.

What is a Propensity Model?“Propensity model” is a broad term describing a set of computations that estimate the probability of an event. We will use propensity models as the tool for making treatment recommendations. Here, the event in question is a limb undergoing a specific surgery based on the historic SOP at our center. We write a propensity model as:

Here is the probability a limb will be prescribed the treatment , the function is a set of “rules”, and are clinically relevant variables (“features”) specific to the treatment being considered, like ankle range-of-motion, age, dynamic equinus, and so on.

We focus on twelve common surgical procedures that collectively account for over 95% of the surgeries documented in the clinical database of our gait center. These surgeries are: selective dorsal rhizotomy, rectus femoris transfer, psoas lengthening, hamstrings lengthening, adductor lengthening, calf muscle lengthening, femoral derotation osteotomy, tibial derotation osteotomy, patellar tendon advancement, distal femoral extension osteotomy, foot and ankle bony reconstruction (e.g., calcaneal osteotomy, talo-navicular fusion), foot and ankle soft tissue reconstruction (e.g., posterior tibialis tendon transfer, split tibialis anterior tendon transfer). The approach we present is not limited to these surgeries. Any treatment — surgical or otherwise — for which there is sufficient baseline and follow up data can be evaluated.

Building and testing treatment recommendation models.

A panel of clinical experts selected a set of features for each of the twelve surgeries analyzed in this study (Appendix 1). The selection process relied on the panelists’ collective expertise and clinical judgment, prioritizing both model parsimony and clinical interpretability. It is important to acknowledge that the selected features may not be optimal, and future refinements — such as adding, removing, or reweighting features — could potentially improve model performance. Feature selection is a critical component of machine learning model development, and extensive research exists on this topic, which lies beyond the scope of this manuscript.

Once the features were chosen, we fit Bayesian Additive Regression Trees (BART) models for each surgery [11]. Other modeling frameworks can be used, ranging from logistic regression to random forests. The purpose of this manuscript is not to explain or promote BART, which is a well-understood standard tool. However, it may be useful for the reader to understand practical reasons why we chose BART as modeling framework. Disinterested readers can skip the following discussion of BART without affecting their ability to understand the rest of this manuscript.

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Why BART? BART is able to model complex response surfaces. This includes nonlinearities, bifurcations, and so on. These are common in real-world treatment decisions. An example of a nonlinearity is the effect of tibial torsion on the decision to perform a tibial derotational osteotomy [Fig 2]. Within the typical torsion range, there is almost no impact of torsion on the decision to derotate. As torsion deviates external or internal to the typical range, the probability of a derotation rises. There are many other situations the reader can imagine where nonlinearities exist. Largely because of its flexibility in modeling complex responses, BART has been shown to be a class-leading causal modeling framework [12].

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Fig 2. Effect of tibial torsion (bimalleolar axis angle) on the probability of a limb undergoing a tibial derotation osteotomy.

The response is highly nonlinear. Close to typical values (~15 degrees) tibial torsion has little effect. As internal (smaller) or external (larger) torsional deviations appear, the probability of surgery rises. A linear model, like logistic regression, would not be able to capture this response.

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BART is able to handle missing data with minimal assumptions. There is no imputation needed and BART treats missingness itself as a feature [13]. This reduces concerns about biases introduced from data that is missing in a systematic manner, such as missing ROM measurements on severely contracted limbs, or missing strength measurements on patients who cannot understand or follow directions. By handling missing data in this principled and user-friendly fashion, BART allows applications of the model to patients with missing data. This is a common, and often non-random occurrence. For example, patients with more severe impairments may not be able to complete or cooperate with instructions, leading to missing data. Excluding these patients would introduce important biases. The ability to competently handle missing data also promotes use of models in centers that may not collect all the measurements described in this study.

BART is Bayesian. As a Bayesian model, BART outputs a probability distribution of results — called a posterior. This allows the uncertainty of any prediction to be observed and reported directly. The Bayesian formulation also provides for built-in regularization — meaning that BART models do not “overreact” to individual features. An important consequence is that BART models generally produce nearly optimal results without the need for manual hyperparameter tuning. This is a stark contrast to many advanced modeling approaches.

