KEY CONCEPTS
• The traditional lambda ratio, long used to determine lubrication regimes, is not applicable to much of today’s machinery.
• A modified lambda ratio has been proposed that takes into account parameters such as running-in wear of asperity summits, micro-EHL deformation of asperities, the 3D nature of roughness and a physics-based lubrication regime criterion.
• Assuming the modified lambda ratio is widely adopted, it will affect areas such as standards, engineering drawings, commercial design software and the evolving field of EV tribology.

Since the 1960s, the traditional lambda ratio (film thickness divided by surface roughness) has been widely used to determine lubrication protocol. However, relatively recent research has shown it often fails to accurately represent lubrication conditions—especially in components that have undergone running-in. Essentially, today’s machinery no longer behaves like the machinery the original lambda thresholds were designed for. This necessitated an alternative approach, referred to here as “modified lambda” but also called “lambda*.”

The traditional lambda ratio has been a cornerstone in tribology since T.E. Tallian’s¹ work in the 1960s. Despite the calculation’s known limitations—particularly its inability to reflect real-world lubrication conditions after running-in—it is still endorsed by standards such as ISO 6336-22²3 It is often used as a quick reference and may owe its continued use to simplicity and familiarity among engineers.

Developed in 2021, modified lambda provides a much more realistic and robust way of evaluating lubrication regimes. The modified parameter considers key factors omitted by traditional lambda, including anisotropic⁴ surface structures, micro-scale elastohydrodynamic lubrication (EHL) behavior and revised transition criterion based on a revised surface roughness parameter reduced peak height (Spk). Despite these adjustments, modified lambda retains traditional lambda’s ease of use and has been validated through experiments and well-established semi-analytical theories. 

Following is a discussion of both approaches, explaining their foundations, limitations and applications with a special focus on electric vehicles (EVs).

Traditional lambda
Per STLE member Dr. Jonny Hansen, Tribology Expert EVs, TRATON Group, the traditional lambda parameter Λ=h_m/S_q (or sometimes Λ=h_m/S_a) was a major achievement when first introduced in the 1960s and has served the scientific and engineering community well over the past 60 years. S_q refers to root-mean-square height; S_a refers to arithmetic mean height. He says, “However, vast progress in rough surface EHL theory and measurement has made it possible to improve the formulation of the film parameter for modern, heavily loaded and mixed lubrication contacts, such as ball bearings in EV drive units.” 

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Spk^A and Sq^B
Independent tests show that Spk changes more than Sq for isotropic surfaces^C,D,E,F,G or anisotropic surfaces, the change is similar, but the issue for traditional Λ is that even if Spk and Sq both change ~30% with running-in (starting from similar initial values), this is not sufficient for the Sq-based Λ; Sq needs to decrease substantially more to reach Λ=3, whereas Spk requires a much smaller decrease to reach Λ*=1. An additional limitation of traditional lambda is that if surfaces are not normal and isotropic, the theoretical justification for using RMS summation (Sq=sqrt(Sq1^2+Sq2^2)) becomes less valid. Therefore, its applicability becomes questionable for surfaces subjected to running-in or for surfaces with pronounced lay.

A. SPK: reduced peak height
B. Sq: root-mean-square height
C. Hansen, J., Björling, M. and Larsson, R. (2020), “Lubricant film formation in rough surface non-conformal conjunctions subjected to GPa pressures and high slide-to-roll ratios,” Scientific Reports, 10, article no. 22250, pp. 1-16. Available at https://doi.org/10.1038/s41598-020-77434-y.
D. Hansen, J., Björling, M. and Larsson, R. (2020), “Topography transformations due to running-in of rolling-sliding non-conformal contacts,” Tribology International, 144, article no. 106126. Available at https://doi.org/10.1016/j.triboint.2019.106126.
E. Hansen, J. (2021), Elasto-hydrodynamic film formation in heavily loaded rolling-sliding contacts, Doctoral dissertation, Luleå University of Technology. Available at http://ltu.diva-portal.org/smash/record.jsf?pid=diva2:1502750.
F. Hansen, J., Prajapati, D. K., Björling, M. and Larsson, R. (2025), “Robustness and sensitivity of the Λ*-ratio in microelastohydrodynamic lubrication,” Tribology Letters, 73, article no. 129. Available at https://doi.org/10.1007/s11249-025-02060-6.
G. Dunaevsky, V. (2017), “A proposed new film thickness-roughness ratio, Λz, in rolling bearings: Notes on an engineer’s experience with surface texture parameters,” SAE Technical Paper, 2017-01-2415. Available at https://doi.org/10.4271/2017-01-2415.

