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The second question is purpose of the separation between Luke and the men described in verses 10-11:

10 Aristarchus my fellow prisoner greets you, and Mark the cousin of Barnabas (concerning whom you have received instructions—if he comes to you, welcome him), 11 and Jesus who is called Justus. These are the only men of the circumcision among my fellow workers for the kingdom of God, and they have been a comfort to me. 12 Epaphras, who is one of you, a servant of Christ Jesus, greets you, always struggling on your behalf in his prayers, that you may stand mature and fully assured in all the will of God. 13 For I bear him witness that he has worked hard for you and for those in Laodicea and in Hierapolis. 14 Luke the beloved physician greets you, as does Demas. Colossians 4:10–14 (ESV).

The Pulpit Commentary states:

Luke the physician, the beloved, saluteth you (Philem. 24; 2 Tim. 4:11). This reference to Luke’s profession is extremely interesting. We gather from the use of the first person plural in Acts 16:10–17, and again from 20:5 to the end of the narrative, that he joined St. Paul on his first voyage to Europe and was left behind at Philippi; and rejoined him six years after on the journey to Jerusalem which completed his third missionary circuit, continuing with him during his voyage to Rome and his imprisonment. This faithful friend attended him in his second captivity, and solaced his last hours; “Only Luke is with me” (2 Tim. 4:11). His being called “the physician” suggests that he ministered to the apostle in this capacity, especially as “his first appearance in St. Paul’s company synchronizes with an attack of St. Paul’s constitutional malady” (Lightfoot: comp. Acts 16:10 and Gal. 4:13–15; the illness referred to in 2 Cor. 1:8–10 and 4:7–5:8 may partly have led to Luke’s rejoining St. Paul in Macedonia). St Luke’s writings testify both to his medical knowledge and to his Pauline sympathies. His companionship probably gave a special colouring to the phraseology and cast of thought of St. Paul’s later Epistles. (On the relations of St. Luke and St. Paul, see a valuable Paper by Dean Plumptre in the Expositor, first series, vol. iv. pp. 134–156.) “The beloved” is a distinct appellation, due partly to Luke’s services to the apostle, but chiefly, one would suppose, to the amiable and gentle disposition of the writer of the third Gospel. It is not unlikely that he is “the brother” referred to in 2 Cor. 8:18, 19. Lucas is a contraction for Lucanus; so that he was not the “Lucius” of Acts 13:1, nor, certainly, the “Lucius my kinsman” of Rom. 16:21, who was a Jew. He was probably, like many physicians of that period, a freedman; and, since freedmen took the name of the house to which they had belonged, may have been, as Plumptre conjectures, connected with the family of the Roman philosopher Seneca and the poet Lucan.

The Pulpit Commentary: Colossians, ed. H. D. M. Spence-Jones (Bellingham, WA: Logos Research Systems, Inc., 2004), 214.