BART is fast, free, and easy to use. There are several packages that implement BART in the R programming language. We use the bartMachine package version 1.3.4.1 with the default parameters (number of trees = 50, number of burn-in samples = 250, number of iterations after burn-in = 1000, alpha = 0.95, beta = 2, k = 2, q = 0.9, nu = 3) [14]. There is vigorous ongoing innovation and development related to BART.

Fitting treatment recommendation models.

We evaluated 7546 gait assessments from 2090 individuals referred to our center by 176 different clinicians. The individuals all had baseline and follow-up CGA visits within a three-year span. We lumped diagnoses into 10 categories. Due to our referral patterns, individuals diagnosed with cerebral palsy dominated the data set (84% cerebral palsy). The mean (sd) baseline age was 8.9 (3.3) years and there were 42% females by sex at birth.

When the treatment recommendation model is a BART model, then treatment probability is a posterior distribution of probabilities (posterior) and the probability of prescribing the treatment is the mean of this posterior. The treatment categories (recommended or not) are defined by the mean probability compared to a critical threshold, which is 0.5 by default.

As is typical in a clinical service, we have an imbalanced data set, with many more untreated limbs than treated. For example, out of 7546 limbs in the database, only 334 of them underwent an adductor release. This makes evaluating model performance challenging. Consider that for an adductor release, with a prevalence of only 4%, a model that predicts no limb gets an adductor release would be 96% accurate. To overcome this challenge, we begin by recognizing that we want a model with good balanced accuracy, which implies good sensitivity and specificity. Alternatives choices would include fitting a model with good accuracy, or high negative predictive value, or high Brier score, etc. By focusing on balanced accuracy, we are implicitly demanding that “If the historical SOP resulted in treatment, we want the treatment recommendation model to recommend treatment. If the historical SOP resulted in no treatment, we want the treatment recommendation model to recommend no treatment.”

We use a standard approach for building a propensity model with imbalanced data. First, we split the dataset into training and testing sets (70% and 30%, respectively). Then we choose the treatment of interest. Next we use under-sampling to generate a balanced sub-sample of the training set with equal numbers of treated and untreated (control) limbs [15]. Finally, we use the balanced sub-sample to fit a BART model to compute the probability that the limb underwent the selected treatment. We test the model performance on the independent test set. Note that the test set is not balanced through undersampling, and therefore exhibits the same imbalanced surgical prevalences that are observed in practice. Using an unbalanced test set gives a realistic estimate of model performance on future clinical data.

Treatment recommendation model performance.

We report standard classifier metrics to give a clear picture of how well the classifier will work in practice (Table 1). Examining the performance, we can make several general statements about the treatment recommendation models. Balanced accuracy is consistently high, ranging from the mid.70s to .90. Specificity and sensitivity are consistently high and balanced, ranging from 0.71 to 0.92. The mean specificity and sensitivity were 0.77 and 0.78, respectively. Note that sensitivity and specificity were even high for low prevalence surgeries with known low inter-surgeon consistency, such as psoas lengthening. Also, keep in mind that the upper limit of model performance is governed by how consistently clinicians made treatment decisions in the past. Treatment decisions on the same data are known to vary widely between clinicians and institutions [1]. Model performance was strong even for the generic “Foot and Ankle …” categories, which are broad and based on almost entirely subjective assessment of foot deformity.

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Table 1. Propensity model performance.

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We can put the treatment recommendation models’ performance in context by examining the specificity and sensitivity estimates for several important and common medical screening tests: Mammography to detect breast cancer exhibits specificity.90 and sensitivity.54 −.94 [16]. Colonoscopy to detect adenomas 6 mm or larger exhibits specificity ~ .90 and sensitivity from.73 −.98 [17]. Pap smears to detect cervical cancer exhibit specificity ~ .75 and sensitivity ~ .68 [18]. Rapid antigen tests for COVID-19 exhibit specificity ~ .99 and sensitivity ~ .65 (.44 in asymptomatic individuals) [19]. Finally, prostate specific antigen testing to detect prostate cancer exhibits specificity ~ .90, sensitivity from.09−.33 [20].

Treatment recommendation profile.

