Hypoglossal Nerve Stimulation on Cardiovascular Outcomes: Complete Statistical Analysis

1. Trial at a Glance

This page covers the registry results for Hypoglossal Nerve Stimulation on Cardiovascular Outcomes (NCT03359096), a randomized na study. The primary outcome is Mean 24-Hour Systolic Ambulatory Blood Pressure Values.

2. Clinical Question

3. Trial Design

This was a randomized, sham-controlled, double-blind, crossover trial, with triple masking listed in the registry. It enrolled 63 participants and is marked completed. The condition was obstructive sleep apnea, and the sponsor was the University of Pennsylvania. The registry lists the start as 2017-12-13 and primary completion as 2022-01-21.

The two device conditions were therapeutic HGNS and subtherapeutic sham HGNS. The therapeutic setting had been established before enrollment as part of clinical care. In a crossover design, participants receive more than one study condition over time, so comparisons may use information from the same participant under both conditions. The sequence assigned to participants, the timing of switching conditions, and any washout approach are not reported in the registry record.

The primary outcome was mean 24-hour systolic ambulatory blood pressure, assessed at Study Week 6, Day 1 and study Week 11, Day 1. The registry describes use of a portable, noninvasive monitor programmed to record blood pressure every 30 minutes, along with recording bedtime and wake time. Primary outcome summaries list 60 participants in each condition. These condition-specific counts do not establish how many participants contributed complete paired observations or how missing measurements were handled.

The registry lists several additional cardiovascular, physiological, and vigilance outcomes. The listed design details do not specify the statistical model used for the crossover comparison, the analysis population, the treatment sequence, or the handling of period and carryover effects. Those details are not reported in the registry record.

4. Participant Flow and Accounting

The registry reports how participants moved through each stage of the study. All counts below are exactly as posted; missing counts are not inferred from other stages.

HGNS Washout (1 Week)

Reasons for not completing this period:

First Intervention (4 Weeks)

Reasons for not completing this period:

HGNS Washout (1 Week)

Second Intervention (4 Weeks)

5. Baseline Characteristics

The table reproduces the baseline characteristics posted in the registry, which describe the participants analysed at baseline and help show whether the randomized groups were comparable.

6. Primary Endpoint

The registry lists the primary outcome as Mean 24-Hour Systolic Ambulatory Blood Pressure Values, measured over Study Week 6, Day 1; study Week 11, Day 1.

The primary endpoint was mean 24-hour systolic ambulatory blood pressure, measured in mmHg. Systolic pressure is the pressure recorded during heart contraction. Ambulatory monitoring collects measurements during ordinary daily activity and sleep rather than relying on a single reading in a clinic. The registry describes a portable monitor programmed to record blood pressure every 30 minutes, with bedtime and wake time recorded to corroborate the monitor’s sleep and wake classifications.

A mean condenses repeated readings into a single continuous outcome for each assessment. Comparing means can describe whether the average recorded systolic pressure differed between therapeutic and sham HGNS during the study periods. The registry lists assessments at Study Week 6, Day 1 and study Week 11, Day 1; the specific method for combining those periods into the posted group summaries is not reported in the registry record.

Ambulatory blood pressure is relevant to the trial’s stated cardiovascular focus because it measures pressure across a day rather than at one isolated visit. However, this endpoint is a blood-pressure measure, not a direct measure of cardiovascular events or overall cardiovascular risk. The result therefore addresses the recorded average systolic pressure, and should not be treated as a direct estimate of other outcomes.

7. Results Posted in the Registry

• Therapeutic HGNS: mean 122.80 versus 123.00 on control; difference -0.20 (95% CI -4.25 to 3.85), standardized effect d = -0.02; z-test p = 0.923.

8. Secondary Efficacy Endpoints

The registry also posts results for secondary outcomes. For each outcome below, the endpoint type, the test applied to the posted summary statistics, the result and its interpretation are given in turn. Secondary endpoints are usually not adjusted for multiple testing, so they support rather than confirm the primary finding.

