To determine the minimum number of days after which the laboratory can be considered safe for use, we need to understand how the radioactive material decays over time.

The half-life of a radioactive material is the time it takes for half of the material to decay. For our given material, the half-life is 18 days. This means every 18 days, the amount of radioactive material is reduced to half of its previous amount.

Given that the initial radiation level is 64 times more than the permissible level, we can represent this situation mathematically. Let $ R $ be the initial radiation level and $ P $ be the permissible level of radiation. We are given:

$$ R = 64P $$

After each half-life period of 18 days, the amount of radioactive material will be halved. If $ n $ is the number of half-life periods required for the radiation to reduce to the permissible level, we can write:

$$ \frac{R}{2^n} = P $$

Using the given information ($ R = 64P $), we get:

$$ \frac{64P}{2^n} = P $$

Dividing both sides by $ P $, we get:

$$ \frac{64}{2^n} = 1 $$

This implies:

$$ 2^n = 64 $$

Knowing that $ 64 $ is a power of $ 2 $:

$$ 64 = 2^6 $$

Therefore, $ n = 6 $. This means it will take six half-life periods for the radiation to decay to a safe level.

Since each half-life period is 18 days, the total number of days required is:

$$ 6 \times 18 = 108 $$

Hence, the minimum number of days after which the laboratory can be considered safe for use is 108 days.

Answer: Option C (108)