Copy the shape, not the size

Ask how big a hospital should be and you will usually get an answer about a hospital somebody has seen. That is not laziness. A building that works is genuine evidence, and the instinct to reproduce it is sound about most of what it reproduces.

But size and shape are answers to different questions, and they transfer differently.

Shape is a response to a parcel, a climate and a technology. It is a solved problem in the sense that the number of good answers is small, the constraints that produce them recur, and a form that resolves a constrained urban site in one city is very likely to resolve one in another. Shape travels.

Size is a response to a population: how many people, how ill, how often, how long they stay, and, in this market, how many of them can pay. None of those is a property of the building. Size does not travel at all, and copying it is the most common way a hospital ends up wrong in a way nobody can fix afterwards.

So the method for this decision is to take the shape and the ratios from the comparator, where they are measurable and transferable, and to derive the size from the catchment, where it is not.

The comparator here is a general hospital on a constrained urban London site, opened in 1993, with four hundred and thirty beds across five floors and a basement, arranged around courtyards with a central atrium. The form was a direct response to the parcel: planning limited the height, so the architects used the site to its edges and brought light into the middle rather than accepting a dark core.

That is a documented, examinable answer to precisely the problem the land page describes, and it arrived there independently of anything on this site. Two constrained parcels reaching courtyards by separate routes is about as strong as architectural evidence gets.

Read down the last column and the pattern is clean. Five of the six comparisons are useful and one is worthless, and the worthless one is the bed number, because of the fourth row.

The chain is short enough to argue with, which is the reason to show it as a chain rather than as a bed number at the end of a report.

Take the population that can reach the site in a tolerable journey. Take the share of them who can actually pay, because in a market where most health spending comes straight out of the patient’s pocket, a person who cannot pay is not demand, however ill they are. Apply an admission rate, multiply by how long people stay, divide by the year, and divide again by the occupancy the hospital can run at.

The fourth term is the one that does not appear in any version of this calculation written in a country with a universal payer, and it is the term with the most leverage. Halve it and the hospital needs twice the catchment for the same number of beds, which means a larger radius, which means a longer drive, which means fewer people willing to make it. It is also the term with the weakest data behind it, and the honest response to that is to run the answer across a range rather than to pick a number and defend it.

It is worth saying plainly what that term also means, because it is uncomfortable. The people excluded from it are ill at the same rate as everybody else. They are absent from this arithmetic not because they do not need a hospital but because they cannot buy one, and the whole of the clinician supply argument elsewhere in this study exists because of what that does to the place.

The occupancy term in that chain is not a policy choice. It is a property of size, and the relationship is strong enough to decide the answer on its own.

Demand does not arrive at its average. If a unit has thirty beds, the day everybody turns up is a far larger proportion of thirty than the equivalent day is of three hundred, so a small unit has to hold much more spare capacity to offer the same chance of a bed being free. The table solves that directly: for each size, the occupancy at which there is a one in fifty chance that an arriving patient finds every bed full.

Read the third column, which is the same fact in money. A thirty bed unit needs about forty per cent more beds per unit of demand than a large one does, permanently, for no clinical benefit whatsoever. That is the arithmetic reason small hospitals are expensive, and it has nothing to do with management.

It compounds with everything that is indivisible. A twenty four hour consultant rota needs roughly the same number of consultants whether the unit has thirty beds or two hundred. A theatre, a scanner, a blood bank, a generator and an oxygen plant are all lumps. Spread across thirty beds they are ruinous; across two hundred they are ordinary.

All of which argues for going bigger, right up until the fourth column.

A hospital feeds on a catchment, and a catchment is an area. Double the beds and you need double the people, but people live on a surface, so the radius grows with the square root and the drive grows with the radius.

Follow the last column of the table and the trade becomes obvious. Going from thirty beds to a hundred and twenty buys sixteen points of occupancy and costs about eighteen minutes of journey. Going from two hundred to four hundred and thirty buys four points and costs another twenty two minutes. The gain is running out while the cost is not.

That is the shape of the answer, and it is the reason there is an optimum at all rather than a preference. Pooling improves with the square root of size, so it flattens. Distance grows with the square root of size too, but it converts into time at a rate set by the road network, and in a city with average speeds like this one that rate is brutal. One curve saturates and the other does not.

Then the costs the table does not price, all of which push the same way. Internal walking distance rises with size, and this study has already put a number on what ten metres costs. Adjacency gets harder, because a bigger hot floor is a bigger plate and eventually it will not fit on the parcel at all. And the marginal patient at the edge of a large catchment is, by construction, the one least likely to come, so the last beds are fed by the weakest demand.

The knee is therefore somewhere in the low hundreds, and on these assumptions it sits around a hundred and twenty to two hundred beds. Above that you are buying catchment radius rather than capability.

The two calculations on this page were arrived at independently and they are worth putting side by side, because a single model agreeing with itself is not evidence.

The chain starts from a catchment and a paying share and produces about a hundred and fifty beds. The scale table starts from queueing and geography, knows nothing about the catchment size, and puts the efficient range at roughly a hundred and twenty to two hundred. They agree, and they agree for unrelated reasons: one is about how much demand exists, the other about where the trade between pooling and distance turns over.

So the working recommendation is a hospital of around a hundred and fifty beds, with the structure, services and land to reach perhaps two hundred and fifty without rebuilding. Not four hundred and thirty. The comparator is a fine hospital and its bed number is an answer to a question about a different city with a different payer.

The convergence is also the warning. Both calculations share the paying share, the admission rate and the length of stay. If those are wrong, both move together and the agreement proves nothing, which is exactly why the shortfalls below name them rather than the arithmetic.

A paying population materially larger or smaller than assumed. This is the dominant term and it moves the answer proportionally. A measured figure from the catchment, rather than a national estimate, would be the single most valuable piece of work anybody could do before the massing gate.

A road network that converts distance into time differently. Every travel figure here assumes an average speed and a detour factor. A site on a route that moves, or a dedicated ambulance approach, changes the fourth column and therefore the knee.

An insurance market that grows. If third party cover expands materially, the paying share rises without the population changing, the catchment tightens and the same building serves a shorter radius. That would argue for building the structure for more beds now even while opening fewer.

A service mix that is mostly day case. The four specialties this project targets are not evenly weighted in bed demand, and a configuration that leans further into day surgery and ambulatory work needs fewer beds and more theatres and recovery bays for the same revenue. That is a different building, not a smaller one.

Four, and the first applies to every number in the scale table.

The occupancy column is a ceiling rather than a target. Queueing theory assumes arrivals are independent and random and that a bed is a server working at a constant rate, and this study has already spent a chapter explaining that the second of those is false. Real occupancy at any given standard sits below what the table says, the gap is largest when the hospital is busiest, and an organisation that reads the column as a target will discover that by filling a corridor.