Mean 24-Hour Diastolic Ambulatory Blood Pressure Values

• Endpoint type: Continuous: a mean value per participant, compared as a difference in means.
• Test: Unadjusted two-sample z-test for the difference in means, using the posted standard deviations.
• Result:
  • Therapeutic HGNS: mean 71.90 versus 72.10 on control; difference -0.20 (95% CI -2.85 to 2.45), standardized effect d = -0.03; z-test p = 0.882.
• Interpretation:
  • The 95% confidence interval includes no difference, so these data do not show a statistically significant difference between Therapeutic HGNS and the control. The interval gives the range of effects that remain compatible with the data.

Mean Pre-Ejection Period (PEP)

• Endpoint type: Continuous: a mean value per participant, compared as a difference in means.
• Test: Unadjusted two-sample z-test for the difference in means, using the posted standard deviations.
• Result:
  • Therapeutic HGNS: mean 105.70 versus 105.40 on control; difference 0.30 (95% CI -6.20 to 6.80), standardized effect d = 0.02; z-test p = 0.928.
• Interpretation:
  • The 95% confidence interval includes no difference, so these data do not show a statistically significant difference between Therapeutic HGNS and the control. The interval gives the range of effects that remain compatible with the data.

Mean Flow-mediated Dilation (FMD)

• Endpoint type: Continuous: a mean value per participant, compared as a difference in means.
• Test: Unadjusted two-sample z-test for the difference in means, using the posted standard deviations.
• Result:
  • Therapeutic HGNS: mean 6.65 versus 6.79 on control; difference -0.14 (95% CI -2.15 to 1.87), standardized effect d = -0.03; z-test p = 0.891.
• Interpretation:
  • The 95% confidence interval includes no difference, so these data do not show a statistically significant difference between Therapeutic HGNS and the control. The interval gives the range of effects that remain compatible with the data.

The posted secondary results follow a broadly similar pattern to the primary result: the recalculated mean differences are close to zero relative to their uncertainty, and the reported unadjusted intervals include zero. These outcomes cover additional blood-pressure measures and physiological measures, rather than offering independent confirmation of a cardiovascular event effect. The secondary results therefore add context about the range of measurements assessed, but do not transform the primary finding into evidence about outcomes that were not measured.

The registry lists nine secondary outcomes, while summary results for three secondary measures appear in the record. Results for the other listed secondary outcomes are not reported in the registry record. The available secondary summaries should not be taken as a complete account of all planned outcome findings, because the record does not explain whether missing results were unmeasured, unavailable, or omitted for another reason.

Secondary comparisons are generally less conclusive when many outcomes are examined without a reported plan for controlling multiplicity. With multiple statistical tests, the chance of at least one apparently notable result can rise even when there is no underlying effect. The registry lists no reported analyses and does not describe a multiplicity adjustment or hierarchy among secondary outcomes. Their unadjusted results consequently carry less weight than a prespecified, fully reported analysis with a clear approach to multiple testing.

In general, secondary outcomes can help assess whether a primary pattern is accompanied by changes in related measurements, but they should be interpreted in light of measurement quality, missing results, and the analysis plan. Here, the record does not report enough analysis detail to establish whether the secondary outcomes were treated as confirmatory or exploratory. Their role is best regarded as descriptive context rather than decisive evidence that strengthens or overturns the primary comparison.

Secondary outcomes registered for this trial, for reference (only those above were recalculated):

• Mean 24-Hour Diastolic Ambulatory Blood Pressure Values
• Mean Nocturnal Systolic and Diastolic Ambulatory Blood Pressure Values
• Mean Muscle Sympathetic Nerve Activity (MSNA) Frequency (Bursts/Minute)
• Mean Muscle Sympathetic Nerve Activity (MSNA) Tonal Activity (Units/Minute)
• Mean Pre-Ejection Period (PEP)
• Mean Flow-mediated Dilation (FMD)
• Mean Peripheral Arterial Stiffness (PAS)
• Mean Psychomotor Vigilance Test (PVT) Reciprocal Reaction Time

9. Statistical Interpretation

10. Statistical Methods Explained

The primary endpoint is continuous, and the registry reports a mean and standard deviation for each condition. A mean describes the central value of the observed measurements. A standard deviation describes how spread out individual measurements are around that mean; it is not the uncertainty in the estimated mean and is not a confidence interval.

The posted primary summaries are 122.8 mmHg for therapeutic HGNS and 123.0 mmHg for sham HGNS, each with 60 participants listed. The unadjusted recalculated mean difference is -0.2 mmHg, using therapeutic minus sham. This direction convention matters: a negative estimate means the therapeutic-group summary is lower than the sham-group summary. It does not by itself indicate whether the difference is important or whether it reflects a treatment effect under a crossover model.

