Achilles Tendon Rupture (ATR) is one of the typical soft tissue injuries. Accurately predicting the rehabilitation outcome of ATR using noisy measurements with missing entries is crucial for treatment decision support. In this work, we design a probabilistic model that simultaneously predicts the missing measurements and the rehabilitation outcome in an end-to-end manner. We evaluate our model and compare it with multiple baselines including multi-stage methods using an ATR clinical cohort. Experimental results demonstrate the superiority of our model for ATR rehabilitation outcome prediction.
Soft tissue injuries, such as Achilles Tendon Rupture (ATR), are
increasing in recent decades [
We use a real-life dataset from the Orthopedic Research Group, aggregated
from multiple previous studies [
D Anm Inm Pnm Xnm wmp
M
Bnp Snp In0p
N
P (a) Using the measurement P^ bp vdm udn wdp
D bp Anm Inm Pnm Xnm Snp
Bnp
In0p M
P
N (b) Using patient traits U^
Age Length Weight : : : DVT 2 : : : ATRS 12 sti ATRS 12 pain
1 27 190 79.8 : : : 1 : : : 8 10
2 36 : : : : : : 8
3 41 172 : : : 0 : : : 10 10
the following. The second category is the scores S, and includes all rehabilitation
outcome tests, such as ATRS [
We design an end-to-end probabilistic model to simultaneously impute the missing entries and predict the rehabilitation outcome as shown in Fig. 2.
We formulate the missing data imputation problem into a collaborative
ltering or matrix factorization problem [
N M " p(PjU; V; P2) = Y Y #Inm ; where Inm is an indicator set to 1 if Pnm is observed and 0 otherwise.
We then predict the score matrix S using the patient information, which can be either the imputed measurement matrix as shown in Fig. 2(a) or the patient trait vector which summaries the patient state as shown in Fig. 2(b). Bayesian linear regression We rst consider a Bayesian linear regression model. The score is modeled as:
N P " p(S j W; b; X) = Y Y
n=1 p=1 p(W) = N (W j 0; w21);
N (Snp j xnwp + bp; S2) #In0p
; p(b) = N (b j 0; b2); where the input X is either the predictors or the patient traits, W and b are the weights and bias parameters for Bayesian linear regression. S indicates the observed rehabilitation scores, that is the rehabilitation outcome B, masked by the boolean observation indicator I0. In the case of the predictors (Fig. 2(a)), we use the observed values so that X = P^ = I P + (1 I) A, where I is the N M measurement observation indicator. In the case of the patient traits (Fig. 2(b)), we have X = U^ .
Bayesian neural network We also consider a Bayesian neural network (BNN)
[
N P "
p(S j ; X) = Y Y
where NN is a BNN parameterized by , the collection of all weights and biases
of the network. We consider fully connected layers with hyperbolic tangent
activations. The graphical model resembles the one in Fig. 2, with the exception
that instead of the weights W and biases b, we have BNN parameters .
Inference We run inference on the entire model in an end-to-end manner. We use
variational inference with the KL divergence [
We present our experimental results, comparing our proposed model with multiple baselines using the ATR dataset. We convert the whole dataset to numerical values and normalize all variables to t in the range of [0; 1]. Baselines We compare four variations of our proposed model against two baselines: mean data imputation, and two-stage model where inference is run in a sequential manner. In the case of mean imputation, the per-patient mean of observations that belong to the training set is imputed to all of their missing measurements. We then use the completed measurement data in the second component of the model. The second baseline is a two-stage version of our model. We rst use the measurement imputation part of our method to impute missing data. Then, we use this completed predictor matrix for the outcome prediction in the second stage. Inference is run separately on each component. Component 2 Input
BLR P BLR S BNN P BNN S
Results We split the training and testing set to re ect the treatment journey. In all of our experiments. For P, we randomly pick 80% of the available data for training and leave the rest for testing. For S, we randomly pick 80% of the patients, take all of their available scores for training and leave all scores of the remaining 20% of patients for testing, since the goal of our work is to predict the outcome using patients' incomplete measurements. We use grid search for hyper-parameter tuning and report the best result for each method in Table 1, using the Mean Absolute Error (MAE).
We can see that our proposed end-to-end model with Bayesian neural network applied on P achieves the best performance for the prediction of rehabilitation outcomes. Our proposed method shows clear improvement over the baselines. We can see that using latent variable models for data imputation gives better performance than the traditional mean imputation method. Additionally, it was established that a di erence of 0.1 is signi cant in the case of ATR in clinical practice. Thus, our result is close to the ideal MAE target 0:1. 5
We developed a probabilistic framework to simultaneously predict the rehabilitation outcome and impute the missing entries in a clinical cohort in the context of Achilles Tendon Rupture rehabilitation. We demonstrated a clear improvement in the accuracy of the predicted outcomes in comparison with traditional data imputation methods. Using the proposed model, we obtained a mean absolute error of 0:156. This result is close to the target 0:1 which is the clinical di erence step. We will thus continue to explore modeling choices to improve the outcome prediction accuracy. Additionally, the proposed method is a general framework that can be used in numerous health-care applications involving a long-term healing process after the treatment. In the future, we would also collaborate with more health-care departments to test and improve our method in these applications.