[Note: This blog post contains two errors. First, Rh factor antigens do not form a heavy coat on solid organs like kidneys. They only do so on red blood cells. Thus, they do not strongly affect tissue compatibility or cause organ rejection.

Second, there are two forms of the A blood type, called Type A1 and non-Type A1. Some patients with Type O have low levels of A antibodies and can accept a kidney from a non-Type A1 donor. Similarly, some patients with Type B can accept a kidney from a non-Type A1B donor.

I will write an updated blog post to replace this one.]

Previous blog posts discussed kidney exchanges and how to improve outcomes by using exact matching and by adding a look-ahead feature. However, there is a fundamental problem with the current composition of traders (kidney patients and their donors) in the exchange that makes it impossible for everyone who enters the exchange to find a partner, no matter how good the software is or how long they wait. There is a solution to this mismatch problem. But first we need to understand why this problem occurs.

Not all pairs enter the exchange

Let’s develop a model of which recipient-donor pairs will enter the exchange and which will not. First we need to define what a match is. We define a match as being blood type compatible and human leukocyte antigen (HLA) compatible. For simplicity, we will ignore HLA compatibility for now. Table 1 below shows the probability that a donor’s kidney will match a recipient. Most of the cells in the table either have a value of 1.00 (shaded green) meaning that donor’s kidney will always match the recipient or 0.00 (unshaded), meaning it never will. The cells in the lower left quadrant with value 0.94 (shaded yellow) indicate the donor’s kidney will usually match, except for a woman who has RhD- blood type and has had a child with RhD+. She will now have antibodies to RhD+ antigens and cannot accept a kidney with RhD+ antigens.

Several assumptions are made in creating this table, as shown in the footnote below the table. As stated above, the most significant is that it ignores HLA sensitivity. About 45 percent of kidney patients have antibodies for foreign HLA. The most common causes are repeated injections of blood-derived EPO as a part of dialysis treatment or from a previous kidney transplant. Women with three or more children with the same father often develop HLA sensitivity as well. HLA sensitivity reduces the chance of a match for the individual and for the recipient population as a whole.

 Table 1. Probability of organ match*

Donor

100.0%

Total

37.4%

35.7%

8.5%

3.4%

6.6%

6.3%

1.5%

0.6%

Total

O+

A+

B+

AB+

O-

A-

B-

AB-

Recipient

37.4%

O+

1.00

0.00

0.00

0.00

1.00

0.00

0.00

0.00

35.7%

A+

1.00

1.00

0.00

0.00

1.00

1.00

0.00

0.00

8.5%

B+

1.00

0.00

1.00

0.00

1.00

0.00

1.00

0.00

3.4%

AB+

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

6.6%

O-

0.94

0.00

0.00

0.00

1.00

0.00

0.00

0.00

6.3%

A-

0.94

0.94

0.00

0.00

1.00

1.00

0.00

0.00

1.5%

B-

0.94

0.00

0.94

0.00

1.00

0.00

1.00

0.00

0.6%

AB-

0.94

0.94

0.94

0.94

1.00

1.00

1.00

1.00

*The following simplifying assumptions are made: 1) Blood type distribution is for U.S. only, all races combined; 2) Probability of having end-stage renal disease (ESRD) is independent of blood type; 3) All recipients have AB blood antigen sensitivity to non-matching blood types; 4) Non-match due to HLA sensitivity is ignored; 5) 40% of recipients are adult women who have had at least one child (if mother is RhD- and father is RhD+, then she will be incompatible with an RhD+ donor); 6) Recipient and donor blood types are independent (i.e., donors are not related to recipient, which would make them more likely to match, and donors to adult women with children are not the spouse, which would make them less likely to match); 7) Recipients stay with their first choice donor regardless of match (i.e., if the first choice donor does not match, the recipient does not continue to search for a compatible donor in order to avoid entering the exchange)

Based on the table above, we can calculate the distribution of matched and unmatched pairs. Table 2 below shows that nearly two-thirds of pairs (total percentage in upper left) will match immediately and do not need to enter the exchange. However, it is useful to divide these matches into two groups. Cells shaded green indicate matched pairs that would not benefit an unmatched pair if they were to enter the exchange. Cells shaded yellow indicate matched pairs that could benefit an unmatched pair if they were to enter the exchange and swap with them. Notice the composition of the cells shaded yellow, they are mostly O donors.

