Do you remember the moment you had the idea for this exhibition?
We started thinking about this five years ago, when we were beginning to plan the restoration of the Virgin Of The Rocks, so an exhibition to celebrate that project seemed like the right thing to do.
I realised that although there'd been great exhibitions of Leonardo's drawings in the past - including one at the Hayward when I was a young lad - there hadn't been one that focussed specifically on his career as a painter.
Really? That seems surprising.
Well, one of the things I came to realise during the course of the research for the show is that every age invents its own Leonardo. For the late 20th Century, his science and his extraordinarily prescient inventions obviously chimed with the times.
It was also important for the National Gallery to provide a sensible corrective to Dan Brown's mystical, heretical Leonardo. It will be interesting to see whether our interpretation will trigger people's imaginations.
How did you decide which paintings you wanted to feature?
In a way, it was a limited shopping list because Leonardo painted extraordinarily little, and even less survives. And of those pictures, it was clear that some would never be lent, whatever happened - I mean, the Mona Lisa was never going to leave Paris even if we wanted it to. Nobody's going to scrape The Last Supper off the walls.
The next step, presumably, is to approach the galleries. How does that work?
You do it in order. For the galleries with which you have existing relationships, you can have quite straightforward frank discussions from the outset. So we started quite early on with The Louvre and the Vatican.
We're very lucky that about a third of the loans are drawings from Windsor and the extraordinary collection of Leonardo drawings that belongs to the Queen. Their generosity in lending has made this whole exhibition possible.
Then, with some of that in place, we then turned to the Czartoryski Foundation about the Lady With The Ermine, which is one of the absolute key loans for the show. We don't expect people to say yes straight away.
So the initial loans create a head of steam that gives you an entry point with other galleries?
That's exactly right. Anyone who makes a loan wants to know what the benefits of doing so are going to be: What are you going to find out about the picture by sending it somewhere else that you wouldn't see if you kept it on your own walls?
So the Hermitage, for example, are lending The Madonna Litta in the awareness that we've gathered all the drawings, by Leonardo and by his followers, that have ever been connected with that work and this will be a moment to see them all together.
To what extent do galleries bargain to get something in return?
It depends, frankly, on what kind of relationship you have. With The Louvre, we have a very informal relationship. We vaguely keep count of how much is going in each direction, but the assumption is that both institutions have a very active exhibition programme and we want to support each other as much as we can.
With other institutions, like the Ambrosiana in Milan, they have a smaller collection and for them, sending the Portrait Of The Musician was an enormous deal which they needed to think about very carefully.
We offered to send Boticelli's mystic nativity as an exchange - but a considered one, because they have a Boticelli of exactly the same moment and they could do a small focus exhibition with the two pictures. So we tried to make these exchanges serious and sensible.
How nervous do other galleries get about transporting their paintings abroad?
All the institutions wanted to be reassured about the way in which we control the ambient climate and humidity and temperature in our exhibition space. Others, on top of that, wanted to create micro-climates for their works, which might be about sealing the picture into its frame or making a case especially for it.
What sort of condition are da Vinci's paintings in?
The fact that his pictures were painted, for the most part, on walnut panels rather than poplar panels is an important factor in making them more movable than other panel paintings of the period.
It's a better wood, it's better grained and it's less likely to be eaten by woodworm, so on the whole it's in better shape.
It's not quite clear why he chose walnut, however. He writes about how that's his support of choice - but his Florentine pictures, including the Mona Lisa, are painted on poplar. It may just have come down to what was available.
Which painting are you proudest of getting?
In the end, the chance to see the two versions of The Virgin Of The Rocks together for the first time in history is a great coup - especially because it will almost certainly never happen again. I try not to think about it too much.
Is there a piece you regret being unable to obtain?
It's probably not tactful to talk about this, but I'm going to anyway. My only real sadness is that there is an incredible drawing in Turin of a young woman's head. It's usually thought to be a sketch for the angel in the Louvre's version of The Virgin Of The Rocks - but some people think it is a portrait of Cecilia Gallerani. The fact that the two Virgins Of the Rocks and Cecilia will be in London would have made its loan wonderful.
But they're doing their exhibition of Leonardo's drawings in Turin from their own collection, and that was something which was long-planned, so it just wasn't going to change.
Leonardo da Vinci: Painter Of The Court Of Milan starts at the National Gallery on 9 November and runs until February next year. Luke Syson was talking to BBC Entertainment and Arts reporter Mark Savage.