When planning an implant-supported restoration, the dentist is faced with surgical and prosthetic technical issues as well as the patient's expectations. Many patients wish an immediate solution to an edentulous condition. This may be especially true in the esthetic zone, and that zone is determined by the patient. The dentist may consider when it is feasible to load the supporting implants with definitive or provisional prosthetics. In this work, many parameters were theoretically assessed for inclusion: bone density, cortical thickness, insertion torque, parafunction, bite load capacity, number of implants under load, implant/crown ratio, implant diameter, and length. After assessment, the most influential parameters were selected. An iteration, using patient age, implant diameter, bite load capacity, and cortical thickness, is now presented to aid the implant dentist in determining the feasibility for immediate functional loading of a just-placed dental implant in a healed site. Extensive testing is required to develop this concept. According to this iteration, most immediate functional loaded implants would fail. A future refined and definitive formula may enable the clinician to safely and immediately functionally load an implant with a definitive prosthesis. For access to the applet, please go to https://implantloading.shinyapps.io/shiny_app/.

## Introduction

There is patient demand for immediate functional loading of just-placed dental implants. There is some evidence that immediate occlusal loading of dental implants is a viable option in some clinical situations.^{1,2 } Clinical factors such as bone quality, implant dimensions, forces of occlusion, prosthetic design, and parafunction have been shown to affect immediate functional loading. These factors may affect the outcome of an immediately loaded dental implant, and some are more important than others. Many of these parameters are measurable with numerical values. There is a need for a method to relate measurable parameters to predict the feasibility for immediate implant functional loading.

Previous work has attempted to describe jaw functioning from a mathematical perspective.^{2 } Reducing clinical functional parameters to mathematics may make treatment planning more predictable, complications fewer, and outcome success rates higher.

Implants placed in immediate functional conditions have been successful.^{3–6 } The parameters for successful immediate functional loading appear to be related to bone quality, having a patient with lesser jaw force capability, the number of implants bearing the occlusal load, a longer implant length and wider diameter, and implants with a rough surface .^{3–6 }

In vitro studies and the clinical experiences of clinicians have determined that bone can successfully resist occlusal forces transmitted by just-placed implants.^{1–6 } Single and multiple implants can be immediately loaded in certain circumstances.^{1–6 } Multiple implants that are splinted distribute the occlusal forces over multiple implants and lessen the per-square-millimeter force transferred to the supporting bone.

Implants placed in high-density bone can support an immediate occlusal load.^{2 } Dense bone is able to keep the implant immobile while under occlusal loads.

Parafunction, intuitively, appears to be a factor detrimental to immediately loading. Implant-supported fixed partial dentures may be intentionally placed about 100 μm shy of occlusal contact. This is an accommodation for natural tooth intrusion so that the implant prosthesis does not bear the full occlusal load during maximum intercuspation and parafunction. Natural teeth do intrude slightly under load, but the intrusion varies from tooth to tooth.^{7 } To avoid a parafunctional overload, 100-μm occlusal relief may be an appropriate dimension.

Forces generated by patients and delivered to the implant/bone–supported prostheses can differ. The individual patient force capability can be determined.^{8 } A patient's jaw force capability can be measured using a transducer measuring device or silicone imprinting, so that a numeric parameter can be identified.^{9,10 } The generated jaw force is related to the position in the arch. Implants placed in more posterior sites are subjected to greater force than those placed in more anterior areas. A greater force may dislodge a just-placed implant.^{9 } There are several devices available for measuring jaw force.^{11 }

Immediate implant functional loading is favorable for maxillary and mandibular fixed splinted complete arch reconstructions and removable mandibular overdentures.^{12 } In addition, for single-tooth reconstructions in the esthetic zone, including premolars and short-span fixed partial dentures, there is a high survival rate but an increased risk for implant loss.^{12 }

Immediate loading with occlusal contacts may not have a greater risk for implant loss than immediate loading without occlusal contacts. There may be less marginal bone loss around immediately loaded implants as compared with delayed loaded implants.^{12 }

There is a higher patient satisfaction with immediate loading, but there are increased complexities in treatment planning. Provisional restorations may be appropriate with an insertion seating torque of greater than 30 Ncm with appropriate prosthetic positioning.^{12 } Immediate loading should be avoided in patients with bruxism and clenching.^{12 } Immediate implant loading requires high patient compliance. Adequate width of the attached mucosa around immediately loaded implants is crucial.^{12 } Nonetheless, there is a greater risk for failure with immediate loading as compared with delayed loading. Grafted sites may pose an even greater risk for failure.^{12 }

The object of this effort is to present an iteration that may theoretically predict the feasibility of immediate implant functional loading.

