Alkeides had indeed been poisoned with hydra venom.
He had used the vile sludge to consume it and prevent his bodily destruction . . . but that deadly poison, which had led to his own death and indirectly driven his third wife to suicide, had still been taken into his body.
It was the influence of King’s Order that kept it from eating away his flesh.
He had drawn out the power that he had taken from the Erymanthian Boar.
It was not, however, the boar’s own power.
The boar was not the greatest thing he had taken and conquered during that expedition.
It was the thing Alkeides, during his penance, had taken from his mentor Kheiron.
It was none other than Kheiron’s immortality.
Kheiron, a centaur bestowed with the power of immortality, had been struck by hydra venom due to a stray shot from Alkeides.
The pain had been so intense that, unable to bear it, Kheiron had surrendered his own immortality to Prometheus and ended his life.
Therefore, while Alkeides had lost the twelve substitute lives that his proper Spirit Origin possessed, he did possess a single substitute lifespan, Kheiron’s immortality until it was ceded to Prometheus, as one of his Noble Phantasms.
Activating it, however, meant nothing less than experiencing the same pain Kheiron had suffered.
At that very moment, that ceaseless agony—the same agony that had led him to choose death while alive—continued to torment him. He was currently cancelling it out by transforming that pain and suffering into strength through the influence of the “mud.”
“Do you regret it? Killing your mentor?”
“. . . Were I to obey my heart, having given myself to vengeance, I ought to rejoice that I was able to liberate my mentor from immortality, a vile curse of the gods.”

Fate/Strange Fake: Volume 5, Chapter 16 ("Breaking Dawn & Wakeless Dreams I")