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Hansen goes on to summarize the fundamental limitations of the traditional Lambda parameter in EV applications as follows:

1. Inadequate roughness representation
Neither Sq nor Sa represents the load-bearing asperity peak heights in a physically meaningful way, i.e., the way a peak parameter such as Spk does. Both Sq and Sa treat the entire height distribution uniformly and do not distinguish valleys from peaks.

In practice, prominent valleys often remain largely unchanged after running-in, while the highest asperities—which govern the earliest contact events and minimum film conditions—blunt or deform first. This makes Sq and Sa insensitive to functional changes in roughness that directly affect film support and regime transitions.

In contrast, a peak-based parameter such as Spk, derived from the Abbott–Firestone bearing curve,⁵ isolates the upper portion of the height distribution that is most relevant to load-carrying and minimum film thickness interactions. Therefore, using Sq or Sa to balance against film thickness is not a physically representative metric for predicting lubrication regime behavior in real contacts that have been run in.

In contrast to the Sa and Sq parameters for isotropic roughness, the reduced peak height, Spk, isolates the protruding summit population above the core roughness by definition. Because these asperities govern initial load-bearing and mixed lubrication behavior, Spk provides a more functionally relevant and sensitive descriptor of wear-induced surface modification associated with fluid film breakdown. For this reason, Spk has been incorporated into the definition of modified lambda.

Comparing the Spk and Sq numerical values pre/post running-in confirms that the Spk is much more involved in running-in wear, making it a better roughness representation for describing lubrication quality:

Spk=0.357/0.195 à -45.4% reduction
Sq=0353/0.300 à-15.0% reduction

2. Arbitrary lubrication regime criteria
The criterion Λ ≈ 3 (or 5) for the transition from full-film EHL (FF-EHL) to mixed lubrication (ML) is based on Gaussian surface assumptions—specifically, three standard deviations (3σ where σ=Spk). Real engineering surfaces, particularly after wear or running-in, are rarely Gaussian; their height distributions are often skewed and anisotropic. Since Sq itself is not a representative measure of the highest asperities and since 3×Sq is a purely statistical construct with no physical linkage to local contact mechanics, Λ ≈ 3 as a regime threshold lacks physical foundation. As a result, it functions as an ad hoc criterion for film collapse and cannot be reliably applied in practical EHL contacts. 

In contrast, the EHL-ML transition criterion for modified lambda is derived from a physically grounded micro-EHL contact mechanics framework. It defines the transition based on whether elastic deformation and hydrodynamic film support at the asperity scale are sufficient to prevent contact interference at the outlet constriction. Under this formulation, modified lambda = 1 corresponds to the point where asperity peaks begin to penetrate the EHL film at the outlet, marking a mechanistically defined onset of ML. This criterion is not an ad hoc multiplier but reflects a specific physical condition in the contact.

3. Lack of 3D nature of roughness
Real surface roughness has three-dimensional lay/texture that affects both hydrodynamic pressures and asperity deformation at multiple scales. A single scalar metric like Sq contains no lateral information and cannot capture how directional texture alters lubrication behavior. 

4. Rigid roughness assumption/neglect of micro-EHL mechanisms
Traditional lambda implicitly assumes that surface asperities are rigid—that the roughness being measured under the microscope is the same as the one passing the EHL conjunction. In reality, under high loads and during the onset of mixed lubrication, asperities elastically deform and blunt, producing a micro-EHL response that materially changes the local film thickness distribution, and consequently the FF-EHL to ML regime transition.

Asperity radius (or roughness wavelength) influences local conformity and hydrodynamic pressure spikes; this gives rise to micro-EHL deformation that the traditional lambda formula cannot represent at all. The surface topography in its relaxed state (as under the microscope) is not the same surface as inside the EHL conjunction. To account for the micro-EHL flattening of roughness inside the contact, either Amplitude Reduction Theory (ART) or the factor fq (correction factor used to combine the roughness of two contacting surfaces into a single composite roughness value) can be used.

