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<!-- DFIN ActiveDisclosure(SM) XBRL Schema Document - http://www.dfinsolutions.com/ -->
<!-- Creation Date      : 2022-07-20T16:21:07.4535034+00:00 -->
<!-- Version            : 5.0.1.747 -->
<!-- Package ID         : b41cc3d76b2443e6a1e91b5393ff3522 -->
<!-- Copyright (c) 2022 Donnelley Financial Solutions, Inc. All Rights Reserved. -->
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  <xsd:element id="pstv_CashPaidForIPRDAssetAcquiredInAnAssetAcquisition" name="CashPaidForIPRDAssetAcquiredInAnAssetAcquisition" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="credit" />
  <xsd:element id="pstv_ProceedsFromWarrantExercises1" name="ProceedsFromWarrantExercises1" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit" />
  <xsd:element id="pstv_UnpaidOfferingCosts" name="UnpaidOfferingCosts" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit" />
  <xsd:element id="pstv_CashPaidDuringPeriodForAbstract" name="CashPaidDuringPeriodForAbstract" type="xbrli:stringItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_UseOfEstimatesTextBlock" name="UseOfEstimatesTextBlock" type="dtr-types:textBlockItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_UseOfEstimatesAbstract" name="UseOfEstimatesAbstract" type="xbrli:stringItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_DisclosureOfLiquidityTextBlock" name="DisclosureOfLiquidityTextBlock" type="dtr-types:textBlockItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_LiquidityAbstract" name="LiquidityAbstract" type="xbrli:stringItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_LicenseAgreementDisclosureTextBlock" name="LicenseAgreementDisclosureTextBlock" type="dtr-types:textBlockItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_LicenseAgreementAbstract" name="LicenseAgreementAbstract" type="xbrli:stringItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_ClosingBidPriceForCommonStockHadFallenBelow" name="ClosingBidPriceForCommonStockHadFallenBelow" type="dtr-types:perShareItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_ConsecutiveBusinessDays" name="ConsecutiveBusinessDays" type="xbrli:durationItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_ClosingBidPrice" name="ClosingBidPrice" type="dtr-types:perShareItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" />
  <xsd:element id="pstv_PeriodForRegainCompliance" name="PeriodForRegainCompliance" type="xbrli:durationItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_WarrantsIssuedUnderwrittenPublicOfferingPeriod" name="WarrantsIssuedUnderwrittenPublicOfferingPeriod" type="xbrli:gYearMonthItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_SeriesUWarrantsMember" name="SeriesUWarrantsMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_DebtInstrumentFloatingInterestRatePercentage" name="DebtInstrumentFloatingInterestRatePercentage" type="dtr-types:percentItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" />
  <xsd:element id="pstv_ClassOfWarrantOrRightExpirationDate" name="ClassOfWarrantOrRightExpirationDate" type="xbrli:dateItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_TermLoanMember" name="TermLoanMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_LoanAndSecurityAgreementMember" name="LoanAndSecurityAgreementMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_SaleOfJapaneseSubsidiaryAndCertainAssetsMember" name="SaleOfJapaneseSubsidiaryAndCertainAssetsMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_MinimumLiquidityCovenantAmount" name="MinimumLiquidityCovenantAmount" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="debit" />
  <xsd:element id="pstv_DebtPrincipalRepaymentStartDate" name="DebtPrincipalRepaymentStartDate" type="xbrli:dateItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_LongTermDebtOutstandingThresholdAmount" name="LongTermDebtOutstandingThresholdAmount" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="credit" />
  <xsd:element id="pstv_OptionsMember" name="OptionsMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_LeaseCommencementDate" name="LeaseCommencementDate" type="xbrli:dateItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_OperatingLeasesExpiryYear" name="OperatingLeasesExpiryYear" type="xbrli:gYearItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_CharlottesvilleMember" name="CharlottesvilleMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_MedidataSolutionsIncMember" name="MedidataSolutionsIncMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_ServicesAgreementMember" name="ServicesAgreementMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_PiramalPharmaSolutionsIncMember" name="PiramalPharmaSolutionsIncMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_PreClinicalResearchStudyObligationsMember" name="PreClinicalResearchStudyObligationsMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_LoremVascularPteLtdMember" name="LoremVascularPteLtdMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_CashPaidForAmountsIncludedInMeasurementOfLeaseLiabilities" name="CashPaidForAmountsIncludedInMeasurementOfLeaseLiabilities" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="credit" />
