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In its Guidelines for Donations of Health Care Equipment (mirror), the WHO says:

The sense among some biomedical engineers and health care professionals who have extensive work experience in these countries is that less than 30%, perhaps as low as 10%, of used equipment ultimately becomes operational.

The number is quite vague and unreferenced. What percentage of donated used medical equipment to developing countries ultimately becomes operational?

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That is on the one hand quite a broad question ("these countries") for not a very vague but a very specific number. On the other hand this is indeed a public health issue very much unfocused by the public eye.

One additional problem not mentioned in the question is that from all the donated equipment some is not suitable, beyond repair, unwanted, without operators or technicians to repair, running out of replacement parts and consumables. That means that it is not only a factor of "what percentage becomes operational", but also of what arrives where, who profits from what along the whole line and how long does it stay operational?

To give you a few examples with references: Effectiveness of medical equipment donations to improve health systems: how much medical equipment is broken in the developing world?

It is often said that most of the medical equipment in the developing world is broken with estimates ranging up to 96% out of service. […] This study examined 112,040 pieces of equipment. An average of 38.3% (42,925, range across countries: 0.83–47%) in developing countries was out of service. The three main causes were lack of training, health technology management, and infrastructure. We hope that the findings will help biomedical engineers with their efforts toward effective designs for the developing world and NGO’s with efforts to design effective healthcare interventions. […]
The analysis may underestimate the number of pieces out of service. We have relied on hospital reported equipment status reports to create the analysis. However, hospitals may ignore some working equipment that never entered their inventory, broken equipment off inventory or they may lack the technical staff to conduct a thorough inventory. In any case, we suspect that many hospitals are underreporting their broken equipment. Unfortunately, we are aware of no independent, peer-reviewed study that tracks a particular shipment of donated equipment from original status to final disposition.

So, how much equipment is out of service in the developing world? The answer is about 40%. We find no evidence to support the statement that most or nearly all the equipment in resource poor settings is broken.

While donations can cause problems and there are undoubtedly some hospitals where the situation is much worse, the analysis does suggest that donating equipment can improve the available healthcare options.

On the other hand, investments in capacity building, health technology management and infrastructure could nearly double the amount of working medical equipment without the expense of collecting, testing, and shipping used medical devices.

Why is this seemingly simple process so ineffective? From Beyond good intentions: lessons on equipment donation from an African hospital

While researching this subject the authors found guidelines for medical equipment donations produced by WHO in 2000. They identify four principles of "good donation practice":

  1. the ensuring of maximum benefit to the recipient,
  2. respect for the wishes and context of the recipient,
  3. the avoidance of quality double standards,
  4. and effective donor-recipient communication and planning.

This case bears out the importance of these principles and the necessity for putting into practice guidelines that appear to have been largely overlooked. The need to strengthen health systems throughout the world is a recognized key to achieving the Millennium Development Goals. This case is illustrative of this challenge and offers support for WHO policy, and indeed support for the contention that the progression from knowledge and policy to practice is the great challenge of international health.

The WHO paper referenced in the question is relatively old (1997) – but apperently not much has changed – Barriers for medical devices for the developing world:

The most important design barrier is the lack of spare parts in the target countries. Any device designed for the developing world will be likely to stop working as soon as the first replacement part is required. […] Other possibilities for the failure to replace parts exists. The cost may be prohibitive or the hospital may lack the expertise or tools required to execute the repair. Most hospitals do not have a technician with more than a high school education. […] One of the most common problems encountered in developing world hospitals is the lack of consumables. Consumables are liquids or supplies required for the use of the equipment, but allowing only limited, or no, reuse […]

  • The developing world represents a market size approximately five-times larger than the developed world.

  • Successfully capturing the developing world market will require more than selling the same devices at lower prices. Most medical devices transplanted from the developed world to the developing world hospital fail.

  • The most important unique design barriers for medical devices in the developing world hospital are the lack of spare parts and the lack of required consumables.

  • Other unique barriers include a lack of reliable power and water, public infrastructure and technical expertise.

  • It is a misconception that designs must be simple and that capital cost is always the primary barrier.

  • Stripped-down designs may be perceived as lesser quality and rejected, despite lower cost.

Looking at just one concrete example of Medical equipment donations in Haiti: flaws in the donation process:

The team found that only 28% of the equipment was working properly and in use for patient care; another 28% was working, but lay idle for technical reasons; 30% was not working, but repairable; and 14% was beyond repair.

Up to this point this answer focussed on just a short list of what in this process usually can go wrong: just dumping some stuff in front of the natives while making money from it is not effective, despite producing some karma-friendly statistics. Speaking of which, some problems are not so much in the design of the process, but in the execution. Equipment donation to developing countries:

There has been one episode when import duty was requested; the charity's response was that the equipment could be ‘safely dumped into the sea’. Fortunately, there was a happy ending to this story and the equipment reached its destination.

On another occasion, a container was ‘lost’ in Somalia, due to a lack of robust arrangements at the destination. Donations to countries which are politically unstable are a particular challenge.

tl;dr

That number of 30% or less does not look very inaccurate today, 20 years after it was identified as a problem, guidelines to reduce this inefficiency published and much money made from this good-doing charitity.

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