A confidence interval gives a range of values compatible with the observed data under the assumptions of the method used. It is not a range containing a known proportion of individual participants’ measurements, nor does it assign a probability to the true value after the data have been observed. The interval for the primary mean difference extends from -4.248 to 3.848. Its width reflects uncertainty in the estimated difference, and the inclusion of zero indicates that the calculation does not clearly separate the estimate from no difference.

The unadjusted p-value for the primary comparison is 0.923. In general, a p-value assesses how unusual data at least as inconsistent with a null hypothesis would be if that hypothesis and the statistical model were correct. It does not give the probability that the treatment has no effect, and a large p-value is not proof of equivalence. The registry does not report a prespecified equivalence margin or a noninferiority margin.

What does a crossover design change?

In a crossover trial, participants can serve as their own comparison because outcomes may be observed under more than one condition. This can reduce the influence of stable differences between participants when the analysis uses paired observations appropriately. The outcome under one condition may nevertheless be related to the outcome under the other condition for the same person. A paired analysis generally requires information about that within-person relationship, which cannot be recovered from separate group means and standard deviations alone.

The registry identifies this trial as crossover, but the posted summaries do not report the treatment sequence, period-specific data, within-person covariance, or a model accounting for period or carryover effects. The summary-based recalculation is therefore an unadjusted comparison, not a substitute for a crossover analysis using participant-level data. Whether the sponsor conducted such an analysis is not reported in the registry record; the registry lists no reported analyses.

Why does over-dispersion matter?

For continuous outcomes, variation among participant measurements affects the precision of a mean difference. Greater variability generally makes an estimate less precise when other features are held constant. General statistical methods must also consider whether the data’s variability is adequately represented by the chosen model. Over-dispersion is a particular concern when a model assumes less variability than the data contain; that mismatch can make uncertainty appear too small. The registry does not report an over-dispersion assessment or a model for the primary endpoint, so no conclusion about such a check can be drawn from the record.

How should the planning figures be interpreted?

The planning illustration uses a two-sided significance level of 0.05, equal allocation, and a normal approximation. Its target effect is a stated fraction of the observed effect, and its entries are participants per arm. Under this illustration, the 80% power column represents the number per arm associated with an 80% chance of meeting the specified statistical criterion if the assumed effect and model are correct. The 90% column represents the corresponding planning target for a 90% chance. These are design calculations, not an assessment of the probability that the completed trial detected an effect.

The registry does not report the original sample-size justification or the trial’s planned statistical analysis. The planning illustration is computed from the observed effect rather than those original design assumptions. It should not be used to infer that the trial was designed around these figures.

What does masking contribute?

Masking can limit the influence of expectations on participant experience, measurement, and assessment. The registry lists triple masking, but the identities of the masked roles and the success of masking are not reported in the registry record. A sham condition can support a controlled comparison, although the incomplete sham description in the record limits assessment of how closely it matched the therapeutic condition.

11. Confidence Intervals and Precision

The primary mean difference was -0.2 mmHg, with an interval from -4.248 to 3.848. The interval spans zero, so the posted estimate is compatible, under the calculation assumptions, with a difference favoring either condition as well as with little or no difference. The point estimate is only one location within that range; it should not be interpreted without the uncertainty around it.

The interval is not especially informative about a narrow range of possible effects: its endpoints are separated by several mmHg. That means the posted summary does not pin down the size or direction of the difference with high precision. It also does not show that every value in the interval is equally plausible, or that values outside it are impossible. The exact confidence level and method used for an interval from a sponsor analysis are not reported in the registry record.

Precision depends on more than the group means. It is affected by measurement variability, the number of usable observations, the statistical model, and, for crossover data, the relationship between a participant’s measurements under the different conditions. The primary summaries list 60 participants per condition, but the record does not describe the number with complete paired outcome data or the analysis used to account for pairing. The unadjusted recalculation therefore cannot fully characterize precision under the intended crossover design.

The interval and p-value answer related but distinct questions: the interval expresses uncertainty across a range of effect values, while the p-value evaluates compatibility with a specified null hypothesis under a model. Neither establishes clinical importance. The registry does not report a threshold for a clinically meaningful change in systolic pressure, so the interval cannot be classified against such a threshold. In general, greater precision would require an interval narrow enough to rule out effects that matter for the scientific question; whether this interval does so cannot be determined from the registry record.

