Prediction: plugin a corelation table (neuron) into a Time-Series Neuron in Keras/ TF
If you just have a normal correlation table (Which I am not sure about) you can do it like this: For every row you can add a correlation of the organ in a row to other organs. So you will have 

Week Baby   Age Organ   Date    IssueFound Problem Type Organ1 Organ2 Organ3 ...
2   Baby2   A0  Organ2  06/24/16    1      F            0.5    0      0.3    ...  


If you do not have a row for a new baby you should then format the data this way together with other parameters. The Corr_Organ1 an aggregation of all correlations that lead to Organ1 failure.

Baby  X-Type_cells Organ1 Organ1_time Organ2 Organ2_time Corr_Organ1 Corr_Organ2 ...
Baby1 1024         Yes    06/24/16    No     nan         0.2         0           ...


Then you make iterations: First forecast everything with zero correlations. Then when you calculate the first forecast and Organ 1 failed you update all other correlations.

Starting data:

Baby  X-Type_cells Organ1 Organ1_time Organ2 Organ2_time Corr_Organ1 Corr_Organ2 ...
Baby1 1024         nan    nan         nan    nan         0           0           ...


After first iteration:

Baby  X-Type_cells Organ1 Organ1_time Organ2 Organ2_time Corr_Organ1 Corr_Organ2 ...
Baby1 1024         yes    xxx         no     nan         0           0.8         ...


Then you predict again based on correlations and get the following result:

Baby  X-Type_cells Organ1 Organ1_time Organ2 Organ2_time Corr_Organ1 Corr_Organ2 ...
Baby1 1024         yes    xxx         yes     nan         0.2         0.8         ...


You repeat those iterations until you do not change anything.

The aggregation of correlations could be just a sum or something more complicated.