## Modeling

The basis of the model comes from ANSYS finite element analyses, which were conducted over the years 2013 to 2015, simulating the stresses placed on the surface of the implant. The analysis concluded that for an average jawbone with 1 to 6 implants, the approximate maximum load allowed was at 30° and 45° to simulate off-axial occlusal loading. In addition, the maximum stresses for different cortical bone thicknesses to refine the approximation of bone strength and loading limitations were investigated.

The loading properties of each patient's bone are unique (Young's modulus, bone density, dimensions). The results of the ANSYS analysis (maximum stresses) was able to deliver a model that considered cortical bone thickness, bone quality, and implant diameter.^{13 } The model was further refined to apply immediate functional loading upon newly placed nonosseointegrated dental implants and to predict the feasibility of immediate occlusal loading.

The model must maximize its predictive power while at the same time minimize the amount of input parameters. Minimizing the input parameters increases computational speed and decreases the amount of data that need to be collected for each patient by the clinician. This allows the implant dentist to focus on the most appropriate data for the immediate loading decision-making process.

## Limitations

The outcome data from this model cannot be directly tested on patients without further study because of the simplifications made in the finite element analysis (FEA) model.^{14–16 }

A verification process can be instituted consisting of FEA bone models with data input from actual patient mandibles. The outputs of the more complex patient-adjacent FEA simulations were compared with the simplified mandible models to ensure a small amount of error existed between the 2 models. Cone-beam computerized tomography (CBCT) measurements were used to build the FEA simulations in the verification process. The FEA model seeks to approximate stress, while the mathematical model predicts the stress on the jaw based on selected input parameters.^{16 } The model was trained on data yielded from the ANSYS simulations.

A comparison was made of the stress values from the FEA simulations using the simplified bone model and the output of the mathematical model to judge the accuracy of the mathematical model.^{16 }

### Development of the model

To protect the anonymity of patients, the Digital Imaging and Communications in Medicine (DICOM) standard was used. The DICOM standard makes anonymization a simple process.^{17 }

A data set was generated based on the range of values for the cortical bone thickness of a given edentulous site and the implant diameter. The cortical bone thickness was found to range from 0.25 mm to 3 mm, and the implant diameter ranged from 3.2 mm to 5.7 mm.

One thousand random points were then generated, and the stress was calculated in megapascals (MPa). Table 1 represents a subset of the points used in the analysis.

Next, the 2 parameters were plotted against the stress to determine who each parameter affects the predicted stress. The plot in Figure 1 shows the relationship between the diameter and the predicted stress of the previous model:

The regression between the predicted stress and the implant diameter had a coefficient of determination of 0.9976; therefore, variations in the implant diameter can explain 99.76% of the variation in the predicted stress.^{18,19 } Theoretically, a coefficient of determination of 1 would demonstrate a perfect linear fit. The value here is less than 1% away from such a fit. Without considering the cortical bone thickness, the implant diameter alone can explain the output of the model. Next, the image in Figure 2 shows the relationship between the predicted stress and the cortical bone thickness.^{20,21 }

The coefficient of determination in this case was 0.00044, which means that the variations in cortical bone thickness obviate the 0.044% of the variation in the predicted stress of this model.

The values of the cortical bone thickness varied between 6 and 2 times as small as those for the implant diameter.^{20,21 } In addition, the coefficient by which the implant diameter was multiplied in the model was a little more than double that multiplied by the cortical bone thickness. Thus, the implant diameter was weighted up to 8 times more heavily than the cortical bone thickness was. However, consideration must be given to extremely thick cortical bone and another important parameter, individual patient bite force capability. This creates a better model.

This more inclusive model used the capability in the programming language R to take the data collected from the simulations and produce a relevant mathematical model. The framework leveraged within R was the generalized additive model, which takes a set of predictive parameters and regresses the values toward the dependent variable to form an output based on a defined type of error distribution. Once enough data points were aggregated, the first version of the model was made, which included the Young's modulus of cortical bone, the bite force of the patient, and a discrete variable indicating the size of the implant (Figure 3).

