Modeling the outcome trajectories in patients with acquired brain injury: a non-linear dynamic evolution approach

doi:10.21203/rs.3.rs-2233535/v1

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Modeling the outcome trajectories in patients with acquired brain injury: a non-linear dynamic evolution approach

https://doi.org/10.21203/rs.3.rs-2233535/v1

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published 18 Apr, 2023

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In this study we provide a dynamic non-linear mathematical approach for modeling the course of disease in acquired brain injury (ABI) patients. Data from a multicentric study was used to evaluate the reliability of the Michaelis-Menten (MM) model applied to well-known clinical variables assessing the outcome of ABI patients. The sample consisted of 156 ABI patients admitted to eight neurorehabilitation subacute units (IRU) and evaluated at baseline (T0), after 4 months from the event (T1) and at discharge (T2).

The MM model was used to characterize the trend of the first PCA dimension (represented by the variables: feeding modality, RLAS, ERBI_A, Tracheostomy, CRS-r and ERBI-B) to predict the most plausible outcome, in terms of positive or negative GOS at discharge. Exploring the evolution over time of the PCA Dimension 1, after day 86 the MM model was able to better discriminate the time course for individuals with positive with respect to negative GOS (Accuracy: 85%; Sensitivity: 90.6%; Specificity: 62.5%).

Using a non-linear dynamic mathematical model, we can provide more comprehensive trajectories of the clinical evolution of ABI patients during the rehabilitation period. Our model can be used to select patients for interventions designed for a specific outcome trajectory.

Biological sciences/Computational biology and bioinformatics/Data processing

Biological sciences/Computational biology and bioinformatics/Predictive medicine

Health sciences/Neurology

Physical sciences/Mathematics and computing

acquired brain injury

outcome prediction

Michaelis-Menten model

dynamic mathematical model

non-linear evolution

Brain injury is the leading cause of death and disability worldwide. The degree of severity varies by a combination of several demographic (i.e., age), etiology (vascular, traumatic or anoxic), clinical (i.e., time from injury, degree of disability at admission), radiological (i.e., side and size of the lesion), cognitive, behavioral, psychosocial, and environmental issues, which can interfere with the effectiveness of rehabilitation interventions and, thus, with final outcome at discharge (Turner-Stokes et al., 2015). The Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) reported that in the last 26 years the global incidence has increased by 8.4% in patients with Traumatic Brain Injury (TBI)(GBD 2016 Traumatic Brain Injury and Spinal Cord Injury Collaborators), whereas declined by 8.1% in stroke patients (GBD 2016 Stroke Collaborators.).

Defining reliable tools for early prediction of outcomes is the main target for physicians to guide care decisions in patients with ABI (Avesani et al., 2011; Steyeberg et al., 2008). In the last ten years, the application of machine learning (ML) algorithms is rapidly increasing in this field of study (Liuzzi et al., 2022), but with a poor translation to clinical practice. This is basically dependent on the uncertainty about the intrinsically ‘‘black-box’’ nature of this approach and the lack of clear advantages with respect to traditional approaches (i.e linear regression models)(Quinn et al., 2022). Indeed, a recent narrative review showed that ML algorithms do not outperform traditional regression approaches for outcome prediction in brain injury (Cerasa et al., 2022).

The current trend to shorten the length of stay in rehabilitation units, as well as the increasing new cases and demand for efficiency in care, imply that knowledge about the prognosis for the outcome is crucial to optimize the management of ABI patients in the first months after brain injury (Maas et al., 2022). For this reason, the definition of a new statistical approach able to better describe the dynamics of clinical changes occurring during the entire neurorehabilitation period is the greatest unmet need for clinical practice. A new computational approach should provide individual predictions of the clinical course depending on solid predictors coupled with every clinical intervention occurring during the neurorehabilitation stay. A dynamic model accounting for the nonlinearity of clinical evolution characterized by high variability in clinical signs, symptoms, and treatments would be the best preferred approach.

