This study will find the highest acceptable treatment dose and timing of infusion of cord blood, culture expanded natural killer (NK) cells, a kind of immune cell, in patients with multiple myeloma.

The NK cells will be given at varying days post autologous stem cell transplant. rhIL-2 is administered after treatment to help the NK cells expand in the body. The safety of this treatment will be studied and researchers want to learn if NK cells will help in treating multiple myeloma.

The primary objective of the study is to assess safety and determine the maximum tolerated dose of PNK-007 as well as the feasibility of treating at various timepoints following ASCT in subjects with multiple myeloma. The secondary objective is to explore the potential clinical efficacy by day 90-100 post ASCT.

Treatment plan includes ASCT followed by PNK-007 which will be administered IV Day 14 post ASCT to determine the maximum tolerated dose. Once the IV Day 14 post ASCT. PNK-007 will be followed by up to six rhIL-2 injections to support the NK cells expansion in the body.

Subjects will be followed for up to 12 months post PNK-007.

Inclusion Criteria:

Subjects must satisfy the following criteria to be enrolled in the study:

1. Subject has eligible disease status:

   1. Newly diagnosed and are undergoing induction therapy prior to undergoing first Autologous stem cell transplant (ASCT) or
   2. Myeloma patients with prior relapse undergoing first ASCT. or
   3. Myeloma patients with relapsed disease after first ASCT who are undergoing second ASCT. Subjects must have achieved at least a partial response (PR) prior to proceeding to ASCT.

2. Subject is > 18 and ≤ 70 years of age at the time of signing the informed consent form (ICF).

3. Subject must understand and voluntarily sign an ICF prior to any study-related assessments/procedures being conducted.

4. Subject is willing and able to adhere to the study schedule and other protocol requirements.

5. Performance status of Karnofsky performance status ≥ 70% or Eastern Cooperative Oncology Group (ECOG) < 2

6. Ability to be off immunosuppressive drugs for at least 3 days prior to the PNK-007 cell infusion. Steroids at the equivalent of no more than 5 mg prednisone per day are permissible.

7. Be a candidate for ASCT based on institutional practices.

8. Subjects must have autologous peripheral blood stem cell graft available in storage for additional transplant in the event of engraftment failure.

9. Female of childbearing potential (FCBP) must:

   1. Have two negative pregnancy tests as verified by the Investigator prior to starting study therapy. She must agree to ongoing pregnancy testing during the course of the study, and after the end of study treatment. This applies even if the subject practices true abstinence* from heterosexual contact.

   2. Either commit to true abstinence* from heterosexual contact (which must be reviewed at applicable study visits and source documented) or agree to use, and be able to comply with, effective contraception without interruption, during the study therapy (including dose interruptions), and for 28 days after discontinuation of PNK-007.

      A female of childbearing potential (FCBP) is a female who:

      1) has achieved menarche at some point, 2) has not undergone a hysterectomy or bilateral oophorectomy or 3) has not been naturally postmenopausal (amenorrhea following cancer therapy does not rule out childbearing potential) for at least 24 consecutive months (ie, has had menses at any time in the preceding 24 consecutive months).

10. Male subjects must:

    1. Practice true abstinence* (which must be reviewed at applicable study visits) or agree to use a condom during sexual contact while participating in the study, during dose interruptions and for at least 28 days following PNK-007 discontinuation, even if he has undergone a successful vasectomy. * True abstinence is acceptable when this is in line with the preferred and usual lifestyle of the subject. (Periodic abstinence [eg, calendar, ovulation, symptothermal, post ovulation methods] and withdrawal are not acceptable methods of contraception]).

Exclusion Criteria:

The presence of any of the following will exclude a subject from enrollment:

1. Subject has plasma cell leukemia.

2. Subject has non-secretory myeloma.

3. Subject has previously undergone allogeneic stem cell transplant.

4. Subject has any significant medical condition, laboratory abnormality, or psychiatric illness that would prevent the subject from participating in the study.

5. Subject has any condition including the presence of laboratory abnormalities which places the subject at unacceptable risk if he or she were to participate in the study.

6. Subject has any condition that confounds the ability to interpret data from the study.

7. Subject has a body weight exceeding 120 kg.

8. Subject has aspartate aminotransferase (AST), alanine aminotransferase (ALT), or alkaline phosphatase ≥ 2.5 x the upper limit of normal (ULN) at screening.

9. Estimated glomerular filtration rate (eGFR) < 30 mL/min/1.73 m2 at screening calculated using the Modification of Diet in Renal Disease Study equation.

10. Subject has a bilirubin level > 2 mg/dL (unless subject has known Gilbert's disease) at screening.

11. Subject has had prior treatment with biologic antineoplastic agents no less than 7 days before PNK-007 infusion and at least 5 half-lives. For agents that have known AEs occurring beyond 7 days after administration (ie, monoclonal antibodies), this period must be extended beyond the time during which acute AEs are known to occur.

12. Subject is pregnant or breastfeeding.

13. Subject has new or progressive pulmonary infiltrates or pleural effusion large enough to be detected by chest x-ray or computed tomography (CT) scan.

14. Subject has active autoimmune disease other than controlled connective tissue disorder or those who are not on active therapy.

15. Subject has human immunodeficiency virus (HIV) are excluded due to increased risk of lethal infections when treated with myeloablative chemotherapy.

16. Subject has history of malignancy, other than multiple myeloma (MM), unless the subject has been free of disease for > 3 years from the date of signing the ICF. Exceptions include the following:

    1. Basal cell carcinoma of the skin
    2. Squamous cell carcinoma of the skin
    3. Carcinoma in situ of the cervix
    4. Carcinoma in situ of the breast
    5. Incidental histological finding of prostate cancer (TNM stage of T1a or T1b)

17. Subject has a history of severe asthma and is presently on chronic medications or has a history of other symptomatic pulmonary disease.

18. Untreated chronic infection or treatment of any infection with systemic antibiotics within 2 weeks prior to melphalan.

19. Subject has any other organ dysfunction that will interfere with the administration of the therapy according to this protocol.

20. Subject has a resting left ventricular ejection fraction (LVEF) of < 35% obtained by echocardiography or multigated acquisition scan (MUGA).

21. Subject was treated with an investigational product no less than 28 days before PNK-007 infusion. Subject must no longer be a participant in the previous interventional study at the time of the PNK-007 infusion.