  <xsd:element id="pstv_FinanceLeaseExpenseAbstract" name="FinanceLeaseExpenseAbstract" type="xbrli:stringItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_CashPaymentInformationAbstract" name="CashPaymentInformationAbstract" type="xbrli:stringItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_SalesOrderTerm" name="SalesOrderTerm" type="xbrli:durationItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_MasterServicesAgreementDate" name="MasterServicesAgreementDate" type="xbrli:dateItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_MasterServicesAgreementInitialTerm" name="MasterServicesAgreementInitialTerm" type="xbrli:durationItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_MasterServicesAgreementTermsDescription" name="MasterServicesAgreementTermsDescription" type="xbrli:stringItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_CompensatoryDamagesAndOperational" name="CompensatoryDamagesAndOperational" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" xbrli:balance="debit" />
  <xsd:element id="pstv_UTHSALicenseAgreementMember" name="UTHSALicenseAgreementMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_PreferredStockDividendRatePercentageEliminatedDate" name="PreferredStockDividendRatePercentageEliminatedDate" type="xbrli:dateItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_SeriesAConvertiblePreferredStockMember" name="SeriesAConvertiblePreferredStockMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_SeriesBConvertiblePreferredStockMember" name="SeriesBConvertiblePreferredStockMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_SeriesCConvertiblePreferredStockMember" name="SeriesCConvertiblePreferredStockMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_TwoThousandEighteenRightsOfferingMember" name="TwoThousandEighteenRightsOfferingMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_NumberOfPreferredStockSharesExercised" name="NumberOfPreferredStockSharesExercised" type="xbrli:sharesItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_RepresentativeSeriesUWarrantMember" name="RepresentativeSeriesUWarrantMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_NumberOfWarrantsExercised" name="NumberOfWarrantsExercised" type="xbrli:sharesItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_PeriodExercisableFromTheDateOfIssuance" name="PeriodExercisableFromTheDateOfIssuance" type="xbrli:durationItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_TwentyTwentyPurchaseAgreementMember" name="TwentyTwentyPurchaseAgreementMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_LincolnParkCapitalFundLLCMember" name="LincolnParkCapitalFundLLCMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_CommitmentFeeMember" name="CommitmentFeeMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_CanaccordGenuityLLCMember" name="CanaccordGenuityLLCMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_AtTheMarketOfferingProgramMember" name="AtTheMarketOfferingProgramMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_SalesAgreementMember" name="SalesAgreementMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_TwoThousandTwentyTwoDistributionAgreementMember" name="TwoThousandTwentyTwoDistributionAgreementMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_DistributionAgreementMember" name="DistributionAgreementMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_PercentageIssuanceOfCommonStock" name="PercentageIssuanceOfCommonStock" type="dtr-types:percentItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="false" />
  <xsd:element id="pstv_CommonStockSalesAvailableForIssuance" name="CommonStockSalesAvailableForIssuance" type="xbrli:monetaryItemType" substitutionGroup="xbrli:item" xbrli:periodType="instant" nillable="true" abstract="false" xbrli:balance="credit" />
  <xsd:element id="pstv_StockBasedCompensation2015EquityIncentivePlan1Member" name="StockBasedCompensation2015EquityIncentivePlan1Member" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_StockBasedCompensation2020EquityIncentivePlanMember" name="StockBasedCompensation2020EquityIncentivePlanMember" type="dtr-types:domainItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
  <xsd:element id="pstv_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsAggregateIntrinsicValueRollForward1" name="ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsAggregateIntrinsicValueRollForward1" type="xbrli:stringItemType" substitutionGroup="xbrli:item" xbrli:periodType="duration" nillable="true" abstract="true" />
</xsd:schema>