The output data of the model in Figure 3 compares the actual stress from the simulations and the predicted stress.^{22 } The green circles are data points, the shaded region is the relative error, and the line is a loess curve that shows the relationship between the actual and predicted stress. Each of the parameters in the model (implant diameter, bite force, Young's modulus, and cortical bone thickness) have *P* values that are much less than .05. The line on the graph represents a loess curve. This is a local regression that is based on a combination of nonlinear polynomial regression and the k-means clustering algorithm.^{23 } Note that this is not the curve of the model; it is based on the location of the data points. The loess curve is meant to help the reader see any obvious trends in the data. The shaded region represents the 95% confidence interval for the loess curve, not the confidence interval of the model. The colored points are based on the relative error of the predicted stress, where the actual stress output from the simulation is the accepted value. Only on the extreme low end of the data set is the error excessively high; the points colored red output relative errors that exceed 90%. These points have bite forces at the extreme low end of 50 N. The rest of the data points have relative errors that are less than 20%. Because these data would be used in patient treatment, a 20% error is not acceptable. The points with the lightest hue of green have less than 5% relative error. While a zero tolerance for error is the goal, an acceptable error rate may be 5%. The mean relative error for this version of the model was 19.09%, which the 2 outlier points disproportionately affected. To mitigate the effect of those points, the median relative error was examined, which was 12.80%, about 33% smaller than the mean. The model was adjusted to mitigate the existence of outputs with such great error.

To see how a range of possible patient virtual parameters would change the output of the model, 1000 theoretical patients were simulated. The mean and standard deviation of both the bite force and Young's moduli were tested and constructed for normal distributions for each parameter and for random data points based on those normal distributions. These were then randomly paired with the Young's modulus distribution and the bite force distribution, which represented a possible patient. The first 500 virtual patients in the data set received small-diameter (3.2-mm) implants, whereas the rest received large-diameter (4.7-mm) implants. Table 2 is a rendering of the simulated data sets.

All of the patients' information put into the model yielded a predicted stress value on each patient's bone. Figure 4 shows the relationship between the predicted stress and bite force, separated by patients who received large- and small-diameter implants.

Unsurprisingly, the stress from the model increases as force increases. Furthermore, the small-diameter implants have larger stress values than the large implants do, which is not surprising considering that stress is the given force divided by area, and area depends on diameter. There is a greater per-square-millimeter of load on the small-diameter implants than the larger diameter implants.^{18,20 } The smaller the denominator, the larger the stress. Figures 5 and 6 separate the large and small implants and show whether the bone meets the failure criterion, stress that exceeds the yield strength of 114 MPa.

In the large-diameter implants, failure begins at about 500 N. Because the per-square-millimeter of stress with large-diameter implants is smaller than that of small-diameter implants with the same force, the yield stress can be exceeded at force figures slightly less than 100 N for small-diameter implants.

Next, the range of Young's moduli for the theoretical patients considered how that quantity was related to the stress. In Figure 7, the points are colored by diameter size.

Unlike the relationship with bite force, the stress output from the model does not necessarily increase as the Young's modulus increases. This was attributed to other factors affecting the stress. This may not be a simple Hooke's law relationship. Hooke's law states that the strain in a solid is proportional to the stress applied to it but within the elastic limit of the solid. The stresses on the small-diameter implants are smaller than those of the large-diameter implants for a given value of Young's modulus. This is due to the surface area presented to the bone. Figures 8 and 9 separate the large- and small-diameter implants and show the failure criterion.

Failure can occur at any value for Young's modulus with the requisite magnitude of force. The opposite of the phenomenon is observed for the stress and force relationship. In addition, for a given value of Young's modulus, the stress from the small-diameter implant is larger than that of the large-diameter implant.

Finally, the effect of the cortical bone thickness on the predicted stress of the simulated patient data set was considered. Figure 10 shows the relationship between the predicted stress and the cortical bone thickness of the mandible for both small- and large-diameter implants:

There was no linear relationship between the cortical bone thickness and the predicted stress. The parameter was statistically significant when predicting stress values, but alone, it did not explain any increases for the decreases in stress for this model. Figures 11 and 12 separate the small- and large-diameter implants.

Just as with the 2 other parameters described above, for a given value of cortical bone thickness, the small-diameter implants yield a larger stress than the large-diameter implants do. After considering all 4 parameters in the model, the small-diameter implants have larger stress values than the large-diameter implants do. The stress increases as the force increases. For the other 2 parameters, the stress value can vary depending on the values of the other parameters. Thus, a patient's individual bite force capability appears to be the most influential parameter.