Table 2. Distribution of matched pairs*

Donor

64.4%

Total

37.1%

16.3%

1.2%

0.1%

6.6%

2.9%

0.2%

0.0%

Total

O+

A+

B+

AB+

O-

A-

B-

AB-

Recipient

16.5%

O+

14.0%

0.0%

0.0%

0.0%

2.5%

0.0%

0.0%

0.0%

30.7%

A+

13.4%

12.7%

0.0%

0.0%

2.4%

2.2%

0.0%

0.0%

4.6%

B+

3.2%

0.0%

0.7%

0.0%

0.6%

0.0%

0.1%

0.0%

3.4%

AB+

1.3%

1.2%

0.3%

0.1%

0.2%

0.2%

0.1%

0.0%

2.8%

O-

2.3%

0.0%

0.0%

0.0%

0.4%

0.0%

0.0%

0.0%

5.1%

A-

2.2%

2.1%

0.0%

0.0%

0.4%

0.4%

0.0%

0.0%

0.8%

B-

0.5%

0.0%

0.1%

0.0%

0.1%

0.0%

0.0%

0.0%

0.6%

AB-

0.2%

0.2%

0.0%

0.0%

0.0%

0.0%

0.0%

0.0%

*Based on same assumptions as first table

The remaining pairs are unmatched and need to enter an exchange to have a chance of getting a match. Table 3 below shows the composition of these unmatched pairs. Cells shaded green indicate that all pairs can match another unmatched pair. These are A/B pair to B/A pair swaps or RhD- recipients who can swap for a RhD- donor. Cells shaded yellow indicate that some but not all pairs can match another unmatched pair. Finally cells shaded red will not match another pair in the exchange.

Table 3. Distribution of unmatched pairs*

Donor

35.6%

Total

0.3%

19.4%

7.3%

3.3%

0.0%

3.4%

1.3%

0.6%

Total

O+

A+

B+

AB+

O-

A-

B-

AB-

Recipient

20.9%

O+

0.0%

13.4%

3.2%

1.3%

0.0%

2.4%

0.6%

0.2%

5.0%

A+

0.0%

0.0%

3.0%

1.2%

0.0%

0.0%

0.5%

0.2%

3.9%

B+

0.0%

3.0%

0.0%

0.3%

0.0%

0.5%

0.0%

0.1%

0.0%

AB+

0.0%

0.0%

0.0%

0.0%

0.0%

0.0%

0.0%

0.0%

3.8%

O-

0.1%

2.4%

0.6%

0.2%

0.0%

0.4%

0.1%

0.0%

1.2%

A-

0.1%

0.1%

0.5%

0.2%

0.0%

0.0%

0.1%

0.0%

0.7%

B-

0.0%

0.5%

0.0%

0.1%

0.0%

0.1%

0.0%

0.0%

0.0%

AB-

0.0%

0.0%

0.0%

0.0%

0.0%

0.0%

0.0%

0.0%

*Based on same assumptions as first table

Table 4 below shows the distribution of unmatched pairs after the exchange. Cells shaded red contain unmatched pairs. Notice that even after the exchange, over one-fourth of recipient-donor pairs are left without a match. And notice the composition. About 92% of these remaining pairs include a recipient with O blood type. There are only two short-term solutions to resolve this mismatch. One is to encourage more people with O blood type to step forward and become altruistic donors (unfortunately, an unlikely behavioral change and one with ethical issues). The other is to encourage the mirror image matched pairs (yellow cells in Table 2) to enter into the exchange pool to facilitate a swap. (Notice also that donors with type AB blood, like me, also have difficulty finding a match.)

Table 4. Distribution of unmatched pairs after exchange*

Donor

26.8%

Total

0.1%

15.8%

3.7%

3.3%

0.0%

2.6%

0.6%

0.5%

Total

O+

A+

B+

AB+

O-

A-

B-

AB-

Recipient

20.8%

O+

0.0%

13.4%

3.2%

1.3%

0.0%

2.2%

0.5%

0.2%

1.4%

A+

0.0%

0.0%

0.0%

1.2%

0.0%

0.0%

0.0%

0.2%

0.3%

B+

0.0%

0.0%

0.0%

0.3%

0.0%

0.0%

0.0%

0.0%

0.0%

AB+

0.0%

0.0%

0.0%

0.0%

0.0%

0.0%

0.0%

0.0%

3.8%

O-

0.1%

2.4%

0.6%

0.2%

0.0%

0.4%

0.1%

0.0%

0.4%

A-

0.0%

0.1%

0.0%

0.2%

0.0%

0.0%

0.0%

0.0%

0.1%

B-

0.0%

0.0%

0.0%

0.1%

0.0%

0.0%

0.0%

0.0%

0.0%

AB-

0.0%

0.0%

0.0%

0.0%

0.0%

0.0%

0.0%

0.0%

*Based on same assumptions as first table

The disadvantage that patients with O blood type face in getting a kidney has long been known. UNOS data shows they typically wait 20 months longer than patients with A blood type for a kidney transplant (some with HLA sensitivity wait for over 100 months) and thus are more likely to die while waiting for an organ. Many in the transplant community are advocating a change in the way cadaver organs are allocated to overcome this problem. For instance, see Nephrology Dialysis Transplantation Jan 2010 (subscription required) for proposed changes in Europe. Eventually, the same pressure may come to bear on the issue of who can and who must enter an exchange.

In the next blog post, I will propose a way out of this dilemma. The solution involves making matched pairs want to enter the exchange for their own benefit.

[Update: The proportion of kidney patients with antibodies to HLA is 45%, not 20% as originally stated.]

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