We will be referring to a case study throughout this manuscript to give an idea of how EB-GAIT looks and is used in practice. All protected health information has been removed or replaced. Some variables — like age — have been altered slightly to preserve the general sense of the individual while maintaining anonymity. The case was selected from a search of our database for individuals who underwent calf muscle lengthening consistent with the treatment recommendation model and had a follow-up visit at around one year post-surgery. The chosen patient had birth, developmental, and treatment history typical for someone diagnosed with hemiplegic cerebral palsy. On the right side, during the same surgical event that included the calf muscle lengthening, the individual also underwent a femoral derotation osteotomy, hamstrings lengthening, and bony foot reconstruction. It may be interesting for readers to know that, when choosing this case, we looked at the follow-up video — so we had a sense of the outcome — but we did not look at the treatment recommendation or outcome prediction model results to see how accurate they were. Only one other patient was screened before selecting this patient. We mention all of this to give the reader some indication of the degree of “cherry-picking” involved in the case-study selection. We are deeply skeptical of case-studies due to the biases they often reflect [21]. A deidentified version of the full EB-GAIT report for this case study is provided as supplementary material for this manuscript.

In practice, we provide a compact display of the surgery recommendations for 12 surgeries. Below is the treatment recommendation profile for the case study [Fig 3]. The categories are divided into probability quintiles and color-coded from high probability (green) to low probability (purple). This color-blind-safe scheme will be maintained in several other places to show indications (green) and counterindications (purple) for surgery. It will also be used to indicate outcome predictions for treated (green) and untreated (purple) limbs.

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Fig 3. Propensities for 12 surgeries.

Probability of surgery is categorized and color coded for quick reference.

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Propensity and outcomes.

Given the skill and experience of the surgeons whose practices shape the SOP, it is reasonable to expect that adherence to the SOP would yield consistently positive outcomes. However, as previously demonstrated [Fig 1], historical outcomes have been modest, stagnant, and unpredictable. It is important to recognize that historical outcomes reflect actual clinical decisions, which do not always align with the SOP. There are many legitimate reasons why the SOP may not have been followed, including the fact that the SOP is not formally documented, as well as critical patient- or clinician-specific factors such as patient goals, family experiences with prior treatments, perceived ability of the patient to engage in rehabilitation, and so on.

The variability in adherence to the SOP provides a unique natural experiment, allowing us to investigate whether following the SOP is leads to improved outcomes. To explore this, we calculated the concordance between administered treatments and the SOP (as estimated by the treatment recommendation models) and examined how outcomes varied with concordance. High concordance indicates that treatments closely followed the SOP, whereas low concordance reflected deviations from the SOP.

We compute concordance for each of the 12 surgeries ) as follows. First, we rescale surgery probability from −1 to 1:

Next, we square the scaled surgery probability to emphasize high and low probabilities while preserving the sign,

This transformation amplifies differences at the extremes while compressing moderate probabilities. Concordance is then defined as if surgery was administered, and 0 otherwise. This approach rewards surgeries performed on limbs with high predicted probabilities, penalizes those performed on limbs with low probabilities, and remains neutral when no surgery is performed or when the predicted probability is near 50%. The overall concordance is the sum of the 12 individual concordances:

We demonstrate the effect of concordance with SOP on outcomes by examining calf muscle lengthening. We first compute the concordance for calf muscle lengthening as described above (). Next we choose relevant outcomes at the body structures level (e.g., Maximum Ankle Dorsiflexion (Knee Extended)) and at the body functions level (e.g., Initial Contact Ankle Angle Sagittal Plane). Finally, we examine the change in these outcome measures after calf lengthening surgery as a function of [Fig 4]. There is a clear and strong benefit to following the historical SOP — though the variation around the trend is wide. Other surgeries and outcomes exhibit similar responses.

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Fig 4. Calf lengthening surgery on limbs that reflect the historical SOP (high concordance) results in large improvements in both static ankle range of motion and dynamic ankle dorsiflexion in gait.

Surgery that does not conform to the SOP leaves these measures unchanged, or even worsened.

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A similar analysis can be conducted at a global level by evaluating the gait kinematic response (GDI) to overall concordance [Fig 5]. As with the isolated surgery results, there is a clear message that following the historical SOP is generally beneficial, although there is significant variability around the mean response.

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Fig 5. Following rather than flouting the historical SOP leads to better GDI outcomes for treated limbs.