On this basis, this multicenter study is aimed to develop a new dynamic model applied to ABI patients characterized by high variability in clinical signs, symptoms, and treatments during the inpatient intensive rehabilitation care period. The proposed approach combines a principal component analysis (PCA) for dimension reduction (allowing to capture of the maximum information reducing the problem dimensionality) and a nonlinear mathematical modeling for describing the evolution of the clinical course in terms of the resulting new PCA dimensions. The nonlinear mathematical formulation employed is a Michaelis-Menten (MM) equation (Michaelis & Menten, 1913) most used in enzyme reactions (Van Slyke & Cullen, 1914). The equation describes the reaction velocity of a system, as a function of a substrate S concentration, where the substrate binds reversibly to enzyme E to create a product P. The MM model can be effective in predicting different saturated dynamics in one independent variable and two parameters and, for the first time, it has been used in this neurological domain for describing the trajectories and predict the clinical evolution of ABI patients.

Clinical Data

As shown in the previous work (Table 1, [Lucca et al., 2021)]), 156 ABI patients (99 males, mean age: 53.7y [range: 22–89]) were enrolled in this study. The mean length of stay in the Intensive Care Unit (ICU) was 48.1 days (range: 10–144), whereas during Intensive Rehabilitation Unit (IRU), ABI patients were discharged after an average period of 167.1 days (range: 46–600). In the entire cohort, most patients were characterized by vascular pathology (n°79; 50.6%), followed by traumatic (n°56; 35.9%), anoxic (n°15; 9.6%) or other pathologies (i.e infections/tumour; n° 6; 3.8%) Twelve patients were discharged before the end of the study (T1 phase) and followed up by weekly telephone interviews.

At discharge, 2.8% were dead, 61.1% had a full recovery of consciousness and 36.1% remained in VS/UWS or MCS. The median GOS value was 3 ranging from 1 to 5. At admission CRS-r (T0 median value: 9 [1–23]; at discharge: 21 [3–23]) and RLAS scores (T0 median value: 3 [1–27]; at discharge: 5 [1–8]) increased, whereas ERBI values decreased (T0 median value: -175 [-50 /-325]; at discharge: -105 [0/-175]) with respect to admission. PSH signs (including possible/probable) showed an evident decrease during the IRU period (see supplementary materials Table 1).

Patient characteristics according to the GOS outcome

Tables 1 and 2 report baseline information and follow-up evaluations according to the GOS binary categorization (Positive/Negative). One hundred and sixteen (81.1%) out of 143 patients were classified with Negative outcome at discharge. Age and ICU days were significantly different between the two groups (p < 0.001 and p = 0.008; respectively). Patients with Negative GOS were older and showed a larger number of days spent in ICU with respect to patients with Positive outcome. The distribution of males and females in the two GOS groups were similar, 35.3% of patients with Negative outcome were females compared to 44.4% of patients with Positive outcome (P = 0.51). Clinical features and use of devices that resulted to be significantly different at baseline were CRS-r (P < 0.001), RLAS (P < 0.001), PSH-AM-CFS (P = 0.04), type of diagnosis (P < 0.001), tracheostomy (P < 0.001), respiratory function (P = 0.03), feeding modality (P < 0.001) and fractures (P = 0.01). Patients with negative GOS presented worse clinical conditions in relationship to the above characteristics.

Table 1

Means and standard deviations of continuous demographic and clinical predictors at baseline (T0), after four months from the event (T1) and at discharge (T2) according to the GOS binary categorization (Positive/Negative)
Variable	Time	Negative GOS	Positive GOS
Variable	Time	Mean ± SD	Mean ± SD
Age	T0	55.8 ± 17.5	41.7 ± 15.9	< 0.001
ICU days	T0	47.8 ± 20.4	34.6 ± 18.1	0.008
CRS-R	T0	10.74 ± 6.72	17.48 ± 5.9	< 0.001
	T1	14.79 ± 7.14	21.78 ± 2.86	< 0.001
	T2	16.38 ± 7.03	22.71 ± 0.81	< 0.001
RLAS	T0	2.92 ± 1.16	4.26 ± 1.32	< 0.001
	T1	3.77 ± 1.45	6.26 ± 1.1	< 0.001
	T2	4.27 ± 1.27	6.85 ± 1.49	< 0.001
ERBI-A	T0	-178.63 ± 32.52	-175 ± 68.3	0.79
	T1	-146.96 ± 69.45	-43.42 ± 55.2	< 0.001
	T2	-109.15 ± 70.23	-9.26 ± 23.15	< 0.001
ERBI-B	T0	0.17 ± 1.13	2.59 ± 12.5	0.32
	T1	2.61 ± 20.68	61.05 ± 33.2	< 0.001
	T2	9.02 ± 17.63	86.67 ± 20.8	< 0.001
PSH-AM-CFS	T0	3.47± 2.98	2.48 ± 1.93	0.04
	T1	2.23 ± 2.62	1.26 ± 1.7	0.05
	T2	1.36 ± 2.12	0.52 ± 1.01	< 0.001
PSH-AM-DLT	T0	2.74 ± 2.46	2.56 ± 2.24	0.71
	T1	2.2 ± 2.17	1.58 ± 0.96	0.05
	T2	1.65 ± 1.54	1.15 ± 0.46	< 0.001
PSH-AM	T0	6.2 ± 5.18	4.96 ± 4.05	0.18
	T1	4.42 ± 4.5	2.84 ± 2.3	0.03
	T2	3.02 ± 3.49	1.67 ± 1.33	< 0.001

