Per my understanding, muscle fibers need to be sufficiently damaged in order for them to later regrow. How does objectively know whether the muscle is sufficiently damaged? Is there an electric machine/sensor tailored for this purpose?
Muscle fibres do not need to be damaged in order for muscle growth to occur. Mechanical tension (detected by mechanotransductors in the muscle and converted to signals that drive hypertrophy) is the primary driver of muscle protein synthesis (the creation of new muscle filaments, resulting in hypertrophy).
Whether or not a muscle grows after exercise is determined by whether the total amount of muscle protein synthesis that occurs following that bout of exercise exceeds the amount of muscle protein breakdown in the same time period. Because muscle protein breakdown rates depend on a huge number of factors, including training status, nutrition, age, sleep quality, and disease state, there's no fixed amount of protein synthesis required to cause muscle growth. So even if we could accurately predict the amount of muscle protein synthesis that a workout would cause, we still wouldn't know if it was enough to cause muscle growth.
We can measure muscle protein synthesis in exercise science labs, but the procedure involves feeding participants 'labelled' protein containing radioactive isotopes, getting them to exercise, and then stabbing them with a horrifyingly large needle in order to extract muscle tissue for analysis, and measuring how much of the labelled proteins were incorporated into new muscle. It's totally impractical for consumer use.
Realistically, the only reliable ways for a person to determine whether hypertrophy is occurring is to try to detect it after the workout, by:
- Whether the limb is increasing in circumference over weeks or months.
- Whether you are able to perform more reps at the same weight or lift more weight for the same number of reps compared to previous workouts.
- Whether your scale weight is increasing without apparent increases in body fat. (e.g. Looking for increases in weight without increases in waist circumference.)
If you weren't detecting improvements though these metrics, that's when you'd need to look why that was the case, with the most likely causes being insufficient quality or volume or training, or insufficient nutrition.