Dose adjustments are necessary when two drugs interact significantly either by pharmacodynamic or pharmacokinetic interaction.
First let’s identify the given two drugs. Lidocaine is a class Ib antiarrhythmic agent that acts by blocking voltage gated sodium channels. It also functions as local anaesthetic by the same mechanism inhibiting local neurotransmission.
On the other hand propranolol is beta blocker mainly blocks beta receptors on the heart thereby blocks actions of catecholamines on the heart. It can be used as class II antiarrhythmic agent, antihypertensive as well as prophylactic in various situations where sympathetic system is overactive. For example, it can be indicated as prophylactic in hyperthyroidism, anxiety and migraine etc.
So both of the drugs act on different targets, one is acting on ion channels and another is acting on receptors. Even though both of these two drugs have antiarrhythmic action they show less significant interaction pharmacodynamically.
Now let's think in view of pharmacokinetic interaction
Lidocaine shows high first pass metabolism therefore it is short acting. Hence it is given by IV infusion. Since its serum levels depend on hepatic metabolism hence hepatic blood flow any drug that modifies hepatic blood flow effects lidocaine serum levels.
Propranolol is a non-selective beta blocker that decreases cardiac output and portal venous pressure. This results in reduction of hepatic blood flow leading to decrease in entry of lidocaine into liver.
In this way, propranolol decreases hepatic metabolism thereby increases levels of lidocaine requiring its dosage adjustment.