Table 2

Absolute frequencies and percentages of categorical demographic and clinical predictors at baseline (T0), after four months from the event (T1) and at discharge (T2) according to the GOS binary categorization (Positive/Negative).
Variables		Groups	Negative GOS		Positive GOS		P-value
Variables		Groups	N	%	N	%	P-value
Sex	T0	Overall: F: M:	116 41 75	81.12 77.36 83.33	27 12 15	18.88 22.64 16.67	0.51
Etiology	T0	Overall: Anoxic: Infectious: Benign neoplasm: TBI: Vascular:	116 13 2 4 35 62	81.12 92.86 100 100 68.63 86.11	27 1 0 0 16 10	18.88 7.14 0 0 31.37 13.89	0.06
PSH crisis in resuscitation	T0	Overall: Presence: Absence:	116 10 106	81.12 66.67 82.81	27 5 22	18.88 33.33 17.19	0.24
Diagnosis	T0	Overall: Emerged from MCS: MCS: VS/UWS:	116 23 43 50	81.12 60.53 82.69 94.34	27 15 9 3	18.88 39.47 17.31 5.66	< 0.001
	T1	Overall: Death: Discharged: Emerged from MCS: MCS: VS/UWS:	114 2 6 51 29 26	82.01 100 50 73.91 96.67 100	25 0 6 18 1 0	17.99 0 50 26.09 3.33 0	< 0.001
	T2	Overall: Death: Emerged from MCS: MCS: VS/UWS:	116 4 66 30 16	81.12 100 70.97 100 100	27 0 27 0 0	18.88 0 29.04 0 0	< 0.001
Tracheostomy	T0	Overall: Presence: Absence:	116 109 7	81.12 85.83 43.75	27 18 9	18.88 14.17 56.25	< 0.001
	T1	Overall: Presence: Absence:	106 68 38	84.8 97.14 69.09	19 2 17	15.2 2.86 30.91	< 0.001
	T2	Overall: Presence: Absence:	112 48 64	80.58 100 70.33	27 0 27	19.42 0 29.67	< 0.001
Respiratory function	T0	Overall: Assisted Autonomous Autonomous + o2	116 4 41 71	81.12 80 70.69 88.75	27 1 17 9	18.88 20 29.31 11.25	0.03
	T1	Overall: Autonomous Autonomous + o2	105 80 25	84.68 80.81 100	19 19 0	15.32 19.19 0	0.01
	T2	Overall: Assisted Autonomous Autonomous + o2	112 1 90 21	80.58 100 76.92 100	27 0 27 0	19.42 0 23.07 0	0.01
Feeding modality	T0	Overall: Oral: Parenteral: PEG: SNG:	116 3 1 64 48	81.12 30 33.33 90.14 81.36	27 7 2 7 11	18.88 70 66.67 9.86 18.64	< 0.001
	T1	Overall: Oral: PEG: SNG:	106 33 63 10	84.8 66 96.92 100	19 17 2 0	15.2 34 3.08 0	< 0.001
	T2	Overall: Oral: Parenteral: PEG: SNG:	112 46 0 63 3	80.58 63.89 0 100 100	27 26 1 0 0	19.42 36.11 100 0 0	< 0.001
Urinary catheter	T0	Overall: Presence: Absence:	116 113 3	81.12 81.29 75	27 26 1	18.88 18.71 25	1
	T1	Overall: Presence: Absence:	106 55 51	84.8 98.21 73.91	19 1 18	15.2 1.79 26.09	< 0.001
	T2	Overall: Presence: Absence:	112 44 68	80.58 100 71.58	27 0 27	19.42 0 28.42	< 0.001
Pressure sores	T0	Overall: Presence: Absence:	116 45 71	81.12 88.23 77.17	27 6 21	18.88 11.76 22.83	0.16
	T1	Overall: Presence: Absence:	106 31 75	84.8 96.88 80.65	19 1 18	15.2 3.12 19.35	0.05
	T2	Overall: Presence: Absence:	112 16 96	80.58 94.12 78.69	27 1 26	19.42 5.88 21.31	0.05
Fractures	T0	Overall: Presence: Absence:	116 22 94	81.12 64.71 86.24	27 12 15	18.88 35.29 13.76	0.01
Fractures	T1	Overall: Presence: Absence:	106 7 99	84.8 77.78 85.34	19 2 17	15.2 22.22 14.66	0.9
Severe spasticity	T0	Overall: Presence: Absence:	116 22 94	81.12 95.65 78.33	27 1 26	18.88 4.35 21.67	0.1
	T1	Overall: Presence: Absence:	106 35 71	84.8 94.59 80.68	19 2 17	15.2 5.41 19.32	0.09
	T2	Overall: Presence: Absence:	112 35 77	80.58 97.22 74.76	27 1 26	19.42 2.78 25.24	0.09
POA	T0	Overall: Presence: Absence:	116 3 113	81.12 75 81.29	27 1 26	18.88 25 18.71	1
	T1	Overall: Presence: Absence:	105 12 93	84.68 80 85.32	19 3 16	15.32 20 14.68	0.88
	T2	Overall: Presence: Absence:	112 7 105	80.58 70 81.4	27 3 24	19.42 30 18.6	0.88
Craniectomy	T0	Overall: Presence: Absence:	116 25 91	81.12 78.13 81.98	27 7 20	18.88 21.87 18.02	0.81
	T1	Overall: Presence: Absence:	106 18 88	84.8 78.26 86.27	19 5 14	15.2 21.74 13.73	0.52
	T2	Overall: Presence: Absence:	112 12 100	80.58 100 78.74	27 0 27	19.42 0 21.26	0.52
Normal-pressure hydrocephalus	T0	Overall: Presence: Absence:	116 13 103	81.12 92.86 79.84	27 1 26	18.88 7.14 20.16	0.41
	T1	Overall: Presence: Absence:	106 24 82	84.8 92.31 82.83	19 2 17	15.2 7.69 17.17	0.37
	T2	Overall: Presence: Absence:	112 26 86	80.58 96.3 76.79	27 1 26	19.42 3.7 23.21	0.37

