In　recent　years,　microfluidic　channels　with　narrow　constrictions　are　extensively　proposed　as　a　new　but　excellent　possibility　for　advanced　delivery　technologies　based　on　either　natural　or　artificial　capsules.　To　better　design　and　optimize　these　technologies,　it　is　essential　and　helpful　to　fully　understand　the　releasing　behavior　of　the　encapsulated　solute　from　capsules　under　various　flow　conditions　which,　however,　remains　an　unsolved　fundamental　problem　due　to　its　complexity.　To　facilitate　studies　in　this　area,　we　develop　a　numerical　methodology　for　the　simulation　of　solute　release　from　an　elastic　capsule　flowing　through　a　microfluidic　channel　constriction,　in　which　the　tension-dependent　permeability　of　the　membrane　is　appropriately　modeled.　Using　this　model,　we　find　that　the　release　of　the　encapsulated　solute　during　the　capsule＇s　passage　through　the　constriction　is　enhanced　with　the　increase　in　the　capillary　number　and　constriction　length　or　the　decrease　in　the　constriction　width.　On　the　other　hand,　a　large　variation　in　the　channel　height,　which　is　generally　larger　than　the　capsule　diameter,　generates　little　effect　on　the　released　amount　of　the　solute.　We　reveal　that　the　effects　of　the　capillary　number　and　constriction　geometry　on　the　solute　release　are　generally　attributed　to　their　influence　on　the　capsule　deformation.　Our　numerical　results　provide　a　reasonable　explanation　for　previous　experimental　observations　on　the　effects　of　constriction　geometry　and　flow　rate　on　the　delivery　efficiency　of　cell-squeezing　delivery　systems.　Therefore,　we　believe　these　new　insights　and　our　numerical　methodology　could　be　useful　for　the　design　and　optimization　of　microfluidic　devices　for　capsule-squeezing　delivery　technologies.　©　2019　Author(s).