12. Planning a Similar Trial: Sample Size

This is a planning illustration, not a re-evaluation of the original trial. It asks: if a new study wanted to detect a difference like the one observed, or a smaller one, how large would it need to be? It is deliberately not “observed power” of the finished trial, which is not a meaningful quantity.

The planning illustration gives participants per arm for the continuous outcomes in this record. It does not give total event counts. Its assumptions are a two-sided significance level of 0.05, equal allocation between the two arms, a normal approximation, and an effect to detect equal to the stated fraction of the observed effect. The original trial’s own sample-size assumptions are not reported in the registry record.

For the primary endpoint, planning for 100% of the observed effect gives 50,210 participants per arm for 80% power and 67,216 participants per arm for 90% power. Planning for 75% of the observed effect gives 89,261 participants per arm for 80% power and 119,496 participants per arm for 90% power. Planning for 50% of the observed effect gives 200,838 participants per arm for 80% power and 268,864 participants per arm for 90% power.

In plain language, the 80% column describes a planning target that would give the specified statistical test an 80% chance of meeting its criterion if the assumed effect and assumptions were correct. The 90% column uses a higher chance target. Smaller effects require more participants because they are harder to distinguish from random variation. The plotted curve is a visual way to follow that relationship across the effect assumptions; it does not change the assumptions behind the figures.

These are planning illustrations based on the observed effect, not observed power. They do not show the probability that this completed trial detected an effect, and they are not evidence that the original study required or recruited those participant counts.

13. How the Sample Size Is Derived

Power is the probability that a study of a given size rejects the null hypothesis when a specific true effect exists. It depends on four things: the effect to detect, the variability of the outcome, the significance level (here a two-sided α of 0.05, so z_1−α/2 = 1.960) and the desired power (z_1−β = 0.842 for 80% and 1.282 for 90%).

Formula

n is the number of participants in each arm, Δ is the difference in means to detect and σ is the common standard deviation.

Worked calculation for this trial

1. Observed difference Δ = -0.200; pooled standard deviation σ = 11.311.
2. n = 2 × (1.96 + 0.842)² × 11.311² / -0.200² = 50,209.3, rounded up to 50,210 participants per arm for 80% power.

The smaller-effect rows in the table above repeat the same calculation with the effect multiplied by 0.75 and 0.5 (for ratios, applied to the logarithm of the ratio).

R code

alpha <- 0.05
z_a   <- qnorm(1 - alpha / 2)          # two-sided
delta <- -0.200000000000003                      # observed difference in means
sigma <- 11.3110565377422                     # pooled standard deviation
n_per_arm <- function(power, frac = 1) {
  ceiling(round(2 * (z_a + qnorm(power))^2 * sigma^2 / (delta * frac)^2, 6))
}
for (f in c(1, 0.75, 0.5))
  cat(sprintf("%3.0f%% of observed effect: n per arm = %s (80%% power), %s (90%% power)\n", 100 * f,
              format(n_per_arm(0.80, f), big.mark = ","), format(n_per_arm(0.90, f), big.mark = ",")))

# Check against R's built-in function (t-based, so very slightly larger)
cat("power.t.test:", round(power.t.test(delta = delta, sd = sigma, power = 0.80)$n, 1), "\n")

# Simulation: exact two-sample t-test using simulated means and variances
set.seed(3359096)
n    <- n_per_arm(0.80)
reps <- 20000
m_diff <- rnorm(reps, delta, sigma * sqrt(2 / n))             # difference in sample means
s2     <- sigma^2 * rchisq(reps, 2 * (n - 1)) / (2 * (n - 1))   # pooled sample variance
tstat  <- m_diff / sqrt(s2 * 2 / n)
cat(sprintf("Simulated power at n = %s per arm: %.3f (%d replicates)\n", format(n, big.mark = ","), mean(abs(tstat) > qt(0.975, 2 * (n - 1))), reps))

Output from running the code

100% of observed effect: n per arm = 50,210 (80% power), 67,216 (90% power)
 75% of observed effect: n per arm = 89,261 (80% power), 119,496 (90% power)
 50% of observed effect: n per arm = 200,838 (80% power), 268,864 (90% power)
power.t.test: 50210.2 
Simulated power at n = 50,210 per arm: 0.800 (20000 replicates)

The simulation draws many sets of group means and variances at the planned sample size and counts how often the two-sample t-test rejects the null. It should land close to the target power; at very small required sample sizes it can fall below it, because the normal approximation is rough there.