Principal Component Analysis

“Severe spasticity”, “pressure sores”, “POA”, “craniectomy” and “normal-pressure hydrocephalus” variables were not considered in the PCA as they did not result significant in a univariable analysis assessing possible differences between GOS groups (Positive/Negative).

The first two components from the PCA, performed with the FAMD method, explaining 55.5% of the total variance (first component, Dimension 1: 39.5%; second component, Dimension 2: 16.0%). Dimension 1, which explained most of the data variability, was considered in the subsequent analyses. Figure 1 shows the contribution of each variable on the first dimension. Dimension 1 was mainly determined by variables Feeding modality, RLAS, ERBI-A, presence of Tracheostomy, CRS-r, and ERBI-B. Dimension 2 was excluded by the modeling approach, as its trend over time did not distinguish between patients with Positive and Negative outcome.

Figure 2 shows the scores that individuals present on each dimension at the three evaluation time points (baseline, T0; after four months from the event, T1; at discharge, T2). Patients were identified according to their outcome (Positive: blue dots, Negative: red dots). Figure shows a general increase of the scores over time along the Dimension 1, with a larger increase observed for patients with Positive outcome. Patients in the two groups show overlapped scores along Dimension 2.

Parameter estimation of Michaelis Menten models

Patient scores on Dimension 1 were represented in function of time expressed in days from the event (Fig. 3). The time course of the Dimension 1 scores was then fitted with a Michaelis-Menten equation on the Training sample separately for individuals with Positive and Negative outcome. Parameters in the Positive model were identified with letter “p”; letter “n” was used for parameters in the Negative model. The estimated values were: a_p= -4.73 (SE = 2.47); m_p= 11.56 (SE = 1.49); K_p= 44.87 (SE = 33.73); a_n= -3.01 (SE = 1.02); m_n= 4.61 (SE = 0.67); K_n= 91.99 (SE = 91.11). While parameters a and K did not result significantly different between the two groups (P = 0.52 and P = 0.63, respectively), parameter m, indicating the maximum value attainable for very large time, was significantly different (P = 0.0001), with a greater value for the Positive group. At baseline, for time equal to zero, individuals with positive and negative outcomes present comparable values for the first dimension. With passing of the days individuals with positive outcomes show gradually increasing values. Parameter K in the Positive group is on average half of that estimated for individuals with negative outcomes, meaning that their improvement is faster. Figure 3, panel C, shows the observed scores and the model fitting, along with the corresponding 95% confidence intervals (dashed lines), for both the groups.