For example, the change in GDI at an overall concordance of 4 is approximately 10 points, whereas at a concordance of −2 it is approximately 0.

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Understanding at an overall level – Partial dependence plots.

The BART models that compute the propensity scores are “black boxes”. The input features for the models are clinically relevant, but it is not easy to discern how the BART model computes the surgery probability based on these features. The actual calculation involves summing the results of many shallow decision trees constructed using Bayesian principles. There are a number of standard tools that can be used for interrogating models like this. The use of a post-processing technique to give a glimpse at the inner workings of a complex machine learning model is called “explainable machine learning”. Note that there are machine learning experts who advocate for simpler models that do not require an explanation, and are instead interpretable in their raw form [22].

At the simplest level we can examine the marginal effect of an individual feature, averaged across values of all other features. This is called a partial dependence plot [23]. Partial dependence plots produce an overview of features’ impact. However, partial dependence plots ignore interactions. For example, restricted ankle range of motion may be an indication for calf muscle lengthening, but the effect may vary with patient age. The interaction of age and ankle ROM on calf lengthening probability is not captured by a partial dependence plot. Furthermore, partial dependence plots average effects across conditions that may be unrealistic. For example, the partial dependence plot for age averages the effect across all heights — so the effect of age = 6 may include a hypothetical 6-year-old who is 2m tall. As such, partial dependence plots can be considered contextless. Nevertheless, partial dependence plots are a standard tool for explaining models such as BART.

Example: Rectus femoris transfer.

Consider the partial dependence plot for rectus femoris transfer [Fig 6]. We describe some, but not all, of the sub-plots. Age: The probability of surgery rises steadily until around age 10, then plateaus. Maximum Swing Knee Angle Sagittal Plane (Max. Knee Flexion): The probability of surgery is high until just over 50 peak knee flexion, steadily decreases until around 60, then plateaus. ROM Swing Knee Angle Sagittal Plane (Knee Flexion ROM): The probability of surgery is high for limited ROM, decreasing slowly until around 35, after which it decreases rapidly. Era: The probability of rectus transfer surgery has steadily decreased over time. Rectus Femoris Spasticity: The probability of surgery increases with increasing spasticity up to modified Ashworth scores of 2, then declines slightly for scores of 3 and 4. Prior Rectus Femoris Transfer: A prior rectus femoris transfer makes a repeat surgery less likely.

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Fig 6. Partial dependence plot for the rectus femoris transfer treatment recommendation model.

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Understanding at an individual level – Shapley values.

In practice, we seek to understand the influence of a specific feature on an individual prediction — for example, how much a patient’s knee flexion ROM contributes to their probability of being prescribed a rectus femoris transfer. Partial dependence plots do not provide this level of insight. Instead, Shapley values offer a rigorous, principled approach to quantifying feature contributions [24]. The Shapley value for a feature represents its contribution to a limb’s surgery probability, relative to the overall mean probability observed in the model’s training data. Since Shapley values are computed with respect to the mean probability of a surgery, they depend on the local SOP and will vary across clinical centers.

For example, suppose the overall probability of a rectus transfer is 15% overall. Consider patient X, whose left knee flexion range-of-motion has a Shapley value of −11%, while their age has a Shapley value of +6%. Assuming all other Shapley values are zero, the probability of patient X undergoing a rectus transfer, based on the historical SOP, would be:

Formally, the Shapley values for the models estimating the SOP-based probability of undergoing Where satisfy the following additive relationship:

Thus, the is the probability that a limb undergoes the treatment, is the mean probability of across all limbs in the training set, and is the Shapley value for the feature for the model predicting .

Thus, the are contributions from features such as age, diagnosis, and so on …). Since is the same for all limbs, only the values determine individual differences in treatment probability.

A positive indicates that feature increases the likelihood of undergoing We call this an indication. Conversely, a negative signifies a counter-indication, meaning the feature reduces the likelihood of treatment. The magnitude of reflects the strength of the indication or counter-indication.

We can compute Shapley values for a limb and present the results in a compact tabular format, sorted from strongest indications to strongest counter-indications for ease of interpretation. Below we show the Shapley values for calf muscle lengthening (Gastroc-Soleus Lengthing) for the case study [Fig 7]. We only show the feature-specific Shapley values, , excluding the mean Shapley value.