In comparing ART versus f_q, although the two curves are inverse in form, they describe the same physical phenomenon. ART quantifies the reduction of roughness amplitude due to elastic micro-EHL deformation, whereas fq represents the corresponding correction to the effective film thickness in the modified lambda formulation. In other words, ART describes how asperities flatten, while fq translates that flattening into an increase in effective film thickness. 

Modified Lambda can be calculated with both approaches (note that h refers to total profile height, h_c refers to core roughness height, and h_m refers to material portion height):


or
 , where Spk* is the Spk taken on the amplitude reduced roughness profile.

STLE member Peter Lee, institute engineer and chief tribologist, Tribology Research and Evaluations, Southwest Research Institute, concurs, explaining, “In any tribological contact, friction and wear are reduced to their minimum when the lubricant film is thick enough to fully separate the two surfaces. Traditionally this has been said to be when the lubricant film thickness is three standard deviations of roughness or more, i.e., three times the thickness of the root-mean-square (RMS) of the combined surface roughness.” 

Lee continues, “This, however, does not always hold true. In some cases, particularly with ball-on-disk testing, full film lubrication can be achieved with the lambda ratios as low as 0.16,^6,7,8 and therefore it can be seen that the traditional lambda ratio equation has its limitations. These limitations include assumptions of the surface by only using Sq, thereby ignoring running in effects—it treats valleys and peaks equally. By ignoring the reduction in peak height and remaining unworn valleys it makes RMS values insensitive to wear. It ignores skewness of the surface height distribution due to running in. By only using Sq as a measurement parameter, the three-dimensional nature of surface roughness is also ignored, including if asperities are sharp or smooth. Finally, the traditional lambda ratio ignores any elasticity of the material.

“If all this is applied to gears and bearings in electric drive units (EDUs), which are often finished to ensure the surfaces are smooth, a lambda ratio of three would greatly over-engineer the lubricant film required to actually provide separation of gear teeth or roller element bearings under load. The modified lambda ratio does take into consideration these surface effects and is therefore a more accurate representation of the actual lambda ratio.”

Hansen adds, “The primary limitation of the traditional Λ-ratio is its reliance on Sq (or Sa), which responds only weakly to running-in. The rigid roughness assumption further contributes to the error, although the choice of Sq dominates. Note that an ART-corrected Λ formulation exists. However, it offers almost no improvement to the traditional lambda parameter—even with ART the error is still substantial.”⁹

Practical consequences of applying traditional lambda
Roland Larsson, professor of machine elements, and chair professor (head of subject) at the Division of Machine Elements, Luleå University of Technology, Sweden, explains that, for lubricated contacts in gears and rolling bearings, the traditional lambda doesn’t yield correct information about the lubrication quality. 

“In all textbooks, even the new ones, you can find the statement that lambda=3 is the threshold between mixed and full-film lubrication,” he says. “And lambda=1 is the threshold between boundary and ML. Those limits are used also in recent publications on EHL. Even though it has been known for a long time that lambda=0.5 may give full-film lubrication. Some companies know this of course, such as the bearing companies.” 

Others agree, including Marcus Björling, associate professor at the Division of Machine Elements, Luleå University of Technology, Sweden. He maintains that the traditional lambda parameter has been shown in many cases to give a completely wrong estimation of the actual lubrication regime. This is especially true for surfaces subject to running-in, which have been shown to operate in a more favorable lubrication regime than was predicted by the traditional lambda parameter. 

“Getting the lubrication regime wrong in EVs has major implications since, for instance, the difference in operating in full-film lubrication or ML will mean that the contact will exhibit ohmic rather than resistive behavior, which in turn leads to very different electrical behaviors,” Björling says. He continues on to explain, what he sees as, the two main flaws of traditional lambda. 

1. The use of simple roughness parameters such as Sq or Sa results in a poor representation of the roughness practical function in a tribological system. These parameters do not distinguish between peaks and valleys; in practice this makes a significant difference in how a tribological system behaves. In addition, using solely Sa and Sq does not take roughness orientation or lay into consideration. In practice, it makes a major difference if the surface, for instance, has a transversal lay in comparison to a longitudinal lay. 
2. The traditional Lambda parameter also does not take the elastic deformations of the surfaces into account. Björling considers this a major shortcoming since in EHL, the pressures are typically in the gigapascal range and the surface roughness is deformed, especially the longer wavelengths. When the surface roughness is deformed, the degree of asperity interaction is reduced and less of the load is then carried by solid-solid contacts. This leads to a more favorable lubrication condition in many cases.