Confidence intervals were used to define cut-offs with the aim of identifying the possible outcome of a patient, based on their clinical assessment over time. Since at baseline (a variable time in days from the event to discharge from ICU and admission in the Intensive Rehabilitation Unit, IRU) the values observed in the two groups are mostly overlapping, a more stringent criterion was used to predict the outcome of a new enrolled patient. From Fig. 3, panel C, it appears that the interval time where observations are mostly superimposed and confidence intervals overlap, ranges from 0 to 86 days (which corresponds to the point where the superior limit of the negative 95% CI and the inferior limit of the positive 95% CI intersect). In this range of days, two cut-offs are used to predict the outcome of a patient. The first cut-off (Threshold 1) is the set of points that lie on the upper limit of the 95% CI estimated for the Negative model and is used to identify subjects who will have a positive outcome if they have a Dimension 1 score higher than it. The second cut-off (Threshold 2) is the set of points that constitute the lower limit of the 95% CI estimated for the positive model and is used to identify subjects who will have a negative outcome if they have a Dimension 1 score smaller than it. Beyond the 86th day, the cut off becomes the lower limit of the 95% CI estimated for the positive model (Threshold 3): individuals with Dimension 1 scores above this curve are expected to have a positive outcome, conversely, they are expected to have a negative outcome if their scores are below the curve. Given the chosen criteria, not all subjects can be classified if evaluated in the initial range (time < 86 days). Due to the superimposition of the two CIs, a certain number of patients (13/42) belongs to a “shadow” area (patients with scores on the first dimension between Threshold 1 and Threshold 2). Subsequent evaluations will allow the prediction of the outcome for these subjects.

Results on Validation model

Table 3 reports the Classification Ability (number of subjects not belonging to the “shadow” area), the Accuracy, Sensitivity and Specificity of the proposed approach in the Validation sample. When patients were classified only according to the first clinical assessment performed at the IRU admission, 29 subjects were classified with an Accuracy of 89.7%, Sensitivity and Specificity of 100% and 57.1%, respectively.

Table 3

Accuracy and classification ability of the approach adopted.
	Time points	Classification Ability: number of classified individuals	Accuracy	^*Sensitivity	^*Specificity
Validation sample N = 42	T0	29/42 (69%)	89.7%	100%	57.1%
	T0 + T1	^#40/42 (95.2%)	85%	90.6%	62.5%
	T1	^#40/42 (95.2%)	82.5%	87.5%	62.5%

* Computed with respect to classified subjects

# The two patients who were not classified belonged to the “shadow” area for the first evaluation (time < 86 days) and presented with missing values after four months from the event (T1)

The subsequent evaluation performed four months from the event provided a classification rate of 95.2% (two individuals not classified at IRU admission presented missing values at T1, so they remained unclassified) with an Accuracy, Sensitivity and Specificity of 85%, 90.6% and 62.5% respectively. If only clinical assessments at T1 were considered, the values of the indices were: 82.5%, 87.5% and 62.5%. Figure 4 shows the observed scores of Dimension 1 on the Validation sample, along with the 95% CIs computed on the Training sample and used as criteria for the outcome prediction.

The approach proposed in this study aims to identify the most determinant predictors in the characterization of the clinical evolution of ABI patients and to determine their outcome trajectories. For this purpose, we combined a PCA, and a mathematical model aimed at early predicting the GOS score, based on the estimated clinical evolution of every single patient.