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Fig 7. Shapley values presented in a tabular form.

Columns correspond to sides. The features are sorted and color-coded from strongest indication to strongest counterindication. The value of the feature is shown. An emoji of a walking person indicates a feature extracted from instrumented gait analysis (as opposed to physical examination). The probability for surgery is indicated at the top of each column with 90% prediction intervals noted parenthetically.

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For the case study, the right limb exhibits a high probability for calf muscle lengthening surgery. The color-coding of indications and counterindications scales with magnitude. The indications (green) for treatment are initial contact ankle angle in the sagittal plane (from the instrumented gait data — indicated by an icon of a walking person), age, maximum ankle dorsiflexion (from the physical examination). There are also several weak indications (light green) and only one counterindication (purple), which is our center’s trend towards less calf muscle lengthening surgery over time. On the left (unaffected) side, surgery is unlikely, and there are numerous counterindications, including the fact that this is the unaffected side of a patient with a diagnosis of hemiplegia.

The probability of surgery for the limb, shown in the top row of Fig 7, is represented as a range rather than a single value. This range corresponds to the 90% prediction interval derived from the Bayesian posterior of the treatment recommendation model. The uncertainty in the surgery probability arises from several factors, such as model imperfections, historical variability in decision-making, measurement errors, and the limited size of the dataset used to train the model. The unaffected side displays a relatively narrow range of uncertainty, whereas the affected side shows a broader range — though it remains entirely above the 50% likelihood threshold.

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What is an outcome prediction model?An treatment outcome model is a set of rules of the form:

Where is a chosen variable, are clinical features deemed relevant to changes in , and are proposed treatments. Note that if the treatment does not occur and if it does.

Fitting outcome prediction models.

) for which we fit treatment recommendation models. For example, changes in femoral torsion and mean stance hip rotation during gait were deemed relevant to a femoral derotation osteotomy, while changes in ankle dorsiflexion ROM and initial contact dorsiflexion during gait were relevant to calf muscle lengthening. We also built models for predicting changes in gait kinematics (Gait Deviation Index) and mobility (Functional Assessment Questionnaire Transform). Currently, we have models for around 20 different outcome variables, with additional outcome models built and tested as needed.

For each model, the same clinical experts chose the predictive features that were felt to be relevant to the treatment outcome. The features consisted of the baseline value of the outcome variable, relevant structural and biomechanical variables, and whether the limb underwent each of the 12 major surgeries of interest. The interval and prior treatment status for each surgery was dichotomous, and did not include any information about dose (e.g., amount of derotation) or technique (e.g., Baker vs. Strayer calf muscle lengthening). The complete list of features for each outcome are listed in Appendix 2.

We used an ad hoc iteration of features to fit models that were accurate and parsimonious. However, there is no guarantee that the current models are optimal. A substantial body of research exists on optimal feature selection for machine learning, and future efforts will include improving the models. Despite the lack of optimization, the models are well-calibrated and provide useful precision, as discussed below.

Outcome prediction model performancePrediction Model Performance.

Clinically useful models must have reasonable accuracy, minimal bias, and reliable precision. An unbiased model ensures that its predictions, on average, align closely with actual observed values, without systematic over- or underestimation. Reliable precision refers to the validity of the model’s prediction intervals (PIs), which should consistently encapsulate the stated proportion of future outcomes. For instance, a 90% PI should contain 90% of the observed values in prospective applications. The outcome models, like the treatment recommendation models, were evaluated retrospectively, using independent test data.

The treatment outcome models performed exceptionally well in terms of low bias and reliable precision. Here we report performance for a representative subset of the treatment outcome models at body structures and body functions levels (Table 2). Accuracy, as measured by mean absolute error (MAE) varied with outcome. For example, the MAE for ankle dorsiflexion ROM was around 6 , which is similar to the test-retest measurement error (8). A similarly strong performance was seen across all physical examination measures. At the kinematic level, MAEs generally reflected a slightly larger fraction of test-retest measurement error. One explanation for these larger errors is that the effects of treatment on causally downstream outcomes is more complex, and thus more difficult to predict. For example, the effect of a femoral derotation osteotomy on femoral torsion (body structure) is reasonably straightforward given the nature of the surgery. But the downstream causal effects on hip rotation (body function) or overall gait pattern (GDI) involves possible patient compensations, rehabilitation details, and so on. Further supporting this conjecture is the fact that the largest observed MAE was for overall mobility as measured by the Functional Assessment Questionnaire Transform (FAQt) [25]. The FAQt measures activities, which are the most causally downstream outcome we predict.