Hansen adds that running-in alters peak heights and promotes elastic micro-EHL film formation by increased asperity radii. “Yet Sq and Sa remain largely unchanged and therefore fail to reflect the functional surface condition,” he says. “This, together with ad hoc lubrication regime criterion, leads to incorrect prediction of the full-film EHL to ML transition and propagates into errors in bearing fatigue life estimation, lubricant viscosity selection and service-interval planning. In EV drive units specifically, RMS-based roughness metrics do not represent the load-bearing asperity peaks or the geometric features that govern minimum film thickness and electrical breakdown conditions. As a result, the onset of full-film EHL and discharge-related behavior can be misidentified.”

Per Hansen, the modified lambda formulation addresses these limitations by:
• Replacing RMS roughness with the peak-sensitive parameter Spk, which reflects the asperities that control contact and film collapse
• Incorporating elastic micro-EHL deformation (f_q) to 3D surface roughness with different lay into the effective film thickness
• Introducing a physically grounded lubrication regime criterion

“By grounding the regime criterion in contact mechanics (modified lambda = 1) rather than statistical multiples of RMS height, modified lambda provides a physically consistent predictor of lubrication transitions in modern EV drive contacts,” he observes. “For engineers, this means that modified lambda can be used as a design tool for predicting and controlling electric discharge behavior in bearing contacts.” 

Why the continued use of traditional Lambda?
Björling believes there are many reasons for the continued widespread use of the traditional Lambda model. He points out, “For starters it is simple to use. It only requires a standard roughness parameter, typically Sq or Sa, in combination with a rough estimate of the lubricant film thickness, normally the minimum film thickness. The traditional Lambda parameter is also found in many places: textbooks, research articles, industry standards, etc. It has also been in use for a long time, more than half a century, so it is found in material spanning a long period of time. Finally, many researchers and engineers that are using the model are probably not aware of the shortcomings.”

Larsson agrees that one of the reasons for its continued application is that the traditional lambda model is very easy to use. “If you just know the Rq (roughness) of the surfaces and can estimate the minimum lubricant film thickness, then you can compute lambda. With its fixed limits (thresholds) it is then easy to make a claim about the lubrication quality. But that claim is wrong in most cases regarding EHL.”

Lee cites the following two reasons for continued use of traditional lambda: 
1. It is referenced in standards. The International Organization for Standardization lists it in its load calculation of load carrying capacity of spur and helical gears, and the Motion and Power Manufacturers Alliance (MPMA) standards also use the traditional lambda ratio. To be fair to both, they were written before the modified lambda ratio was introduced.
2. The traditional lambda ratio has been in use for decades and is well known and well documented, whereas modified lambda was released in 2021 and has not yet made it into textbooks and is not well known.

“Once it becomes better known and is incorporated into standards it will help in more robust lubrication design and improved failure predictions,” Lee adds. 

On a more philosophical level, Hansen notes that traditional lambda’s long historical use has created institutional inertia where engineers, researchers and standards bodies rely on a familiar metric that has served as a first-order lubrication indicator for more than 50 years. 

“In many conventional applications, it has been considered ‘good enough’ for approximate regime classification,” he explains. “However, as modern systems become more demanding—particularly in electrified drivetrains and EV motors where lubrication quality interacts with electrical phenomena from parasitic stray or bearing currents—the limitations of this simplified statistical formulation become more apparent. The traditional lambda ratio persists primarily due to simplicity, legacy practice and standardization—not because it is universally or physically rigorous for modern tribological systems.”

How traditional and modified lambda differ
The traditional lambda ratio compares the smooth minimum film thickness to a statistical roughness measure:

 

Hansen explains that the transition from full-film EHL to ML is typically defined by a statistical criterion (Λ ≈ 3-5), corresponding to three standard deviations of a Gaussian height distribution. This makes the regime threshold heuristic rather than physically derived. In contrast, the modified lambda ratio compares an effective film thickness to a peak-sensitive roughness parameter with roughness texture/lay:



In the previously modified lambda equation, Spk represents the load-bearing asperity summits that first penetrate the lubricant. The term h_cf_q accounts for elastic micro-EHL deformation of roughness, where f_q depends on asperity radius and therefore captures the three-dimensional nature of surface texture/lay.