The Michaelis Menten model was used to characterize the trend of the first PCA dimension (represented by feeding modality, RLAS, ERBI_A, Tracheostomy, CRS-r and ERBI-B) useful to predict the most plausible outcome (in terms of positive or negative GOS) of ABI patients upon entry into the IRU. The proposed methodology allows us to identify the possible evolution of the patient based on the results from the first to the subsequent clinical evaluation. Despite the greater accuracy and sensitivity (89.7% and 100%, respectively) obtained from the entrance visit to the IRU (visit at T0), for the 31% of patients it was not possible to predict what their presumed outcome would be. The subsequent assessment (4-months post-event, visit at T1), added to the first, however, allowed physicians to assign all patients to one of the two identified evolutions with an accuracy of 85%, a sensitivity of 90.6% and an increased specificity of 62.5% compared to 57.1% obtained with the only evaluation at T0. The only two patients for whom it was not feasible to establish the possible outcome had missing values at T1. The reported performance is in line with several previous studies using machine learning algorithms to predict outcomes in acquired brain injury (Cerasa et al., 2022). In a previous study (Bruschetta et al., 2022) different ML approaches were compared with the classic Linear Model (LM) to predict the final evolution of TBI patients according to 2 and 4 GOS classes based on the clinical assessment at T0. The accuracy obtained in the case of a binary outcome was similar to that obtained when the classical linear model is used. The LM approach however is underperforming in terms of sensitivity compared to our results (66.7% versus 90.6%). The difference could be due to the additional information at T1 which determines the membership trend.

The evolution over time of impairments, disabilities and recovery after ABI is characterized by a large amount of diversity (Campbell, 2004). Some patients show any improvement even in the long term, whereas other patients fully recover within hours or days post stroke. Specific demographical (age) and clinical factors (i.e, initial severity of disability, radiological markers; comorbidities) as well as the extent of improvement observed within the first days/weeks post-injury are considered the main indicators of the outcome at six-months (Kwakkel & Kollen, 2013), although their magnitude in influencing predictive models change as a function of statistical approach (Cerasa et al., 2022). In our study, we demonstrated that the ensemble of feeding modality, RLAS, ERBI_A, Tracheostomy, CRS-r and ERBI-B is able to define the time course of clinical evolution in ABI patients who will be discharged with a Positive or Negative outcome, reaching the best classification of the subjects and a high accuracy rate starting from the 86th days post-injury. Before the 86th day, the classification rate is low despite a very high accuracy. Even though the outcome of ABI patients is heterogeneous and individual recovery patterns differ, clear mathematical regularities (i.e. logistics and sigmoidal) have been found in these nonlinear patterns of recovery, making the outcome in terms of body functions and activities highly predictable (Koyama et al., 2005; Heller et al., 1987; Zarahn et al., 2011). In a recent study, van der Vliet et al., (2020), developed a longitudinal mixture model of motor recovery that describes the time course of the Fugl–Meyer motor upper extremity (FM-UE) scale after a first-ever ischemic stroke. Analyzing data from 412 first-ever ischemic stroke patients, they identified the FM-UE trajectories of 5 subgroups, organized into 3 clinically relevant clusters of poor, moderate, and good motor recovery. Their model can provide a satisfactory prognosis to patients as early as 1-week poststroke. There are some methodological differences with respect to our study. First, we analyzed data from a large and heterogenous group of acquired brain injuries including vascular (50.6%), traumatic (35.9%), anoxic (9.6%) or other pathologies (3.8%). Next, our model predicts the final outcome (GOS) on the basis of measurements of different clinical features: the level of consciousness (measured by CRS-r); the level of cognitive functioning (measured by the RLAS); the level of clinical complexity and disability (measured by the ERBI); the rate of paroxysmal hyperactivity of the sympathetic system (evaluated by the PSH-AM scale). The outcome is therefore estimated by relying on a complete clinical picture of the patient. Conversely, Vliet et al. (2020) predicted motor recovery, measured only by the FM-UE scale, and no assessment of the reliability of the model was carried out on an external cohort of subjects.

In conclusion, our combined PCA-MM approach provides an excellent prediction of patient outcome, together with the relative trend over time which demonstrates that at admission, a specific ensemble of clinical indicators is useful to predict the outcome at discharge, but a large part of dataset (n° 31) remained undefined. The prediction of time course improves at 86 days and any new information after this time point should contribute to the understanding of functional recovery patterns in the first 3 months after injury. In future work, we plan to perform further external validations on other datasets for capturing dynamic changes in prognosis during intensive care courses extending the current model with new objective predictors, such as neuroimaging data (EEG, PET, fMRI).

