 1 peripheral neuropathy per NCI CTCAE v5.0
5. Participants with active or chronic corneal disorders, history of corneal transplantation, or active ocular conditions requiring ongoing treatment/monitoring such as uncontrolled glaucoma, wet age-related macular degeneration requiring intravitreal injections, active diabetic retinopathy with macular edema, macular degeneration, presence of papilledema, and/or monocular vision
6. Participants with serious concurrent illness or clinically relevant active infection, including, but not limited to the following:
1. Active hepatitis B or C infection (whether or not on active antiviral therapy)
2. HIV infection
3. Active cytomegalovirus infection
4. Active COVID-19/SARS-CoV-2 infection. Although SARS-CoV-2 testing is not mandatory for study entry, testing should follow local clinical practice guidelines and standards.
5. Any other concurrent infectious disease requiring IV antibiotics within 2 weeks before starting study drug Note: Testing at screening for hepatitis (a) is required, while not required for the remaining infections above (b-e) unless clinically indicated.
Participants with known hepatitis B surface antigen seropositivity and/or detectable hepatitis C virus RNA will be excluded. Participants who have positive hepatitis B core antibody and/or hepatitis B surface antibody can be enrolled but must have an undetectable serum hepatitis B virus DNA.
Participants who have positive hepatitis C virus antibody must have an undetectable hepatitis C virus RNA serum level. Participants will be monitored and managed according to Guideline for the prevention of immunosuppressive therapy or chemotherapy-induced reactivation of hepatitis B virus infection (The Japan Society of Hepatology 2022).
7\. Participants with history of multiple sclerosis or other demyelinating disease and/or Lambert-Eaton syndrome (paraneoplastic syndrome) 8. Participants with clinically significant cardiac disease including, but not limited to, any one of the following:
1. Myocardial infarction =\< 6 months prior to first dose of study medication
2. Unstable angina pectoris
3. Uncontrolled congestive heart failure (New York Heart Association \> class II)
4. Uncontrolled \>= Grade 3 hypertension (per NCI CTCAE v5.0)
5. Uncontrolled cardiac arrhythmias 9. Participants with a history of hemorrhagic or ischemic stroke within six months prior to first dose of TAK-853 10. Participants with a history of cirrhotic liver disease (Child-Pugh Class B or C) 11. Participants with a previous clinical diagnosis of ILD, including pneumonitis 12. Participants with required use of folate-containing supplements (e.g., folate deficiency) 13. Participants with prior hypersensitivity to monoclonal antibodies or maytansinoids 14. Participants with prior treatment with TAK-853 or other FR alpha-targeting agents 15. Participants with untreated or symptomatic CNS metastases 16. Participants with a history of other malignancy within 3 years prior to first dose of TAK-853 Note: does not include tumors with a negligible risk for metastasis or death (e.g., adequately controlled basal-cell carcinoma or squamous-cell carcinoma of the skin, or carcinoma in situ of the cervix or breast) 17. Prior known hypersensitivity reactions to study drugs and/or any of their excipients