“The key theoretical difference is that modified lambda is derived from micro-EHL contact mechanics,” Hansen says. “The transition criterion (modified lambda = 1) corresponds to a specific physical condition: the onset of contact interference at the EHL outlet constriction. It is not based on statistical multiples of RMS roughness.”

In application, this means modified lambda:
• Reflects running-in wear of asperity summits
• Accounts for micro-EHL deformation of asperities
• Accounts for the 3D nature of roughness
• Provides a physically defined regime boundary

Running-in is important. Larsson stresses, “One way to achieve high modified lambda is to run in the surfaces in such way that short wavelengths are plastically deformed and removed from the topography. This may, however, lead to residual stresses that causes problems later in the component’s service life.”

“Modified lambda is a mechanism-based and couples roughness geometry with fluid–solid elastic interaction,” Hansen adds. “As a result, it improves regime classification and provides a more reliable framework for both research and engineering applications where knowledge of EHL quality is critical.”

Björling says, with modified lambda, the shortcomings of traditional Lambda are to a large extent addressed by the use of a peak-based parameter, Spk, in combination with a parameter denoted fq, taking into account the directional nature of the roughness and its deformation under pressure. “Engineering systems that comprise non-conformal contacts, like gears, rolling element bearings and cam followers, are systems where the difference between traditional lambda and modified lambda can potentially be very large, and it is therefore possible to optimize to a greater extent using modified lambda,” he adds.

Ultimately, Larsson explains, the difference is that modified lambda takes the deformation of the roughness into account. In EHL, the pressures are in the gigapascal range, and the surface roughness is deformed, it is smoothed out—at least the longer wavelengths. Meaning that the surface roughness measured as Ra or Rq is much lower inside the contact (where the lubrication takes place) than in the open air and that the traditional lambda is much greater than if based on the non-deformed roughness. 

“Modified lambda takes this into account in a relatively simple way,” he says. “The other good thing with modified lambda is that there is an easy way to grasp threshold at modified lambda=1: Below 1 it is ML, above 1 it is full film.”

Applications: The systems that benefit most
So which systems are the most logical applications for modified lambda? Larsson says it’s gear teeth contact, cam-follower contacts and rolling bearing contacts—essentially gear transmissions. This is assuming that roughness is selected with care.

Lee says, “All systems that experience EHL will benefit from using the modified lambda ratio over the traditional lambda ratio. Examples of these systems are gears when they mesh, rolling element bearings and cam followers. EDUs contain both gears and bearings.”

Hansen says the biggest beneficiaries are systems whose surface topography, running-in and elastic deformations from elasto-hydrodynamics materially influence lubrication quality. These include:

• Electrified drivetrains and EV motor bearings, where lubrication quality interacts with electrical phenomena and discharge behavior
• Applications with significant running-in, where peak modification and micro-EHL deformation change the effective lubrication condition
• Contacts with engineered or anisotropic textures (e.g., ground, honed or directionally finished surfaces), where RMS roughness does not capture functional peak geometry and lay
• Rolling contact fatigue (RCF) predictions, where accurate identification of mixed versus full-film regimes affects life estimations
• Lubricant selection and viscosity optimization, where regime misclassification can lead to incorrect design margins

“In essence, modified lambda is most beneficial in heavily loaded EHL contacts where roughness summits and micro-EHL effects govern film formation,” Hansen says. “Even in its simplified form (f_q=0), modified lambda significantly improves regime predictions compared to the traditional Λ-ratio.”

