Why transplant pathways need explicit states
After transplantation, a patient may live with a functioning graft, experience graft failure, receive another graft, experience failure again, or die. These events do not all have the same meaning for every study. A question about time to failure of the first graft differs from a question about a patient’s path through repeated transplantation and death. A useful state diagram makes the relevant clinical statuses explicit and shows which changes are possible under the study protocol.
The diagram is not just a list of events. Each state describes a status that a patient can occupy, and each arrow represents a permitted change in that status. The state definitions should be based on the patient’s observable clinical condition, not merely on a convenient event label. In particular, “graft failure” must be defined by the protocol: it might mean irreversible loss of graft function, return to dialysis, or another prespecified criterion. The study must also specify how it handles a failure and retransplantation recorded at the same visit.
For a pathway that follows an initial graft and permits one retransplant, a clear starting set of states is:
- F0: alive with the initial graft functioning;
- F: alive after failure of the current graft, with no functioning graft (for example, awaiting retransplantation);
- F1: alive with a functioning retransplant graft;
- D: dead.
The label F is deliberately about the patient’s current status, rather than a claim that the patient can experience failure only once. The states are mutually exclusive at a time: a living patient cannot simultaneously have a functioning graft and be in the no-functioning-graft state. They should also cover every status relevant to the analysis. If the protocol includes another status, such as alive after graft removal but before a defined failure criterion is met, the state set may need to be expanded.
Constructing the core transplant diagram
Suppose the question is: “From the initial transplant onward, how do patients move among functioning graft, graft failure, retransplantation, and death?” If the protocol permits one retransplant, the corresponding transition set is:
The first arrow records failure of the initial graft. The second allows death while the initial graft is still functioning. From F, a patient may receive a retransplant and enter F1, or die before retransplantation. From F1, the patient may die while the retransplant is functioning. If the study also follows failure of the retransplant, add \(F1 \to F\). That return arrow makes the failure-and-retransplant pathway recurrent; it should be included only if such histories are part of the study’s state definition and follow-up.
A diagram’s arrows are determined by the question and protocol, not by a wish to include every clinically imaginable event. For example, \(F0 \to F1\) skips the failed-graft status. It is not appropriate if the analysis records failure before retransplantation and needs to distinguish that interval. If failure and retransplantation are effectively recorded as one combined transition, or the intermediate status is outside the target question, a direct arrow may be a defensible simplification—but its meaning must be stated.
Worked Example: Following an Initial Graft Through Retransplantation
Hypothetical protocol. A study begins follow-up on the day of an initial transplant. It records irreversible graft failure, retransplantation, and death. A retransplant can occur only after the initial graft has failed. The study ends follow-up at death or administrative study end. The target is the sequence of clinical statuses, including death before or after retransplantation.
First define the states: F0 means alive with the initial graft functioning; F means alive after failure and before a functioning retransplant; F1 means alive with a functioning retransplant; and D means dead. These definitions separate patients who are alive but have no functioning graft from those who do.
The protocol allows \(F0 \to F\), because the initial graft can fail; \(F0 \to D\), because a patient can die before documented failure; \(F \to F1\), because retransplantation can restore a functioning graft; and \(F \to D\), because death can occur before retransplantation. It also allows \(F1 \to D\). Thus the diagram is:
Now classify three hypothetical patient records. Patient A follows \(F0 \to F \to F1\) and remains alive at study end. Patient B follows \(F0 \to D\). Patient C follows \(F0 \to F \to D\). All three paths use permitted arrows. The diagram distinguishes death with a functioning initial graft from death after failure while awaiting retransplantation, even though both patients die. No transition out of D is needed because the study does not follow any status after death.
Let the study question determine the detail
A state diagram should be detailed enough to preserve distinctions that matter to the target question, but not so detailed that it creates irrelevant categories. This can be justified by considering what the analysis needs to tell apart. If the question is whether the initial graft fails, then the initial graft’s status must remain identifiable. If the question is whether a patient ever receives a second transplant, retransplantation must be represented as a transition or a distinct status. If the outcome is overall death, separating deaths by graft status may not be needed for the endpoint itself, although it may still be useful for describing pathways to death.
A short construction argument makes the logic precise. Let \(H(t)\) denote the patient’s observed event history up to time \(t\), and let \(S(t)\) be the state assigned from that history. A state definition is adequate for a study question if histories assigned to the same state do not need to be distinguished to answer that question. If the question concerns whether the current graft is the original graft, then histories with a functioning initial graft and histories with a functioning retransplant cannot both be coded simply as “functioning graft.” They are different states for that question. If the question concerns only whether the patient currently has any functioning graft, the two histories may be combined, provided graft identity is not otherwise needed.
This is a design principle, not a guarantee that a chosen state process satisfies a Markov assumption. Even after states are defined, the adequacy of the current state as a summary of relevant past history is a separate modeling issue. For example, patients with a functioning retransplant may have different future risks depending on how many previous grafts failed. If that history matters, a generic functioning-graft state may hide relevant heterogeneity.
Worked Example: When “Functioning Graft” Is Too Broad
Hypothetical question. Investigators want to estimate and describe failure of the first graft separately from failure of any later graft. Their records distinguish graft number, but an initial draft diagram uses only “functioning graft,” “failed,” and “dead.”