## The Applet

An applet was built so that users can interface and interact with clinical data. A written equation cannot be provided because the model is a general additive model. This means that the model is a linear sum of nonparametric smoothing functions. Nonparametric functions are functions that do not have a form specified by a parametric formula. Instead, the data set at hand dictates the form of the function. The downside to this type of modeling is that the user needs a robust data set. If a user fits only a few data points to a nonparametric smoothing function, there is a high risk of overfitting, especially compared with parametric functions. One can attribute this fact to the phenomenon of the data providing the impetus for both the model structure and the model estimates. The smoothing functions are unknown by definition. The software develops them in such a way that they are uniquely fit to the data set at hand. There are no formulae or algorithms that apply to all smoothing functions. An example of a structure for a general additive model (that is not applicable to this specific case) is the linear sum of two nonparametric smoothing functions, in which the software bases the first function on a the weighted mean of different parts of the data set and bases the second function on a discrete variable. In Figure 13, one can observe what the applet looks like before the user inputs any values. The model has 4 inputs: the bite force of the patient, the cortical bone thickness, the implant diameter, and the Young's modulus of the patient's bone. The implant dentist can measure 3 properties but not the Young's modulus of a patient's cortical bone.

To account for the lack of a Young's modulus, a range of possible values for the Young's modulus of cortical bone (18.5–46.2 GPa, a difference of 27.7 GPa) was considered. It was assumed that the maximum and minimum values of the range were within 6 standard deviations of the mean. Assuming that the range of Young's moduli is normal, this assumption has a 99.99966% chance of holding true. This degree of certainty makes this assumption cogent.

Given the assumption of the maximum and minimum was within 6 standard deviations, a normal distribution of possible values for Young's modulus was constructed. The standard deviation of the distribution was one-sixth of the range of the Young's modulus values. Since the range was 27.7 GPa, 1 standard deviation of this distribution is about 4.62 GPa. The mean of the distribution is the average of the maximum and minimum, 32.35 GPa. The distribution construction was based on 10 000 normally distributed random numbers, with the mean and standard deviation from above. The distributions were further adjusted based on the aging effect of bone outlined in a literature review (see the Appendix). After the age of 35 years, the Young's modulus of bone decreases by about 10% every 10 years. Thus, age was used to extrapolate possible values of the Young's modulus, and so one of the inputs into the applet is the patient's age. The age input indicates to the software how to alter the distribution of the Young's modulus values. If the patient is older than 45 years, the software discounts values of the distribution by 10%. If the patient is older than 55 years, that figure changes to 20%. For ages 65 and up, the distribution values decrease by 30%.

Figure 14 shows the changes in the distribution for age. The dashed line in the density plots on the right represents the average for the given distribution. The horizontal lines on the boxplots on the left are also the means of the distribution. The edges of the boxes represent the 25th and 75th percentiles. The outlier points are those above 3 standard deviations.

When the user presses the enter button in the applet, the software applies the bite force, cortical bone thickness, implant diameter, and Young's modulus distribution to the model. Because the distribution of Young's modulus values is an input, there is a corresponding distribution of stress values as an output. What the applet then indicates is the probability that the implant will fail with immediate functional loading. In the output, we show the 5th, 25th, 50th, 75th, and 95th percentile outcomes for the stress on the bone. This provides an approximate probability of whether the immediate loading will fail. Figure 15 shows what the filled-out applet looks like, whereas Figure 16 shows the percentile table.

To gauge the probability of failure, the distribution of stress values that the model outputs is show in Figure 17. The red dashed line in the plot represents the failure criterion of 114 MPa. Values to the right of the dashed line are those in which immediate implant loading will fail, and those to the left indicate the stress values where the immediate implant loading should be successful. An example of the distribution plot, in which the inputs used are from those from Figure 15, is demonstrated in Figure 17.

For access to the applet, please go to https://implantloading.shinyapps.io/shiny_app/.

## Conclusions

The objective of this project has been to develop an iteration that generates a Boolean output that specifies detailed parameters that combines terms and excludes others to determine whether immediate dental implant functional loading is feasible. A basic framework was first established correcting fundamental flaws accounting for physiological loading conditions. This included the range of possible bite force capabilities and correctly accounting for the bone quality of a patient. This iteration assumes a healed site, a rough surface implant, accurate bite force measurements, and accurate cortical bone measurements by CBCT.

The first principal stress as the dependent variable is bite force capability, and this is the primary determiner of whether the bone would fail upon immediate loading.

By generating models from actual patient data taken from CBCT scans, a robust validation process was established. From the research models created, an iteration in the form of an applet was developed for implant dentists to easily interface with and obtain a real-time indication of whether or not a prospective implant site is eligible for immediate implant functional loading. According to this iteration, most immediate functional loaded implants will probably fail. Nonetheless, future innovations may reduce failures.

## Note

No conflicts of interest or commercial or financial interests were claimed by any of the authors.

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