The potential impact of widespread adoption 
How will modified lambda impact standards, design practices and failure prediction in tribological systems? Hansen points to four areas: 
• Standards. Incorporating modified lambda into bearing and gear standards (e.g., ISO 16281, ISO/TR 6336-31, AGMA 925) would modernize lubrication regime assessments and improve their accuracy.
• Engineering drawings. Including lubrication-relevant parameters such as Spk and asperity radius/wavelength on technical drawings would shift surface specifications from convention-driven to function-driven. Rather than defaulting to Sa, Sq or some other less relevant parameter, engineers would specify the features that govern film formation. Modified lambda clarifies which roughness metrics are physically relevant and helps establish a more consistent basis for surface specification on drawings.
• Commercial design software. Updating bearing and gear design tools with modified lambda would improve the accuracy of computational models that use lubrication quality as a governing parameter.
• EV tribology. In electrified drivetrains, accurate identification of the full-film to ML transition is critical when assessing and controlling bearing current discharge phenomena. Modified lambda provides a much more reliable basis for predicting and managing discharge-related behavior when compared with the traditional lambda ratio. It could help bearing and gear manufacturers to design surfaces that protect against electric discharge damage.

Per Björling, introducing modified lambda into bearing and gear standards will improve the estimation of the lubrication regime and deliver better prediction in areas such as the wear and fatigue performance of components. 

“If engineers were to start using modified lambda instead of traditional lambda it would result in more accurate predictions of the lubrication regimes, and a greater awareness among engineers that the nature and lay of the roughness of the surfaces plays a major role, not only the roughness amplitude,” he concludes. “A more accurate prediction of the lubrication regime would improve failure predictions in tribological systems since the lubrication regime largely governs the type of failure mechanism that is dominating in that particular lubrication regime.” 
Ultimately, the potential consequences of the model choice (traditional lambda versus modified lambda) can be significant. Björling explains that, for example, if the traditional lambda model makes the prediction that a tribological system is working in severe ML while it is in fact working in full film, it may lead to the assumption that abrasive and adhesive wear will take place, when instead RCF will be the main wear mechanism.

Modified lambda will help to design the surface roughness of machine components. Larsson says, “For example, it will help you select surface finishing methods. Upcoming studies will, furthermore, show relationships between modified lambda and typical failure modes such as pitting, scuffing and electric discharge. It might even be possible to characterize the roughness by using the modified lambda concept. Today there are a hundred different parameters. No one of them is really sufficient to describe tribological performance. But if the fq parameter is defined for a surface, that alone can be a much better parameter than the hundred others put together.”

Per Hansen, work is ongoing to extend modified lambda to line contacts, which will broaden its applicability to gear contacts and other wide elliptical EHL components. Adoption of modified lambda would modernize lubrication assessment by integrating peak roughness, surface texture and micro-EHL effects into standards, design practice and failure prediction frameworks.
REFERENCES
1. T. E. Tallian is a well-known tribology expert, who made major contributions in areas such as rolling bearing life prediction, surface fatigue (spalling) and elastohydrodynamic lubrication contacts. 
2. ISO 6336-22 is an international standard specifying methods for calculating the micropitting load capacity of cylindrical gears. 
3. AGMA 925 is an American standard that provides methods for evaluating the effects of lubrication and operating conditions on gear surface distress mechanisms, including micropitting, scuffing and wear in cylindrical gears.
4. Anisotropic refers to a material or surface that has different properties depending on direction.
5. The Abbott–Firestone bearing curve (also called the material ratio curve) is a graphical representation of a surface profile showing the percentage of material present at different depths. It is used to evaluate how a surface will support load and retain lubricant.
6. Hansen, J., Björling, M. and Larsson, R. (2019), “Mapping of the lubrication regimes in rough surface EHL contacts,” Tribology International, 131, pp. 637-651. DOI: 10.1016/j.triboint.2018.11.015.
7. Hansen, J., Björling, M. and Larsson, R. (2020), “Topography transformations due to running-in of rolling–sliding non-conformal contacts,” Tribology International, 144, Article 106126. DOI: 10.1016/j.triboint.2019.106126.
8. Hansen, J., Björling, M. and Larsson, R. (2020), “Lubricant film formation in rough surface non-conformal conjunctions subjected to GPa pressures and high slide-to-roll ratios,” Scientific Reports, 10, Article 22250.
9. Hansen, J., Prajapati, D. K., Björling, M. and Larsson, R. (2025), “Robustness and sensitivity of the Λ*-ratio in microelastohydrodynamic lubrication,” Tribology Letters, 73, article no. 129. Available at https://doi.org/10.1007/s11249-025-02060-6.
Jeanna Van Rensselar heads her own communication/public relations firm, Smart PR Communications, in Naperville, Ill. You can reach her at jeanna@smartprcommunications.com.