This is basically an analysis of change model.
2 measurements on each subject were taken at baseline, and 2 more at follow-up. I will refrain from calling this "control" and "intervention" as that could be somewhat misleading.
We have repeated measures within patients. So we could consider a model that fits random intercepts for patients, to control for this. There are also repeated measures within each kidney of each patient. I would suggest the following model:
measure ~ time + LR + (1 | PatientID)
In order to fit this model, it is necessary to "reshape" the data as follows:
PatientID time LR measure
1 -0.5 L 19
1 -0.5 R 29
1 0.5 L 27
1 0.5 R 20
2 -0.5 L 14
2 -0.5 R 13
2 0.5 L 13
2 0.5 R 11
The estimate for time will answer the research question: What is the change in measure associated with the intervention, while controlling for the repeated measures within patients, and within the same kidney's of each patient.
Another approach is to fit nested random effects, and treat LR as a random factor:
measure ~ time + LR + (1 | PatientID/LR)