14. Multiplicity, Missing Data and Analysis Populations

The registry lists a primary outcome and multiple secondary outcomes, but it does not report a statistical analysis plan, a hierarchy for testing outcomes, or an adjustment for multiple comparisons. Multiplicity matters because each additional hypothesis test creates another opportunity for a result to appear unusual by chance. Without a stated strategy, nominal unadjusted p-values across outcomes should not be treated as if they came from a single isolated test.

The registry lists no reported analyses. The posted arm summaries and recalculated comparisons do not identify whether the analyses were prespecified, whether outcome definitions changed, or how deviations from any protocol were handled. The original statistical plan, interim analyses, and sample-size justification are not reported in the registry record. No conclusions about those design or reporting choices can be drawn from their absence.

The record also does not describe the analysis population, exclusions after randomization, or how missing outcome data were handled. Although the primary condition summaries list 60 participants in each group, the total enrollment was 63. The record does not explain the relationship between those figures or whether all enrolled participants contributed to each posted outcome. Secondary outcome summaries have different group counts, but reasons for those differences are not reported in the registry record.

For a crossover trial, missingness can affect paired comparisons if a participant contributes an outcome under only one condition. The record does not report how incomplete pairs were treated, whether the analysis accounted for treatment sequence or period, or whether carryover was assessed. These details are important for judging whether the comparison preserves the benefits of randomization and crossover measurement. Their absence limits interpretation of both precision and the validity of the unadjusted summary-based comparison.

15. Safety Results

The registry reports serious adverse events as affected participants divided by participants at risk.

Deaths

Other (non-serious) adverse events

Participants with at least one non-serious event at or above the registry's 0% reporting threshold.

The registry lists 60 participants at risk for serious adverse events in each condition. It records zero serious adverse events in the therapeutic HGNS group and zero in the subtherapeutic sham group. The record also lists zero other adverse events in each group. These counts describe the events entered in the registry; they do not show that every possible harm was actively assessed or that no unrecorded events occurred.

The death count is listed separately: zero in the therapeutic group and one in the sham group. The registry record does not report a relationship between that death and study participation or the device condition. The count alone cannot establish cause or attribute the event to either condition.

No serious adverse-event terms are listed. This should be read as no terms listed in the record, not as a detailed account of event types or severity. The registry does not report additional information about timing, duration, adjudication, or participant-level exposure relevant to interpreting safety. Small event counts and limited reporting make these entries descriptive; they do not support a broad conclusion about comparative safety.

16. What the Result Does, and Does Not, Mean

The posted primary data show similar mean 24-hour systolic ambulatory blood pressure summaries in the therapeutic and sham conditions: 122.8 mmHg and 123.0 mmHg, respectively. The unadjusted recalculated mean difference was -0.2 mmHg, and its interval ranged from -4.248 to 3.848. This is a small point estimate accompanied by uncertainty that includes differences in either direction.

That result is not proof that the conditions are equivalent. Demonstrating equivalence requires a prespecified margin defining what differences are small enough to count as equivalent, along with an analysis designed for that question. No equivalence margin is reported in the registry record. Nor does a p-value of 0.923 establish the absence of a difference; it indicates that this unadjusted comparison did not distinguish the observed data from the null under its assumptions.

The endpoint is a blood-pressure measure, not a heart attack, stroke, or other cardiovascular event outcome. The trial’s title and stated cardiovascular focus do not change what was measured in the posted primary result. The result cannot establish whether the device changes future cardiovascular risk, other clinical outcomes, or outcomes beyond the recorded assessment periods.

The crossover design may allow informative within-participant comparisons, but the posted record does not describe the participant-level analysis needed to assess that advantage. The reported comparison is a recalculated unadjusted comparison from group summaries, and the registry lists no reported analyses. It is therefore not a substitute for a prespecified analysis accounting for the crossover structure.