Patients

This study was a secondary analysis of data from a prospective multicenter observational study (Lucca et al., 2021), where machine learning algorithms were employed to evaluate the prognostic value of paroxysmal sympathetic hyperactivity (PSH), together with other well-known clinical features, on the outcome of ABI patients. Details of the main study are provided elsewhere (Lucca et al., 2021), but, in brief, eight neurorehabilitation units screened consecutively and admitted ABI patients with DoC from March 2018 to January 2019. Patients fulfilling the following inclusion criteria were enrolled: i) adult age (≥ 18 years); ii) clinical diagnosis of vegetative state/unresponsive wakefulness syndrome (VS/UWS)( Laureys et al., 2010) or in the minimally conscious state (MCS, i.e. patients exhibiting minimal but definitive behavioral evidence of self-awareness or environmental awareness), and patients emerged from MCS (reliable demonstration of either interactive communication or functional object use), based on standardized clinical criteria of Coma Recovery Scale – revised (CRS-r=)[8–10]; iii) etiology: traumatic, vascular (i.e., ischemic or hemorrhagic), anoxic and other (i.e., infective or benign tumor); iv) time post-injury ≤ 3 months calculated at admission, v) first admission in intensive rehabilitation unit (IRU). Patients with mixed etiology (e.g. anoxic and vascular), previous history of acquired brain injury, psychiatric or neurodegenerative diseases, were excluded from the study.

The present study was conducted after approval of the Institutional Ethics Committee of the coordinator center (S.Anna Institute, Protocol Register no. 244, 10/24/2017). The surrogate decision-makers of the patients enrolled in the study provided their written informed consent. The original forms were collected and stored at each participant center. All the experimental procedures were conducted according to the policies and ethical principles of the Declaration of Helsinki. Data supporting the findings of this study are included in this published article.

Procedure and Outcome Measures

This was a multicenter prospective observational study. All patients underwent three specific clinical evaluations: at baseline (T0), after 4 months from the event (T1) and at discharge (T2). The examination was performed by neurologists, physiatrists or neuropsychologists with experience in disorders of consciousness who were blind to any other result. At study entry all participating centres collected: a) demographic information, b) etiology, c) clinical features and presence of devices (assessment of respiratory function, tracheostomy, feeding modality, urinary catheter, pressure sores, severe spasticity, fractures, craniectomy, normal-pressure hydrocephalus). As predictive measures, we also considered the: i) level of consciousness as measured with the Italian version of the CRS-r (Estraneo et al., 2015); ii) the level of cognitive functioning as measured by the Rancho Los Amigos Scale (RLAS)(Hagen et al., 1972); iii) level of clinical complexity and disability as measured by the Early Rehabilitation Barthel Index (ERBI)( Scho¨nle, 1995). The rate of paroxysmal hyperactivity of the sympathetic system was recorded using the PSH assessment measure (PSH-AM) scale (Lucca et al., 2019). The GOS (Jennett et al., 1981) was used as the main outcome measure.

Longitudinal Michaelis-Menten Model

The GOS score, originally coded into 5 levels (death = 1; vegetative = 2; severe disability = 3; moderate disability = 4; good recovery = 5) was categorized into a binary variable: “Positive outcome” (GOS ≥ 4) and “Negative outcome” (GOS < 4).

For the analysis purpose, the entire sample was split into a Training (70% of cases) and a Validation (30% of cases) sample, by using the data-splitting method balanced according to the GOS variable. The Training sample was used as the fitting population for model development. The performance of the model was tested in the Validation sample.

The set-up procedure employed, as a first step, a Principal Component Analysis (PCA) in the Training sample with the aim of reducing the dimensionality of the dataset. A preliminary univariable analysis was performed to identify predictors significantly associated with GOS outcome at each evaluation time point. Given the presence of both continuous and categorical variables, a Factorial Analysis of Mixed Data (FAMD) approach was used. Continuous variables were scaled to unit variance and the categorical variables were transformed into a set of disjunctive variables (0 in absence of the modality; 1 in presence of the modality) and then scaled. The first two principal components, explaining most of the dataset variability, were then interpreted based on the scores that each predicting factor exhibited on each component. The variables entered the PCA with all the available clinical evaluations (baseline, T0; after 4 months from the event, T1; at discharge, T2). This allowed us to explore the evolution over time of the PCA dimensions. The trend of the dimensions was evaluated according to the Positive/Negative outcome. Subjects were represented on a two-dimensional space according to their values on the first 2 PCA components. The first component was then evaluated over time and fitted with a Michaelis-Menten function, which exhibits a saturated trend with increasing dependent variable ${x}$:

$${f}\left({x}\right)={a}+\frac{{m}{*}{x}}{{K}+{x}}$$

where parameter ${a}$ is the value of the dependent variable (Dimension 1) at time 0 (${x}=0$); ${m}$ is the increment that the dependent variable takes to achieve its maximum value (${a}+{m}$) as x goes to infinity; K, the Michaelis constant, is the value of variable x at which the function reaches the value ${a}+{m}/2$. In the function ${f}\left({x}\right)$ the independent variable ${x}$ represents the time.

