Islamic Ruling on Cleaning Teeth After Eating and Swallowing Food Particles
According to this Islamic ruling, food removed from the teeth with a toothpick should be thrown away, and swallowing it is haram if blood is mixed with it.

Islamic scholars state in this fatwa that after finishing food, a person should carefully clean the food particles stuck between the teeth with a toothpick. Leaving these particles in the mouth can cause tooth disease and bad smell.
If food comes out by using a toothpick, it should be thrown away. It should not be swallowed. If there is blood mixed with those particles, then swallowing them is haram.
However, if something comes out naturally with the tongue, then it can be swallowed. If someone throws away what comes out with the toothpick, then this is better. If someone does not throw it away, then there is no sin, as long as there is no blood.
This Islamic ruling is based on a hadith found in Sunan Abi Dawood. It is reported by Abu Huraira that Prophet Muhammad said that whoever cleans between his teeth after eating should throw away what comes out, and whatever comes out with the tongue may be swallowed. Whoever does this has done well, and whoever does not, there is no harm on him.
Therefore, according to this fatwa and the view of Islamic scholars, it is better to throw away the particles removed by toothpick, and swallowing them with blood is not allowed.