Overall, the registry supports a cautious descriptive statement about the recorded group means and the uncertainty around their unadjusted difference. It does not justify a treatment recommendation or a definitive claim of benefit, lack of benefit, equivalence, or harm. Such conclusions would require appropriate analysis details and evidence addressing the relevant clinical outcomes.

17. Why This Trial Matters Statistically

This trial is statistically interesting because it combines randomization and masking with a crossover comparison. In general, crossover designs can make efficient use of within-participant information when the same person is measured under both conditions. That potential advantage depends on appropriate handling of paired observations, treatment sequence, period effects, and possible carryover. The registry identifies the design but does not report those analysis details, making the distinction between the intended design and the posted summary-based comparison important.

The primary outcome is a repeated-measure cardiovascular risk marker rather than a clinical event. This highlights a common interpretive issue: a carefully measured surrogate or physiological endpoint can answer a narrower question than whether participants experience fewer cardiovascular events. The posted result should be read at the level of the endpoint actually recorded.

The record also illustrates why a near-zero point estimate and a large p-value do not establish equivalence. The interval remains broad enough to include differences in either direction, and no clinically meaningful or equivalence threshold is reported. Precision and the scientific question must be considered together rather than treating statistical significance as the sole criterion.

Finally, the listed planning illustration is based on the observed effect and is not the trial’s original design justification. It makes visible how planning estimates depend on the assumed effect size, but it cannot retrospectively determine whether the study was adequately planned or powered. The registry record does not report the original justification or a complete analysis plan. These features make transparent reporting of design assumptions and participant-level analyses especially important for interpreting this crossover trial.

18. Limitations

• The posted summary statistics are the only data these results rest on; patient-level data, full protocols and statistical analysis plans are not available.
• Recalculated results are simple, unadjusted calculations from posted summaries. They do not replace the sponsor's prespecified analysis.

Several limitations specific to this record constrain interpretation. Although the trial is identified as randomized, triple-masked, and crossover, the sequence of conditions, timing of switching, and washout procedures are not reported in the registry record. Without these details, readers cannot assess how treatment order or lingering effects were addressed.

The record contains group means and standard deviations but no participant-level paired data or reported sponsor analysis. The recalculated unadjusted comparison therefore does not account for the within-person correlation that may be central to a crossover analysis. The method used to combine measurements from the two assessment periods is also not reported.

The primary summaries list 60 participants in each condition, whereas enrollment is 63. The record does not explain this relationship, identify the analysis population, or describe exclusions and missing-data handling. Secondary outcome counts vary, and reasons for those variations are not reported. The registry lists nine secondary outcomes but posts summaries for only three; the status of the others is not explained.

The complete sham-arm description is not reported in the registry record, and the roles covered by triple masking are not specified. The record also does not report the original sample-size justification, interim analyses, multiplicity plan, or clinically meaningful threshold for the primary endpoint. Finally, safety entries are limited to counts, with no listed serious-event terms and no reported attribution for the recorded death. These omissions restrict evaluation of both statistical validity and clinical meaning.

19. Related Tutorials

20. Related Statistical Calculators

21. Sources

• ClinicalTrials.gov: Hypoglossal Nerve Stimulation on Cardiovascular Outcomes — NCT03359096.
• Registry record last updated 2023-03-30; status last verified 2023-03.
• Results first posted 2023-03-30.
• Data retrieved from ClinicalTrials.gov on 2026-11-09.
• Group sizes, means, event counts and sponsor-reported p-values are reproduced as posted. Effect estimates, confidence intervals and p-values not attributed to the sponsor are independently calculated from the posted summary statistics.

22. Record Summary

The registry describes a completed randomized, sham-controlled, triple-masked crossover trial of therapeutic versus subtherapeutic sham HGNS in people with obstructive sleep apnea. It enrolled 63 participants and assessed mean 24-hour systolic ambulatory blood pressure at the stated study-week visits.

The posted primary means were 122.8 mmHg with therapeutic HGNS and 123.0 mmHg with sham. The recalculated unadjusted mean difference was -0.2 mmHg, with an interval from -4.248 to 3.848 and an unadjusted p-value of 0.923. The record lists no reported analyses, and key details for crossover analysis, missing data, multiplicity, and the original sample-size rationale are not reported in the registry record.

These results describe the posted blood-pressure summaries and do not establish an effect on cardiovascular events or equivalence between conditions.