Model Fitting

The parameter vector (${a},{m},{K}$) was estimated by minimizing the sum of squared residuals by using the adaptive non-linear least squares algorithm. The model was adapted separately to observations from individuals with Positive and Negative outcome with the aim of assessing two possible different evolutions for these two subgroups of patients.

Parameter estimates were identified for each subgroup (Positive/Negative outcome), average trends and the corresponding 95% confidence intervals were then obtained. Confidence intervals for the fitted values were computed by means of a Monte Carlo approach by drawing 10,000 samples from multinormal distributions on the model parameters.

Model-Validation

The coordinates on the PCA components (built on the Training dataset) were predicted on the Validation sample. Confidence intervals for the fitted values of the two Michalis -Mentel models (Positive/Negative GOS outcome) on the Training dataset were used to identify thresholds for predicting the evolution over time of the individuals in the Validation dataset. Thresholds were established at different time points and their predictive ability in determining a Positive/Negative GOS outcome was assessed by computing Accuracy, Sensitivity and Specificity.

Statistical Analysis

Means and standard deviations and counts and percentages were used to summarize continuous and categorical variables, respectively. Chi-squared test, or Fisher exact test when appropriate, was used to study the possible association between factorial variables and the Positive/Negative GOS outcome at each time point. For continuous variables a two-independent samples t-test was used to assess possible differences between the two groups. All statistical analyses were performed with R package version 4.1.3.

GOS: Glasgow Outcome Scale

CRS-r: Coma Recovery Scale-revised

ERBI: Early Rehabilitation Barthel Index

ICU: intensive care unit

IRU: intensive rehabilitation unit

RLAS: Rancho Los Amigos Scale

MCS: minimally conscious state

ML: Machine learning

MM: Michaelis-Menten

PSH-AM: Paroxysmal Sympathetic Hyperactivity–Assessment Measure

ABI: acquired brain injury

VS/UWS: vegetative state/unresponsive wakefulness syndrome

ACKNOWLEDGEMENTS

The study was carried out at the S. Anna Institute with support from the institute. This study has been devised by the Special Interest Group on acquired brain injury and DoC (ABI & DoC SIG) of the Italian Society of Neurological Rehabilitation (SIRN, see http://www.sirn.net).

AUTHOR DISCLOSURE STATEMENT

The authors declare no conflict of interest.

AUTHOR CONTRIBUTIONS

Simona Panunzi: Data Analysis; writing—review and editing, meethdology

Lucia Francesca Lucca: Conceptualization; investigation

Antonio De Tanti: data curation,

Francesca Cava: data curation,

Annamaria Romoli: data curation,

Rita Formisano: data curation,

Federico Scarponi: data curation,

Anna Estraneo: data curation,

Diana Frattini: data curation,

Paolo Tonin: resources and project administration

Ilaria Piergentilli: Data Analysis; methdology

Giovanni Pioggia: project administration

Andrea De Gaetano: project administration

Antonio Cerasa: Conceptualization and writing—review and editing

FUNDING

Nothing to declare

DATA AVAILABILITY STATEMENT

the dataset has been uploaded as supplementary material

STUDY REGISTRATION

This study was approved by the Ethical standards Committee of the coordinator center as a multicenter observational study (Protocol Register n. 244, 10/24/2017)

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No competing interests reported.

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Modeling the outcome trajectories in patients with acquired brain injury: a non-linear dynamic evolution approach

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Abstract

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Introduction

Results

Clinical Data

Patient characteristics according to the GOS outcome

Principal Component Analysis

Parameter estimation of Michaelis Menten models

Results on Validation model

* Computed with respect to classified subjects

Discussion

Methods

Patients

Procedure and Outcome Measures

Longitudinal Michaelis-Menten Model

Model Fitting

Model-Validation

Statistical Analysis

Glossary

Declarations

References

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Supplementary Files

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