The necessary expenses of such care and treatment, including maintenance, personal necessities, artificial limbs, appliances and accessories and their upkeep, and of conveyance to and from the places designated for such services, shall be borne in whole or in part by the parent or guardian if financially able, or as funds are available by the department of health; provided, that the department may charge to the county in which such children with a physical disability reside a portion of the cost, the portion to be determined by the formula as authorized in § 68-12-107(4).