That draft merges two statuses: alive with a functioning first graft and alive with a functioning retransplant. Once merged, a transition from “functioning graft” to “failed” does not reveal whether the first graft or a later graft failed. The diagram therefore cannot directly distinguish the two outcomes required by the question.
Refine the states to F0 (initial graft functioning), F (current graft failed and no functioning graft), F1 (first retransplant functioning), and, if a second retransplant is followed, F2 (second retransplant functioning). For a study following up to two grafts, the feasible arrows include \(F0 \to F\), \(F \to F1\), \(F1 \to F\), and \(F \to F2\); death arrows are added from each living state in which death can occur. The repeated \(F\) state is acceptable only if the analysis does not need to distinguish which graft failed on entry to it. If that distinction is required, label failure states by graft number, such as \(F^{(0)}\) and \(F^{(1)}\), and connect each to the appropriate retransplant state.
The refinement adds states because the question requires graft-specific interpretation. It is not merely a cosmetic relabeling: without the refinement, the desired failure type is lost when the records are translated into the diagram.
Assumptions, diagnostics, and common failures
Before drawing arrows, establish the population and observation rules. State whether time begins at transplantation, at discharge, or at another prespecified origin; define graft failure and retransplantation; and specify whether death is recorded regardless of current graft status. The diagram assumes each patient can be assigned to one relevant state at a time, that the listed statuses cover the follow-up of interest, and that each included arrow represents a transition the protocol can observe or define. If a study uses a different time origin or records events only at widely spaced visits, the apparent ordering of failure, retransplantation, and death may be uncertain.
The diagram describes permitted pathways; it does not assert that every permitted pathway occurs in the sample, nor does it assign probabilities to arrows. Check it against the protocol and example event histories. A useful audit is to ask, for each state, what clinical event moves a patient into it, what events can move the patient out, and whether every observed record can be represented without inventing an unobserved transition.
- Missing death arrows: If death can occur from a living state during follow-up, an arrow to D is needed when death is part of the diagram’s scope. Omitting it can imply that death is impossible there.
- Impossible retransplantation arrows: A transition directly from a functioning initial graft to a functioning retransplant is inconsistent with a protocol requiring failure first, unless the study deliberately combines the intermediate events.
- Ambiguous failure state: “Failed” should say whether the patient is alive, whether a functioning graft remains, and whether the state applies after failure of any graft or a specific graft.
- Unjustified cycles: A return from failure to functioning graft represents retransplantation, not recovery of the failed graft, unless the protocol defines recovery as possible. Label the event clearly.
- Over-aggregation: Merging initial and retransplant graft states may prevent graft-specific conclusions or hide dependence on the number of previous failures.
One common mistake is to treat every patient who has died as having followed the same route, and therefore to draw a single death arrow from “graft failure.” That reasoning is incorrect if patients may die while a graft is functioning or while waiting for retransplantation. The correction is to add a death arrow from each living state in which death can occur, so the diagram preserves the clinically distinct routes into the absorbing death state.
Choosing among simpler and richer representations
A transplant multi-state diagram is not always the only useful representation. The main alternative depends on the endpoint:
| Representation | What it preserves | What it can leave out |
|---|---|---|
| Competing endpoints from the initial functioning-graft state | Which first event occurs, such as graft failure or death before failure | Later retransplantation and subsequent graft history |
| Composite endpoint such as graft failure or death | Whether either component has occurred | Which component occurred and the pathway after failure |
| Multi-state transplant pathway | Ordered changes among functioning graft, failure, retransplantation, and death | History distinctions not represented in the chosen states |
| Recurrent event history | Repeated failures or transplant events as events over time | May not directly represent the patient’s current clinical status |
A simpler endpoint representation can be appropriate when the scientific question stops at the first event. It is insufficient when the question concerns what happens after failure or how patients move through retransplantation. Conversely, a more elaborate multi-state diagram does not automatically improve an analysis: extra states need clear definitions, reliable ascertainment, and enough data to support the intended comparisons. A diagram also cannot resolve ambiguity in event timing or make an inadequately defined failure criterion clinically meaningful.
Check Your Understanding
For each question, identify the relevant diagram choice. Then check the brief solutions below.
- A study records initial graft failure, death before failure, and no events after the first event. What arrows are needed from the initial functioning-graft state?
- A protocol requires failure of the current graft before retransplantation. Is a direct arrow from initial functioning graft to functioning retransplant appropriate if failure is a required, separately recorded status?
- A patient dies after graft failure while waiting for retransplantation. Which arrow represents this route?
- Why might a study of graft-specific failure need separate states for a functioning initial graft and a functioning retransplant?
- A study follows repeated retransplants but uses one shared “functioning graft” state. Name one distinction that may be lost.
Solutions. (1) Include an arrow from the initial functioning-graft state to failure and an arrow from that state to death. The diagram stops at the first event, so it need not represent later transitions. (2) No. Use the path from functioning initial graft to failure, then from failure to functioning retransplant. A direct arrow would skip a required, separately recorded status. (3) Use the arrow from the alive-after-failure state to death. (4) The combined state would not identify which graft was functioning, so a later failure could not be attributed to the initial graft or a retransplant. (5) It may hide graft number or the number of earlier failures, either of which could matter to the study question or future transitions.