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Table 2. Outcome model performance.

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Case study outcomes.

We return to the case study introduced earlier. The patient underwent surgery at four levels during a single session, all on the right side. These were a femoral derotation osteotomy, hamstrings lengthening, calf muscle lengthening, and bony foot reconstruction. We note that the actual surgery was in concordance with the treatment recommendation models.

Single-level treatment.

. The difference between these two predictions is the estimated treatment effect. We estimate the 50% and 90% prediction intervals for the treatment effect from the posterior distribution produced by the BART model.

In clinical practice, we generally report outcome predictions from each of 12 single-level surgeries. At the body structure level, most surgeries have isolated effects. For example, a femoral derotation osteotomy affects femoral torsion, but does not directly affect ankle dorsiflexion ROM. The reverse is true for a calf muscle lengthening. At the kinematic and activities levels, several surgeries can affect a single outcome. For example, both femoral and tibial derotation osteotomies may affect foot progression during gait. In situations like this, we generally recommend examining the predictions from each single-level surgery and summing the effects as an approximation. This approach makes the reasonable assumption that interactions are small.

Body structure level.

For the femoral derotation osteotomy we estimated the change in femoral torsion. For the hamstrings lengthening we estimated the change in popliteal angle. For the calf muscle lengthening we estimated the change in passive ankle dorsiflexion ROM. For the bony foot surgery we estimate the change in overall weightbearing foot deformity severity [Fig 8]. We observe that the predicted outcomes are extremely accurate for change in ankle dorsiflexion ROM and anteversion, well within the 50% PI for change in weightbearing foot deformity, and at the edge of the 50% PI for change in popliteal angle. Note that by comparing the treated prediction (top, green range) to the untreated control prediction (bottom, purple range) we can estimate the treatment effect (middle, grey range). In this case, each surgery was predicted to have a relatively high probability of producing positive treatment effect at the body structures level. Keep in mind that treatment effect is an unobservable quantity, computed with the aid of an estimated counterfactual.

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Fig 8. Body structure outcome.

Measured outcomes are white circles. Prediction intervals are thick (50%) and thin (90%) lines, and a barely-visible solid point indicates the mean prediction.

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Body function level (Kinematics).

We also estimated kinematic outcomes, which are at the level of body functions. For the femoral derotation osteotomy we estimated the change in hip rotation. For the hamstrings lengthening we estimated the change in initial contact knee flexion. For the calf muscle lengthening we estimated the change in initial contact ankle dorsiflexion. For the bony foot surgery we estimated the change in mean stance foot progression [Fig 9]. The measured changes in ankle dorsiflexion at initial contact, hip rotation during stance, and foot progression during stance were all within the 50% prediction interval, while the measured change in knee flexion at initial contact was larger than predicted, falling outside the 50% prediction interval but well within the 90% prediction interval. Note that this is sensible, given the slightly-larger-than-expected improvement in popliteal angle [Fig 8].

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Fig 9. Body function (gait kinematics) outcome.

Measured is white circle. Prediction intervals are thick (50%) and thin (90%) lines, and a small solid point indicates the mean prediction.

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Multi-level treatment.

In addition to modeling the outcomes of single-level treatment, the outcome prediction models can be used to estimate the effects of multi-level surgery. For the treated right limb of the case study, we estimated the structural and kinematic outcomes of interest for the multi-level surgery administered (femoral derotation osteotomy, hamstrings lengthening, calf muscle lengthening, and bony foot and ankle surgery). Estimated changes were all within the 50% prediction intervals [Fig 10]. The outcome estimates from multi-level surgery were similar to those obtained from single-level surgery, demonstrating — for this case — the claim that individual surgeries often affect specific body structures and functions with limited interactions.

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Fig 10. Multi-level (Multi) and single-level (Single) surgery predictions compared to measured outcomes for the treated side of the case-study individual.

The multi- and single-level predictions are similar for all chosen outcomes.

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