The main focus of the present computational modelling work is to determine the extensional rheological response of some model biofluids, with a view to ultimately aiding experimentally based analyses and clinical practice. This is accomplished in the present study through model extensional flows and rheological investigation, addressing filament stretching and contraction flows, and upon which significant advances are presented. As such, two biofluid flow systems within the human body are of current interest: (i) respiratory disorders and sputum in the lung airways (associated with filament stretching), where stretchiness of mucus sputum in situ is vital, with clinical focus on chronic obstructive pulmonary disease (COPD/sputum); and (ii) bile flow in the biliary system (contraction flow), with clinical focus on disorders of primary sclerosis cholangitis and common bile duct narrowing. Both sputum and bile biofluid systems are represented through kinetic theory rheological fluid modelling, with capability to represent material structure entanglement, branching and anisotropy. This is practically achieved by appealing to the class of pom-pom differential constitutive models, extracted from polymer melt physics and deployed here through a single extended pom-pom (SXPP) approximation. This class of models is sufficiently rich to enable description of both network structure and rheological properties, exhibiting viscoelastic response (memory), with strain-hardening/softening and shear-thinning properties.